Contents

1 Set-Valued Mappings 1 1.1 Lower Semicontinous Set-Valued Mappings . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Upper Semicontinuous Set-Valued Mappings . . . . . . . . . . . . . . . . . . . . . . 5

2 Fixed Point Theorems of Set-Valued Mappings 11 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Knaster-Kuratowski-Mazurkiewicz Principle . . . . . . . . . . . . . . . . . . . . . . 12 2.3 Ky Fan Minimax Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.4 Ky Fan Minimax Inequality-I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.5 Ky Fan Minimax Inequality-II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.6 Maximal Element and Browder-Fan Fixed Point in Product Spaces . . . . . . . . . 28

2.6.1 Maximal Element and Fixed Point of Couple-Majorized Mappings . . . . . 28 2.6.2 Maximal Element and Fixed Point of L_{s}-Majorized Mappings . . . . . . . . 34

2.7 Fan-Glicksberg Fixed Points in G-Convex Spaces . . . . . . . . . . . . . . . . . . . 43 2.8 Nonlinear Analysis of Hyperconvex Metric Spaces . . . . . . . . . . . . . . . . . . . 47

2.8.1 Fan-Glicksberg Fixed Points in Hyperconvex Metric Spaces . . . . . . . . . 47 2.8.2 KKM Theory in Hyperconvex Metric Spaces . . . . . . . . . . . . . . . . . 49 2.8.3 Intersection Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 2.8.4 Fixed Point Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 2.8.5 Saddle Points and Nash Equilibria . . . . . . . . . . . . . . . . . . . . . . . 57 2.8.6 Generic Stability of Fixed Points in Hyperconvex Metric Spaces . . . . . . . 59

Chapter 1

Set-Valued Mappings

In this chapter, we report some new results on the study of both lower and upper semicontinuous set-valued mappings. In section 1, we focus in the study of intersection of both lower semicontinuous set-valued mappings. In section 2, we emphasize the study of upper semicontinuous set-valued mappings and some characterization of set-valued mappings being upper semicontinuous is given.

1.1 Lower Semicontinous Set-Valued Mappings

In this section, we discuss the property of intersection for two set-valued mappings being lower semicontinuous. Before our discuss, we first introduce some definitions and notions.

Suppose X is any given set, throughout this discussion we denote by 2^{X} and F(X) the families of all subsets and all non-empty finite subsets of X, respectively. If S is a non-empty subset of a topological space X, we also denote by \overline{S} (or clS) and intS the closure and interior of the set S in X.

Let X and Y be two topological spaces and F : X → 2^{Y} a set-valued mapping. As usual, we say that F is upper (resp., lower) semicontinuous on X if for each x ∈ X and for each open set G in Y with F (x) ⊂ G (resp., F (x) ∩ G ≠ ∅), there exists an open neighborhood N_{x} of x in X such that F (z) ⊂ G (resp., F (z) ∩ G ≠ ∅) for all z ∈ N_{x}. A set-valued mapping is said to be USCO if it is an upper semicontinuous mapping with non-empty and compact values. Let E^{∗} be the dual space (i.e., the vector space of all continuous linear functionals) of a real Hausdorff topological vector space E. A set-valued mapping F : X → 2^{E} is said to be upper demicontinuous on X if for each x ∈ X and any open half-space (or say, hyper-plane) H in E containing F (x)(i.e., F (x) ⊂ H), there exists an open neighborhood N_{x} of x in X such that F (z) ⊂ H for all z ∈ N_{x}. We recall that an open half-plane H in E is set of the form H := {x ∈ E : ϕ(x) < λ} for some ϕ ∈ E^{∗} and λ ∈ R. A set-valued mapping F : X → 2^{E} is said to be upper hemicontinuous on X if for each ϕ ∈ E^{∗} and for each λ ∈ R, the set {x ∈ X : sup_{u∈F (x)} ϕ(u) < λ} is open in X.

Before the study of intersection for two set-valued mappings being lower semicontinuous, we first have the following result.

Theorem 1.1.1. Let X and Y be two topological spaces and A be a closed (resp., open) subset of X. Suppose F_{1} : X → 2^{Y} and F_{2} : A → 2^{Y} are lower semicontinuous (resp., upper semicontinuous) such that F_{2}(x) ⊂ F_{1}(x) for all x ∈ A. Then the mapping F : X → 2^{Y} defined by

F (x) := \begin{cases}F_{1}(x), & if x ∉ A ; \\ F_{2}(x), & if x ∈ A \end{cases}

is lower (resp., upper) semicontinuous.

Proof. Let U be any closed (resp., open) subset of Y. Clearly

{x ∈ X : F (x) ⊂ U} = {x ∈ A : F_{2}(x) ⊂ U} ∪ {x ∈ X \ A : F_{1}(x) ⊂ U}
⊂ {x ∈ A : F_{2}(x) ⊂ U} ∪ {x ∈ X : F_{1}(x) ⊂ U}.

Conversely, if x ∈ A and F_{2}(x) ⊂ U, then F (x) = F_{2}(x) ⊂ U and if x ∈ X and F_{1}(x) ⊂ U, then F (x) = F_{2}(x) ⊂ F_{1}(x) ⊂ U whenever x ∈ A and F (x) = F_{1}(x) ⊂ U when x ∈ X \ A. This shows that we also have {x ∈ X : F (x) ⊂ U} ⊃ {x ∈ A : F_{2}(x) ⊂ U} ∪ {x ∈ X : F_{1}(x) ⊂ U}. Hence

{x ∈ X : F (x) ⊂ U} = {x ∈ A : F_{2}(x) ⊂ U} ∪ {x ∈ X : F_{1}(x) ⊂ U}.

As A and U are closed (resp., open) and F_{1} and F_{2} are lower semicontinuous (resp., upper semicontinuous ), the set {x ∈ X : F (x) ⊂ U} is also closed (resp., open). Therefore, F is lower semicontinuous (resp., upper semicontinuous). □

In what follows, we shall dosciss some properties for the intersection of two lower semicontinuous set-valued mappings. We first have the following result which was given by Tulcea [?] (see also Lemma 4.1 of Chang [?]).

Theorem 1.1.2. Let X be a topological space, E be a topological vector space and Y be a non-empty subset of E. Suppose ϕ : X → 2^{Y} is a lower semicontinuous mapping and V is any non-empty open subset of E. Then the set-valued mapping H : X → 2^{Y} defined by H(x) := (ϕ(x) + V) ∩ Y for each x ∈ X has an open graph in X × Y.

Proof. Suppose (x_{0}, y_{0}) ∈ Graph H; then y_{0} ∈ (ϕ(x_{0})+V)∩Y. Choose z_{0} ∈ ϕ(x_{0}) and v_{0} ∈ V such that y_{0} = z_{0} + v_{0} ∈ Y. Since V is open in E, there exists an open neighborhood W of zero in E such that v_{0} + W + W ⊂ V. Since ϕ is lower semicontinuous and ϕ(x_{0}) ∩ (z_{0} − W) ∩ Y ≠ ∅, the set U_{x_{0}} = {x ∈ X : ϕ(x) ∩ (z_{0} − W) ∩ Y ≠ ∅} is an open neighborhood of x_{0} in X. Let V_{y_{0}} := (y_{0} + W) ∩ Y, then V_{y_{0}} is an open neighborhood of y_{0} in Y. Now if (x, y) ∈ U_{x_{0}} × V_{y_{0}}, then ϕ(x)∩(z_{0}−W)∩Y ≠ ∅ and y ∈ (y_{0}+W)∩Y. Choose any z ∈ ϕ(x)∩(z_{0}−W)∩Y, then z ∈ ϕ(x) and z_{0} ∈ z +W so that y ∈ (y_{0} +W)∩Y = (z_{0} +v_{0} +W)∩Y ⊂ (z +v_{0} +W +W)∩Y ⊂ (ϕ(x)+V)∩Y. Thus U_{x_{0}} × V_{y_{0}} ⊂ Graph H. Therefore H has an open graph in X × Y. □

By the definition of the lower semiocntinuity for set-valued mappings we have the following result.

Theorem 1.1.3. Let X and Y be two topological spaces and ψ, ϕ : X → 2^{Y} be two correspondences such that ψ is lower semicontinuous and ϕ has open graph in X × Y. Then the map θ(x) : X → 2^{Y} defined by θ(x) := ψ(x) ∩ ϕ(x) for each x ∈ X is lower semicontinuous.

Proof. For any open subset V of Y, we have

{x ∈ X : ϕ(x) ∩ ψ(x) ∩ V ≠ ∅} = {x ∈ X : Graph ϕ ∩ ({x} × ψ(x)) ∩ (X × V) ≠ ∅}.

Since the mapping x ↦ {x}×ψ(x) from X to 2^{X×Y} is also lower semicontinuous on X and X ×V is open in X ×Y, the set {x ∈ X : ψ(x)∩ϕ(x)∩V ≠ ∅} is open in X. Hence θ is lower semicontinuous. □

Let X and Y be topological spaces and ψ, ϕ : X → 2^{Y} be lower semicontinuous. We observe that if Graph ϕ is open in X × Y, then it is shown in Theorem 1.1.3 that ϕ ∩ ψ is also lower semicontinuous. Clearly, if Graph ϕ is open in X × Y, then ϕ is open-valued. The following example shows that (a): the Graph ϕ need not be open in X × Y even though ϕ is open-valued and (b): even when ϕ(x) ∩ ψ(x) ≠ ∅ for all x ∈ X, if the condition ‘Graph ϕ is open in X × Y ’ is weakened and replaced by the condition that ‘ϕ is open-valued’, then the mapping ϕ ∩ ψ need not be lower semicontinuous.

Example 1.1.4. Let F_{1}, F_{2} : [0, 1] → 2^{[0,1]} be defined by

F_{1}(x) = \begin{cases}[0, 1], & if x = 0; \\ (0, 1] \ \{1/n : n = 2, 3, ⋯, \}, & if x ≠ 0 \end{cases}

and

F_{2}(x) = \begin{cases}\{0\}, & if x = 0; \\ \{1/n : n = 1, 2, ⋯ \}, & if x ≠ 0. \end{cases}

Then it is easy to see that (i) F_{1} and F_{2} are lower semicontinuous, (ii) Graph F_{1} is not open in [0, 1] × [0, 1] and (iii) F_{1} is open-valued such that for each x ∈ [0, 1],

F_{1}(x) ∩ F_{2}(x) = \begin{cases}\{0\}, & if x = 0; \\ \{1\}, & if x ≠ 0. \end{cases}

It follows that F_{1} ∩ F_{2} is not lower semicontinuous.

Since the translation of a mapping (resp., convex set, open set) preserves its lower semicontinuity at a point (resp., convexity, openness), we observe that the proof of Proposition 1.1 of Marano [121, p.286] actually produced the following slightly strengthened version (where the original assumption that ϕ is lower semicontinuous is replaced by the weaker assumption that ϕ is lower semicontinuous at s_{0}) and we state it as:

Lemma 1.1.5. Let S be a topological space and ϕ : S → 2^{ℝ^{n}} be a mapping with non-empty convex values. Suppose ϕ is lower semicontinuous at some s_{0} ∈ S and y_{0} ∈ int_{ℝ^{n}}ϕ(x_{0}), then there exists a neighborhood U_{0} of s_{0} in S such that y_{0} ∈ int_{ℝ^{n}}ϕ(s) for all s ∈ U_{0} .

As a review of Lemma 1.1.5, a natural question raised is as follows:

Question 1.1.6. Let S be a topological space and ϕ be a mapping from S to ℝ^{n} with convex values. Let x_{0} ∈ S and ϕ be lower semicontinuous at x_{0}. Assume that B(y_{0}, r) ⊂ ϕ(x_{0}), where B(y_{0}, r) is the open ball of radius r centered at y_{0}. Then is there a neighborhood U of x_{0} such that B(y_{0}, r) ⊂ ϕ(x) for all x ∈ U ?

Unfortunately, the following simple example shows that the above statement is not valid:

Example 1.1.7. Let ϕ : ℝ → 2^{ℝ} be defined by ϕ(x) = [x − 1, x + 1] (or (x − 1, x + 1)) for each x ∈ ℝ. Clearly ϕ is lower semicontinuous. Fixed x_{0} ∈ ℝ and let y_{0} = x_{0} and B(y_{0}, 1) = (x_{0} − 1, x_{0} + 1). It is easy to see that for any neighborhood U of x_{0} in ℝ, B(y_{0}, 1) ⊄ ϕ(x) for any x ∈ U with x ≠ x_{0}.

By following the idea in the proof of Lemma 1 from Monteiro [?], we have the following result. ion:

Lemma 1.1.8. Let S be a topological space and ϕ : S → 2^{ℝ^{n}} be a set-valued mapping with convex values. Suppose ϕ is lower semicontinuous at some x_{0} ∈ S and B(y_{0}, t) ⊂ ϕ(x_{0}) where B(y_{0}, t) is the open ball of positive radius t centered at y_{0}. Then for any number r with 0 < r < t, there exists a neighborhood U_{0} of x_{0} such that B(y_{0}, r) ⊂ ϕ(x) for all x ∈ U_{0}.

Proof. By restricting ϕ to the set {x ∈ X : ϕ(x) ∩ int_{ℝ^{n}}ϕ(x_{o}) ≠ ∅}, which is an open neighborhood of x_{0} in S by the lower semicontinuity of ϕ at x_{0}, we may assume without loss of generality that ϕ is non-empty valued. Choose any r′ with r < r′ < t, then we have \overline{B(y_{0}, r′)} ⊂ B(y_{0}, t) ⊂ ϕ(x_{0}) where \overline{B(y_{0}, r′)} is the closure of B(y_{0}, r′) in ℝ^{n}. Note that \overline{B(y_{0}, r′)} is compact. Choose any ϵ with 0 < ϵ < r′ − r. Let \{B(y_{i}, ϵ/2) : i = 1, ⋯, m\} be a finite cover of \overline{B(y_{0}, r′)} with y_{i} ∈ B(y_{0}, r′) for all i = 1, ⋯, m. By Lemma 1.1.5, there exists an open neighborhood U_{1} of x_{0} in S such that y_{0} ∈ int_{ℝ^{n}}ϕ(x) for all x ∈ U_{1}; in particular, int_{ℝ^{n}}ϕ(x) ≠ ∅ for all x ∈ U_{1}. By lower semicontinuity of ϕ at x_{0} again, there exists an open neighborhood U_{0} of x_{0} in S such that for each x ∈ U_{0}, ϕ(x) ∩ B(y_{i}, ϵ/2) ≠ ∅ for all i = 1, ⋯, m. By replacing U_{0} by U_{1} ∩ U_{0}, we may also assume that U_{0} ⊂ U_{1}. Now given any non-empty subsets A and B of ℝ^{n}, define e(A, B) = sup{d(a, B); a ∈ A}, where d is the Euclidean metric on ℝ^{n} and d(a, B) = inf{d(a, b) : b ∈ B}. It is easy to see that for each x ∈ U_{0},

e(B(y_{0}, r′), ϕ(x)) ≤ e(∪_{i=1}^{m}B(y_{i}, ϵ/2), ϕ(x)) ≤ ϵ.

By the Lemma of Moreau [?], for each x ∈ U_{0},

d(y_{0}, ℝ^{n} \ B(y_{0}, r′)) ≤ d(y_{0}, ℝ^{n} \ ϕ(x)) + e(B(y_{0}, r′), ϕ(x)),

so that

d(y_{0}, ℝ^{n} \ ϕ(x)) ≥ d(y_{0}, ℝ^{n} \ B(y_{0}, r′)) − e(B(y_{0}, r′), ϕ(x)) ≥ r′ − ϵ > r.

It follows that B(y_{0}, r) ⊂ ϕ(x) for all x ∈ U_{0}. □

As a consequence of Lemma 1.1.8, we have the following fact.

Corollary 1.1.9 Let S be a topological space and ϕ : S → 2^{ℝ^{n}} be a lower semicontinuous set-valued mapping with non-empty open convex values. Then the graph of ϕ is open.

Remark 1.1 of Marano in [?] also shows that Lemma 1.1.8 does not hold if ϕ takes its values in an infinite dimensional Hilbert space.

We now also have the following result.

Theorem 1.1.10. Let S be a topological space and F_{1}, F_{2} : S → 2^{ℝ^{n}} be lower semicontinuous at x_{0} ∈ S such that F_{1} is open and convex-valued. Then F_{1} ∩ F_{2} is lower semicontinuous at x_{0}.

Proof. Let U be an open subset of ℝ^{n} such that F_{1}(x_{0}) ∩ F_{2}(x_{0}) ∩ U ≠ ∅. Choose any y_{0} ∈ F_{1}(x_{0}) ∩ F_{2}(x_{0}) ∩ U. Since F_{1}(x_{0}) ∩ U is open, y_{0} is an interior point of F_{1}(x_{0}) ∩ U. By Lemma 1.1.9, there is an open neighborhood V_{1} of x_{0} in S and r > 0 such that B(y_{0}, r) ⊂ F_{1}(x) for all x ∈ V_{1}, where B(y_{0}, r) is an open ball in ℝ^{n} of radius r and centered at y_{0}. Since U is open in ℝ^{n}, we may assume without loss of generality that B(y_{0}, r) ⊂ U. Since F_{2} is lower semicontinuous at x_{0}, there is an open neighborhood V_{2} of x_{0} in S such that F_{2}(x) ∩ B(y_{0}, r) ≠ ∅ for all x ∈ V_{2}. Let V = V_{1} ∩ V_{2}, then V is an open neighborhood of x_{0} in S such that for each x ∈ V, F_{1}(x) ∩ F_{2}(x) ∩ U ⊃ F_{2}(x) ∩ B(y_{0}, r) ≠ ∅. Hence F_{1} ∩ F_{2} is lower semicontinuous. □

The following example from Ledicki and Spakowski [?] shows that even if F_{2} has closed (or open) convex values, the conclusion of Theorem 1.1.10 fails to hold if ℝ^{n} is replaced by an infinite dimensional Banach space:

Example 1.1.11. Let Y = l^{∞}, the Banach space of all bounded sequences x = (x_{n})_{n=1}^{∞} of real numbers with ∥x∥_{∞} = sup_{n∈N} |x_{n}| < ∞ and S = [0, 1]. Define G_{1}, G_{2} : S → 2^{Y} by

G_{1}(t) = {x ∈ Y : x_{1} ≥ t, x_{k} ≤ k − t for k ≥ 2}

and

G_{2}(t) = {x ∈ Y : x_{1} ≤ 1 − t, x_{k} ≤ k(1 − x_{1} − t) and x_{k} ≤ k + x_{1}/k − t/k for k ≥ 2}.

Then G_{1} and G_{2} are both lower semicontinuous at zero with closed convex values and int_{Y} (G_{1}(0)∩ G_{2}(0)) ≠ ∅, but G_{1} ∩ G_{2} is not lower semicontinuous at 0 (e.g., see the Example 3 from [?]. Now we define F_{1}, F_{2} : S → 2^{Y} by F_{1}(t) = int_{Y} G_{1}(t) and F_{2}(t) = int_{Y} G_{2}(t) (or G_{2}(t)) for each t ∈ S. Since

\overline{F_{1}(0) ∩ F_{2}(0)} = \overline{(int_{Y} G(0)) ∩ (int_{Y} G_{2}(0))} = \overline{(int_{Y} G_{1}(0)) ∩ G_{2}(0)} = G_{1}(0) ∩ G_{2}(0),

F_{1} ∩ F_{2} is also not lower semicontinuous at t = 0 by Proposition 7.3.3 of Klein and Thompson [?]. (or Proposition 2.3 of Michael [?].

Theorem 1.1.12. Let S be a topological space and F_{1}, F_{2} : S → 2^{ℝ^{n}} be lower semicontinuous at x_{0} ∈ S such that for each x ∈ S, F_{1}(x) and F_{2}(x) are both convex and (intF_{1}(x)) ∩ F_{2}(x) ≠ ∅ or F_{1}(x) ∩ (intF_{2}(x)) ≠ ∅ whenever F_{1}(x) ∩ F_{2}(x) ≠ ∅. Then F_{1} ∩ F_{2} is also lower semicontinuous at x_{0} ∈ S.

Proof. Without loss of generality, we may assume that (intF_{1}(x_{0})) ∩ F_{2}(x_{0}) ≠ ∅. Since F_{1}(x_{0}) and F_{2}(x_{0}) are convex, by Theorem of Dolecki [?], \overline{(intF_{1}(x_{0})) ∩ F_{2}(x_{0})} = \overline{F_{1}(x_{0}) ∩ F_{2}(x_{0})} = \overline{F_{1}(x_{0})} ∩ \overline{F_{2}(x_{0})}. In order to prove F_{1} ∩ F_{2} is lower semicontinuous at x_{0}, it suffices to prove that (intF_{1}) ∩ F_{2} is lower semicontinuous at x_{0}. Now let U be any non-empty open subset of ℝ^{n} such that (intF_{1}(x_{0})) ∩ F_{2}(x_{0}) ∩ U ≠ ∅. Choose any y_{0} ∈ (intF_{1}(x_{0})) ∩ F_{2}(x_{0}) ∩ U. Since intF_{1}(x_{0}) ∩ U is open, y_{0} is an interior point of intF_{1}(x_{0}) ∩ U. Since F_{1} is lower semicontinuous at x_{0} and intF (x_{0}) ≠ ∅, we have \overline{F_{1}(x_{0})} = \overline{intF_{1}(x_{0})}. Therefore intF_{1} is also lower semicontinuous at x_{0}. By Lemma 1.1.8, there is an open neighborhood V_{1} of x_{0} in S and r > 0 such that B(y_{0}, r) ⊂ intF_{1}(x) for all x ∈ V_{1}, where B(y_{0}, r) is an open ball in ℝ^{n} of radius r and centered at y_{0}. Since U is open in ℝ^{n}, we may assume without loss of generality that B(y_{0}, r) ⊂ U. Since F_{2} is lower semicontinuous at x_{0}, there is an open neighborhood V_{2} of x_{0} in S such that F_{2}(x) ∩ B(y_{0}, r) ≠ ∅ for all x ∈ V_{2}. Let V = V_{1} ∩ V_{2}, then V is an open neighborhood of x_{0} in S such that for each x ∈ V, (intF_{1}(x)) ∩ F_{2}(x) ∩ U ⊃ F_{2}(x) ∩ B(y_{0}, r) ≠ ∅. Hence (intF_{1}) ∩ F_{2} is lower semicontinuous at x_{0}, so that F_{1} ∩ F_{2} is also lower semicontinuous at x_{0}. □

Theorem 1.1.13. Let S be a topological space, X be a non-empty subset of a finite dimensional Hausdorff topological vector space and F_{1}, F_{2} : S → 2^{aff(X)} be lower semicontinuous at x_{0} ∈ S. If for each x ∈ S, F_{1}(x) is convex and open in aff(X), then F_{1} ∩ F_{2} is also lower semicontinuous at x_{0}.

Proof. Choose any a_{0} ∈ aff(X) and let Y =aff(X) −a_{0}, then Y is a finite dimensional Hausdorff topological vector space and is therefore topologically isomorphic to some ℝ^{n}. Define F̃_{1}, F̃_{2} : S → 2^{Y} by F̃_{1}(x) = F_{1}(x)−a_{0}, F̃_{2}(x) = F_{2}(x)−a_{0} for all x ∈ S. Then F̃_{1} and F̃_{2} are lower semicontinuous at x_{0} such that for each x ∈ S, F̃_{1}(x) is convex and is open in Y. By Theorem 1.1.12, F̃_{1} ∩ F̃_{2} is lower semicontinuous at x_{0}. It follows that F_{1} ∩ F_{2} is also lower semicontinuous at x_{0}. □

By the same proof of Theorem 1.1.13, we have the following result.

Theorem 1.1.14. Let S be a topological space, X be a non-empty subset of a finite dimensional Hausdorff topological vector space and F_{1}, F_{2} : S → 2^{aff(X)} be lower semicontinuous at x_{0} ∈ S. If for each x ∈ S, F_{1}(x) and F_{2}(x) are convex and ri(F_{1}(x)) ∩ F_{2}(x) ≠ ∅ or F_{1}(x) ∩ ri(F_{2}(x)) ≠ ∅ whenever F_{1}(x) ∩ F_{2}(x) ≠ ∅, then F_{1} ∩ F_{2} is also lower semicontinuous at x_{0}.

For other study of set-valued analysis and its applications, we refer the interested reader to Aubin and Franksowska [?], Beer [?] or Klein and Thompson’s book [?] and references therein.

1.2 Upper Semicontinuous Set-Valued Mappings

When studying nonlinear set-valued problems such as existence of fixed points for set-valued mappings, the existence of solutions for differential inclusions, and others, we often face the problem of dealing with different kinds of continuity for set-valued mappings. In the case of the single-valued mapping, the continuity defined by neighborhood is equivalent with that definition by using language of sequences or nets. However, in the case of strict set-valued mappings, these two definitions are no longer equivalent. As a result, a few concepts about continuity of set-valued mappings, for example, upper semicontinuity, upper demicontinuity and upper hemicontinuity have been introduced and often used in the study of nonlinear set-valued analysis (e.g., see Aubin and Ekeland [?], Aubin and Frankowska [?], Fan [?], Klein and Thompson [?], and related references therein). Thus, it is necessary to illustrate the intrinsic quality among these different continuities.

It is obvious that each upper semicontinuous set-valued mapping is upper demicontinuous and each upper demicontinuous set-valued mapping is upper hemicontinuous, but the inverse is not true in general. However, if set-valued mapping is compact-valued, the upper hemicontinuity implies the upper demicontinuity. In this case, the concepts of upper demicontinuity and upper heimcontinuity coincide. Furthermore, when the image space of a set-valued mapping with compact-valued is compact, all three definitions above are equivalent. But, under what kind of conditions is the upper hemicontinuity equivalent to the upper demicontinuity without assumption of compactness for set-valued mappings? To the best of our knowledge, there are not much discussion in this direction.

The purpose of this section is to give a characterization of an upper hemicontinuous mapping with non-empty convex values being upper demicontinuous, i.e., we show that an upper hemicontinuous set-valued mapping with non-empty convex values (not necessarily compact-valued) is upper demicontinuous if and only if the set-valued mapping has no interior asymptotic plane. Then we give another characterization for a set-valued mapping being upper semicontinuous if and only if the underlying space is finite dimensional.

Throughout this section, let X be a Hausdorff topological space and E a real Hausdorff locally convex topological linear space. For the sake of convenience, we first introduce the definition of asymptotic plane which could give us the characterization of upper hemicontinuity being upper demicontinuity for set-valued mappings with non-compact values. Now we have the following property for upper semicontinuous set-valued mappings.

Theorem 1.2.1. Let X be a topological space, Y be a regular subset of a topological vector space E and ϕ : X → 2^{Y} be a compact-valued upper semicontinuous. Then for any given nonempty subset V of E, the mapping H_{V} : X → 2^{Y} defined by H_{V} (x) := (ϕ(x) + \overline{V}) ∩ Y for each x ∈ X has a closed graph in X × Y.

Proof. Suppose {(x_{α}, y_{α})}_{α∈Γ} is a net in GraphH such that (x_{α}, y_{α}) ↦ (x, y) in X × Y. It suffices to prove that y ∈ H_{V} (x). For each α ∈ Γ, as y_{α} ∈ H_{V} (x_{α}) = ϕ(x_{α}) + \overline{V}, there exists z_{α} ∈ ϕ(x_{α}) and v_{α} ∈ \overline{V} such that y_{α} = z_{α} + v_{α}. Let K := \{x, x_{α} : α ∈ Γ\}. Then K is a compact subset of X. By the upper semicontinuity of ϕ and the assumption that Y is regular, the set ϕ(K) := ∪_{x∈K}ϕ(x) is compact by Theorem 7.1.15 of Klein and Thompson [?]. Without loss of generality, we may assume that there exists a subset (z_{α′})_{α′∈Γ′} of (z_{α})_{α∈Γ} such that z_{α′} ↦ z ∈ ϕ(K). As the graph of ϕ is closed in X × Y, we have z ∈ ϕ(x). Note that \{v_{α′}\}_{α′∈Γ′} ⊂ \overline{V} and v_{α′} = v_{α′} + z_{α′} − z_{α′} = y_{α′} − z_{α′} → y − z, so that y − z ∈ \overline{V}. It follows that y = z + y − z ∈ ϕ(x) + \overline{V} = H_{V} (x). Therefore (x, y) ∈ GraphH and we have proved that the graph of H_{V} is closed in X × Y and hence the proof is complete. □

As an application of Theorem 1.2.1, we have the following corollary:

Corollary 1.2.2. Let X be a topological space, Y a non-empty compact subset of a topological vector space E and ϕ : X → 2^{Y} be a set-valued mapping. Suppose the mapping clϕ : X → 2^{Y} defined by clϕ(x) := cl_{Y} (ϕ(x)) for each x ∈ X is upper semicontinuous. Then for each given non-empty subset V of E, the mapping H : X → 2^{Y} defined by H(x) := (clϕ(x) + \overline{V}) ∩ Y is upper semicontinuous.

Proof. Note that each compact topological space is regular. Thus the conclusion follows from Theorem 1.2.1 and Theorem 7.1.16 of Klein and Thompson [?]. □

Proposition 1.2.3. Let X be a topological space and Y be a locally convex topological space. If F : X → 2^{Y} is upper semicontinuous, then the mapping coF : X → 2^{Y} defined by coF (x) := co(F (x)) for each x ∈ X is upper semicontinuous.

Proof. From the definition of upper semicontinuity, it suffices to prove that for each open neighborhood V of Y, the set U := {x ∈ X : coF (x) ⊂ V } is open in X. Now suppose x_{0} ∈ U, then coF (x_{0}) ⊂ V. As Y is a locally convex topological vector space, without loss of generality, we may assume that V is convex. As F is upper semicontinuous and F (x_{0}) ⊂ coF (x_{0}) ⊂ V, there exits a non-empty open neighborhood N(x_{0}) of x_{0} such that F (w) ⊂ V for all w ∈ N(x_{0}). Thus coF (w) ⊂ co(V) = V for all w ∈ N(x_{0}) as V is convex. Therefore N(x_{0}) ⊂ U, which implies that U is open in X. Hence the conclusion follows and we complete the proof.□

As a review of Proposition 1.2.3, we have the following:

Theorem 1.2.4. Let X be a topological space and Y be a non-empty subset of a normed space. If F : X → 2^{Y} is upper semicontinuous, then the mapping \overline{coF} : X → 2^{Y} defined by \overline{coF} (x) := \overline{coF (x)} for each x ∈ X is upper semicontinuous.

Proof. Note that each subset of normed space is normal and the composition of upper semicontinuous mappings is upper semicontinuous, the conclusion now follows from Proposition 1.2.3 and Proposition 7.3.5 of Klein and Thompson [?]. □

Definition 1.2.5. Let E^{∗} be the dual space of a Hausdorff locally convex topological linear space E and A a non-empty subset of E. For each given ϕ ∈ E^{∗} and λ ∈ R, the set M_{ϕ,λ} = {x ∈ E : ϕ(x) = λ} is said to be an asymptotic plane of A with respect to (ϕ, λ) ∈ E^{∗} × R (in short, M_{ϕ,λ} is said to be an asymptotic plane of A if there is no confusion in the content) if M_{ϕ,λ} ∩ A = ∅ and there exists a positive number δ ∈ R such that M_{ϕ,μ} ∩ A ≠ ∅ for all μ ∈ (λ − δ, λ).

Remark 1.2.6. From Definition 1.2.5 above if follows that all compact sets and weakly convex compact sets in locally topological vector space E have no asymptotic planes.

Definition 1.2.7. Let X be a Hausdorff topological space, E a Hausdorff locally convex topological linear space and F : X → 2^{E} \ {∅} a set-valued mapping with non-empty values. For any given point x_{0} ∈ X, the asymptotic plane M_{ϕ,λ} of F (x_{0}) is said to be a boundary asymptotic plane of F at x_{0}, if there exists an open neighborhood N(x_{0}) of x_{0} such that

F (x) ⊂ H_{ϕ,λ} = {y ∈ E : ϕ(y) < λ}

for all x ∈ N(x_{0}). The asymptotic plane M_{ϕ,λ} of F (x_{0}) is said to be an ‘interior asymptotic plane’ of F at x_{0}, if for any neighborhood N(x_{0}) of x_{0}, there exists x ∈ N(x_{0}) such that

F (x) ⊄ H_{ϕ,λ} := {y ∈ E : ϕ(y) < λ}.

If for all x ∈ X, the set F (x) has no interior asymptotic plane, then the mapping F is said to have no interior asymptotic plane.

Remark 1.2.8. By Definition 1.2.7, it follows that each asymptotic plane of a given set A in E is either a boundary or interior asymptotic plane. Now we have the following characterization for a set-valued upper hemicontinuous mappings being upper demicontinuous.

Theorem 1.2.9. Let X be a Hausdorff topological space and E be a Hausdorff locally convex topological linear space. Suppose F : X → 2^{E} is an upper hemicontinuous set-valued mapping with non-empty convex values (not necessarily compact-valued). Then F is upper demicontinuous if and only if the mapping F has no interior asymptotic plane on X.

Proof. \underline{Necessity}: Suppose the conclusion is false, then there exists x_{0} ∈ X and an asymptotic plane M_{ϕ,λ} at x_{0} such that M_{ϕ,λ} is an interior asymptotic plane of F (x_{0}) for some ϕ ∈ E^{∗} and λ ∈ R. By Definition 1.2.1, there exists some δ > 0, such that for any μ ∈ (λ − δ, λ), we have that H_{ϕ,λ} ∩ F (x_{0}) ⊃ M_{ϕ,μ} ∩ F (x_{0}) ≠ ∅. Next we want to show that F (x_{0}) ⊂ H_{ϕ,λ}. In fact, if there exists y ∈ F (x_{0}) such that y ∉ H_{ϕ,λ}, it follows that ϕ(y) ≥ λ. Taking y_{0} ∈ H_{ϕ,λ} ∩ F (x_{0}), we have that ϕ(y_{0}) < λ. Let

y_{1} := \frac{ϕ(y) − λ}{ϕ(y) − ϕ(y_{0})}y_{0} + \frac{λ − ϕ(y_{0})}{ϕ(y) − ϕ(y_{0})}y.

We know that y_{1} ∈ F (x_{0}) and ϕ(y_{1}) = λ. Thus M_{ϕ,λ}∩F (x_{0}) ≠ ∅. This contradicts our assumption that M_{ϕ,λ} is an asymptotic plane of F (x_{0}). Therefore, we must have that F (x_{0}) ⊂ H_{ϕ,λ}. Secondly by the definition of interior asymptotic planes, for neighborhood N(x_{0}) of x_{0}, there exists x ∈ N(x_{0}) such that F (x) ⊄ M_{ϕ,λ}. This contradicts with our assumption that F is demicontinuous at x_{0} ∈ X. Hence F has no interior asymptotic plane at x_{0}.

\underline{Sufficiency}: Suppose that F is not demicontinuous on X. Then there exists x_{0} ∈ X and an open half space H_{ϕ,λ} = {y ∈ E : ϕ(y) < λ} with F (x_{0}) ⊂ H_{ϕ,λ} for some ϕ ∈ E^{∗} and λ ∈ R, such that for any neighborhood N(x_{0}) of x_{0}, there exists x ∈ N(x_{0}) such that F (x) ⊄ H_{ϕ,λ}. Therefore, for any neighborhood N(x_{0}) of x_{0}, there exist x_{N(x_{0})} ∈ N(x_{0}) and y_{N(x_{0})} ∈ F (x_{N(x_{0})}) such that ϕ(y_{N(x_{0})}) ≥ λ. In what follows, we want to show that M_{ϕ,λ} is not a boundary asymptotic plane of F (x_{0}). In order to do so, it suffices to show that M_{ϕ,λ} is an asymptotic plane of F (x_{0}). Now let the net {x_{v}}_{v∈V} be a direct family of the set {N_{V} (x_{0})}_{v∈V} with that lim_{v∈V} x_{v} := x_{0}, where {N_{v}(x_{0})}_{v∈V} is the family of all neighborhood N(x_{0}) at the point x_{0}. As we just discussed above, for each neighborhood N_{v}(x_{0}) of x_{0}, there exists x_{v} ∈ N_{v}(x_{0}) and y_{v} ∈ F (x_{v}) such that sup_{u∈F (x_{v})} ϕ(u) ≥ ϕ(y_{v}) ≥ λ. By the hemicontinuity of F, we do have that

sup_{u∈F (x_{0})}(u) ≥ \overline{lim_{v}} sup_{u∈F (x_{v})} ϕ(u) ≥ λ.

Note that F (x_{0}) ⊂ H_{ϕ,λ}, it follows that that sup_{u∈F (x_{0})} ϕ(u) = λ. Now take y_{0} ∈ F (x_{0}), it follows that

ϕ(y_{0}) < λ as F (x_{0}) ⊂ H_{ϕ,λ}. Let δ = λ−ϕ(y_{0}). We then have that δ > 0 and (λ−δ, λ) = (ϕ(y_{0}), λ). For any μ ∈ (λ − δ, λ) and σ ∈ (μ, λ), as sup_{u∈F (x_{0})} ϕ(u) = λ, there exists y_{1} ∈ F (x_{0}) such that

ϕ(y_{1}) > σ. This implies that μ ∈ (ϕ(y_{0}), ϕ(y_{1})). By the convexity of F (x_{0}), we know that F (x_{0}) is a path connected set. By the intermediate-value theorem, there exists x̂ ∈ F (x_{0}) such that ϕ(x̂) = μ. It follows that M_{ϕ,μ} ∩ F (x_{0}) ≠ ∅ for all μ ∈ (λ − δ, λ), i.e., M_{ϕ,λ} is an asymptotic plane of F (x_{0}). Therefore the set M_{ϕ,λ} is not a boundary asymptotic plane of F at x_{0} and thus M_{ϕ,λ} is an interior asymptotic plane of F at x_{0}. This contradicts our assumption that F has no any interior asymptotic plane on X and thus the proof is completed. □

Example 1.2.10. Let E := R^{2}, X := [0, 1] and define a set-valued mapping from X to E by F (x) = {(u, v) : u > x, v ≥ 1/(u − x)} for each x ∈ [0, 1]. Then F: X → 2^{E} is a setvalued mapping with non-empty and convex values. For any given ϕ ∈ E^{∗}, without loss of its generality, we might assume that the functional ϕ has the form as ϕ(u, v) := au + bv, where both a and b are two constants determined by the functional ϕ. If either a or b is positive, it is clear that sup_{(u,v)∈F (x)} ϕ(u, v) = +∞. In the case both a and b are negative, we can also check

out that sup_{(u,v)∈F (x)} ϕ(u, v) = ax − 2\sqrt{ab}. Thus the function x ↦ sup_{(u,v)∈F (x)} ϕ(u, v) is continuous,

which means that the set-valued mapping F is upper hemicontinuous. However, for any x ∈ X, let u := x. Then u is an interior asymptotic plane of F (x). By applying Theorem 1.2.9, F is not upper demicontinuous.

Corollary 1.2.11. Let X be a Hausdorff topological space and E a Hausdorff locally convex topological vector space. If the set-valued mapping F : X → 2^{E} \ {∅} is upper hemicontinuous with non-empty and compact convex values, then F is upper demicontinuous.

Proof. We first show that each compact set A of E has no asymptotic plane as pointed out in the previous Remark 1.2.6. Indeed, if A has an asymptotic plane M_{ϕ,λ}, where ϕ ∈ E^{∗} and λ ∈ R, by Definition 1.2.5, for any u ∈ A, we have that ϕ(u) ≠ λ and there exists δ > 0 such that for any μ ∈ (λ − δ, λ), there is x̂ ∈ A such that ϕ(x̂) = μ. Now define B := {u ∈ A : ϕ(u) ≤ λ}. Then B is also a compact subset of E. Therefore, there exists u_{0} ∈ B such that ϕ(u_{0}) = sup_{u∈B} ϕ(u). By the arbitrariness of μ ∈ (λ − δ, λ), it follows that ϕ(u_{0}) = λ. Note that M_{ϕ,λ} is an asymptotic plane of A, and by the definition of set B, it follows that for each u ∈ B, ϕ(u) < λ. Therefore ϕ(u_{0}) < λ. This is impossible as we just showed that ϕ(u_{0}) = λ. Thus the compact set A in E has no asymptotic planes. By Theorem 1.2.9, F is upper demicontinuous and the proof is completed. □

As another application of Theorem 1.2.9, we have the following result which says that each upper hemicontinuous mapping with non-empty convex and weakly compact values is upper demicontinuous.

Corollary 1.2.12. Let X be a Hausdorff topological space and E a Hausdorff locally convex topological vector space. If the set-valued mapping F : X → 2^{E} \ {∅} is hemicontinuous with non-empty weakly compact convex values, then F is demicontinuous.

Proof. If there exists x ∈ X such that F (x) has an asymptotic plane M_{ϕ,λ}, then for any u ∈ F (x), we have that ϕ(u) ≠ λ and secondly, there exists δ > 0 such that for each μ ∈ (λ − δ, λ), there is u ∈ F (x) such that ϕ(u) = μ. Now define a set B := {u ∈ F (x) : ϕ(u) ≤ λ}. Then for any u ∈ B, it follows that ϕ(u) < λ. Let {μ_{n}}_{n∈N} be a sequence of the set (λ − δ, λ) such that lim_{n→∞} μ_{n} = λ. Then by the definition of the set B and our discussion above, there exists {u_{n}} ⊂ B such that ϕ(u_{n}) = μ_{n}. As each weakly compact set in the Hausdorff locally convex topological vector space E is weakly closed, B is also convex weakly compact, and thus there exists a directed subfamily {u_{v}}_{v∈D} ⊂ {u_{n}} and a point u_{0} ∈ B such that for any ϕ ∈ E^{∗}, we have that lim_{v} ϕ(u_{v}) = ϕ(u_{0}). Therefore, ϕ(u_{0}) = lim_{v} ϕ(u_{v}) = lim_{v} μ_{v} = λ. This contradicts with that for each u ∈ B, ϕ(u) < λ as shown above. Therefore F has no asymptotic planes. By Theorem 1.2.9, F is demicontinuous and the proof is completed. □

Corollary 1.2.13. Let X be a Hausdorff topological space and E a reflexive Banach space. If the set-valued mapping F : X → 2^{E} \ {∅} is hemicontinuous with non-empty bounded and closed convex set, then F is demicontinuous.

Proof. Indeed, it is a special case of Corollary 1.2.12. Here we do give a short proof by using Theorem 1.2.9 as follows. If there exists x ∈ X such that F (x) has an asymptotic plane M_{ϕ,λ}, then on one hand, we know that the sets F (x) and M_{ϕ,λ} can’t be strongly separated by any real hyperplane; on the other hand, note that E is reflexive Banach space, F (x) is bounded closed convex and so is M_{ϕ,λ}, then the sets F (x) and M_{ϕ,λ} can be strongly separated by a real hyper-plane. This is impossible. Finally the conclusion is derived by Theorem 1.2.9. □

Remark 1.2.14. We would like to note that all results so far in this section still hold if the locally topological vector space E is replaced by a topological vector space (not necessarily having locally convex structure) with many sufficient continuous linear functionals.

Before we end this section, we will give another characterisation for the set-valued mapping F_{r} defined below on a normed space being upper semicontinuous.

Let X be a normed space and for any given r > 0, we define a set-valued mapping F_{r} : X → 2^{X} by

F_{r}(x) = {y ∈ X : ∥x − y∥ ≤ r}

for each x ∈ X. For each sets A and B in X, the distance of A and B is defined by dist(A, B) := inf_{a∈A,b∈B} ∥a − b∥. We first need the following result.

Lemma 1.2.15. Let (E, ∥ · ∥) be a (finite-dimensional) normed space and U a non-empty closed subset of E. Let B_{r}(x) := {z ∈ E : ∥z − x∥ ≤ r} and d := dist(B_{r}(x), U), where r ∈ ℝ and x ∈ E. If U ∩ B_{r}(x) = ∅, then for each y ∈ U, we have ∥y∥ ≥ r + d.

Proof. For each y ∈ U, as y ∉ B_{r}(x), so that ∥y − x∥ > r. Now we rewrite y as y = (1 − \frac{r}{∥y−x∥})(y − x) + \frac{r}{∥y−z∥}(y − x) + x. Let b := (1 − \frac{r}{∥y−x∥})(y − x) and c := \frac{r}{∥y−z∥}(y − x). As \frac{r}{∥y−x∥} < 1, we have that ∥y − x∥ = ∥b∥ + ∥c∥. Note that c ∈ B_{r}(x) and b ∈ U − B_{r}(x). Thus ∥b∥ ≥ d and we have ∥y − x∥ ≥ r + d.

We also need the following classic result from functional analysis, for example, see Wong [?, p.67].

Riesz’s Lemma. Let X_{0} be a proper and closed subspace of a normed space X. Then for any r ∈ (0, 1), there exists x_{r} ∈ X such that ∥x_{r}∥ = 1 and ∥x − x_{r}∥ ≥ r for all x ∈ X_{0}.

Now we have the following characterisation for the mapping F_{r} being upper semicontinuous defined on a normed space if and only if the underlying space is finite dimensional.

Theorem 1.2.16. For any given r > 0, the set-valued mapping F_{r} is upper semicontinuous if and only if X is a finite dimensional space.

Proof. \underline{Sufficiency}: Suppose it is not true. As the translation of the mapping F_{r} preserves its upper semicontinuity at a point, without loss of generality, we may assume that F_{r} is not upper semicontinuus at 0. ¿From the definition of upper semicontinuity, there exist a non-empty open subset O of E with F_{r}(0) ⊂ O and a sequence {x_{n}}_{n∈ℝ} with lim_{n→∞} x_{n} = 0 such that for each n ∈ ℕ, there exists y_{n} ∈ F_{r}(x_{n}) with y_{n} ∉ O. Let U be the complement of u in E, i.e., U = {x ∈ E : x ∉ O}. Then U is a closed subset of E. Let d := dist(B_{r}(0), U). We have that d > 0 as B_{r}(0) ∩ U = ∅ and both B_{r}(0) and U are non-empty and closed. Moreover ∥y_{n}∥ ≥ d + r for all n ∈ ℕ from Lemma 1.2.15. On the other hand, y_{n} ∈ B_{r}(x_{n}), so that ∥y_{n} − x_{n}∥ ≤ r for all n ∈ ℕ. Therefore

lim sup_{n→∞} ∥y_{n}∥ ≤ lim sup_{n→∞} ∥y_{n} − x_{n}∥ + lim_{n→∞} ∥x_{n}∥ ≤ r.

This conclusion contradicts with that ∥y_{n}∥ ≥ r + d for all n ∈ ℕ. Thus F must be upper semicontinuous at 0.

\underline{Necessity}: By following Kucia’s idea [?], without loss of generality, we may assume that r = 1. Indeed, if X is infinite dimensional, by considering the point x = 0, and Riesz’s Lemma, there exists b_{n} ∈ F_{1}(0) such that ∥b_{n}∥ = 1 and ∥b_{n} − b_{m}∥ ≥ \frac{1}{2} for all n ≠ m. Let a_{n} = (1 + \frac{1}{n})b_{n} and x_{n} = \frac{1}{n}b_{n}. Then

\frac{1}{2} ≤ ∥b_{n} − b_{m}∥ = ∥a_{n} − a_{m} − \frac{1}{n}b_{n} + \frac{1}{m}b_{m}∥ ≤ ∥a_{n} − a_{m}∥ + \frac{1}{n} + \frac{1}{m}.

Note that \frac{1}{4} ≤ ∥a_{n} − a_{m}∥ for all m > n ≥ 8, it follows that the set C = \{a_{n} : n = 1, 2, 3, ⋯ \} is closed in X and F_{1}(0) ⊂ X \ C = U. It is clear that lim_{n→∞} x_{n} = 0, a_{n} ∈ F_{1}(x_{n}) and F_{1}(V) ⊂ U as a_{n} ∉ U (i.e., for any neighbourhood V of the point 0, F_{1}(V) ⊄ U). This means that the mapping x ↦ F_{1}(x) is not upper semicontinuous at x = 0.

Chapter 2

Fixed Point Theorems of Set-Valued Mappings

In this Chapter, we focus in the discussion of fixed point theory for set-valued mappings by using Knaster-Kuratowski-Mazurkiewicz (KKM) principle in both topological vector spaces and hyperconvex metric spaces. In particular, the fixed point theory of set-valued mappings for Browder-Fan and Fan-Glicksberg type fixed points have been extensively studied in the setting of locally convex spaces, H-spaces, G-convex spaces and metric hyperconvex spaces. By using its own feature of hyperconvex metric spaces being a special class of H-spaces, we also establish its general KKM theory and then its various applications. In section 2, we first discuss some recent development of KKM theory itself and the general Ky Fan minimax principle is given in section 3. In sections 4 and 5, two types of Ky Fan minimax inequalities and their equivalent fixed point forms for setvalued mappings are given. In section 6, Browder-Fan fixed point theorems in production spaces for couple-majorized and L_{s}-majorized set-valued mappings are discussed in terms of maximal elements. In section 7, the general Fan-Glicksberg type fixed point theorem is discussed in G-Convex spaces which include locally convex H-spaces, locally convex topological vecstor spaces and metric hyperconvex metric spaces as special cases. Finally, the general KKM theory and its various application in metric hyperconvex spaces and the generic stability of fixed points are discussed in section 8.

2.1 Introduction

The classical Knaster-Kuratowski-Mazurkiewicz principle (often called KKM theorem, or KKM lemma [?]) has numerous applications in various fields of pure and applied mathematics. These studies and applications are called the KKM Theory today.

In 1961, Ky Fan proved the generalization of the classical KKM theorem in infinite dimensional Hausdorff topological vector spaces and established an elementary but very basic geometric lemma for set-valued mappings. In 1968, Browder gave a fixed point form of Fan’s geometric lemma and it is now called the Browder-Fan fixed point theorem. Since then there have been numerous generalizations of the Browder-Fan fixed point theorem and their applications in coincidence and fixed point theory, minimax inequalities, variational inequalities, nonlinear analysis, convex analysis, game theory, mathematical economics, and so on. By applying his geometric lemma in 1972, Ky Fan obtained a minimax inequality which plays a fundamental role in nonlinear analysis and mathematical economics and has been applied to potential theory, partial differential equations, monotone operators, variational inequalities, optimization, game theory, linear and nonlinear programming, operator theory, topological group and linear algebra. In particular, by using Ky Fan’s minimax inequality, a more general form of the Fan-Glicksberg fixed point theorem is derived for set-valued mappings which are inward (or outward) as defined by Fan in 1969 which are more general than Halpern’s definitions for inward (or outward) mappings in 1965.

Recently, Horvath obtained some generalizations of Fan’s geometric lemma and his minimax inequality in 1983 and 1987 by replacing convexity assumption with topological properties: pseudoconvexity and contractibilty. By extending Horvath’s concepts, Bardaro and Ceppitelli [?] in 1988 obtained generalizations of Ky Fan minimax inequalities to topological spaces which have a so called H-structure (also called H-spaces). Following this line, a number of generalizations of Ky Fan’s minimax inequalities are given by Horvath [?], Baradaro and Ceppitelli [?], Ding and Tan [?], Ding et al. [?], Chang and Ma [?], Park [?], Tarafdar [?], and Tan et al. [?] in topological spaces which need not have a linear structure but with an H-structure.

Throughout this Chapter all spaces are assumed to be Hausdorff unless otherwise specified. Let X and Y be non-empty sets. We shall denote by 2^{Y} the family of all non-empty subsets of Y, F(X) the family of all non-empty finite subsets of X. Let X be a topological space. For each non-empty subset A of X, we denote the closure of A in X by cl_{X}A (in short, clA) or \overline{A} if there is no confusion. A subset A of X is said to be compactly closed (resp., open) if A ∩ C is closed (resp., open) in each non-empty compact subset C of X.

2.2 Knaster-Kuratowski-Mazurkiewicz Principle

In this section, we will discuss some recent development of KKM theory in H-spaces.

A topological space X is said to be contractible if the identity mapping I_{X} of X is homotopic to a constant function. Let ℕ and ℝ denote the set of all natural numbers and the set of all real numbers, rspectively. Let v_{0}, v_{1}, ⋯, v_{n} be n + 1 points in a Euclidean space which are not contained in any linear manifold (i.e., a translation of a vector subspace) of dimension less than n. The convex hull of these n + 1 points is called an n-dimensional simplex and is denoted by v_{0}v_{1} ⋯ v_{n}. The points v_{0}, v_{1}, ⋯, v_{n} are also called vertices of the simplex. For 0 ≤ k ≤ n and 0 ≤ i_{0} < i_{1} < ⋯ < i_{k} ≤ n, the k-simplex v_{i_{0}}v_{i_{1}} ⋯ v_{i_{k}} is a subset of the n-simplex v_{0}v_{1} ⋯ v_{n}. It is called a k-dimensional face (or say, simply k-face) of the simplex v_{0}v_{1} ⋯ v_{n}.

For the convenience of our discussion, for n ∈ ℕ and N = {0, 1, ⋯, n}, we also denote by Δ_{N} = co{e_{0}, ⋯, e_{n}} the standard simplex of dimension n, where {e_{0}, ⋯ e_{n}} is the canonical basis of ℝ^{n+1} and for J ∈ F(N), Δ_{J} = co{e_{j} : j ∈ J}. The points e_{0}, e_{1}, ⋯, e_{n} are called vertices of the simplex Δ_{N}. Suppose that T = {τ_{0}, τ_{1}, ⋯, τ_{m}} is finite collection of n-simplexes in v_{0}v_{1} ⋯ v_{n} such that (i): ∪_{i=1}^{m}τ_{i} = v_{0}v_{1} ⋯ v_{n}; and (ii): τ_{i} ∩ τ_{j} is either empty or a common face of any dimension of τ_{i} and τ_{j}, where i, j ∈ {1, 2, ⋯, m}. Then T is called a triangulation of v_{0}v_{1} ⋯ v_{n}. Let T be a triangulation. A vertex of a simplex in T is called a vertex of T. An (n − 1)-face of a simplex from T is called a boundary (n − 1) -simplex of T if it is a face of exactly one n-simplex of T.

In this section, we first present an extremely useful result published in 1929 due to Knaster, Kuratowski, and Mazurkiewicz [?], which is regarded as one of the most important principles in nonlinear analysis, and also provides the foundation for our study in this part. Its proof is based on the celebrated Sperner’s combinatorial lemma [?] published in 1928. There are several versions of Sperner’s Lemma and its extensions, for more details of its proofs, see Border [?, p.23-24], Cohen [?], Dugundji and Granas [?], Fan [?], Shashkin [?, p.9-14], Shih and Lee [?], Wong [?], Yuan [?], Zeidler [?] and many other books for its numerous and different proofs.

KKM Principle. Let C_{0}, ⋯, C_{n} be closed subsets of the standard n-dimensional simplex Δ_{N} and let {e_{0}, ⋯, e_{n}} be the set of its vertices. If for each J ∈ F(N), Δ_{J} ⊂ ∪_{j∈J}C_{j}. Then ∩_{i=0}^{n}C_{i} ≠ ∅.

We would like to note that KKM principle above can used proved by using the following celebrated Sperner’s combinatorial lemma appeared in 1928 which is stated as follows without proof (see Yuan [?] for more details).

Sperner’s Lemma. Let T be a triangulation of an n-simplex Δ = v_{0}v_{1} ⋯ v_{n}. To each vertex v of T, let ψ(v) ∈ {0, 1, ⋯, n} be such that whenever v ∈ v_{i_{0}}v_{i_{1}} ⋯ v_{i_{k}}, ψ(v) ∈ {i_{0}, i_{1}, ⋯, i_{k}} for 0 ≤ i_{0} < i_{1} < ⋯ i_{k} ≤ n. Then the number of n-simplexes in T whose vertices receive n + 1 different values is odd.

On the other hand, Shih and Tan [?] (see also Kim [?] or Lassonde [?]) provided the following dual form of KKM theorem in the sense that the word closed could be replaced by the word open.

The Dual Form of KKM Principle. Let C_{0}, ⋯, C_{n} be open subsets of the standard ndimensional simplex Δ_{N} and let {e_{0}, ⋯, e_{n}} be the set of its vertices. If for each J ∈ F(N), Δ_{J} ⊂ ∪_{j∈J}C_{j}. Then ∩_{i=0}^{n}C_{i} ≠ ∅.

By following Shih and Tan [?], a set-valued mapping B : F → 2^{Δ_{N}} is said to be Shapleymapping if for each τ ∈ F, we have

τ ⊂ ∪_{τ⊃ρ∈F}B(ρ).

Let C_{0}, C_{1}, ⋯, C_{n} be subsets of a standard n-simplex Δ_{N}. If we define a mapping B : F → 2^{Δ_{N}} by B(i) = C_{i} for each i ∈ N = {0, 1, ⋯, n} and B(ρ) = ∅ for all non 0-dimensional faces of Δ_{N}. Then it is easy to see that B is a Shapley mapping if and only if the following condition is satisfied:

Δ_{J} ⊂ ∪_{j∈J}C_{j} = ∪_{j∈j}B(j)

for each J ∈ F(N).

Now we state the following selection theorem due to Shih and Tan [?].

Theorem 2.2.1. If B : F → 2^{Δ_{N}} is a Shapley-mapping such that each B(ρ) is an open subset of Δ_{N} for each ρ ∈ F. Then B admits a Shapley-selection A : F → 2^{Δ_{N}} such that each A(ρ) is a closed subset of Δ_{N} and A(ρ) ⊂ B(ρ) for all ρ ∈ F.

As an application of Shih and Tan’s selection theorem for Shapley mapping and the KKM principle, we can now prove the dual form of the KKM principle above.

The Proof of the dual form of the KKM principle: We define a mapping B : F → 2^{Δ_{N}} by B(i) = C_{i} for each i ∈ N = {0, 1, ⋯, n} and B(ρ) = ∅ for all non-0-dimensional faces of Δ_{N}. As we note above that B is a Shapley mapping if and only if Δ_{J} ⊂ ∪_{j∈J}C_{j} = ∪_{j∈j}B(j) for each J ∈ F(N). By Theorem 2.2.1, there exists another Shapley mapping A : F → 2^{Δ_{N}} such that A(ρ) is closed and A(ρ) ⊂ B(ρ) for each ρ ∈ F. Let D_{i} = A(i) for each i ∈ N = {0, 1, ⋯, n}. Then we have that D_{i} is closed and D_{i} ⊂ C_{i} for all i ∈ N. Since A is also a Shapley mapping, it follows that Δ_{J} ⊂ ∪_{j∈J}D_{j} = ∪_{j∈j}A(j) for each J ∈ F(N). By the KKM principle, we have that ∩_{i=0}^{n}A(i) ≠ ∅. Thus it implies that ∩_{i=0}^{n}B(i) ≠ ∅ and the proof is completed. □.

The following notions introduced by Bardaro and Ceppitelli in [?] were motivated by the earlier work of Horvath [?].

Definition 2.2.2. A pair (X, {Γ_{A}}) (also called an H-structure is said to be an H-space (also called C-space according to Horvath [?]) if X is a topological space and {Γ_{A}}_{A∈F(X)} a given family of non-empty contractible subsets Γ_{A} of X, indexed by A ∈ F(X) such that Γ_{A} ⊂ F_{B} whenever A ⊂ B. Let (X, {Γ_{A}}) be an H-space. A non-empty subset D of X is said to be: (i) H-convex (also called an F -set by Horvath [?]) if Γ_{A} ⊂ D for each A ∈ F(D); (ii) weakly H-convex if Γ_{A} ∩ D is contractible for each A ∈ F(D) (or equivalently, (D, {Γ_{A} ∩D}) is an H-space); and (iii) H-compact in X if for each A ∈ F(X), there exists a compact, weakly H-convex subset D_{A} of X such that D ∪ A ⊂ D_{A}. It is clear that the product space of a family of H-spaces is also an H-space.

The following example from Horvath [?, p.345] shows that an H-space may not be a convex subset of a topological vector space.

Example 2.2.3. Let X be a convex set in a topological vector space E and Y any topological space. Suppose that f : X → Y is a continuous bijection. For given A ∈ F(Y), let D_{A}:= co{x ∈ X : f(x) ∈ A}. Then D_{A} is convex, so that D_{A} is contractible. Since D_{A} is also compact, so that f : D_{A} → f(D_{A}) is an homeomorphism. Let Γ_{A} = f(D_{A}). Then Γ_{A} is contractible and Γ_{A} ⊂ Γ_{A′} whenever A ⊂ A′ ∈ F(Y). Therefore (Y, {Γ_{A}}) is an H-space. Note that the space Y itself may be a torus, the Mobius band, or the Klein bottle. This example shows that an H-space does not have to be contractible.

Definition 2.2.4. Let D be a non-empty subset of an H-space (X, {Γ_{A}}). By following Tarafdar [?], H-convex hull of D, denoted by Hco(D), is defined by

Hco(D) = ∩{B ⊂ X : B is H-convex and D ⊂ B}.

Clearly, Hco(D) is the smallest H-convex subset of X containing D and the intersection of any family of H-convex set is also H-convex. Let X be a topological space. By following Horvath [?], an n-dimensional singular face structure on X is a mapping F : F(N) → 2^{X} such that (a) for each J ∈ F(N), F (J) is not-empty and contractible and (b) for any J, J′ ∈ F(N), J ⊂ J′ implies F (J) ⊂ F (J′).

Let N = {0, 1, ⋯, n}. We denote by ∂Δ_{N} and Δ_{N}^{k} the boundary and kth-skeleton of ndimensional simplex Δ_{N}, respectively. By the fact that if X is a contractible topological space, each (single-valued) continuous function f : ∂Δ_{N} → X can be continuously extended to Δ_{N}, i.e., there exists a continuous function g : Δ_{N} → X such that g(x) = f(x) for all x ∈ ∂Δ_{N}, we have the following continuous selection result for H-structures.

Lemma 2.2.5. let X be a topological space. For each non-empty subset J of N = {0, 1, ⋯, n}, let F_{J} be a non-empty contractible subset of X with F_{J} ⊂ F_{J′} whenever ∅ ≠ J ⊂ J′ ⊂ {0, 1, ⋯, n}. Then there exists a continuous function f : Δ_{N} → X such that f(Δ_{J}) ⊂ F_{J} for each non-empty subset J of {0, 1, ⋯, n}.

Proof. For each singleton {i}, take any fixed point x_{i} ∈ F_{{i}}. Then f^{0} : Δ_{N} → X defined by f^{0}(e_{i}) := x_{i} is a continuous function on the 0th-skeleton Δ_{N}^{0} of Δ_{N}. Now assume that a continuous function f^{k} : Δ_{N}^{k} → X on the kth-skeleton of Δ_{N} has been constructed in a such way such that f^{k}(Δ_{J}) ⊂ F_{J} if cardJ ≤ k + 1. Let Δ_{J} be a face of dimension k + 1 of Δ_{N} and for each i ∈ J, let J_{i} := J \ {i}. Then the boundary set ∪_{i∈J}Δ_{J_{i}} of Δ_{J} is contained in the kth-skeleton of Δ_{N} and f^{k}(∂Δ_{J}) ⊂ ∪_{i∈J}f^{k}(Δ_{J_{i}}) ⊂ ∪_{i∈J}F_{J_{i}} ⊂ F_{J}. By the fact that F_{J} is contractible, f^{k} can be extended to a continuous function f_{J}^{k+1} : Δ_{J} → F_{J}. If Δ_{J} and Δ_{J′} are two different k+1 dimensional faces of Δ_{N} with non-empty intersection, then f_{J}^{k+1} and f_{J′}^{k+1} coincide with the continuous function f^{k} on Δ_{J} ∩Δ_{J′}. By gluing the functions f_{J}^{k+1} together, we obtain a continuous function f^{k} : Δ_{N}^{k+1} → X, which has the desired property. □

Proposition 2.2.6. Let X be a topological space. Let F : F(N) → 2^{X} be a singular face structure on X and {M_{i} : i = 0, ⋯, n} be a family of closed (resp., open) subsets of X such that for any J ∈ F(N), F (J) ⊂ ∪_{i∈J}M_{i}. Then ∩_{i=0}^{n}M_{i} ≠ ∅.

Proof. By Lemma 2.2.5, there is a continuous function f : Δ_{N} → X such that for each J ∈ F(N), f(Δ_{J}) ⊂ F (J). Let C_{i} = f^{−1}(M_{i}) for each i = 0, ⋯, n. Then {C_{i}}_{i=0}^{n} is a family of closed (resp., open) subsets of Δ such that for any J ∈ F(N), Δ_{J} ⊂ ∪_{i∈J}C_{i}. By the KKM principle (resp., its dual form), ∩_{i=0}^{n}C_{i} ≠ ∅. Take any x_{0} ∈ ∩_{i=0}^{n}C_{i}, then f(x_{0}) ∈ ∩_{i=0}^{n}M_{i} ≠ ∅. □

As an application of Proposition 2.2.6, we have the following result.

Theorem 2.2.7. Let X be a contractible topological space, {M_{i} : i = 0, ⋯, n} be a closed (resp., open) covering of X and {F_{i} : i = 0, ⋯, n} a family of contractible subsets of X such that (i) for any i ∈ {0, ⋯, n}, F_{i} ∩M_{i} = ∅ and (ii) for any J ∈ F(N) with J ≠ N, ∩_{i∈J}F_{i} is non-empty and contractible. Then ∩_{i=0}^{n}M_{i} ≠ ∅.

Proof. Define F : F(N) → 2^{X} by F (N) = X and F (J) = ∩_{i∉J}F_{i} if J ∈ F(N) with J ≠ N. Then F is a singular face structure on X. Note that for any J ∈ F(N), F (J) ⊂ ∪_{i∈J}M_{i} since F (J) ⊂ ∪_{i∈J}M_{i} ∪ ∪_{i∉J}M_{i} and F (J) ∩ M_{i} = ∅ whenever i ∉ J. Then Proposition 2.2.7 implies that ∩_{i=0}^{n}M_{i} ≠ ∅. □

Theorem 2.2.7. Let X be a contractible topological space and Y be a topological space, {M_{i} : i = 0, ⋯, n} be an open (resp., closed) covering of Y and {F_{i} : i = 0, ⋯, n} be a family of contractible subsets of X. Let S : X → Y be continuous such that

(a) for each i ∈ {0, 1, ⋯, n}, F_{i} ⊂ S^{−1}(M_{i}); and

(b) for each J ∈ F(N) with J ≠ N, ∩_{i∈J}F_{i} is non-empty and contractible.

Then ∩_{i=0}^{n}M_{i} ≠ ∅.

Proof. Suppose the contrary, so that ∪_{i=0}^{n}M_{i}^{C} = Y, where M_{i}^{C} = Y \ M_{i} denotes the complement of M_{i} in Y for each i = 0, 1, ⋯, n. Thus {M_{i}^{C} : i = 0, ⋯, n} is an open (resp., closed) covering of Y. So that {S^{−1}(M_{i}^{C}) : i = 0, ⋯, n} is an open (resp., closed) covering of X and by condition of (a), for each i = 0, 1, ⋯, n, F_{i} ∩ S^{−1}(M_{i}^{C}) = ∅. Therefore F_{i} and S^{−1}(M_{i}^{C}) for i = 0, ⋯, n satisfy all hypotheses of Theorem 2.2.7. By Theorem 2.2.7, ∩_{i=0}^{n}S^{−1}(M_{i}^{C}) ≠ ∅ which contradicts that {M_{i} : i = 0, ⋯, n} is a covering of Y. Thus ∩_{i=0}^{n}M_{i} ≠ ∅. □

Since Chang and Yang [?] gave a generalization of the KKM theorem in which the domain need not be a subset of its range, there are several generalizations in this direction. For example, Chang and Ma [?] extended this definition into H-spaces and later Zhou [?] gave a more generalized definition and obtained a characterization of the generalized HKKM mapping which is also a generalization of the corresponding result given by Chang and Zhang [?].

Definition 2.2.9. Let X be a non-empty set and Y a topological space. A mapping G : X → 2^{Y} ∪ {∅} is said to be transfer closed valued (e.g., see Zhou and Tian [?]) if for each x ∈ X and y ∉ G(x), there exist x′ ∈ X and an open neighborhood N(y) of y in Y such that y′ ∉ G(x′) for each y′ ∈ N(y). It is obvious that if a mapping G : X → 2^{Y} is transfer closed valued, then for each x ∈ X and y ∈ Y with y ∉ G(x), there exists some x′ ∈ X such that y ∉ clG(x′).

The following lemma was first proved by Zhou and Tian [?] for the case when the domain X is a topological space and in the present form by Zhou [?]; for completeness, we include its proof here.

Lemma 2.2.10. Let X be a non-empty set, Y a topological space and G : X → 2^{Y}. Then ∩_{x∈X}G(x) = ∩_{x∈X}clG(x) if and only if the mapping G is transfer closed valued.

Proof. \underline{Sufficiency}: It is clear that ∩_{x∈X}G(x) ⊂ ∩_{x∈X}cl_{Y} G(x). It is sufficient to show that ∩_{x∈X}clG(x) ⊂ ∩_{x∈X}G(x). Suppose that y ∉ ∩_{x∈X}G(x). Then there exists some x ∈ X such that y ∉ G(x). Since G is transfer closed valued on X, there exists some x′ ∈ X such that y ∉ cl_{Y} G(x′), so that y ∉ ∩_{x∈X}clG(x). Therefore ∩_{x∈X}clG(x) = ∩_{x∈X}G(x).

\underline{Necessity}: Suppose (x, y) ∈ X × Y such that y ∉ G(x); then y ∉ ∩_{z∈X}G(z) = ∩_{z∈X}clG(z) so that there exists x′ ∈ X such that y ∉ clG(x′). But then there exists an open neighborhood N(y) of y in Y such that N(y) ∩ G(x′) = ∅ so that y′ ∉ G(x′) for all y′ ∈ N(y). Thus G is transfer closed valued. □.

Definition 2.2.11. Let D be a non-empty subset of an H-space (X, {Γ_{A}}). A mapping F : D → 2^{X} is called HKKM if Γ_{A} ⊂ ∪_{x∈A}F (x) for each A ∈ F(X). When X is a non-empty convex subset of a topological vector space and Γ_{A} = coA, the convex hull of A for each A∈ F(X), then (X, {Γ_{A}}) becomes an H-space. In this case, the notion of an HKKM mapping F : D → 2^{X} coincides with the notion of a KKM mapping F, i.e., coA ⊂ ∪_{x∈A}F (x) for each A ∈ F(D).

Definition 2.2.12. Let X be a non-empty set and Y a topological space. A mapping G : X → 2^{Y} is said to be a generalized HKKM (in short, GHKKM) if for each finite subset A = {x_{1}, ⋯, x_{n}} of X, there exists a corresponding finite subset B = {y_{1}, y_{2} ⋯, y_{n}} (y′_{i}s need not be distinct here) in Y and a family {Γ_{C}}_{C∈F(B)} of non-empty contractible subsets of Y such that Γ_{C} ⊂ Γ_{C′} whenever C ⊂ C′ ∈ F(Y) such that

Γ_{{y_{i_{j}}:j∈J}} ⊂ ∪_{j=1}^{s}G(x_{i_{j}})

for ∅ ≠ J ⊂ {0, 1, ⋯, n}.

It is clear that each HKKM mapping is GHKK and the following example from Chang and Yang [?] shows that the converse does not hold.

Example 2.2.13. Let E = (−∞, +∞) and X = [−2, +2]. Define a set-valued mapping G : X → 2^{E} by G(x) = [−(1 + \frac{x^{2}}{5} ), \frac{x^{2}}{5} ], for each x ∈ X. Since ∪_{x∈X}H(x) = [−9/5, 9/5] and x ∉ G(x) for each x ∈ [−2, −9/5) ∪ (9/5, 1]. This shows that G is not a KKM mapping. Next we prove that G is a generalized KKM mapping. In fact, for any finite subset {x_{1}, ⋯, x_{n}} ⊂ X, take {y_{1}, ⋯, y_{n}} ⊂ [−1, 1], then for any finite subset {y_{i_{1}}, ⋯, y_{i_{k}}} ⊂ {y_{1}, ⋯, y_{n}}, we have

co{y_{i_{1}}, ⋯, y_{i_{k}}} ⊂ [−1, 1] = ∩_{x∈X}G(x) ⊂ ∪_{j=1}^{k}G(x_{i_{j}})

Thus G is GHKKM.

Theorem 2.2.14. Let X be a non-empty set and both Y and Z two topological spaces. Suppose S : Y → Z is continuous and G : X → 2^{Z} is such that:

(1) the composition mapping S^{−1} ∘ G : X → 2^{Y} defined by (S^{−1} ∘ G)(x) = ∪_{z∈G(x)}{y ∈ Y : z = S(y)} for each x ∈ X, is a generalized HKKM mapping; and

(2) for each x ∈ X, G(x) is closed (resp., open) in Y.

Then the family {G(x) : x ∈ X} has the finite intersection property, i.e., for each A ∈ F(X), ∩_{x∈A}G(x) ≠ ∅.

Proof. For any finite subset {x_{0}, x_{1}, ⋯, x_{n}} of X, since S^{−1} ∘ G : X → 2^{Y} is a generalized HKKM mapping, there exists a finite subset B = {y_{0}, y_{1}, ⋯, y_{n}} of Y and a family {Γ_{C}}_{C∈F(B)} of non-empty contractible subsets of Y such that Γ_{C} ⊂ Γ_{C′} whenever C ⊂ C′ such that

Γ_{{y_{i_{0}} ,y_{i_{1}} ,··· ,y_{is} }} ⊂ ∪_{j=0}^{s}(S^{−1} ∘ G)(x_{i_{j}})

for each finite subset {y_{i_{0}}, y_{i_{1}}, ⋯, y_{i_{s}}} of {y_{0}, y_{1}, y_{2}, ⋯, y_{n}}, where (0 ≤ s ≤ n). Let M_{i} = S^{−1}(G(y_{i})) for each i = 0, 1, ⋯, n; and define a mapping F : F(N) → Y by F (J) = Γ_{{y_{k}:k∈J}} for each J ∈ F(N). Since S is continuous, M_{i} is closed (resp., open) in Y for i = 0, 1, ⋯, n by the assumption (2). Moreover the mapping F is a singular face structure on Y. Therefore all hypotheses of Proposition 2.2.6 are satisfied. By Proposition 2.2.6, ∩_{i=0}^{n}M_{i} ≠ ∅. Take any y_{0} ∈ ∩_{i=0}^{n}M_{i}, then S(y_{0}) ∈ ∩_{i=0}^{n}G(x_{i}) ≠ ∅. □

As an application of Theorem 2.2.14, we have the following result due to Zhou [?].

Theorem 2.2.15. Let X be a non-empty set and Y a topological space. Let G : X → 2^{Y} be such that

(a) G is transfer closed valued on X; and

(b) there exists a non-empty finite subset X_{0} of X such that the set Y_{0} = ∩_{x∈X_{0}}clG(x) is non-empty and compact in Y.

Then the intersection ∩_{x∈X}G(x) is non-empty and compact if and only if the mapping clG is a generalized HKKM mapping.

Proof. \underline{Necessity}: Suppose ∩_{x∈X}G(x) is non-empty and compact. Take any y_{0} ∈ ∩_{x∈X}G(x). Note that the singleton set {y_{0}} is contractible. For each A = {x_{1}, ⋯, x_{n}} ∈ F(X), take B = {y_{1}, ⋯, y_{n}} with y_{i} = y_{0} for all i = 1, 2, ⋯, n and let Γ_{B′} = {y_{0}} for all B′ ∈ F(B). Since y_{0} ∈ clG(x) for all x ∈ X, it is clear that the mapping clG is generalized HKKM.

\underline{Sufficiency}: Since the mapping clG is a generalized HKKM by Theorem 2.2.14 with Y = Z and S being the identity mapping on Y, the family {clG(x) : x ∈ X} has the finite intersection property. Now define a mapping G′(x) = cl_{Y} G(x) ∩ Y_{0} for each x ∈ X. Then the family of non-empty compact subsets {G′(x) : x ∈ X} has the finite intersection property, so that ∩_{x∈X}cl_{Y} G(x) = ∩_{x∈X}G′(x) ≠ ∅. Since G is transfer closed, ∩_{x∈X}G(x) = ∩_{x∈X}clG(x) by Lemma 2.2.10. Therefore ∩_{x∈X}G(x) ≠ ∅ □

An an immediate consequence of of Theorem 2.2.15, we have the following characterization of non-empty intersections for generalized HKKM mappings.

Theorem 2.2.16. Let X be a non-empty set and Y a compact topological space. Let G : X → 2^{Y} be transfer closed valued on X. Then the intersect ∩_{x∈X}G(x) is non-empty if and only if the mapping clG is a generalized HKKM mapping.

Theorem 2.2.16 is a generalization of the corresponding results given by Chang and Yang [?] and Chang and Ma [?].

As special cases of Theorem 2.2.16, we have the following famous extension of the classical KKM principle (KKM Lemma) in topological vector spaces to Fan [?] which plays a very important role in the study of nonlinear analysis today.

Theorem 2.2.17. Let X be a non-empty subset of a topological vector space E and F : X → 2^{E} be a set-valued KKM mapping. Then the family {F (x) : x ∈ X} has the finite intersection property, i.e., ∩_{x∈A}F (x) ≠ ∅ for each A ∈ F(X).

As a finite dimensional version of Theorem 2.2.17, we have the following classical FKKM principle.

Corollary 2.2.18 (FKKM Principle). Let X be a non-empty subset of a finite dimensional space ℝ^{n}, where n ∈ ℕ. Suppose F : X → 2^{ℝ^{n}} is a set-valued KKM mapping. Then the family {F (x) : x ∈ X} has the finite intersection property.

2.3 Ky Fan Minimax Principle

In this section, as an application of FKKM lemma, we will prove Ky Fan minimax principle in topological vector spacces. For other results in more general setting such as H-spaces and G-convex spaces, please see Yuan [?] and references wherein for more details.

The minimax inequality of Fan [?] is fundamental in proving many existence theorems in nonlinear analysis. There have been numerous generalizations of Fan’s minimax inequality by weakening the compactness assumption or the convexity assumption. In [?], using Horvath’s approach [?], Bardaro and Ceppitelli obtained some minimax inequalities in topological spaces which have H-space structure. Following this line, there are many generalizations given by Horvath [?], Tarafdar [?], Ding and Tan [?], Ding et al. [?], Chang and Ma [?], Park [?], Tan et al. [?]. These results generalize most of the corresponding results given by Fan [?] and [?], Degundji and Granas [?], Lassonde [?], and Zhou and Chen [?] to topological spaces which have the so-called H-structure.

Theorem 2.3.1. Let X be a non-empty set and Y a compact topological space and ϕ : X × Y → ℝ ∪ {−∞, +∞} be such that:

(a) the mapping x ↦ {y ∈ Y : ϕ(x, y) ≤ 0} is transfer closed valued;

(b) the mapping x → cl_{Y} {y ∈ Y : ϕ(x, y) ≤ 0} is generalized HKKM on X.

Then there exists y^{∗} ∈ Y such that ϕ(x, y^{∗}) ≤ 0 for all x ∈ X.

Proof. Define a mapping G : X → 2^{Y} by G(x) = {y ∈ X : ϕ(x, y) ≤ 0} for each x ∈ X. Then we have: (1) the mapping G is transfer closed valued and (2) the mapping clG is generalized HKKM. By Theorem 2.2.16, ∩_{x∈X}G(x) ≠ ∅. Take any y^{∗} ∈ ∩_{x∈X}G(x), then sup_{x∈X} ϕ(x, y^{∗}) ≤ 0 for all x ∈ X. □

Remark 2.3.2. It is clear that condition (a) of Theorem 2.3.1 is equivalent to the following condition which first appeared in Tan et al. [?]:

The fact (a)′: for each y ∈ Y with {x ∈ X : ϕ(x, y) > 0} ≠ ∅, there exists x′ ∈ X such that y ∈ int_{Y} {y′ ∈ Y : ϕ(x′, y′) > 0}.

Theorem 2.3.2. Let X be a non-empty subset of a compact H-space (Y, {Γ_{A}}) and ϕ : X × Y → ℝ ∪ {−∞, +∞} be such that:

(a) the mapping x ↦ {y ∈ Y : ϕ(x, y) ≤ 0} is transfer closed valued on X; and

(b) the mapping x ↦ cl_{Y} {y ∈ Y : ϕ(x, y) ≤ 0} is HKKM on X.

Then there exists y^{∗} ∈ Y such that ϕ(x, y^{∗}) ≤ 0 for all x ∈ X.

Proof. Since (Y, {Γ_{A}}) is an H-space, each HKKM mapping is automatically a generalized HKKM mapping. Therefore all hypotheses of Theorem 2.3.1 are satisfied. By Theorem 2.3.1, there exists y^{∗} ∈ Y such that ϕ(x, y^{∗}) ≤ 0 for all x ∈ X. □.

Corollary 2.3.4. Let X be a non-empty subset of a compact H-space (Y, {Γ_{A}}) and ϕ : X × Y → ℝ ∪ {−∞, +∞} be such that:

(a) for each x ∈ X, y ↦ ϕ(x, y) is lower semicontinuous on Y; and

(b) the mapping x ↦ cl_{Y} {y ∈ Y : ϕ(x, y) ≤ 0} is HKKM on X.

Then there exists y^{∗} ∈ Y such that ϕ(x, y^{∗}) ≤ 0 for all x ∈ X.

Proof. Suppose y ∈ Y is such that {x ∈ X : ϕ(x, y) > 0} ≠ ∅. Fix any x′ ∈ X with ϕ(x′, y) > 0. By (a), there exists an open neighborhood N(y) of y such that ϕ(x′, y′) > 0 for each y′ ∈ N(y). Hence y ∈ int_{Y} {y′ ∈ X : ϕ(x′, y′) > 0}. Now the conclusion follows from Theorem 2.3.3 and the fact (a)′ preceding it. □

Corollary 2.3.5. Let X be a non-empty subset of a non-empty compact convex set Y in a topological vector space and ϕ : X × Y → ℝ ∪ {−∞, +∞} be such that:

(a) the mapping x ↦ {y ∈ Y : ϕ(x, y) ≤ 0} is transfer closed valued on X; and

(b) the mapping x ↦ cl_{Y} {y ∈ Y : ϕ(x, y) ≤ 0} is KKM on X.

Then there exists y^{∗} ∈ Y such that ϕ(x, y^{∗}) ≤ 0 for all x ∈ X.

Proof. For each A ∈ F(Y), let Γ_{A} = co(A). Then (Y, {Γ_{A}}) is an H-space and the mapping x ↦ cl_{Y} {y ∈ Y : ϕ(x, y) ≤ 0} is HKKM on X. Thus the conclusion follows from Theorem 2.3.3. □

Corollary 2.3.6. Let X be a non-empty subset of a non-empty compact convex set Y in a topological vector space and ϕ : X × Y → ℝ ∪ {−∞, +∞} be such that:

(a) the mapping x ↦ {y ∈ Y : ϕ(x, y) ≤ 0} is transfer closed valued on X;

(b) for each A ∈ F(X) and each y ∈ co(A), min_{x∈A}ϕ(x, y) ≤ 0.

Then there exists y^{∗} ∈ Y such that ϕ(x, y^{∗}) ≤ 0 for all x ∈ X.

Proof. By Corollary 2.3.5, we only need to prove that the map x → cl_{Y} {y ∈ Y : ϕ(x, y) ≤ 0} is KKM on X. Suppose not, then there exist A ∈ F(X) and y ∈ co(A) such that y ∉ ∪_{x∈A}cl_{X}{y ∈ X : ϕ(x, y) ≤ 0}. It follows that ϕ(x, y) > 0 for each x ∈ A, so that min_{x∈A}ϕ(x, y) > 0 which is a contradiction. □.

As seen from the proof of Corollary 2.3.4, the condition ‘for each x ∈ X, y → ϕ(x, y) is lower semicontinuous’ implies the condition ‘for each y ∈ X with {x ∈ X : ϕ(x, y) > 0} ≠ ∅, there exists x′ ∈ X such that y ∈ int_{X}{y′ ∈ X : ϕ(x′, y′) > 0}’. Thus Corollary 2.3.5 and hence Theorem 2.3.1 and Theorem 2.3.3 generalize Theorem 1 of Yen [?] (see also Theorem 2.2 of Simons [?]) and Theorem 2.11 of Zhou and Chen [?]. The following is an example for which Theorem 2.3.3 is applicable while Theorem 1 of Yen [?] and Theorem 2.11 of Zhou and Chen [?] are not.

Example 2.3.7. Let Y = [0, 1] and X be the set of all rational numbers in [0, 1]. Define ϕ : X × Y → ℝ by

ϕ(x, y) = \{ \begin{matrix}x − y, & if y is rational, \\ 2, & if y is irrational \end{matrix}

for each (x, y) ∈ X×Y. Suppose (x, y) ∈ X×Y and ϕ(x, y) > 0. If y is irrational, then clearly y < 1. If y is rational, then since ϕ(x, y) = x − y > 0, we also have y < x ≤ 1. In either case, take x′ = 1 and note that {y′ ∈ Y : ϕ(x′, y′) > 0} = [0, 1), so that y ∈ [0, 1) = int_{Y} {y′ ∈ Y : ϕ(x′, y′) > 0}. Thus condition (a)′ and hence condition (a) of Theorem 2.3.3 is satisfied. Moreover, for each x ∈ X, cl_{X}{y ∈ Y : ϕ(x, y) ≤ 0} = cl_{X}{y ∈ Y : y is rational and y ≥ x} = [x, 1]. It follows that the map x → cl_{Y} {y ∈ Y : ϕ(x, y) ≤ 0} is KKM on X. Thus condition (b) of Theorem 2.3.3 is also satisfied. Therefore Theorem 2.3.3 is applicable. However, for each x ∈ X, the map y → ϕ(x, y) is not lower semicontinuous and hence Theorem 1 of Yen [?] and Theorem 2.11 of Zhou and Chen [?] are not applicable.

The Example 2.3.7 above shows that for each x ∈ X, the lower semicontinuity of the mapping y ↦ ϕ(x, y) is not essential for the existence of solutions for minimax inequalities.

Corollary 2.3.8 (Ky Fan Minimax Inequality). Let X be a non-empty subset of a nonempty compact convex set Y in a topological vector space and ϕ : X × Y → ℝ ∪ {−∞, +∞} be such that:

(a) the mapping y ↦ f(x, y) is lower semicontinuous for each given x ∈ X;

(b) for each A ∈ F(X) and each y ∈ co(A), min_{x∈A}ϕ(x, y) ≤ 0.

Then there exists y^{∗} ∈ Y such that ϕ(x, y^{∗}) ≤ 0 for all x ∈ X.

2.4 Ky Fan Minimax Inequality-I

Ky Fan’s minimax inequality [?] has become a versatile tool in nonlinear and convex analysis. In this part, we shall first give a minimax theorem which generalizes those generalizations of Ky Fan’s minimax inequality due to Aubin [?], Aubin and Ekeland [?], Ben-El-Mechaiekh et al [?], Chang [?], Deguire [?], Ding and Tan [?], Ding and Tarafdar [?], Dugundji and Granas [?], Fan [?], Granas [?], Lassonde [?], Park [?], Shih and Tan [?], Tan and Yuan [?], Tarafdar [?], Zhang [?] and Yen [?]. Then several equivalent fixed point forms are then formulated.

For the convenience of our discussion, we first state the following FKKM Lemma again which is a special case of Theorem 2.2.17.

FKKM Lemma. Let X and Y be non-empty sets in a topological vector space E and F : X → 2^{Y} be such that

(i) for each x ∈ X, F (x) is closed in Y;

(ii) for each A ∈ F(X), co(A) ⊂ ∩_{x∈A}F (x);

(iii) there exists an x_{0} ∈ X such that F (x_{0}) is compact. Then ∩_{x∈X}F (x) ≠ ∅.

Theorem 2.4.1. Let X be a nonempty convex subset of a topological vector space E and f, g : X × X → ℝ ∪ {−∞, +∞} be such that

(1) f(x, y) ≤ g(x, y) for each (x, y) ∈ X × X;

(2) for each A ∈ F(X) and y ∈ coA, min_{x∈A} g(x, y) ≤ 0;

(3) there exist a nonempty compact convex subset X_{0} of X and a nonempty compact subset K of X such that for each ŷ ∈ X \ K, there is an x ∈ co(X_{0} ∪ {ŷ}) with ŷ ∉ cl_{X}{y ∈ X : g(x, y) ≤ 0}.

Then ∩_{x∈X}cl_{X}{y ∈ X : f(x, y) ≤ 0} ∩ K ≠ ∅.

Proof. Define F, G : X → 2^{X} by

F (x) := cl_{X}{y ∈ X : f(x, y) ≤ 0} and G(x) := cl_{X}{y ∈ X : g(x, y) ≤ 0}

for each x ∈ X. Then G(x) ⊂ F (x) for each x ∈ X by (1). In order to derive the conclusion, it suffices to prove that the family {F (x) ∩ K : x ∈ X} has the finite intersection property. For each A ∈ F(X), let D := co(X_{0} ∪ A). Then D is non-empty compact and convex. We first note that by (2), g(x, x) ≤ 0 for each x ∈ X. Define H : D → 2^{D} by H(x) = {y ∈ D : g(x, x) ≤ 0} for each x ∈ D. Then we have (a): D(x) ≠ ∅ for each x ∈ D and (b): coA ⊂ ∪_{x∈A}D(x) by (2) again. Otherwise there exists x ∈ coA such that g(x, x) > 0, which is impossible. By FKKM Lemma above, ∩_{x∈D}cl_{D}H(x) ≠ ∅. Taking any ŷ_{0} ∈ cl_{D}H(x). Then the condition (3) implies that y_{0} inK, so that ∩_{x∈A}H(x)∩K ≠ ∅. As cl_{D}H(x) ⊂ F (x) for each x ∈ D, we have ∩_{x∈A}F (x)∩K ≠ ∅. That is, the family {F (x) ∩ K : x ∈ X} has the finite intersection property. Therefore ∩_{x∈X}F (x) ≠ ∅ and we complete the proof. □

In order to study Ky Fan type minimax inequalities for functions which may not be lower semicontinuous, we have the following definition:

Definition 2.4.1. Let X be a non-empty set (without any topology), Y be a topological space and λ ∈ ℝ. A fucntion f : X × Y → ℝ ∪ {−∞, +∞} is said to be λ-transfer lower (resp., upper) semiconinuous on Y if for each (x, y) ∈ X × Y with f(x, y) > λ (resp., f(x, y) < λ), there exists x′ ∈ X and a non-empty open neighborhood N(y) of y in Y such that f(x′, w) > λ (resp., f(x′, w) < λ) for all w ∈ N(y). Moreover f is said to be λ-transfer compactly lower (resp., upper) semicontinuous on Y if f is λ-transfer lower (resp., upper) semicontinuous on each non-empty compact subset of Y.

It is clear that if f is lower (resp., upper) semicontinuous on Y, then f is λ-transfer lower (resp., upper) semicontinuous on Y for each λ ∈ ℝ.

We also recall the following definition which is indeed equivalent to the definition 8 of Zhou and Tian [?].

Definition 2.4.2. Let X be a non-empty set (without any topology) and Y a topological space. A set-valued mapping F : X → 2^{Y} is then (1): said to be transfer (resp., compactly) closed valued on X if for each (x, y) ∈ X × Y with y ∉ F (x), there exist an x′ ∈ X and a non-empty (resp., compactly) open neighborhood N(y) of y in Y such that w ∉ F (x′) for all w ∈ N(y); (2): F is said to be transfer (resp., compactly) open valued if the mapping x → Y \ F (x) is transfer (resp., compactly) closed valued and (3): F is said to be transfer (resp., compactly) open values if the mapping F_{1} : X → 2^{Y} defined by F_{1}(x) := Y \ F (x) for each x ∈ X is transfer (resp., compactly) closed valued.

Of course, each set-valued mapping with closed (resp., open) values is compactly transfer (resp., open) closed valued. For the convenience of our discussion, we have the following definition which is a dual version of Definition 2:

Definition 2.4.3. Let X be a non-empty set (without any topology) and Y be a topological space. A set-valued mapping G : Y → 2^{X} is said to be transfer (resp., compactly) open inverse valued provided for each (x, y) ∈ X × Y with y ∈ G^{−1}(x), there exist x′ ∈ X and a non-empty (resp., compactly) open neighborhood N(y) of y such that N(y) ⊂ G^{−1}(x′). Moreover F is said to be transfer (resp., compactly) closed inverse valued if the mapping F_{1} : Y → 2^{X} defined by F_{1}(y) := X \ F (y) for each y ∈ Y is transfer (resp., compactly) open inverse valued.

Let X be a non-empty set (without any topology) and Y be a topological space. From definitions, it is clear that a set-valued mapping G : Y → 2^{X} is transfer (resp., compactly) open inverse valued if and only if the mapping F : X → 2^{Y} defined by F (x) := Y \ G^{−1}(x) for each x ∈ X is transfer (resp., compactly) closed valued on X.

Before we prove some general minimax inequalities which are improvements of Ky Fan minimax inequality, we need the following lemma:

Lemma 2.4.2. Let X be a non-empty set, Y a topological space and f : X × Y → ℝ∪{−∞, +∞} be an extended real-valued function. For each given λ ∈ ℝ, define a set-valued mapping F_{λ} : X → 2^{Y} by F_{λ}(x) := {y ∈ Y : f(x, y) ≤ λ} for each x ∈ X. Then ∩_{x∈X}F_{λ}(x) = cl_{Y} F_{λ}(x) if and only if the mapping f is λ-transfer lower semicontinuous on Y. In particular, if ∩_{x∈X}cl_{Y} F_{λ}(x) is compact, then ∩_{x∈X}F_{λ}(x) = cl_{Y} F_{λ}(x) if and only if f is λ-transfer compactly lower semicontinuous on Y.

Proof. Sufficiency: It is clear that ∩_{x∈X}F_{λ}(x) ⊂ ∩_{x∈X}cl_{Y} F_{λ}(x). It suffices to prove that ∩_{x∈X}clF_{λ}(x) ⊂ ∩_{x∈X}F_{λ}(x). Suppose that y ∉ ∩_{x∈X}F_{λ}(x). Then there exists some x ∈ X such that y ∉ F_{λ}(x), i.e., f(x, y) > λ. As f is λ-transfer lower semicontinuous on Y, there exist x′ ∈ X and a non-empty open neighborhood N(y) of y such that f(x′, w) > λ for all w ∈ N(y). Then w ∉ F_{λ}(x′) for all w ∈ N(y) and we have y ∉ cl_{Y} F_{λ}(x′). Note that N(y) is a non-empty open neighborhood of y, so that y ∉ ∩_{x∈X}cl_{Y} F_{λ}(x). Therefore ∩_{x∈X}F_{λ}(x) = ∩_{x∈X}cl_{Y} F_{λ}(x).

Necessity: Suppose there exists (x, y) ∈ X × Y such that f(x, y) > λ, i.e., y ∉ F_{λ}(x). Then y ∉ ∩_{x∈X}F_{λ}(x) = ∩_{x∈X}cl_{Y} F_{λ}(x), so that there exists x′ ∈ X such that y ∉ cl_{Y} F (x′). But then there exists an open neighborhood N(y) of y in Y such that N(y) ∩ F_{λ}(x′) = ∅. Therefore w ∉ F_{λ}(x′) for all w ∈ N(y). This means that f(x′, w) > λ for all w ∈ N(y), so that f is λ-transfer lower semicontinuous on Y.

In the case that ∩_{x∈X}cl_{Y} F_{λ}(x) is compact, it is easy to check that the same arguments above are still applicable for the func tion f is λ-transfer compactly lower semicontinuous. Thus we complete the p roof. □.

We note that the similar result of Lemma 2.4.2 was also given by Zhou [?]. Now as an immediately consequence of Theorem 2.4.1, we have the following result.

Theorem 2.4.3. Let X be a non-empty convex subset of a topological vector space and ϕ, ψ : X × X → R ∪ {−∞, ∞} be such that

(a) ϕ(x, y) ≤ ψ(x, y) for each (x, y) ∈ X × X;

(b) ϕ is 0-transfer compactly lower semicontinuous on its second variab le; (c) for each A ∈ F(X) and for each y ∈ co(A), min_{x∈A}ψ(x, y) ≤ 0;

(d) there exist a non-empty closed and compact subset K of X and x_{0} ∈ X such that ψ(x_{0}, y) > 0 for all y ∈ X \ K. Then there exists y ∈ K such that ϕ(x, y) ≤ 0 for all x ∈ X.

Proof. From the condition (d), ŷ ∉ {y ∈ X : ϕ(x_{0}, y) ≤ 0} for each ŷ ∈ X \ K. Hence {y ∈ X : ϕ(x_{0}, y) ≤ 0} ⊂ K. As K is closed, we must have that cl_{X}{y ∈ X : ϕ(x_{0}, y) ≤ 0} ⊂ K. Therefore for each ŷ ∈ X \ K, ŷ ∉ cl_{X}{y ∈ X : ϕ(x_{0}, y) ≤ 0}. Now let X_{0} := {x_{0}} in Theorem 2.4.1, it follows that ∩_{x∈X}cl_{X}{y ∈ X : ϕ(x, y) ≤ 0} ≠ ∅. As ϕ is 0-transfer lower semicontinuous on its second variable, Lemma 1.2 implies that

∩_{x∈X}{y ∈ X : ϕ(x, y) ≤ 0} = ∩_{x∈X}cl_{X}{y ∈ X : ϕ(x, y) ≠ 0} ≠ ∅.

Thus there exists ŷ ∈ K such that ϕ(x, y_{0}) ≤ 0 for all x ∈ X and the proof is complete. □

The following are equivalent formulations of Theorem 2.4.3.

Theorem 2.4.3′ (First Geometric Form). Let X be a non-empty convex subset of a topological vector space and B, D ⊂ X × X be such that

(a) B ⊂ D;

(b) the mapping B_{1} : X → 2^{X} defined by B_{1}(x) := {y ∈ X : (x, y) ∈ B} for each x ∈ X is transfer compactly open valued;

(c) for each A ∈ F(X) and for each y ∈ co(A), there exists x ∈ A such that (x, y) ∉ D;

(d) there exist a non-empty closed and compact subset K of X and x_{0} ∈ X such that (x_{0}, y) ∈ D for all y ∈ X \ K. Then there exists ŷ ∈ K such that {x ∈ X : (x, ŷ) ∈ B} = ϕ.

Theorem 2.4.3′′ (Second Geometric Form). Let X be a non-empty conve x subset of a topological vector space and M, N ∈ X × X be such that

(a) N ⊂ M;

(b) the mapping M_{1} : X → 2^{Y} defined by M_{1}(x) := {y ∈ X : (x, y) ∈ M} for each x ∈ X is transfer compactly closed valued;

(c) for each A ∈ F(X) and for each y ∈ co(A), there exists x ∈ A such that (x, y) ∈ N;

(d) there exist a non-empty closed and compact subset K of X and x_{0} ∈ X such that (x_{0}, y) ∉ N for all y ∈ X \ K. Then there exists ŷ ∈ K such that X × {ŷ} ⊂ M.

Theorem 2.4.3′′′ (Maximal Element Version). Let X be a non-empty convex subset of a topological vector space and P, Q : X → 2^{X} ∪ {ϕ} such that

(a) for each x ∈ X, P (x) ⊂ Q(x);

(b) P is transfer compactly open inverse valued;

(c) for each A ∈ F(X) and for each y ∈ co(A), there exists x ∈ A such that x ∉ Q(y);

(d) there exist a non-empty closed and compact subset K of X and x_{0} ∈ X such that X \ K ⊂ Q^{−1}(x_{0}). Then there exists ŷ ∈ K such that P (ŷ) = ϕ.

Sketch of proofs:

(1) Theorem 2.4.3 ⟹ Theorem 2.4.3′: Let ϕ, ψ : X × X → ℝ be the characteristic function of B, D respectively. □

(2) Theorem 2.4.3′ ⟹ Theorem 2.4.3: Define B := {(x, y) ∈ X × X : ϕ(x, y) > 0} and D := {(x, y) ∈ X × X : ψ(x, y) > 0}. □

(3) Theorem 2.4.3′ ⟹ Theorem2.4.3′′: Let B := X × X \ M and D := X × X \ N. □

(4) Theorem 2.4.3′′ ⟹ Theorem 2.4.3′: Let M := X × X \ B and N := X × X \ D. □

(5) Theorem 2.4.3′′ ⟹ Theorem 2.4.3′′′: Let N := {(x, y) ∈ X × X : x ∉ Q(y)} and M := {(x, y) ∈ X × X : x ∉ P (y)}. □

(6) Theorem 2.4.3′′′ ⟹ Theorem 2.4.3′′: Define P, Q : X → 2^{X} by P (y) := {x ∈ X : (x, y) ∉ M} and Q(y) := {x ∈ X : (x, y) ∉ N} for each y ∈ X respectively. □

Lemma 2.4.4. Let X be a non-empty convex subset of a topological vector space and ϕ, ψ : X × X → R ∪ {−∞, +∞} be such that (i) ψ(x, x) ≤ 0 for each x ∈ X; (ii) for each y ∈ X, the set {x ∈ X : ψ(x, y) > 0} is convex.

Then for each A ∈ F(X) and for each y ∈ co(A), min_{x∈A}ψ(x, y) ≤ 0.

Proof. Suppose the conclusion were false, then there exist A ∈ F(X) and y ∈ co(A) such that min_{x∈A}ψ(x, y) > 0. It follows that A ⊂ {x ∈ X : ψ(x, y) > 0} so that y ∈ co(A) ⊂ {x ∈ X : ψ(x, y) > 0} by (ii) and ψ(y, y) > 0, which contradicts (i). Therefore the conclusion must hold. □

In view of Lemma 2.4.4, Theorem 2.4.3 implies the following

Theorem 2.4.5. Let X be a non-empty convex subset of a topological vector space and ϕ, ψ : X × X → R ∪ {−∞, +∞} be such that

(a) ϕ(x, y) ≤ ψ(x, y) for each (x, y) ∈ X × X and ψ(x, x) ≤ 0 for each x ∈ X;

(b) ϕ is 0-transfer compactly lower semicontinuous on its second varia ble;

(c) for each fixed y ∈ X, the set {x ∈ X : ψ(x, y) > 0} is convex;

(d) there exist a non-empty closed and compact subset K of X and a point x_{0} ∈ X such that ψ(x_{0}, y) > 0 for all y ∈ X \ K. Then there exists ŷ ∈ K such that ϕ(x, ŷ) ≤ 0 for all x ∈ X.

As a special case of Theorem 2.4.5, we have the following corollary.

Corollary 2.4.6. Let X be a non-empty convex subset of a topological vector space and ϕ, ψ : X × X → R ∪ {−∞, +∞} be such that

(a) ϕ(x, y) ≤ ψ(x, y) for each (x, y) ∈ X × X;

(b) for each fixed x ∈ X, y ↦ ϕ(x, y) is compact lower semicontinuous on its second variable;

(c) for each fixed y ∈ X, the set {x ∈ X : ψ(x, y) > sup_{x∈X} ψ(x, x)} is convex;

(d) there exist a non-empty closed and compact subset K of X and a point x_{0} ∈ X such that ψ(x_{0}, y) > sup_{x∈X} ψ(x, x) for all y ∈ X \ K. Then there exists ŷ ∈ K such that ϕ(x, ŷ) ≤ sup_{x∈X} ψ(x, x) for all x ∈ X.

The following are fixed point versions of Theorem 2.4.5:

Theorem 2.4.5′. Let X be a non-empty convex subset of a topological vector space and P, Q : X → 2^{X} be such that

(a) for each x ∈ X, P (x) ⊂ Q(x);

(b) P is transfer compactly open inverse valued;

(c) for each y ∈ X, Q(y) is convex;

(d) there exists a non-empty closed and compact subset K of X and x_{0} ∈ X such that X \K ⊂ Q^{−1}(x_{0});

(e) for each y ∈ K, P (y) ≠ ∅. Then there exists a point x ∈ X such that x ∈ Q(x).

Theorem 2.4.5′′. Let X be a non-empty convex subset of a topological vector space and P, Q : X → 2^{X} be such that

(a) for each x ∈ X, P (x) ⊂ Q(x);

(b) P is transfer compactly open inverse valued;

(c) there exist a non-empty closed and compact subset K of X and x_{0} ∈ X such that X \ K ⊂ (coQ)^{−1}(x_{0});

(d) for each y ∈ K, P (y) ≠ ∅. Then there exists x ∈ X such that x ∈ coQ(x).

Theorem 2.4.5′′′. Let X be a non-empty convex subset of a topological vector space and P, Q : X → 2^{X} be such that

(a) for each x ∈ X, P (x) ⊂ Q(x);

(b) P is transfer compactly open inverse valued;

(c) there exist a non-empty closed and compact subset K of X and x_{0} ∈ X such that X \ K ⊂ Q^{−1}(x_{0});

(d) for each y ∈ K, P (y) ≠ ∅. Then there exists x ∈ X such that x ∈ coQ(x).

Theorem 2.4.5′′′′. Let X be a non-empty convex subset of a topological vector space and Q : X → 2^{X} be such that

(a) for each y ∈ X, Q^{−1}(y) contains an subset O_{y} which is open in each non-empty compact subset C of X and ∪_{y∈X}O_{y} = X;

(b) there exist a non-empty closed and compact subset K of X and a point x_{0} ∈ X such that x_{0} ∈ co(Q(y)) for all y ∈ X \ K. Then there exists a point x̂ ∈ X such that x̂ ∈ co(Q(x̂)).

Theorem 2.4.5′′′′′. Let X be a non-empty convex subset of a topological vector space and Q : X → 2^{X} be such that

(a) for each x ∈ X, Q(x) is convex;

(b) for each y ∈ X, Q^{−1}(y) contains a subset O_{y} (which may be empty) of X which is compactly open in X;

(c) there exist a non-empty closed and compact subset K of X and a point x_{0} ∈ X such that x_{0} ∈ Q(y) for all y ∈ X \ K and K ⊂ ∪_{y∈X}O_{y}. Then there exists a point x̂ ∈ X such that x̂ ∈ Q(x̂).

Indication of proofs:

(1) Theorem 2.4.5 ⟹ Theorem 2.4.5′: Define ϕ, ψ : X × X → R by

ϕ(x, y) := \begin{cases}1, & if x ∈ P (y), \\ 0, & if x ∉ P (y); \end{cases} and ψ(x, y) := \begin{cases}1, & if x ∈ Q(y), \\ 0, & if x ∉ Q(y). \end{cases}

(2) Theorem 2.4.5′ ⟹ Theorem 2.4.5: Define P, Q : X → 2^{X} by P (y) := {x ∈ X : ϕ(x, y) > 0} and Q(y) := {x ∈ X : ψ(x, y) > 0} for each y ∈ X respectively. □

(3) Theorem 2.4.5′ ⟺ Theorem 2.4.5′′ ⟺ Theorem 2.4.5′′′ is obvious. □

(4) Theorem 2.4.5′′ ⟹ Theorem 2.4.5′′′′: Define P : X → 2^{X} by P (x) := {y ∈ X : x ∈ O_{y}} for each x ∈ X. □

(5) Theorem 2.4.5′′′′ ⟹ Theorem 2.4.5′′: Let O_{y} := P^{−1}(y) for each y ∈ X. □

(6) Theorem 2.4.5′′′′ ⟺ Theorem 2.4.5′′′′′: Obvious. □

2.5 Ky Fan Minimax Inequality-II

In part, we will discuss another type of minimax inequalities and its fixed point forms.

As an application of Theorem 2.4.1 and Lemma 2.4.2, we have the following Ky Fan type minimax inequality.

Theorem 2.5.1. Let X be a nonempty convex subset of a topological vector space E and f, g : X × X → ℝ ∪ {−∞, +∞} be such that

(1) f(x, y) ≤ g(x, y) for each (x, y) ∈ X × X;

(2) f is 0-transfer compactly lower semicontinuous on its second variable;

(3) for each A ∈ F(X) and y ∈ coA, min_{x∈A} g(x, y) ≤ 0;

(4) there exist a nonempty compact convex subset X_{0} of X and a nonempty compact subset K of X such that for each ŷ ∈ X \ K, there is an x ∈ co(X_{0} ∪ {ŷ}) with ŷ ∉ cl_{X}{y ∈ X : g(x, y) ≤ 0}.

Then there exists a ŷ ∈ K such that f(x, ŷ) ≤ 0 for all x ∈ X.

Proof. By Theorem 2.4.1 and our condition (4), ∩_{x∈X}cl_{X}{y ∈ X : f(x, y) ≤ 0} ∩ K ≠ ∅. As f is 0-transfer compactly lower semicontinuous on its second variable, Lemma 2.4.2 implies that ∩_{x∈X}cl_{X}{y ∈ X : f(x, y) ≤ 0} ∩ K = ∩_{x∈X}{y ∈ X : f(x, y) ≤ 0} ∩ K. Take any fixed ŷ ∈ ∩_{x∈X}{y ∈ X : f(x, y) ≤ 0}. Then sup_{x∈X} f(x, ŷ) ≤ 0 and we complete the proof. □

Theorem 2.5.2. Let X be a non-empty convex subset of a topological vector space and ϕ, ψ : X × X → R ∪ {−∞, +∞} be such that

(a) ϕ(x, y) ≤ ψ(x, y) for each all x, y ∈ X, and ψ(x, x) ≤ 0 for all x ∈ X;

(b) ϕ is 0-transfer compactly lower semicontinuous on its second variable ;

(c) for each y ∈ X, the set {x ∈ X : ψ(x, y) > 0} is convex;

(d) there exist a non-empty compact convex subset X_{0} of X and a non-empty compact subset K of X such that for each y ∈ X \ K, there exists an x ∈ co(X_{0} ∪ {y}) with y ∉ cl_{X}{w ∈ X : ψ(x, w) ≤ 0}. Then there exists a ŷ ∈ K such that ϕ(x, ŷ) ≤ 0 for all x ∈ X.

Proof. By (a) and (c), Lemma 2.4.4 implies that for each A ∈ F(X) and for each y ∈ co(A), min_{x∈A}ϕ(x, y) ≤ 0. Then ϕ and ψ satisfy all hypotheses of Theorem 2.1. By Theorem 2.1, there exists a ŷ ∈ K such that ϕ(x, ŷ) ≤ 0 for all x ∈ X. □

As an immediately consequence of Theorem 2.5.2, we have the following minimax inequality.

Corollary 2.5.3. Let X be a non-empty convex subset of a topological vector space and ϕ : X × X → R ∪ {−∞, +∞} be such that

(a) ϕ(x, x) ≤ 0 for all x ∈ X;

(b) for each fixed x ∈ X, y ↦ ϕ(x, y) is compactly lower semicontinuous;

(c) for each y ∈ X, the set {x ∈ X : ϕ(x, y) > 0} is convex;

(d) there exist a non-empty compact convex subset X_{0} of X and a non-empty compact subset K of X such that for each y ∈ X \K, there exists an x ∈ co(X_{0}∪{y}) with y ∉ {w ∈ X : ϕ(x, w) ≤ 0}. Then there exists a ŷ ∈ K such that ϕ(x, ŷ) ≤ 0 for all x ∈ X.

Proof. Let ψ(x, y) := ψ(x, y) in Theorem 2.5.2. Then the conclusion follows from Theorem 2.5.2. □

Example 2.5.1. Let X = [0, ∞) and ϕ, ψ :→ R be defined by

ϕ(x, y) = \begin{cases}1, & if (x, y) ∈ X × X : x < y < x + \frac{1}{2+x} for 0 ≤ x < 1, \\ 0, & otherwise; \end{cases}

and

ψ(x, y) = \begin{cases}1, & if (x, y) ∈ X × X and x > y, \\ 0, & otherwise. \end{cases}

Then all the conditions (a), (b), (c) and (d) in Theorem 2.4.3 are satisfied with K = {0} = {x_{0}} so that there exists ŷ = 0 ∈ X such that ϕ(x, ŷ) ≤ 0 for all x ∈ X. However, there does not exist a non-empty compact convex subset X_{0} of X and a non-empty compact subset K of X such that for each y ∈ X \ K, there exists an x ∈ co(X_{0} ∪ {y}) with ϕ(x, y) > 0. Thus Corollary 2.5.3 is not applicable.

Example 2.5.2. Let X := [0, ∞) and ϕ, ψ : X × X → R be defined by

ϕ(x, y) = ψ(x, y) = \begin{cases}1, & if y > x or y > x + \frac{1}{3+x}, \\ 0, & otherwise. \end{cases}

Then all the conditions (a), (b), (c) and (d) of Corollary 2.5.3 are satisfied with X_{0} = K = {0} and ψ(x, y) = ϕ(x, y) for each (x, y) ∈ X × X. So that there exists ŷ ∈ X such that ϕ(x, ŷ) ≤ 0 for all x ∈ X. However, there does not exist a non-empty subset K of X and x_{0} ∈ X such that ψ(x_{0}, y) > 0 for all y ∈ X \ K. Therefore Theorem 2.4.3 is not applicable and hence above two examples show that Theorem 2.4.3 and Corollary 2.5.3 are not comparable.

The following are equivalent formulations of Theorem 2.5.1.

Theorem 2.5.1′ (First Geometric Form). Let X be a non-empty convex subset of a topological vector space and B, D ⊂ X × X be such that

(a) B ⊂ D;

(b) the mapping B_{1} : X → 2^{X} defined by B_{1}(x) := {y ∈ X : (x, y) ∈ B} for each x ∈ X is transfer compactly open valued;

(c) for each A ∈ F(X) and for each y ∈ co(A), there exists x ∈ A such that (x, y) ∉ D;

(d) there exist a non-empty compact and convex subset X_{0} of X and a non-empty compact subset K of X such that for each y ∈ X \ K, there exists x ∈ co[X_{0} ∪ {y}] with y ∉ cl_{X}{w ∈ X : (x, w) ∉ D}. Then there exists ŷ ∈ K such that {x ∈ X : (x, ŷ) ∈ B} = ϕ.

Theorem 2.5.1′′ (Second Geometric Form). Let X be a non-empty convex subset of a topological vector space and M, N ∈ X × X be such that

(a) N ⊂ M;

(b) the mapping M_{1} : X → 2^{Y} defined by M_{1}(x) := {y ∈ X : (x, y) ∈ M} for each x ∈ X is transfer compactly closed valued;

(c) for each A ∈ F(X) and for each y ∈ co(A), there exists x ∈ A such that (x, y) ∈ N;

(d) there exist a non-empty compact and convex subset X_{0} of X and a non-empty compact subset K of X such that for each y ∈ X \ K, there exists x ∈ co[X_{0} ∪ {y}] with y ∉ cl_{X}{w ∈ X : (x, w) ∈ N}. Then there exists ŷ ∈ K such that X × {ŷ} ⊂ M.

Theorem 2.5.1′′′ (Maximal Element Version). Let X be a non-empty convex subset of a topological vector space and P, Q : X → 2^{X} ∪ {ϕ} such that

(a) for each x ∈ X, P (x) ⊂ Q(x);

(b) P is transfer compactly open inverse valued;

(c) for each A ∈ F(X) and for each y ∈ co(A), there exists x ∈ A such that x ∉ Q(y);

(d) there exist a non-empty compact and convex subset X_{0} of X and a non-empty compact subset K of X such that for each y ∈ X \K, there exists x ∈ co[X_{0}∪{y}] with y ∉ cl_{X}{X \Q^{−1}(x)}. Then there exists ŷ ∈ K such that P (ŷ) = ϕ.

The following are fixed point versions of Theorem 2.5.2:

Theorem 2.5.2′. Let X be a non-empty convex subset of a topological vector space and P, Q : X → 2^{X} be such that

(a) for each x ∈ X, P (x) ⊂ Q(x);

(b) P is transfer compactly open inverse valued;

(c) for each y ∈ X, Q(y) is convex;

(d) there exist a non-empty compact and convex subset X_{0} of X and a non-empty compact subset K of X such that for each y ∈ X \K, there exists x ∈ co[X_{0} ∪{y}] with y ∉ cl_{X}[X \Q^{−1}(x)].

(e) for each y ∈ K, P (y) ≠ ∅. Then there exists a point x ∈ X such that x ∈ Q(x).

Theorem 2.5.2′′. Let X be a non-empty convex subset of a topological vector space and P, Q : X → 2^{X} be such that

(a) for each x ∈ X, P (x) ⊂ Q(x);

(b) P is transfer compactly open inverse valued;

(c) there exist a non-empty compact and convex subset X_{0} of X and a non-empty compact subset K of X such that for each y ∈ X \ K, there exists x ∈ co[X_{0} ∪ {y}] with y ∉ cl_{X}[X \ (coQ)^{−1}(x)].

(d) for each y ∈ K, P (y) ≠ ∅. Then there exists x ∈ X such that x ∈ coQ(x).

Theorem 2.5.2′′′. Let X be a non-empty convex subset of a topological vector space and P, Q : X → 2^{X} be such that (a) for each x ∈ X, P (x) ⊂ Q(x);

(b) P is transfer compactly open inverse valued;

(c) there exist a non-empty compact and convex subset X_{0} of X and a non-empty compact subset K of X such that for each y ∈ X \K, there exists x ∈ co[X_{0} ∪{y}] with y ∉ cl_{X}[X \Q^{−1}(x)].

(d) for each y ∈ K, P (y) ≠ ∅. Then there exists x ∈ X such that x ∈ coQ(x).

Theorem 2.5.2′′′′. Let X be a non-empty convex subset of a topological vector space and Q : X → 2^{X} be such that

(a) for each y ∈ X, Q^{−1}(y) contains an subset O_{y} which is open in each non-empty compact subset C of X and ∪_{y∈X}O_{y} = X;

(b) there exist a non-empty compact and convex subset X_{0} of X and a non-empty compact subset K of X such that for each y ∈ X \ K, there exists x ∈ co[X_{0} ∪ {y}] with y ∉ cl_{X}[X \ (coQ)^{−1}(x)]. Then there exists a point x̂ ∈ X such that x̂ ∈ co(Q(x̂)).

Theorem 2.5.2′′′′′. Let X be a non-empty convex subset of a topological vector space and Q : X → 2^{X} be such that

(a) for each x ∈ X, Q(x) is convex;

(b) for each y ∈ X, Q^{−1}(y) contains a subset O_{y} (which may be empty) of X which is compactly open in X;

(c) there exist a non-empty compact and convex subset X_{0} of X and a non-empty compact subset K of X such that for each y ∈ X \ K, there exists x ∈ co[X_{0} ∪ {y}] with y ∉ cl_{X}[X \ (coQ)^{−1}(x)]. Then there exists a point x̂ ∈ X such that x̂ ∈ Q(x̂).

As an equivalent form of Theorem 2.5.2′′, we have the following maximal element theorem which will be used later to derive the existence of maximal elements for majorized set-valued mappings in next Section.

Theorem 2.5.4. Let X be a non-empty convex subset of a topological vector space and P, Q : X → 2^{X} be such that (a) for each x ∈ X, P (x) ⊂ Q(x);

(b) P is transfer compactly open inverse valued;

(c) there exist a non-empty compact and convex subset X_{0} of X and a non-empty compact subset K of X such that for each y ∈ X \ K, there exists x ∈ co[X_{0} ∪ {y}] with y ∉ cl_{X}[X \ (coQ)^{−1}(x)].

(d) for each x ∈ X, x ∉ coQ(x). Then there exists x ∈ K such that Q(x) = ∅.

Proof. By Theorem 2.5.2′′, the conclusion follows. □

The following example shows that a transfer open inverse mapping may not have open inverse values.

Example 2.5.3. Let X := [0, 1] and F : X → 2^{X} is defined by

F (x) := \begin{cases}[x, 1], & if x is rational; \\ [0, 1], & if x is irrational. \end{cases}

Then it is clear that F is not open inverse valued but F is transfer open inverse valued.

2.6 Maximal Element and Browder-Fan Fixed Point in Product Spaces

In this section, we discuss the generalization of the Browder-Fan fixed point theorem in product space for both couple-majorized and L_{s}-majorized set-valued mappings by use the concept of maximal elements.

2.6.1 Maximal Element and Fixed Point of Couple-Majorized Mappings

In this part, we study Browder-Fan fixed point theorems in product spaces for coupe-majorized set-valued mappings. In order to do so, we need the concept of maximal elements which plays a very important role in mathematics economics.

Let X and Y be two sets and G : X → 2^{Y} a set-valued mapping. A point x ∈ X is said to be a maximal element of G if G(x) = ∅. Suppose A : X → 2^{Y} is a set-valued mapping and S : Y → X is a (single-valued) map (throughout this section, map always means a single-valued mapping). The composition of the mapping A ∘ S from Y to X is defined by A ∘ S(y) := A(S(y)) for each y ∈ Y. Let U be a non-empty of a topological space X. Then int_{X}U denotes the interior of U in X and we shall denote simply by intU for convenience.

Definition 2.6.1. Let X be a topological space and Y be a non-empty set in a topological vector space. Then a set-valued mapping A : X → 2^{Y} is said

(1): to have convex values if A(x) is convex in Y for all x ∈ X; and

(2): to have open fibers (or, open inverse, or open lower sections) if the set A^{−1}(y) := {x ∈ X : y ∈ A(x)} is open in X for each y ∈ Y.

We recall the following definition which was first given by Deguire and Yuan [?].

Definition 2.6.2. Let X := Π_{j∈J}X_{j} and Y = Π_{i∈I}Y_{i} be two product spaces of non-empty convex subsets {X_{j}}_{j∈J} and {Y_{i}}_{i∈I} in topological vector spaces E and F, respectively. The families {A_{i}}_{i∈I} and {B_{j}}_{j∈J} are said to be couple-majorized if that for any given (i, j) ∈ I × J, and for each (x, y) ∈ X × Y with A_{i}(x) ≠ ∅ and B_{j}(y) ≠ ∅, there exist mappings A_{i(x)} : X → 2^{Y_{i}}, B_{j(y)} : Y → 2^{X_{j}} and a non-empty open neighborhood N(x) of x in X and a non-empty open neighborhood N(y) of y in Y such that:

(a): both A_{i(x)} and B_{j(y)} have (maybe empty) convex values and open fibers;

(b): A_{i}(z) ⊂ A_{i(x)}(z) and B_{j}(w) ⊂ B_{j(y)}(w) for all (z, w) ∈ N(x) × N(y); and

(c): (u_{j}, v_{i}) ∉ (B_{j(y)}(v), A_{i(x)}(u)) for all (u, v) ∈ X × Y.

We also call the mapping pair (A_{i(x)}, B_{j(y)}) which satisfies conditions (a)-(c) above a couplemajorant of the mapping pair (A_{i}, B_{j}) at the point (x, y) ∈ X × Y.

Let E be a topological vector space. Then a polytope of E is the convex hull of a non-empty finite subset of E. By the partition of unity for paracompact, the following result can be easily proved, for example, see Dugundji and Granas in [?], Michael [?], Yannelis and Prabhakar [?] and Deguire and Lassonde [?] for more recent results in this direction.

Lemma 2.6.3. Let X be a paracompact topological space and Y be a non-empty convex subset of a topological vector space E. Suppose F : X → 2^{Y} is a set-valued mapping with nonempty convex values and open fibers. Then there exists a continuous mapping f : X → Y such that f(x) ∈ F (x) for each x ∈ X. In particular, if X is compact, there exists a polytope D in Y such that f(X) := ∪_{x∈X}f(x) ⊂ D.

Proof. For each y ∈ Y, the set F^{−1}(y) is open in X and note that F (x) ≠ ∅ for all x ∈ X. It follows that {F^{−1}(y)}_{y∈Y} is an open covering of X. Since X is paracompact, there is a locally finite and open refinement {O_{i}}_{i∈I} of {F^{−1}(y)}_{y∈Y}. Hence for each ∈ I, there is y(i) ∈ Y such that O_{i} ⊂ F^{−1}(y(i)). Let {P_{i}}_{i∈I} be a partition of unity subordinated to {O_{i}}_{i∈I}. The function f : X → Y defined by

f(x) := Σ_{i∈I}P_{i}(x)y(i)

for each x ∈ X is continuous as it is locally a finite sum of continuous functions {P_{i}}_{i∈I}. For each x ∈ X, P_{i}(x) ≠ ∅ implies that x ∈ O_{i} ⊂ F^{−1}(y(i)), so that y(i) ∈ F (x). Note that F (x) is non-empty and convex. It follows that f(x) ∈ F (x) for all x ∈ X. If X is compact, by the proof above, there exists a finite number of y_{1}, ⋯, y_{n} ∈ Y such that X ⊂ ∪_{i=1}^{n}F^{−1}(y_{i}). Let D be the convex hull of {y_{1}, ⋯, y_{n}}. Then it is clear that the continuous selection f of F has the property that f(x) ∈ D for all x ∈ X. □

We now have the following existence theorem of maximal elements for a pair of mappings which may not have convex values nor open fibers.

Theorem 2.6.4. Let X be a non-empty compact and convex subset of a topological vector space E and Y another non-empty convex subset of a topological vector space F. Suppose A : X → 2^{Y} and B : Y → 2^{X} are couple-majorized. Then there exists either x_{0} ∈ X such that A(x_{0}) = ∅ or y_{0} ∈ Y such that B(y_{0}) = ∅.

Proof. Suppose the contrary that for each x ∈ X and y ∈ Y, A(x) ≠ ∅ and B(y) ≠ ∅. Then for each (x, y) ∈ X × Y, there exists a non-empty open neighborhood N(x) of x in X and an open neighborhood N(y) of y in Y and two mappings A_{x} : X → 2^{Y} and B_{y} : Y → 2^{Y} such that:

(a) A(z) ⊂ A_{x}(z) for each z ∈ N(x) and B(z) ⊂ B_{y}(z) for each z ∈ N(y);

(b) (u, v) ∉ (B_{y}(v), A_{x}(u)) for each (u, v) ∈ X × Y;

(c) A_{x}^{−1}(y) and B_{y}^{−1}(x) are open in X and Y, respectively, for each y ∈ Y and x ∈ X.

Since X is regular, for each x ∈ X (resp., y ∈ Y), let O(x) (resp., O(y)) be an open neighborhood of x in X (resp., y ∈ Y) such that \overline{O(x)} ⊂ N(x) (resp., \overline{O(y)} ⊂ N(y)). Note that X is compact and X = ∪_{x∈X}O(x), then there exist x_{1}, ⋯, x_{n} ∈ X such that X = ∪_{i=1}^{n}O(x_{i}). Define A_{i} : X → 2^{Y} for i = 1, 2, ⋯, n by

A_{i}(x) = \begin{cases}A_{x_{i}}(x), & if x ∈ \overline{O(x_{i})}; \\ Y, & if x ∈ X \ \overline{O(x_{i})}. \end{cases}

Then we have:

(a′) A_{i} is convex-valued;

(b′) A(x) ⊂ A_{i}(x) for each x ∈ X; and

(c′) for each y ∈ Y, A_{i}^{−1}(y) is open in X as

A_{i}^{−1}(y) = \{x ∈ \overline{O(x_{i})} : y ∈ A_{x_{i}}(x)\} ∪ (X \ \overline{O(x_{i})})
= (A_{x_{i}}^{−1}(y) ∩ \overline{O(x_{i})}) ∪ (X \ \overline{O(x_{i})})
= [A_{x_{i}}^{−1}(y) ∩ \overline{O(x_{i})})] ∩ [\overline{O(x_{i})} ∪ (X \ \overline{O(x_{i})}]
= A_{x_{i}}^{−1}(y) ∪ (X \ \overline{O(x_{i})}),

which is open by (c). Now define A′ : X → 2^{Y} by A′(x) = ∩_{i=1}^{n}A_{i}(x) for each x ∈ X. Clearly, we have:

(a′′) A′ is (maybe empty) convex-valued;

(b′′) A(x) ⊂ A′(x) for each x ∈ X; and

(c′′) for each y ∈ Y, the set (A′)^{−1}(y) is open in X since

(A′)^{−1}(y) = {x ∈ X : y ∈ ∩_{i=1}^{n}A_{i}(x)} = ∩_{i=1}^{n}A_{i}^{−1}(y).

Note that Y is also compact and Y = ∪_{y∈Y} O(y), there exist y_{1}, ⋯, y_{m} ∈ Y such that Y = ∪_{i=1}^{m}O(y_{i}). Define B_{i} : Y → 2^{X} for i = 1, 2, ⋯, m by

B_{i}(y) = \begin{cases}B_{y_{i}}(y), & if y ∈ \overline{O(y_{i})}; \\ X, & if y ∈ Y \ \overline{O(y_{i})}. \end{cases}

Now, similarly to the arguments used for the mapping A′, we have:

(i′) B_{i} is (maybe empty) convex-valued;

(ii′) B(y) ⊂ B_{i}(y) for each y ∈ Y; and

(iii′) for each x ∈ X, B_{i}^{−1}(x) is open in Y. Now we define B′ : Y → 2^{X} by B′(x) = ∩_{i=1}^{m}B_{i}(y) for each y ∈ Y. Then we have:

(i′′) B′ is convex-valued;

(ii′′) B(y) ⊂ B′(y) for each y ∈ Y; and

(iii′′) for each x ∈ X, the set (B′)^{−1}(x) is open in Y since

(B′)^{−1}(x) = {y ∈ Y : x ∈ ∩_{i=1}^{n}B_{i}(y)} = ∩_{i=1}^{n}B_{i}^{−1}(x).

Moreover, we also have:

(d) for each x ∈ X and y ∈ Y, x ∉ B′(y) and y ∉ A′(x) by the definition of A′ and B′; and (e) both mappings A′ and B′ have non-empty values as ∅ ≠ A(x) ⊂ A′(x) and ∅ ≠ B(y) ⊂ B′(y)

for each x ∈ X and y ∈ Y.

As X is compact and A′ has non-empty convex values and open fibers, Lemma 2.6.3 implies that there exists a polytope D in Y and a continuous (single-valued) function f : X → D such that f(x) ∈ A′(x) for each x ∈ X. Now the mapping F : X → 2^{X} defined by F (x) := B′(f(x)) for each x ∈ X has the following properties:

(1) F (x) is non-empty convex for each x ∈ X; and

(2) F^{−1}(y) := {x ∈ X : y ∈ (B′)^{−1}(f(x))} = f^{−1}((B′)^{−1}(y)) is open in X for each y ∈ X by (iii′′).

Thus Lemma 2.6.3 implies that there exists x_{0} ∈ X such that x_{0} ∈ F (x_{0}) = B′(f(x_{0})). Let y_{0} = f(x_{0}). Then we have that x_{0} ∈ B′(y_{0}) and y_{0} ∈ A′(x_{0}) as f(x_{0}) ∈ A′(x_{0}), but this contradicts with (d). Therefore the conclusion follows and we complete the proof. □

Let X = Y in Theorem 2.6.4. As a direct application of the Browder-Fan fixed point theorem, we have the following exsitence of maximal elements for a set-valued mapping in topological vector spaces.

Theorem 2.6.5. Let X be a non-empty compact and convex subset of a topological vector space E and A : X → 2^{X} a set-valued mapping. Suppose that for each x ∈ X with A(x) ≠ ∅, there exists a non-empty open neighborhood N(x) of x in X and a set-valued mapping A_{x} : X → 2^{X} which has (maybe empty) convex values such that:

(1) A(z) ⊂ A_{x}(z) for all z ∈ N(x);

(2) A_{x}^{−1}(y) := {z ∈ X : y ∈ A_{x}(z)} is open for each y ∈ X; and

(3) z ∉ A_{x}(z) for all z ∈ X.

Then there exists x_{0} ∈ X such that A(x_{0}) = ∅.

Proof. By following the same idea used in the proof of Theorem 2.6.4, we will prove the conclusion. Suppose contrary that for each x ∈ X, A(x) ≠ ∅. Then for each x ∈ X, there exists a non-empty open neighborhood N(x) of x in X and a mapping A_{x} : X → 2^{X}, which satisfies conditions (1)-(3) of Theorem 2.6.5. Note that X is compact. For each x ∈ X, let O(x) be an open neighborhood of x in X such that \overline{O(x)} ⊂ N(x). Then X = ∪_{x∈X}O(x). By the compactness of X, there exist x_{1} ⋯ x_{n} ∈ X such that X = ∪_{i=1}^{n}O(x_{i}). Define A_{i} : X → 2^{X} for i = 1, 2 ⋯ n by

A_{i}(x) = \begin{cases}A_{x_{i}}(x), & if x ∈ \overline{O(x_{i})} \\ X, & if x ∈ X \ \overline{O(x_{i})}. \end{cases}

Then by the similar arguments used in the proof of Theorem 2.6.4, we have:

(a′) A_{i} is non-empty convex-valued;

(b′) A(x) ⊂ A_{i}(x) for each x ∈ X; and

(c′) for each y ∈ X, A_{i}^{−1}(y) is open in X. Now define A′ : X → 2^{X} by A′(x) = ∩_{i=1}^{n}A_{i}(x) for each x ∈ X. Clearly we also have:

(a′′) A′ is non-empty convex-valued;

(b′′) A(x) ⊂ A′(x) for each x ∈ X; and (c′′) for each y ∈ X, the set (A′)^{−1}(y) is open in X since

(A′)^{−1}(y) = {x ∈ X : y ∈ ∩_{i=1}^{n}A_{i}(x)} = ∩_{i=1}^{n}A_{i}^{−1}(y).

As X is compact and A′ has non-empty convex values and open fibers, Lemma 2.6.3 implies that A′ has a fixed point. On the other hand, it is clear that x ∉ A′(x) for all x ∈ X by its definition. This contradiction shows that there must exist x_{0} ∈ X such that A(x_{0}) = ∅. □

As another application of Theorem 2.6.4, we have the following existence theorem of maximal elements for two families which are coupled-majorized mappings in product spaces.

Theorem 2.6.6. Let X = Π_{j∈J}X_{j} and Y = Π_{i∈I}Y_{i}, where {X_{j}}_{j∈J} and {Y_{i}}_{i∈I} are two families of non-empty compact convex subsets of topological vector spaces E_{j} and F_{i}, respectively for each (i, j) ∈ I × J, where I and J are two arbitrary index sets. Suppose A_{i} : X → 2^{Y_{i}} and B_{j} : Y → 2^{X_{j}} are couple-majorized families such that

∪_{j∈J}{y ∈ Y : B_{j}(y) ≠ ∅} = ∪_{j∈J}int{y ∈ Y : B_{j}(y) ≠ ∅}

and

∪_{i∈I}{x ∈ X : A_{i}(x) ≠ ∅} = ∪_{i∈I}int{x ∈ X : A_{i}(x) ≠ ∅}.

Then there exists at least one x ∈ X such that A_{i}(x) = ∅ for all i ∈ I or there exists y ∈ Y such that B_{j}(y) = ∅ for all j ∈ J.

Proof. For each i ∈ I, define A′_{i} : X → 2^{Y} by

A′_{i}(x) = Π_{j∈I,j≠i}Y_{j} × A_{i}(x)

for each x ∈ X. For each x ∈ X, define I(x) = {i ∈ I : A_{i}(x) ≠ ∅}. Now we define A : X → 2^{Y} by

A(x) = \begin{cases}∩_{i∈I(x)}A′_{i}(x), & if I(x) ≠ ∅ \\ ∅, & if I(x) = ∅. \end{cases}

Similarly, for each j ∈ J, define B′_{j} : Y → 2^{X_{j}} by

B′_{i}(y) = Π_{i∈J,i≠j}X_{j} × B_{i}(y)

for each y ∈ Y. For each y ∈ Y, define J(y) = {j ∈ J : B_{j}(y) ≠ ∅}. Now we define B : Y → 2^{X} by

B(y) = \begin{cases}∩_{j∈J(y)}B′_{i}(y), & if J(y) ≠ ∅ \\ ∅, & if J(y) = ∅. \end{cases}

Then we shall show that the pair (A, B) is couple-majorized. Let (x, y) ∈ X × Y. We may assume that I(x) ≠ ∅ and J(y) ≠ ∅, otherwise the conclusion follows. Then x ∈ ∪_{i∈I}{x ∈ X : A_{i}(x) ≠ ∅} and y ∈ ∪_{j∈J}{y ∈ Y : B_{j}(y) ≠ ∅}. Since (A_{i}, B_{j}) is couple-majorized, there exists an open neighborhood N(x) of x and an open neighborhood N(y) of y in Y and a pair of a couple-majorant C_{i} : X → 2^{Y_{i}} and D_{j} : Y → 2^{X_{j}} at (x, y) such that:

(a) A_{i}(z) ⊂ C_{i}(z) for each z ∈ N(x) and B_{j}(z) ⊂ D_{j}(z) for each z ∈ N(y);

(b) (u_{j}, v_{i}) ∉ (D_{j}(v), C_{i}(u)) for each (u, v) ∈ X × Y; and

(c) the set C_{i}^{−1}(v_{i}) and D_{j}^{−1}(u_{j}) are open in X and Y, respectively, for each v_{i} ∈ Y_{i} and u_{j} ∈ X_{j}.

By our assumptions, without loss of generality, we may assume that N(x) ⊂ int{x ∈ X : A_{i}(x) ≠ ∅} and N(y) ⊂ int{y ∈ Y : B_{j}(y) ≠ ∅}. Thus A_{i}(z) ≠ ∅ for each z ∈ N(x) and B_{j}(z) ≠ ∅ for each z ∈ N(y). Now we define C_{x} : X → 2^{Y} and D_{y} : Y → 2^{X} by

C_{x}(z) = Π_{j∈I,j≠i}Y_{j} × C_{i}(z)

for each z ∈ X and

D_{y}(z) = Π_{i∈J,i≠j}X_{j} × D_{j}(z)

for each z ∈ Y. Then we claim that (C_{x}, D_{y}) is a couple-majorant of the pair (A, B) at (x, y) ∈ X × Y. Indeed, we have

(i) A(z) = ∩_{k∈I(z)}A′_{k}(z) ⊂ A′_{i}(z) ⊂ C_{x}(z) for each z ∈ N(x);

(ii) B(z) = ∩_{k∈J(z)}B′_{k}(z) ⊂ B′_{i}(z) ⊂ D_{y}(z) for each z ∈ N(y);

(iii) C_{x}^{−1}(v) = (C′_{i})^{−1}(v) and D_{y}^{−1}(u) = (D′_{j})^{−1}(u) are open in X and Y, respectively, for each u ∈ X and v ∈ Y; and

(iv) (u, v) ∉ D_{y}(v) × C_{x}(u) for each u ∈ X and v ∈ Y. Therefore (C_{x}, D_{y}) is a couple-majorant of (A, B) at (x, y). Thus (A, B) satisfies all hypotheses of Theorem 2.6.4. By Theorem 2.6.4, there exists at least one x ∈ X such that A(x) = ∅ or y ∈ Y such that B(y) = ∅, i.e., A_{i}(x) = ∅ for all i ∈ I or B_{j}(y) = ∅ for all j ∈ J and the proof is completed. □

As an application of Theorem 2.6.5, we have the following existence of maximal elements for a family of set-valued mappings in topological vector spaaces.

Theorem 2.6.7. Let X = Π_{i∈I}X_{i} be a family of non-empty compact convex subsets {X_{i}}_{i∈I} of topological vector spaces E_{i} and I an arbitrary index set. Let {A_{i}}_{i∈I} be a family of set-valued mappings such that ∪_{i∈I}{x ∈ X : A_{i}(x) ≠ ∅} = ∪_{i∈I}int{x ∈ X : A_{i}(x) ≠ ∅}, where for each i ∈ I, A_{i} : X → 2^{X_{i}} is a set-valued mapping. Suppose that for each i ∈ I and x ∈ X with A_{i}(x) ≠ ∅, there exists a non-empty open neighborhood N(x) of x in X and a set-valued mapping A_{i,x} : X → 2^{X_{i}} which takes (maybe empty) convex-values such that:

(1) A_{i}(z) ⊂ A_{i,x}(z) for all z ∈ N(x);

(2) z_{i} ∉ A_{i,x}(z) for all z ∈ X; and

(3) A_{i,x}^{−1}(y) := {z ∈ X : y ∈ A_{i,x}(z)} is open in X for each y ∈ X_{i}.

Then there exists at least x̂ ∈ X such that A_{i}(x̂) = ∅ for all i ∈ I.

Proof. By following the similar used in the proof of Theorem 2.6.6, for each i ∈ I we define A′_{i} : X → 2^{X} by

A′_{i}(x) = Π_{j∈I,j≠i}X_{j} × A_{i}(x)

for each x ∈ X. For each x ∈ X, define I(x) = {i ∈ I : A_{i}(x) ≠ ∅}. Now define A : X → 2^{X} by

A(x) = \begin{cases}∩_{i∈I(x)}A′_{i}(x), & if I(x) ≠ ∅ \\ ∅, & if I(x) = ∅. \end{cases}

Then we shall show that A satisfies all conditions of Theorem 2.6.5. Without loss of the generality, we may assume that I(x) ≠ ∅ (otherwise the conclusion follows). Then x ∈ ∪_{i∈I}{x ∈ X : A_{i}(x) ≠ ∅} and there exists i ∈ I such that A_{i}(x) ≠ ∅. Thus there exists an open neighborhood N(x) of x in X and a set-valued mapping A_{i,x} : X → 2^{X_{i}} which satisfies all conditions (1)-(3) of Theorem 2.6.7. By our assumptions, we can also assume that N(x) ⊂ int{x ∈ X : A_{i}(x) ≠ ∅}. Thus A_{i}(z) ≠ ∅ for each z ∈ N(x). Now define A_{x} : X → 2^{X} by

A_{x}(z) = Π_{j∈I,j≠i}X_{j} × A_{i,x}(z)

for each z ∈ X. Then it is clear that the mapping A_{x} has the following properties:

(i) A(z) = ∩_{k∈I(z)}A′_{k}(z) ⊂ A′_{i}(z) ⊂ A_{x}(z) for each z ∈ N(x), and A_{x} takes (maybe empty) convex values;

(ii) A_{x}^{−1}(v) = (A′_{i,x})^{−1}(v) which is open in X for each v ∈ X; and

(iii) z ∉ A_{x}(z) for all z ∈ X. Therefore we have shown that all hypotheses of Theorem 2.6.5 are satisfied. By Theorem 2.6.5, there exists x̂ ∈ X such that

We recall that a coincidence theorem asserts that if S : X → 2^{Y} and T : Y → 2^{X} have nonempty values and satisfy certain other conditions then there exist x_{0} ∈ X and y_{0} ∈ Y such that y_{0} ∈ Sx_{0} and x_{0} ∈ T y_{0}.

Remark 2.6.8. When X is a compact topological space instead of having a linear structure and the mapping B_{j} is replaced by a continuous single-valued mapping S in Theorems 2.6.4 and 2.6.6, we will establish existence theorems of maximal elements for L_{S}-majorized mappings (see Definition 2.6.13) in (not necessarily compact) product spaces. For the convenience, we first recall the following definition.

Definition 2.6.9. Let X be a Hausdorff topological space, and I any index set. For each i ∈ I, let Y_{i} be a non-empty convex subset of a Hausdorff topological vector space and A_{i} : X → 2^{Y_{i}}. Then {A_{i}}_{i∈I} is said to be a KF family if the following assumptions are satisfied:

(i) for each i ∈ I, A_{i}(x) is convex for each x ∈ X; and

(ii) for each x ∈ X, there exists i ∈ I such that A_{i}(x) ≠ ∅; and

(iii) the set A_{i}^{−1}(y_{i}) is open in X for all y_{i} ∈ Y_{i} and i ∈ I.

Now we arrive in the place to have the following coincidence theorem in product spaces.

Theorem 2.6.10. Let I and J be any two index sets, X = Π_{j∈I}X_{j} and Y = Π_{i∈I}Y_{i} be product spaces of non-empty compact convex subset X_{j} and non-empty convex subset Y_{i}, each in a Hausdorff topological vector space. Suppose that both {A_{i}}_{i∈I} and {B_{j}}_{j∈J} are KF families. Then there exist (i_{0}, j_{0}) ∈ I × J and (x̂, ŷ) ∈ X × Y such that ŷ_{i_{0}} ∈ A_{i_{0}}(x̂) and x̂_{j_{0}} ∈ B_{j_{0}}(ŷ).

Proof. Suppose the conclusion is not true. Then {A_{i}}_{i∈I} and {B_{j}}_{J∈J} are couple-majorized. By Theorem 2.7.6, there exist either x̂ ∈ X such that A_{i}(x̂) = ∅ for all i ∈ I or there exists ŷ ∈ Y such that B_{j}(ŷ) = ∅ for all j ∈ J, which contradicts that both {A_{i}}_{i∈I} and {B_{j}}_{j∈I} are KF families. Thus the conclusion follows. □

As an application of Theorem 2.7.7, we have the following component fixed point theorem which could be regarded as an extension of the Browder-Fan fixed point theorem in product spaces.

Theorem 2.6.11. Let I be any index set and X = Π_{i∈I}X_{i} the product space of non-empty compact convex sets X_{i} in a topological vector space E_{i} for each i ∈ I. Suppose that {A_{i} : A_{i} : X → 2^{X_{i}}}_{i∈I} is a KF family. Then there exist i_{0} ∈ I and x̂ ∈ X such that x̂_{i_{0}} ∈ A_{i_{0}}(x̂).

Proof. Suppose the conclusion were not true. By Theorem 2.6.7, there exists x̂ ∈ X such that A_{i}(x̂) = ∅ for all i ∈ I. Of course, this contradicts with that {A_{i}}_{i∈I} is a KF family. Thus there must exist x̂ ∈ X and i_{0} ∈ I such that x̂_{i_{0}} ∈ A_{i_{0}}(x̂) and the proof is completed. □

An an immediate consequence of Theorem 2.6.11, we have the following Sion’s minimax inequality for the version of two functions of Sion in [?].

Corollary 2.6.12 (Sion’s Minimax Inequality). Let X be a non-empty compact convex subset of a topological vector space E and Y a non-empty compact convex subset of a topological vector space F. Suppose f, g : X × Y → ℝ ∪ {−∞, +∞} satisf that:

(i) f(x, y) ≤ g(x, y) for each (x, y) ∈ X × Y;

(ii) for each fixed x ∈ X, y ↦ f(x, y) is quasi-concave, and for each fixed y ∈ Y, x ↦ f(x, y) is lower semicontinuous; and

(iii) for each fixed x ∈ X, y ↦ g(x, y) is upper semicontinuous, and for each fixed y ∈ Y, x ↦ g(x, y) is quasi-convex.

Then we have

min_{x∈X} sup_{y∈Y} f(x, y) ≤ sup_{y∈Y} min_{x∈X} g(x, y).

Proof. Let λ_{0} := min_{x∈X} sup_{y∈Y} f(x, y). For each given ϵ > 0, let λ_{ϵ} = λ_{0} − ϵ and define A : X → 2^{Y} and B : Y → 2^{X} by

A(x) = {y ∈ Y : f(x, y) > λ_{ϵ}}

for each x ∈ X and

B(y) = {x ∈ X : g(x, y) < λ_{ϵ}}

for each y ∈ Y. Note that both A and B have (maybe empty) convex values and open fibers, and moreover they do not have any coincident point. Otherwise there exists (x_{0}, y_{0}) ∈ X × Y such that λ_{ϵ} < f(x_{0}, y_{0}) ≤ g(x_{0}, y_{0}) < λ_{ϵ}, which is impossible. By Theorem 2.6.10, we must have that either A or B is not a KF mapping. As A(x) ≠ ∅ for each x ∈ X, there must exist y_{0} ∈ Y such that B(y_{0}) = ∅. Therefore inf_{x∈X} g(x, y_{0}) ≥ λ_{0} − ϵ, so that sup_{y∈Y} inf_{x∈X} g(x, y) ≥ λ_{0} − ϵ. As ϵ is arbitrary, we have sup_{y∈Y} inf_{x∈X} g(x, y) ≥ λ_{0}. Then

sup_{y∈Y} inf_{x∈X} g(x, y) ≥ min_{x∈X} sup_{y∈Y} f(x, y),

and the proof is complete. □

So far we have established Browder-Fan fixed point theorems in compact product spaces. In next part, we will discuss non-compact versions of the Browder-Fan fixed point theorems in product spaces for the so-called L_{s}-majorized set-valued mappings.

2.6.2 Maximal Element and Fixed Point of L_{s}-Majorized Mappings

The aim of this part is to discuss the Browder-Fan fixed point theorems in product spaces for L_{s}-majorized set-valued mappings in the setting of non-compact production spaces through the concept of maximal elements. As applications, the general minimax inequality theory for a family of functions also is developed. We first recall some definitions and notions.

Definition 2.6.13. Let X be a topological space and Y = Π_{i∈I}Y_{i} be a product of non-empty convex sets Y_{i}, each in a Hausdorff topological vector space, where I is any index set. Suppose S : Y → X is a (single-valued) mapping and A_{i} : X → 2^{Y_{i}} is a set-valued mapping for each i ∈ I. Then A_{i} is said to be

(i): of class L_{S} if (a) A_{i} is convex valued; (b) y_{i} ∉ A_{i}(S(y)) for each for each y ∈ Y; and (c) A_{i}^{−1}(y_{i}) := {x ∈ X : y_{i} ∈ A_{i}(x)} is open in X for each y_{i} ∈ Y_{i}; and

(ii): L_{S}-majorized if for each x ∈ X, there exists an open neighborhood N(x) of x in X and a convex-valued mapping B_{x} : X → 2^{Y_{i}} such that (a): A_{i}(z) ⊂ B_{x}(z) for each z ∈ N(x); (b): y_{i} ∉ B_{x}(S(y)) for each y ∈ Y and (c): B_{x}^{−1}(y_{i}) is open in X for each y_{i} ∈ Y_{i}. Here the mapping B_{x} is called an L_{S}-majorant of A_{i} at x.

In this section, we shall denote by M_{S}(X, Y_{i})_{i∈I} (resp., L_{S}(X, Y_{i})_{i∈I}) the set of all families of {A_{i}}_{i∈I} such that for each i ∈ I, A_{i} : X → Y_{i} is L_{S}-majorized (resp., is of class L_{S}). In the case that X = Π_{i∈I}X_{i} and Y_{i} = X_{i} and the map S is the identity mapping on Y, i.e., S = I_{X}, we shall write M(X, X_{i})_{i∈I} (resp., L(X, X_{i})_{i∈I}) ) in place of M_{I_{X}} (X, X_{i})_{i∈I} (resp., L_{I_{X}} (X, X_{i})_{i∈I}) throughout this section. Moreover if the index set I is a singleton, the mapping A in M_{S}(X, Y) (resp., A in L_{S}(X, Y)) is said to be L_{S}-majorized (resp., of class L_{S}) from X to Y.

It is clear that each class of L_{S} mapping is L_{S}-majorized, but the converse is not true in general. Let A_{i} be a non-empty subset of Y_{i} for any given but fixed i ∈ I, we define Π_{j≠i,j∈I}Y_{j} ⊗ A_{i} := {y = (y_{j})_{j∈I} ∈ Π_{j∈I}Y_{i} with the property that y_{i} ∈ A_{i}}.

We now establish an existence theorem of maximal elements for L_{S}-majorized families. In this part, we first give an existence theorem for an L_{S}-majorized mapping defined in a compact topological space. Then as applications, the existence theorem for maximal elements for any arbitrary families which are L_{S}-majorized is derived.

Theorem 2.6.14. Let X be a compact Hausdorff topological space and Y be a non-empty convex subset of a Hausdorff topological vector space E, S : Y → X be continuous and A : X → 2^{Y} be L_{S}-majorized. Then there exists x ∈ X such that A(x) = ∅.

Proof. Suppose the contrary that A(x) ≠ ∅ for all x ∈ X. Then for each x ∈ X, then there exists a non-empty open neighborhood N(x) of x in X and a convex-valued mapping B_{x} : X → 2^{Y} such that:

(a) A(z) ⊂ B_{x}(z) for each z ∈ N(x);

(b) y ∉ B_{x}(S(y)) for each y ∈ Y; and

(c) B_{x}^{−1}(y) is open in X for each y ∈ Y.

Since X is regular, for each x ∈ X, let O(x) be an open neighborhood of x in X such that \overline{O(x)} ⊂ N(x). Note that X is compact and X = ∪_{x∈X}O(x), there exist x_{1}, ⋯, x_{n} ∈ X such that X = ∪_{i=1}^{n}O(x_{i}). Fix an arbitrary i ∈ {1, ⋯, n}. Define B_{i} : X → 2^{Y} by

B_{i}(x) = \begin{cases}B_{x_{i}}(x), & if x ∈ \overline{O(x_{i})}; \\ Y, & if x ∈ X \ \overline{O(x_{i})}, \end{cases}

then we have (a′) B_{i} is convex-valued; (b′) A(x) ⊂ B_{i}(x) for each x ∈ X; and (c′) for each y ∈ Y, B_{i}^{−1}(y) is open in X since

B_{i}^{−1}(y) = \{x ∈ \overline{O(x_{i})} : y ∈ B_{x_{i}}(x)\} ∪ (X \ \overline{O(x_{i})})
= (B_{x_{i}}^{−1}(y) ∩ \overline{O(x_{i})}) ∪ (X \ \overline{O(x_{i})})
= [B_{x_{i}}^{−1}(y) ∪ (X \ \overline{O(x_{i})})] ∩ [\overline{O(x_{i})} ∪ (X \ \overline{O(x_{i})}]
= B_{x_{i}}^{−1}(y) ∪ (X \ \overline{O(x_{i})})

which is open by (c). Now we define B : X → 2^{Y} by

B(x) = ∩_{i=1}^{n}B_{i}(x) for each x ∈ X. Clearly,

(a′′) B is convex-valued;

(b′′) A(x) ⊂ B(x) for each x ∈ X;

(c′′) for each y ∈ Y, the set B^{−1}(y) is open in X since

B^{−1}(y) = {x ∈ X : y ∈ ∩_{i=1}^{n}B_{i}(x)} = ∩_{i=1}^{n}B_{i}^{−1}(y);

and

(d′′) for each y ∈ Y, y ∉ B(S(y)): indeed if y ∈ Y, then S(y) ∈ X = ∪_{i=1}^{n}O(x_{i}), so there exists i ∈ {1, ⋯, n} such that S(y) ∈ O(x_{i}). Thus B_{i}(S(y)) = B_{x_{i}}(S(y)) by the definition of B_{i} and hence y ∉ B(S(y)) since B(S(y)) ⊂ B_{x_{i}}(S(y)).

By (c′′), {B^{−1}(y) : y ∈ Y } is an open cover of X; thus there exist y_{1}, ⋯, y_{n} ∈ Y such that X = ∪_{j=1}^{m}B^{−1}(y_{j}). Let {β_{j} : j = 1, ⋯, m} be a continuous partition of X subordinated to {B^{−1}(y_{j}) : j = 1, ⋯, m}. Define ϕ : X → Y by

ϕ(x) = Σ_{j=1}^{m}β_{j}(x)y_{j}

for each x ∈ X. Then ϕ is continuous. If x ∈ X and β_{j}(x) ≠ ∅, then x ∈ B^{−1}(y_{j}) which implies that y_{j} ∈ B(x) so that ϕ(x) = Σ_{i=1}^{m}β_{j}(x)y_{j} ∈ B(x) for each x ∈ X. Therefore ϕ is a continuous selection of B. Let Y_{0} = co{y_{1}, ⋯, y_{m}}. Then Y_{0} is a non-empty compact subset of Y. Note that ϕ ∘ S : Y → Y_{0} is continuous, so that ϕ ∘ S|_{Y_{0}} : Y_{0} → Y_{0} is a continuous self-map. By the Brouwer fixed point theorem, there exists ŷ ∈ Y_{0} such such that ŷ = ϕ ∘ S(ŷ) ∈ B(S(ŷ)), which contradicts (d′′) that y ∉ B(S(y)) for each y ∈ Y. Hence there must exist x̂ ∈ X such that A(x̂) = ∅. □

As a consequence of Theorem 2.6.14, we have the following existence result for maximal elements of L_{S}-majorized mappings.

Theorem 2.6.15. Let X be a Hausdorff topological space and Y be a non-empty convex subset of a Hausdorff topological vector space E. Let S : Y → X be a compact and continuous map and A : X → 2^{Y} be L_{S}- majorized. Then there exists x̂ ∈ X such that A(x̂) = ∅.

Proof. Let X_{0} ⊂ X be non-empty and compact such that S(Y) ⊂ X_{0}. Consider the restriction of the mapping A to X_{0}, i.e., the mapping A|_{X_{0}} : X_{0} → 2^{Y} which is L_{S}-majorized. By Theorem 2.6.14, there exists x̂ ∈ X_{0} such that A(x̂) = ∅. □

Theorem 2.6.16. Let X be a Hausdorff topological space and I any index set. For each i ∈ I, let Y_{i} be a non-empty convex subset of a topological vector space. Let Y = Π_{i∈I}Y_{i} and S : Y → X be a compact continuous (single-valued) map and {A_{i}}_{i∈I} ∈ M_{S}(X, Y_{i})_{i∈I}. Suppose that ∪_{i∈I}{x ∈ X : A_{i}(x) ≠ ∅} = ∪_{i∈I}int{x ∈ X : A_{i}(x) ≠ ∅}. Then there exists x̂ ∈ X such that A_{i}(x̂) = ∅ for each i ∈ I.

Proof. For each i ∈ I, define A′_{i} : X → 2^{Y} by A′_{i}(x) := Π_{j≠i,j∈I}Y_{j} ⊗ A_{i}(x) for each x ∈ X. Since A_{i} is L_{S}-majorized from X to Y_{i}, so is A′_{i} from X to Y. For each x ∈ X, define I(x) = {i ∈ I : A_{i}(x) ≠ ∅}. Now we define A : X → 2^{Y} by

A(x) = \begin{cases}∩_{i∈I(x)}A′_{i}(x), & if I(x) ≠ ∅; \\ ∅, & if I(x) = ∅. \end{cases}

To complete the proof, it suffices to show that I(x) = ∅ for some x ∈ X. Suppose the contrary that I(x) ≠ ∅ for each x ∈ X.

Fix an arbitrary x ∈ X. Since I(x) ≠ ∅ by the assumption, x ∈ ∪_{i∈I}{x ∈ X : A_{i}(x) ≠ ∅}. Thus there exists i ∈ I such that A_{i}(x) ≠ ∅. As A_{i} is L_{S}-majorized, there exists an open neighborhood N(x) of x in X and an L_{S} majorant B_{i} of A_{i} at x such that:

(a) A_{i}(z) ⊂ B_{i}(z) for each z ∈ N(x);

(b) y_{i} ∉ B_{i}(S(y)) for each y ∈ Y; and

(c) the set B_{i}^{−1}(y_{i}) is open in X for each y_{i} ∈ Y_{i}. Without loss of generality, we may assume that N(x) ⊂ int{x ∈ X : A_{i}(x) ≠ ∅}. Thus A_{i}(z) ≠ ∅ for each z ∈ N(x). Now define B_{x} : X → 2^{Y} by B_{x}(z) = π_{i}^{−1}(B_{i}(z)) for each z ∈ X. We claim that B_{x} is an L_{S}-majorant of A at x. Indeed, we have

(i) A(z) = ∩_{j∈I(x)}A′_{j}(z) ⊂ A′_{i}(z) ⊂ B_{x}(z) for each z ∈ N(x);

(ii) y_{i} ∉ B(S(y) since y_{i} ∉ B_{i}(S(y)) for each y ∈ Y by (b); and

(iii) B^{−1}(y) = (B′_{i})^{−1}(y_{i}) which is open in X. Therefore B_{x} is an L_{S}-majorant of A at x.

Thus A satisfies all hypotheses of Theorem 2.6.15. By Theorem 2.6.15, there exists x̂ ∈ X such that A(x̂) = ∅, i.e., A_{i}(x̂) = ∅ for each i ∈ I. It follows that I(x̂) = ∅ which contradicts our assumption. The conclusion follows. □

We remark that for a given (single-valued) map S : Y → X and {A_{i}}_{i∈I} ∈ KF (X, Y_{i})_{i∈I}, for each i ∈ I, if y_{i} ∉ A_{i}(S(y)) for all y ∈ Y, then the family {A_{i}}_{i∈I} ∈ M_{S}(X, Y_{i})_{i∈I}. In the case I is a singleton set and X = Y, the mapping A ∈ KF (X, X) is also called a KF mapping.

Before we give the existence of maximal elements for L_{S}-majorized family {A_{i}}_{i∈I} ∈ M_{S}(X, Y_{i})_{i∈I} when both X and S are not compact, we first need the following simple fact from [?].

Lemma 2.6.17. Let X be a compact Hausdorff topological space and {A_{i}}_{i∈I} ∈ KF (X, Y_{i})_{i∈I}. Then there is a subset C = Π_{i∈I}C_{i} of Y such that for each x ∈ X, there exists i ∈ I such that A_{i}(x) ∩ C_{i} ≠ ∅. Moreover for each i ∈ I, C_{i} is a polytope and all of those polytope, except a finite number, consist of a single point.

Proof. Note that for each x ∈ X, there exists i ∈ I such that A_{i}(x) ≠ ∅, so that X = ∪_{i∈I} ∪_{y_{i}∈Y_{i}} A_{i}^{−1}(y_{i}). Since X is compact, there exists a finite subset J of I such that for each j of J, there exists a finite subset J_{j} = {y_{j}^{1}, ⋯, y_{j}^{m_{j}}} of Y_{j} with X = ∪_{j∈J} ∪_{i=1}^{m_{j}} A_{j}^{−1}(y_{j}^{i}). Take any fixed y^{0} = {y_{i}^{0}}_{i∈I} ∈ Y and define

C_{j} = \begin{cases}co(J_{j}), & if j ∈ J; \\ \{y_{j}^{0}\}, & if j ∉ J. \end{cases}

Then C = Π_{i∈I}C_{i} is the required set. □

The following selection theorem which shows that each L_{S}-majorized mapping in a paracompact space is majorized by a mapping of class L_{S}.

Lemma 2.6.18. Let X be a Hausdorff regular and paracompact topological space, I any index set, and Y = Π_{i∈I}Y_{i}, where each Y_{i} is a non-empty convex subset of a Hausdorff topological vector space for each i ∈ I. Let S : Y → X be a map and {A_{i}} ∈ M_{S}(X, Y_{i})_{i∈I}. Then there exists a family {B_{i}}_{i∈I} ∈ L_{S}(X, Y_{i})_{i∈I} such that for each i ∈ I, A_{i}(x) ⊂ B_{i}(x) for all x ∈ X.

Proof. Fix any i ∈ I. For each x ∈ X and note that A_{i} is L_{S}-majorized, there exists a non-empty open neighborhood N(x) of x in X and a mapping B_{x} : X → 2^{Y_{i}} such that:

(a) A_{i}(z) ⊂ B_{x}(z) for each z ∈ N(x);

(b) y_{i} ∉ B_{x}(S(y)) for each y ∈ Y; and

(c) B_{x}^{−1}(y_{i}) is open in X for each y_{i} ∈ Y_{i}. Then ∪_{x∈X}N(x) is an open cover of X. Note that X is paracompact and regular, let {O(x)}_{x∈X} be a locally finite open cover of X such that cl_{X}O(x) ⊂ N(x) for each x ∈ X and X = ∪_{x∈X}O(x). For each x ∈ X, define B′_{x} : X → 2^{Y_{i}} by

B′_{x}(z) = \begin{cases}B_{x}(z), & if z ∈ cl_{X}O(x); \\ Y_{i}, & if z ∈ X \ cl_{X}O(x). \end{cases}

Then it is easy to verify that

(B′_{x})^{−1}(y_{i}) = (cl_{X}O(x) ∩ B_{x}^{−1}(y_{i})) ∪ (X \ cl_{X}O(x)) = B_{x}^{−1}(y_{i}) ∪ (X \ cl_{X}O(x))

which is open in X for each y_{i} ∈ Y_{i}. Now define B_{i} : X → 2^{Y_{i}} by

B_{i}(z) = ∩_{x∈X}B′_{x}(z)

for each z ∈ X. Then we can verify that B has the following properties:

(i) B_{i}(x) is convex valued for each x ∈ X;

(ii) y_{i} ∉ B_{i}(S(y)) for each y ∈ Y;

(iii) B_{i}^{−1}(y_{i}) is, indeed, open in X by the locally finite property of the family {O(x)}_{x∈X} (e.g., see the proof of Lemma 2 of Ding and Tan [?] or Yuan [?]); and

(iv) A_{i}(x) ⊂ B_{i}(x) for each x ∈ X. Thus B_{i} is of class L_{S} and the family {B_{i}} ∈ L_{S}(X, Y_{i})_{i∈I} and by the definition of B, we have that for each i ∈ I, A_{i}(x) ⊂ B_{i}(x) for all x ∈ X. Thus the conclusion follows. □

In order to give the existence theorem of maximal elements for a family of class L_{S}-mappings which the map S is not compact, we need the following result which is an fixed point theorem as an application of Theorem 2.6.16.

Theorem 2.6.19. Let X be a Hausdorff topological space and I be any index set. For each i ∈ I, let Y_{i} be a non-empty convex subset of a Hausdorff topological vector space. Let Y = Π_{i∈I}Y_{i}, S : Y → X be compact continuous and {A_{i}}_{i∈I} ∈ KF (X, Y_{i})_{i∈I}. Then there exists ŷ ∈ Y and i_{0} ∈ I such that ŷ_{i_{0}} ∈ A_{i_{0}}(S(ŷ)).

Proof. Suppose the conclusion were false. Then {A_{i}}_{i∈I} ∈ L_{S}(X, Y_{i})_{i∈I} and the set {x ∈ X : A_{i}(x) ≠ ∅} is also open in X, so that the family {A_{i}}_{i∈I} satisfies all hypotheses of Theorem 2.6.16. By Theorem 2.6.16, there exists x̂ ∈ X such that A_{i}(x̂) = ∅ for all i ∈ I, which contradicts our assumption that {A_{i}}_{i∈I} ∈ KF (X, Y_{i})_{i∈I}. Therefore there must exist ŷ ∈ Y and i_{0} ∈ I such that ŷ_{i_{0}} ∈ A_{i_{0}}(S(ŷ)). □

Now we have the following existence theorem of maximal elements for a family in which domains are not compact and the map S also need not be compact from Deguire et al. [?].

Theorem 2.6.20. Let X be a Hausdorff topological space and I be any index set. Let Y = Π_{∈I}Y_{i} and S : Y → X continuous, where Y_{i} is a non-empty convex subset of a topological vector space for each i ∈ I and {A_{i}}_{i∈I} ∈ L_{S}(X, Y_{i})_{i∈I}. Suppose there exists a non-empty compact subset K of X and a non-empty compact convex subset C_{i} of Y_{i} for each i ∈ I such that for each x ∈ X \ K, there exists i ∈ I such that A_{i}(x) ∩ C_{i} ≠ ∅. Then there exists x ∈ K such that A_{i}(x) = ∅ for all i ∈ I.

Proof. Suppose it is not true. Then for each x ∈ K, there exists i ∈ I such that A_{i}(x) ≠ ∅. By Lemma 2.6.17, there exists a subset D = Π_{i∈I}D_{i} such that for each x ∈ K, there exists i ∈ I such that A_{i}(x) ∩ D_{i} ≠ ∅, where D_{i} is a polytope for each i ∈ I. Note that S is continuous, so that the set X_{0} = S(D) is a non-empty compact subset of X. For each i ∈ I, we set H_{i} = co(C_{i} ∪ D_{i}), then H_{i} is a non-empty compact convex subset of Y_{i}. If x ∈ X \ K, there exists i ∈ I such that ∅ ≠ A_{i}(x) ∩ C_{i} ⊂ A_{i}(x) ∩ H_{i} by (I). If x ∈ K, by our assumption there exists i ∈ I such that ∅ ≠ A_{i}(x) ∩ D_{i} ⊂ A_{i}(x) ∩ H_{i}. Thus in any case, for each x ∈ X, there exists i ∈ I such that A_{i}(x) ∩ H_{i} ≠ ∅. For each i ∈ I, define A′_{i} : X_{0} → 2^{H_{i}} by

A′_{i}(x) = A_{i}(x) ∩ H_{i}

for each x ∈ X_{0} and let S_{1} = S|_{H}, which is the restriction of the continuous map S to H := Π_{i∈I}H_{i}. Let X := X_{0} and Y_{i} := H_{i} for each i ∈ I in Theorem 2.6.19. Then the family {A′_{i}}_{i∈I} and S_{1} satisfy all hypotheses of Theorem 2.6.19. By Theorem 2.6.19, there exists y ∈ H and i_{0} ∈ I such that y_{i_{0}} ∈ A′_{i_{0}}(S_{1}(y)), i.e., y_{i_{0}} ∈ A_{i_{0}}(S(y)) ∩ H_{i_{0}}. Therefore y_{i_{0}} ∈ A_{i_{0}}(S(y)) which contradicts that y_{i_{0}} ∉ A_{i_{0}}(S(y)) for all y ∈ H since A_{i_{0}} is of class L_{S}. This contradiction shows that there must exist x ∈ K such that A_{i}(x) = ∅ for all i ∈ I. □

As applications of Lemma 2.6.18 and Theorem 2.6.20, we have the following existence theorem of maximal elements for L_{S}-majorized family by assuming paracompactness instead of compactness for the domains.

Theorem 2.6.21. Let X be a Hausdorff regular and paracompact topological space and I any index set. For each i ∈ I, let Y_{i} be a non-empty convex subset of a Hausdorff topological vector space. Let Y = Π_{i∈I}Y_{i}, S : Y → X be a continuous mapping and {A_{i}}_{i∈I} ∈ M_{S}(X, Y_{i})_{i∈I}. Suppose there exists a non-empty compact subset K of X and a non-empty compact convex subset C_{i} of Y_{i} for each i ∈ I such that for each x ∈ X \ K, there exists i ∈ I such that A_{i}(x) ∩ C_{i} ≠ ∅. Then there exists x ∈ K such that A_{i}(x) = ∅ for all i ∈ I.

Proof. By Lemma 2.6.18, for each i ∈ I, there exists B_{i} : X → 2^{Y_{i}} which is of class L_{S} such that A_{i}(x) ⊂ B_{i}(x) for each x ∈ X. Hence the family {B_{i}}_{i∈I} satisfies all hypotheses of Theorem 2.6.20. By Theorem 2.6.20, there exists x ∈ X such that B_{i}(x) = ∅ for each i ∈ I, so that A_{i}(x) = ∅ for each i ∈ I. □

Finally we are in turn to give component versions of Browder-Fan fixed point theorems and coincidence theorems in non-compact product spaces. In this part, as applications of existence theorems of maximal element for class L_{S} mappings in topological spaces, we shall give some component versions of Browder-Fan type fixed point theorems in product spaces for the family of KF mappings.

As an equivalent fixed point version of Theorem 2.6.20, we have the following fixed point theorem in product spaces.

Theorem 2.6.22. Let X be a Hausdorff topological space and I be any index set. For each i ∈ I, let Y_{i} be a non-empty convex subset of a Hausdorff topological vector space and Y = Π_{i∈I}Y_{i}. Suppose S : Y → X is continuous and A_{i} : X → 2^{Y_{i}} is convex-valued for each i ∈ I. Moreover, suppose there exists a non-empty compact subset K of X and a non-empty compact convex subset C_{i} of Y_{i} for each i ∈ I such that:

(a) for each x ∈ X \ K, there exists i ∈ I such that A_{i}(x) ∩ C_{i} ≠ ∅;

(b) the set A_{i}^{−1}(y) is open in X for each y ∈ Y_{i} and i ∈ I; and

(c) for each x ∈ K, there exists i ∈ I such that A_{i}(x) ≠ ∅. Then there exists ŷ ∈ Y and i_{0} ∈ I such that ŷ_{i_{0}} ∈ A_{i_{0}}(S(ŷ)).

Proof. Theorem 2.6.20 ⇒ Theorem 2.6.22: Suppose the conclusion of Theorem 2.6.22 were false. Then all hypotheses of Theorem 2.6.20 are satisfied. By Theorem 2.6.20, there exists x ∈ K such that A_{i}(x) = ∅ for all i ∈ I, which contradicts condition (c). Hence the conclusion must be true. □

Theorem 2.6.22 ⇒ Theorem 2.6.20: Suppose the conclusion of Theorem 2.6.20 were false. Then all hypotheses of Theorem 2.6.22 are satisfied. By Theorem 2.6.22, there exists ŷ ∈ Y and i_{0} ∈ I such that ŷ_{i_{0}} ∈ A_{i_{0}}(S(ŷ)) which contradicts the assumption that A_{i_{0}} is of class L_{S}. □

We wish to point out that the conclusion of Theorem 2.6.22 could not guarantee in general that ŷ is a fixed point of the product mapping Π_{i∈I}A_{i}(S) as we can only assure that there exists i_{0} ∈ I (not necessarily for all i ∈ I) such that ŷ_{i_{0}} ∈ A_{i_{0}}(S(ŷ)). As an application of Theorem 2.6.22, we have the following coincidence theorem for two families of set-valued mappings in product spaces.

Theorem 2.6.23. Let I be any index set, X = Π_{i∈I}X_{i} and Y = Π_{i∈I}Y_{i} be products of non-empty convex subsets X_{i} and Y_{i}, each in a Hausdorff topological vector space. For each i ∈ I, let A_{i} : X → 2^{Y_{i}} and B_{i} : Y → 2^{X_{i}}. Suppose there exists a non-empty compact subset K of X and a non-empty compact subset L of Y and non-empty compact convex subsets C_{i} of Y_{i} and D_{i} of X_{i} for each i ∈ I such that:

(i) for each i ∈ I, A_{i}^{−1}(y) and B_{i}^{−1}(x) are open in X and Y, respectively for each (x, y) ∈ Y_{i}×X_{i} for each i ∈ I;

(ii) for each (x, y) ∈ (X × Y) \ (K × L), there exists i ∈ I such that A_{i}(x) ∩ C_{i} ≠ ∅ and B_{i}(y) ∩ D_{i} ≠ ∅; and

(iii) for each (x, y) ∈ K × L, there exists i ∈ I such that A_{i}(x) ≠ ∅ and B_{i}(y) ≠ ∅. Then there exist (x̂, ŷ) ∈ X × Y and i_{0} ∈ I such that ŷ_{i_{0}} ∈ A_{i_{0}}(x̂) and x̂_{i_{0}} ∈ B_{i_{0}}(ŷ).

Proof. Let M := K × L. For each i ∈ I, let E_{i} = C_{i} × D_{i} and define W_{i} : X × Y → 2^{Y_{i}×X_{i}} by

W_{i}(x, y) = A_{i}(x) × B_{i}(y)

for each (x, y) ∈ X ×Y. Also define S : X ×Y → Y ×X by S(x, y) = (y, x) for each (x, y) ∈ X ×Y. Then all hypotheses of Theorem 2.6.22 are satisfied. By Theorem 2.6.22, there exists (x̂, ŷ) ∈ X ×Y and i_{0} ∈ I such that (x̂_{i_{0}}, ŷ_{i_{0}}) ∈ W_{i_{0}}(x̂, ŷ). Thus the conclusion follows. □

Before we close this section, we will give some minimax inequalities for families of functions. In this part, as applications of existence theorems of maximal elements for a family of set-valued mappings, we shall first give an existence theorem of solutions for a system version of Ky Fan minimax inequalities in which functions are lower semicontinuous it its first variable and quasiconcave in its second variable. By introducing a new concept called L-majorized, we establish the system of Ky Fan minimax inequalities for a family of L-majorized functions which are not lower semicontinuous nor quasi-concave.

As an application of Theorem 2.6.16, we have the following result Ky Fan minimax inequality for a family of functions.

Theorem 2.6.24. Let X be a Hausdorff topological space, I any index set and Y = Π_{i∈I}Y_{i} be the product of non-empty convex set Y_{i}, each in a Hausdorff topological vector space. Suppose S : Y → X is compact continuous and for each i ∈ I, the function f_{i} : X × Y_{i} → ℝ satisfies the following conditions:

(i) for each fixed x ∈ X, y_{i} ↦ f_{i}(x, y_{i}) is quasi-concave; and

(ii) for each fixed y_{i} ∈ Y_{i}, x ↦ f_{i}(x, y_{i}) is lower semicontinuous. Then we have

(A): For each λ ∈ ℝ, at least one of the following statement is satisfied:

(1) there exists x ∈ X such that sup_{i∈I} sup_{y_{i}∈Y_{i}} f_{i}(x, y_{i}) ≤ λ, or

(2) there exists y ∈ Y and i ∈ I such that f_{i}(S(y), y_{i}) > λ.

and (B): The following minimax inequalities holds:

inf_{x∈X} sup_{i∈I} sup_{y_{i}∈Y_{i}} f_{i}(x, y_{i}) ≤ sup_{i∈I} sup_{y∈Y} f_{i}(S(y), y_{i}).

Proof. (A): Fix arbitrary i ∈ I and λ ∈ ℝ. Define A_{i} : X → 2^{Y_{i}} by

A_{i}(x) = {y_{i} ∈ Y_{i} : f_{i}(x, y_{i}) > λ}

for each x ∈ X. Then for each y_{i} ∈ Y_{i}, the set A_{i}^{−1}(y_{i}) is open in X by (ii) and for each x ∈ X, A_{i}(x) is convex valued by (i).

If the case (2) of (A) does not hold, then for each y ∈ Y and i ∈ I, y_{i} ∉ A_{i}(S(y)). Thus {A_{i}}_{i∈I} ∈ L_{S}(X, Y_{i})_{i∈I}. By Theorem 2.6.16, there exists x ∈ X such A_{i}(x) = ∅ for each i ∈ I, i.e.,

(1) of (A) holds.

(B): Let λ_{0} = sup_{i∈I} sup_{y∈Y} f_{i}(S(y), y_{i}), then (2) of (A) fails to hold, so that the (1) of (A) must hold, i.e., there exists x_{0} ∈ X such that sup_{i∈I} sup_{y_{i}∈Y_{i}} f_{i}(x_{0}, y_{i}) ≤ λ_{0}. Therefore we have

inf_{x∈X} sup_{i∈I} sup_{y_{i}∈Y_{i}} f_{i}(x, y_{i}) ≤ sup_{i∈I} sup_{y∈Y} f_{i}(S(y), y_{i}). □

As an analytical formulation of Theorem 2.6.20, we have the following result.

Theorem 2.6.25. Let X be a Hausdorff topological space, I any index set and Y = Π_{i∈I}Y_{i} be the product of non-empty convex sets Y_{i}, each in a Hausdorff topological vector space, and S : Y → X be continuous. Suppose for each i ∈ I, the function f_{i} : X × Y_{i} → ℝ satisfies the following conditions:

(i) for each fixed x ∈ X, y_{i} ↦ f_{i}(x, y_{i}) is quasi-concave; and

(ii) for each fixed y_{i} ∈ Y_{i}, x ↦ f_{i}(x, y_{i}) is lower semicontinuous. Then for each λ ∈ ℝ, at least one of the following statement is satisfied:

(1) there exists x ∈ X such that sup_{i∈I} sup_{y_{i}∈Y_{i}} f_{i}(x, y_{i}) ≤ λ, or

(2) there exist y ∈ Y and i ∈ I such that f_{i}(S(y), y_{i})) > λ, or

(3) for each given non-empty compact subset K of X and for each given non-empty compact convex subset C_{i} of Y_{i}, where i ∈ I, there exists x ∈ X \ K such that

sup_{i∈I} sup_{y_{i}∈C_{i}} f_{i}(x, y_{i}) ≤ λ.

Proof. Theorem 2.6.20 ⇒ Theorem 2.6.25: For each fixed i ∈ I and each given λ ∈ ℝ, define A_{i} : X → 2^{Y_{i}} by

A_{i}(x) = {y_{i} ∈ Y_{i} : f_{i}(x, y_{i}) > λ}

for each x ∈ X. Then conclusion follows from Theorem 2.6.20.

Theorem 2.6.25 ⇒ Theorem 2.6.20: For each i ∈ I, define f_{i} : X × Y_{i} → ℝ by

f_{i}(x, y) = \begin{cases}1, & if y_{i} ∈ A_{i}(x); \\ 0, & otherwise \end{cases}

for each (x, y_{i}) ∈ X × Y_{i}. By Theorem 2.6.25, with λ = 0, as (2) and (3) do not hold, thus (1) must hold, i.e., there exists x ∈ X such that sup_{i∈I} sup_{y_{i}∈Y_{i}} f_{i}(x, y_{i}) ≤ 0. It follows that x ∈ K and by (1) and A_{i}(x) = ∅ for all i ∈ I. □

We note that when I is a singleton set, Theorems 2.6.24 and 2.6.25 are non-linear alternative improved version of Ky Fan minimax inequality.

Before we study the system of Ky Fan minimax inequalities for a family of real-valued functions which may not be lower semicontinuous nor quasi-concave, we need the following definition.

Let X be a Hausdorff topological space and I be any index set. Let Y = Π_{i∈I}Y_{i} be a product of non-empty convex sets Y_{i}, each in a Hausdforff topological vector space. Suppose for each i ∈ I, f_{i} : X × Y_{i} → ℝ ∪ {−∞, +∞} is a function. Then the family {f_{i}}_{i∈I} is said to be L-majorized if the following condition (LC) is satisfied for each i ∈ I.

The Condition (LC): For each given λ ∈ ℝ, if there exists (x, y_{i}) ∈ X × Y_{i} such that f_{i}(x, y_{i}) > λ, then there exists a non-empty open neighborhood N(x) of x in X and a real-valued function f_{i}^{x} : X × Y_{i} → ℝ ∪ {−∞, +∞} such that:

(a_{1}): f_{i}(z, w_{i}) ≤ f_{i}^{x}(z, w_{i}) for each (z, w_{i}) ∈ N(x) × Y_{i};

(a_{2}): for each fixed z ∈ X, w_{i} ↦ f_{i}^{x}(z, w_{i}) is quasi-concave on Y_{i};

(a_{3}): for each fixed w_{i} ∈ Y_{i}, z ↦ f_{i}^{x}(z, w_{i}) is lower semicontinuous on X; and

(a_{4}): for each w ∈ Y, f_{i}(S(w), w_{i}) ≤ λ implies that f_{i}^{x}(S(w), w_{i}) ≤ λ.

Now as an application of Theorem 2.6.16, we have the following existence theorem of solutions for Ky Fan minimax inequalities for a family of real-valued functions which do not have lower semicontinuity nor quasi-concavity.

Theorem 2.6.26. Let X be a Hausdorff topological space, I any index set and Y = Π_{i∈I}Y_{i} the product of non-empty convex sets Y_{i}, each in a Hausdorff topological vector space. Suppose S : Y → X is compact continuous and for each i ∈ I, the function f_{i} : X × Y_{i} → ℝ is L-majorized such that for each given λ ∈ ℝ, ∪_{i∈I} ∪_{y_{i}∈Y_{i}} {x ∈ X : f_{i}(x, y_{i}) > λ} = ∪_{i∈I}int[∪_{y_{i}∈Y_{i}}{x ∈ X : f_{i}(x, y_{i}) > λ}]. Then we have:

(A): For each λ ∈ ℝ, at least one of the following statement is satisfied:

(1) there exists x ∈ X such that sup_{i∈I} sup_{y_{i}∈Y_{i}} f_{i}(x, y_{i}) ≤ λ, or

(2) there exists y ∈ Y and i ∈ I such that f_{i}(S(y), y_{i}) > λ.

(B): The following minimax inequality holds:

inf_{x∈X} sup_{i∈I} sup_{y_{i}∈Y_{i}} f_{i}(x, y_{i}) ≤ sup_{i∈I} sup_{y∈Y} f_{i}(S(y), y_{i}).

Proof. (A): For each given λ ∈ ℝ and fixed i ∈ I, define A_{i} : X → 2^{Y_{i}} by

A_{i}(x) = {y_{i} ∈ Y_{i} : f_{i}(x, y_{i}) > λ}

for each x ∈ X. For each given x ∈ X, if A_{i}(x) ≠ ∅, then there exists an non-empty open neighborhood N(x) of x in X and a function f_{i}^{x} : X × Y_{i} → ℝ ∪ {−∞, +∞} having the properties (a_{1})-(a_{4}). Now define another mapping A_{i}^{x} : X → 2^{Y_{i}} by

A_{x}^{i}(z) = {y_{i} ∈ Y_{i} : f_{i}^{x}(z, y_{i}) > λ}

for each z ∈ X. Then we have (i): A_{i}(z) ⊂ A_{x}^{i}(z) for each z ∈ N_{x} by (a_{1}); (ii): A_{x}^{i}(z) is convex for each z ∈ X by (a_{2}); and (iii): (A_{x}^{i})^{−1}(y_{i}) is open in X for each y_{i} ∈ Y_{i} by (a_{3}).

If the case (2) of (A) does not hold, then for each y ∈ Y and i ∈ I, y_{i} ∉ A_{i}(S(y)). By (a_{4}), y_{i} ∉ A_{x}^{i}(S(y)) for each y ∈ Y. Thus A_{x}^{i} is a majorant of A_{i} at x. Therefore A_{i} is L_{S}- majorized and the family {A_{i}}_{i∈I} ∈ M_{S}(X, Y_{i})_{i∈I}. Moreover, the set ∪_{i∈I}{x ∈ X : A_{i}(x) ≠ ∅} = ∪_{i∈I}int ∪_{y_{i}∈Y_{i}} {x ∈ X : f_{i}(x, y_{i}) > λ} is open in X by our assumption. Thus {A_{i}}_{i∈I} satisfies all hypotheses of Theorem 2.6.16. By Theorem 2.6.16, there exists x ∈ X such A_{i}(x) = ∅ for each i ∈ I, i.e., (1) of (A) holds.

(B): Let λ_{0} = sup_{i∈I} sup_{y∈Y} f_{i}(S(y), y_{i}), then (2) of (A) fails to hold, so that the (1) of (A) must hold, i.e., there exists x_{0} ∈ X such that sup_{i∈I} sup_{y_{i}∈Y_{i}} f_{i}(x_{0}, y_{i}) ≤ λ_{0}. Therefore we have

inf_{x∈X} sup_{i∈I} sup_{y_{i}∈Y_{i}} f_{i}(x, y_{i}) ≤ sup_{i∈I} sup_{y∈Y} f_{i}(S(y), y_{i}). □

When S is not compact, by a similar argument used in proving Theorems 2.6.25 and 2.6.26 and as an application of Theorem 2.6.21 instead of Theorem 2.6.20, we have the following existence theorem of Ky Fan minimax inequalities for a family of L-majorized functions, which is an analytical formulation of Theorem 2.6.21.

Theorem 2.6.27. Let X be a Hausdorff regular and paracompact topological space, I any index set, and Y = Π_{i∈I}Y_{i} the product of non-empty convex sets Y_{i}, each in a Hausdorff topological vector space. Suppose S : Y → X is continuous and for each i ∈ I and the function f_{i} : X ×Y_{i} → ℝ is L-majorized.

Then for each λ ∈ ℝ, at least one of the following statements is satisfied:

(1) there exists x ∈ X such that sup_{i∈I} sup_{y_{i}∈Y_{i}} f_{i}(x, y_{i}) ≤ λ, or

(2) there exists y ∈ Y and i ∈ I such that f_{i}(S(y), y_{i})) > λ, or

(3) for each given non-empty compact subset K of X and for each given non-empty compact convex subset C_{i} of Y_{i}, where i ∈ I, there exists x ∈ X \ K such that

sup_{i∈I} sup_{y_{i}∈C_{i}} f_{i}(x, y_{i}) ≤ λ.

Proof. By combining arguments used in both Theorem 2.6.25 and Theorem 2.6.26, and by applying Theorem 2.6.21 instead of Theorem 2.6.20, the conclusions follows. □

Let I be a singleton set, as a consequence of Theorem 2.6.24 above, we have the following:

Corollary 2.6.28. Let X be a Hausdorff topological space and Y be a non-empty convex set in a Hausdorff topological vector space. Suppose S : Y → X is compact continuous and f : X × Y → ℝ satisfies the following conditions:

(i) for each fixed x ∈ X, y ↦ f(x, y) is quasi-concave;

(ii) for each fixed y ∈ Y, x ↦ f(x, y) is lower semicontinuous. Then we have:

(A): For each λ ∈ ℝ, at least one of the following statement is satisfied:

(1) there exists x ∈ X such that sup_{y∈Y} f(x, y) ≤ λ, or

(2) there exists y ∈ Y such that f(S(y), y) > λ.

(B): The following minimax inequality holds:

inf_{x∈X} sup_{y∈Y} f(x, y) ≤ sup_{y∈Y} f(S(y), y).

As a special case of Theorem 2.6.25, we have the following minimax inequality.

Corollary 2.6.29. Let X be a Hausdorff topological space and Y be a non-empty convex set in a Hausdorff topological vector space, and S : Y → X be continuous. Suppose the function f : X × Y → ℝ satisfies the following conditions:

(i) for each fixed x ∈ X, y ↦ f(x, y) is quasi-concave; and

(ii) for each fixed y ∈ Y, x ↦ f(x, y) is lower semicontinuous. Then for each λ ∈ ℝ, at least one of the following statements is satisfied:

(1) there exists x ∈ X such that sup_{y∈Y} f(x, y) ≤ λ, or

(2) there exist y ∈ Y such that f(S(y), y)) > λ, or

(3) for each given non-empty compact subset K of X and for each given non-empty compact convex subset C of Y, there exists x ∈ X \ K such that

sup_{y∈C} f(x, y) ≤ λ.

As special cases of Theorems 2.6.26 and 2.6.27, we also have the following result.

Corollary 2.6.30. Let X be a Hausdorff topological space and Y be a non-empty convex set in a Hausdorff topological vector space. Suppose S : Y → X is compact continuous and f : X × Y → ℝ is L-majorized such that for each given λ ∈ ℝ, the set ∪_{y∈Y} {x ∈ X : f(x, y) > λ} is open in X. Then we have that

(A): For each λ ∈ ℝ, at least one of the following statement is satisfied:

(1) there exists x ∈ X such that sup_{y∈Y} f(x, y) ≤ λ, or

(2) there exists y ∈ Y such that f(S(y), y) > λ.

(B): The following minimax inequality holds:

inf_{x∈X} sup_{y∈Y} f(x, y) ≤ sup_{y∈Y} f(S(y), y).

Corollary 2.6.31. Let X be a Hausdorff regular and paracompact topological space and Y a non-empty convex set in a Hausdorff topological vector space. Suppose S : Y → X is continuous and f : X × Y → ℝ is L-majorized.

Then for each λ ∈ ℝ, at least one of the following statement is satisfied:

(1) there exists x ∈ X such that sup_{y∈Y} f(x, y) ≤ λ, or

(2) there exist y ∈ Y such that f(S(y), y)) > λ, or

(3) for each given non-empty compact subset K of X and for each given non-empty compact convex subset C of Y, there exists x ∈ X \ K such that sup_{y∈C} f(x, y) ≤ λ.

2.7 Fan-Glicksberg Fixed Points in G-Convex Spaces

In this section, we will establish Fan-Glicksberg type fixed point theorem for upper semicontinuous set-valued mappings with closed acyclic values in the setting of another abstract convex structure - called a locally G-convex space which generalizes usual convexity such as locally convex H-spaces, l.c.spaces (locally H-convex spaces), hyperconvex metric spaces and locally convex topological spaces.

A topological space X is said to be contractible if the identity mapping I_{X} of X is homotopic to a constant function. We recall that a non-empty space is said to be acyclic if all of its reduced Čech homology groups over rationals vanish. In particular, each contractible space is acyclic and thus any non-empty convex or star-shaped set is acyclic.

We have seen in last few years, many papers have been devoted to prove the existence of fixed points for set-valued mappings in nonlinear analysis without traditional linear structures. As a result, several generalizations of convexity notions such as hyperconvex metric spaces, H-spaces, G-convex spaces have been introduced and studies by Aronszajn and Panitchpakdi [?], Bardaro and Ceppitelli [?], Bielawaski [?], Horvath [?]-[?], Kirk et al. [?], Park [?]-[?], Park and Kim [?], Tarafadr [?], Tarafdar and Watson [?], Wu [?], Yuan [?] and many others. Recently, in order to cover the general economic models without linear convex structures, Park and Kim [?] introduces another abstract convexity notion-called G-convex spaces which include many abstract convexity notions such as H-convex spaces as a special case. Formally we recall the following definition which was first given by Park and Kim [?].

Definition 2.7.1. A generalized convex space, or say, a G-convex space (X, D; Γ) consists of a topological space X, a non-empty subset D of X and a function Γ : F(X) → X \ {∅} such that (1): for each A, B ∈ F(X), Γ(A) ⊂ Γ(B) if A ⊂ B; and (2): for each A ∈ F(X) with |A| = n + 1, there exists a continuous function ϕ_{A} : Δ_{n} → Γ(A) such that ϕ_{A}(Δ_{J}) ⊂ Γ(J) for each ∅ ≠ J ⊂ {0, 1, ⋯, n}, where A = {x_{0}, x_{1}, ⋯, x_{n}}, and Δ_{J} denotes the face of Δ_{n} the corresponding to the sub-index of J in {0, 1, 2, ⋯, n}; and (3): A subset C of the G-convex space (X, D; Γ) is said to be G-convex if for each A ∈ F(D), Γ_{A} ⊂ C for all A ⊂ C.

For the convenience of our discussion, we also denote Γ(A) by Γ_{A} or Γ_{N} if there is no confusion throughout this section for each A = {x_{0}, x_{1}, x_{2}, ⋯, x_{n}} ∈ F(X), where N is the set of all index for the set A, i.e., N = {0, 1, 2, ⋯, n}. A space X is said to have a G-convex structure if and only of X is a G-convex space. By Lemma 2.2.5, it is clear that the notion of G-convex spaces includes the corresponding H-spaces as a special class, which, in turn, includes topological vector spaces as special cases. In order to establish our fixed point theorem, we now introduce another notion called locally G-convex which include locally H-convex spaces, hyperconvex metric spaces (their definitions will be given in next section shortly), and locally convex spaces as special cases.

Definition 2.7.2. A G-convex X is said to be a locally G-convex space if X is a uniform topological space with uniformity U which has an open base β := {V_{i} : i ∈ I} of symmetric encourages such that for each V ∈ β, the set V (x) := {y ∈ X : (y, x) ∈ V } is a G-convex set for each x ∈ X.

Recall that a non-empty space is said to be acyclic if all of its reduced Čech homology groups over rational vanish. In particular, any contractible space is acyclic, and thus any non-empty convex or star-shaped set is acyclic. Let X and Y be two topological spaces. We recall that a setvalued mapping F : X → 2^{Y} is said to have transfer open inversed values, if X = ∪_{y∈Y} int(F^{−1}(y)), where int(F^{−1}(y)) denotes the relative interior of the set F^{−1}(y) := {x ∈ X : y ∈ F (x)} for each y ∈ Y in X; and F is said to be upper semicontinuous if the set {x ∈ X : F (x) ⊂ U} is open for each open neighborhood U in Y. When Y is a G-convex space, the mapping T : X → 2^{Y} is said to have G-convex values if T (x) is a G-convex set for each x ∈ X.

We first have the following selection result of for a set-valued mapping with transfer open inversed values.

Lemma 2.7.3. Let Y be a compact topological space and X be a G-convex space. Suppose T : Y → 2^{X} \ {∅} is an open inverse valued mapping with G-convex values. Then there exist some positive integer n ∈ N and two single-valued continuous mappings ϕ : Δ_{n} → X and ψ : Y → Δ_{n} such that the mapping f : Y → X defined by f(y) := ϕ(ψ(y)) for each x ∈ Y is a continuous selection of T, i.e., f(y) ∈ T (y) for all y ∈ Y, where Δ_{n} is the standard n-dimensional simplex with vertices e_{0}, e_{1}, ⋯, e_{n}.

Proof. Since Y is compact and T has open inversed values, it follows that Y has a finite open cover {O_{i} : i = 0, 1, ⋯, n}, where O_{i} ⊂ int(F^{−1}(x_{i})) for some x_{i} ∈ X. Let A = {x_{0}, x_{1}, ⋯, x_{n}}. Suppose {ψ_{i}}_{i=0}^{n} is a partition of unity the corresponding to the finite covering {O_{i} : i = 0, 1, ⋯, n}, that is, ψ_{i}(y) ≥ 0 and Σ_{i}ψ_{i}(y) = 1 and if ψ_{i}(y) > 0, then y ∈ O_{i}. Set J(y) := {i ∈ I : ψ_{i}(y) > 0} for each y ∈ Y. Then for each i ∈ J(y), it follows that y ∈ O_{i} ⊂ T^{−1}(x_{i}), so that x_{i} ∈ T (y). This means A ∩ T (x) ≠ ∅ for all x ∈ X. Secondly, the mapping ψ : Y → Δ_{n} defined by ψ(y) := Σ_{i=0}^{n}ψ_{i}(y)e_{i} for each y ∈ Y is continuous. Since X is a G-convex space, there exists a continuous mapping ϕ : Δ_{n} → Γ_{A} such that ϕ(Δ_{J}) ⊂ Γ_{J} for each non-empty finite subset J in {0, 1, ⋯, n}. Then the mapping f : Y → X defined by f(y) := ϕ(ψ(y)) for each y ∈ Y, is continuous. Since T (y) is G-convex, it follows that f(y) := ϕ(ψ(y)) ⊂ ϕ(Δ_{J(y)}) ⊂ Γ_{J(y)} ⊂ T (y) for each y ∈ Y (as Γ_{J(y)} ⊂ T (y)), i.e., f(y) ∈ T (y) for each y ∈ Y and thus this completes the proof. □

By employing the same arguments of Eilenberg and Montgomery [?, p.106-107], and using Theorem 6.3 of Gòrniewicz [?, p.111] in his paper published in 1975, instead of the coincidence theorem used in [?, p.106-107], the following lemma was obtained by Shioji [?, p.188]. As it involves a long proof parallel to the one used by Eilenberg and Montgomery in their paper published in 1946, we thus omit its details here (see also Lemma 2.1 of Park et al. [?]).

Lemma 2.7.4 (Eilenberg and Montgomery, Gòrniewicz and Shioji and Park). Let Δ_{N} be an n-dimensional simplex with the Euclidean topology and Y be a compact topological space. Let ψ : Y → Δ_{N} be a single-valued continuous mapping and T : Δ_{N} → 2^{Y} be a set-valued upper semicontinuous mapping with non-empty compact acyclic (e.g., contractible) values. Then there exists x_{0} ∈ Δ_{N} such that x_{0} ∈ ψ ∘ T (x_{0}), where ψ ∘ T denotes the composition of mapping T with ψ.

We now have the following fixed point theorem in locally G-convex spaces given by Yuan [?], which is a generalization of the Fan-Glicksberg type fixed point theorems for upper semicontinuous set-valued mappings with non-empty closed acyclic values given in several existing literature (e.g., see Kirk and Shin [?], Park [?]-[?], Tarafdar [?], Wu [?] and others in locally convex spaces).

Theorem 2.7.5. Let X be a compact locally G-convex space and F : X → 2^{X} be an upper semicontinious set-valued mappings with non-empty closed acyclic values. Then F has a fixed point, i.e., there exists x_{0} ∈ X such that x_{0} ∈ F (x_{0}).

Proof. Since (X, Γ_{A}) is a locally Hausdorff G-convex space, X has a uniform structure U and without loss of generality, suppose V := {V_{i} : i ∈ I} is a symmetric and open base family for the uniform structure U. Then for each base V_{i} and x ∈ X, the set V_{i}(x) := {y ∈ X : (y, x) ∈ V_{i}} = {y ∈ X : (x, y) ∈ V_{i}} is an open G-convex set, and the ∩_{i∈I}V_{i} = Δ = {(x, x) : x ∈ X}. For each V_{i}, we define a set-valued mapping T_{i} : X → 2^{X} by T_{i}(x) := {y ∈ X : (y, x) ∈ V_{i}) = V_{i}(x) for each x ∈ X. Then it is clear T_{i} is a set-valued mapping from X to X with non-empty G-convex values and ∪_{y∈X}int(T_{i}^{−1}(y)) ⊂ X by the fact that for each y ∈ X, the set T_{i}^{−1}(y) = {x ∈ X : (x, y) ∈ V_{i}} is open in X, so that int(T_{i}^{−1}(y)) = T_{i}^{−1}(y). By Lemma 2.7.3, there exist some positive integer n ∈ N and two (single-valued) continuous mappings g_{i} : Δ_{n} → X and ϕ_{i} : X → Δ_{n} such that f_{i}(x) := g_{i}(ϕ_{i}(x)) ∈ T_{i}(x) for each x ∈ X, where Δ_{n} is a standard n-dimensional simplex. Now we define another set-valued mapping P_{i} : Δ_{n} → 2^{X} by P_{i}(x) := F (g_{i}(x)) for each x ∈ Δ_{n}. Then P_{i} is an upper semicontinuous set-valued mapping with non-empty closed and acyclic values. By Lemma 2.7.4, the mapping ϕ_{i} ∘ P_{i} : Δ_{n} → 2^{Δ_{n}} has a fixed point, i.e., there exists u_{v_{i}} ∈ Δ_{n} such that u_{v_{i}} ∈ ϕ_{i}(P_{i}(u_{v_{i}})). Then there exists x_{v_{i}} ∈ P_{i}(u_{v_{i}}) = F (g_{i}(u_{v_{i}})) ⊂ X and u_{v_{i}} = ϕ_{i}(x_{v_{i}}). Let y_{v_{i}} = g_{i}(u_{v_{i}}) = g_{i}(ϕ_{i}(x_{v_{i}})) = f_{i}(x_{v_{i}}) ∈ T_{i}(x_{v_{i}}). Thus x_{v_{i}} ∈ F (y_{v_{i}}), i.e., (x_{v_{i}}, y_{v_{i}}) ∈ graph F and (x_{v_{i}}, y_{v_{i}}) ∈ V_{i}, where graph F denotes the graph of the mapping F. Since X×X is compact, we may assume that {x_{v_{i}}, y_{v_{i}}}_{i∈I} converges to (x_{0}, y_{0}). Note that F is an upper semicontinuous mapping with non-empty closed values, it follows that the graph of F is closed and thus (x_{0}, y_{0}) ∈ graph F, i.e., y_{0} ∈ F (x_{0}). In order to finish the proof, it suffices to show that x_{0} = y_{0}. Since {V_{i} : i ∈ I} is an open symmetric base family for the uniform structure U and lim_{i∈I}(x_{v_{i}}, y_{v_{i}}) = (x_{0}, y_{0}), without loss of generality, there exists V_{0} ∈ {V_{i} : i ∈ I} such that (x_{v_{j}}, x_{0}) ∈ V_{j} and (y_{v_{j}}, y_{0}) ∈ V_{j} for all V_{j} ∈ {V_{i} : i ∈ I} with V_{j} ⊂ V_{0}. Now for any U ∈ {V_{i} : i ∈ I} with U ⊂ V_{0}, let V_{u} ∈ {V_{i} : i ∈ I} with V_{u} ⊂ V_{0} be such that V_{u} ∘V_{u} ∘V_{u} ⊂ U. Since (x_{v_{u}}, y_{v_{u}}) ∈ V_{u}, it follows that (x_{0}, y_{v_{u}}) ∈ V_{u} ∘V_{u} and thus (x_{0}, y_{0}) ⊂ V_{u} ∘ V_{u} ∘ V_{u} ⊂ U. Therefore we have show that (x_{0}, y_{0}) ∈ U for all U ∈ {V_{i} : i ∈ I} with U ⊂ V_{0}. Note that ∩{U : U ∈ V, and U ⊂ V_{0}} = ∩{V_{i} : i ∈ I} = Δ. This implies that x_{0} = y_{0} and thus x_{0} is a fixed point of F. This completes the proof. □

It is clear each locally convex H-space is a locally G-convex space. Note that if X is a compact convex subset of a locally convex Hausdorff topological vector space E and let Γ_{A} := co(A) for each A ∈ F(X). Then (X, Γ_{A}) is a locally convex H-space. Thus Theorem 2.7.5 is a really generalization of celebrated Fan-Glicksberg fixed point theorem for upper semicontinuous set-valued mappings with non-empty closed and acyclic (e.g., contractible) values in both locally convex H-spaces, l.c.spaces (locally H-convex spaces) and locally convex topological vector spaces. Theorem 2.7.5 shows that Theorem 2.1 of Tarafdar [?] is true when mapping F is only upper semicontinous. Theorem 2.7.5 also generalizes Theorem 3.2 of Tarafdar and Watson [?] and the corresponding Fan-Glicksberg type fixed point theorems given by Wu [?] to locally G-convex spaces which include locally H-convex space and locally convex H-spaces as special classes.

Definition 2.7.6. An H-space (X, Γ_{A}) is said to be a locally convex H-space if X is a uniform topological space with the uniform structure U and there exists a base family {V_{i} : i ∈ I} for U such that for each V_{i} ∈ {V_{i} : i ∈ I}, the set V_{i}(x) := {y ∈ X : (y, x) ∈ V_{i}} is H-convex for each x ∈ X.

We remark that Definition 2.7.6 above includes so-called l.c.-space given by Horvath [?] as a special case (see also Tarafdar [?] and [?]). We do know that each non-empty convex subset S in locally convex spaces has a uniform structure U induced by the family of seminorms; on the other hand, each convex S in topological vector spaces has a natural H-structure Γ_{A} given by Γ_{A} := co(A) for each non-empty finite subset A ∈ F(S). Thus locally convex H-spaces include locally convex topological vector spaces as a special case.

In the setting of metric spaces, we have the following definition of locally convex H-spaces for metric spaces the corresponding to Definition 2.7.6.

Definition 2.7.7. Let the metric space (Y, Γ) be an H-space with the metric d. Then Y is said to be a locally metrically convex H-space if for each r > 0, the set {y ∈ Y : d(y, E) < r} is H-convex for each H-convex set E in Y and also all open balls are H-convex.

It is clear each locally convex H-space is a locally G-convex space. Note that if X is a compact convex subset of a locally convex Hausdorff topological vector space E and let Γ_{A} := co(A) for each A ∈ F(X). Then (X, Γ_{A}) is a locally convex H-space. Thus Theorem 2.7.5 is a really generalization of celebrated Fan-Glicksberg fixed point theorem for upper semicontinuous set-valued mappings with non-empty closed and acyclic (e.g., contractible) values in both locally convex H-spaces, l.c.spaces (locally H-convex spaces) and locally convex topological vector spaces. Theorem 2.7.5 shows that Theorem 2.1 of Tarafdar [?] is true when mapping F is only upper semicontinous. Theorem 2.7.5 also generalizes Theorem 3.2 of Tarafdar and Watson [?] and the corresponding Fan-Glicksberg type fixed point theorems given by Wu [?] to locally G-convex spaces which include locally H-convex space and locally convex H-spaces as special classes.

Thus we have the following Fan-Glickberg fixed point theorm which is first given by Wu [?]

Theorem 2.7.9. Let (X, Γ_{A}} be a compact Hausdorff locally convex H-space. Suppose F : X → 2^{X} is an upper semicontinuous set-valued mapping with non-empty closed acyclic values. Then F has at least one fixed point, i.e., there exists x ∈ X such that x ∈ F (x).

Remark 2.7.10. Though we have established several fixed point theorems for upper semicontinuous (resp., hemicontinuous) set-valued mappings in topological vector spaces which have sufficient enough continuous linear functionals, it is clear that a Hausdorff topological vector space may not have enough continuous linear functionals (e.g, the Hardy space H^{p} for p ∈ (0, 1)), thus Schauder’s famous conjecture which asks ‘if a continuous (even single-valued) mapping defined on a non-empty compact and convex subset of a Hausdorff topologocal vector space has a fixed point’ remains open since 1934. For more historic details on Schauder’s conjecture and its development, the interested readers are referred to Dugundji and Granas [?], Granas [?], Idzik [?], Park [?] and references therein.

The celebrated Ky Fan minimax inequality has numerous applications in various and diverse fileds of mathematics and applied science. Also, there have appeared a number of equivalent formulations and generalizations. For example, this chapter gives us a picture and relationships among Brouwer fixed point theorem [?], Sperner lemma [?], KKM lemma, Fan’s geometric lemma [?], Browder-Fan fixed point theorem, Fan-Glicksberg fixed point theorem, Ky Fan type minimax theorem and the equivalent forms to Ky Fan minimax inequality, which is regarded as one of the most important principles in nonlinear analsysis. In this line, more details have been illustrated by Aubin [?], Aubin and Ekeland [?], Border [?], Park [?], [?] and references therein. We all know that nonlinear analysis began early this century with the birth of Brouwer fixed point. However, Ky Fan minimax inequality is a more convenient form as it can be used to impliy a whole series of existence theorems for solutions of nonlinear equations or inclusions (see Aubin [?, p.135]) in a very general form. Such theorems are very useful in many applications, in particularly in optimization theory, mathematical economics, and game theory as we can see from Yuan [?]. The book of Singh et al. [?] also gives an extensively study among fixed point theory and best approximations by using KKM-map principle.

2.8 Nonlinear Analysis of Hyperconvex Metric Spaces

The aim of this section is to discuss general KKM theory and its various application in metric hyperconvex spaces.

2.8.1 Fan-Glicksberg Fixed Points in Hyperconvex Metric Spaces

Sine [?] and Soardi [?] proved independently that the fixed point property for nonexpansive mappings holds in bounded hyperconvex spaces. Ever since hyperconvex metric spaces have been used widely, and many interesting results for nonexpansive mappings have been established in the framework of hyperconvex structures, e.g., see Baillon [?], Goebel and Kirk [?], Khamsi [?], Sine [?]-[?] and others. In this section, we first show that each hyperconvex metric space is indeed a complete locally metrically H-space (thus, it is a locally convex H-space) and then using fixed point theorem established in the previous section, we give a hyperconvex version of fixed point theorem for upper semicontinuous set-valued mappings with closed acyclic values in hyperconvex metric spaces, which, in turn, unifies the corresponding fixed point theory of set-valued mappings in existing literatures.

We first recall the following definition of a hyperconvex metric space due to Aronszajn and Panitchpakdi [?].

Definition 2.8.1. A metric space (X, d) is said to be a hyperconvex space if for any connection of points {x_{α}} of X and any collection {r_{α}} of non-negative real numbers with d(x_{α}, x_{β}) ≤ r_{α} +r_{β}, we have ∩_{α}B(x_{α}, r_{α}) ≠ ∅, where B(x, r) denotes the closed ball centered at x ∈ X with the radius r.

This definition can be seen as equivalently a binary intersection property plus metric convexity in the sense that for each given x, y ∈ X and α ∈ [0, 1], there exists z ∈ X such that d(x, z) = αd(x, y) and d(y, z) = (1 − α)d(x, y). The corresponding linear theory for hyperconvex spaces can be found in Lacey [?]. Sine [?] points out the nonlinear theory is still developing. Hyperconvex spaces can have quite strange aspects. For example, a hyperconvex subset need not be convex in even ℝ^{2} (with the l_{∞} norm). Also convex sets in linear spaces may fail to be hyperconvex (but for this one must go to at least ℝ^{3}, e.g., see Sine [?]). Some nice spaces such as Hilbert space may fail to be hyperconvex. However hyperconvexity enjoys some properties similar to convexity and others much like compactness. More convincing analogies hold for hyperconvex sets which are ball intersections. The Nachbin-Kelley-Goodner and Hasumi Theorem in Lacey [?, p.92] (see also Isbell [?], Nachbin [?] or Kelley [?]) says that a Banach space is hyperconvex if and only if it is linearly isometric to C(K), where C(K) is the space of all continuous real functions defined on some Stonian space K (i.e., K is Hausdorff, compact and extremely disconnected). Thus the space l_{∞}(I) for any set I and the space L_{∞}(μ) for a finite measure μ are examples of hyperconvex spaces. Secondly, order intervals in L_{∞} are hyperconvex, but weak compact convex sets of L_{∞} need not be hyperconvex. Moreover we note that hyperconvex sets of L_{∞} may not be convex (e.g., see Lin and Sine [?, p.943]). We also remark that any hyperconvex space (indeed any metric space) embeds isometrically in some l_{∞}(I), e.g., see Lacey [?] or Khamsi [?, p.300-301].

In order to study the KKM theory in hyperconvex spaces, we first recall some notions and basic facts about hyperconvex spaces which will be used later in this chapter.

Definition 2.8.2. Let (M, d) be a metric space. Assume that a subset A of M is bounded. We will use the following notions throughout this chapter.

(1) co(A) = ∩{B ⊂ M : B is a closed ball such that A ⊂ B}.

(2) A(M) = {A ⊂ M : A = co(A)}, i.e., A ∈ A(M) if and only if A is an intersection of closed balls. In this case, we shall say that A is an admissible subset of M. We also note that if M is a hyperconvex space, then each admissible set in M is also hyperconvex.

For the convenience of our study, we also have the some facts about hyperconvex metric spaces whose proofs can be found from Aronszajn and Panitchpakdi [?], and Lacey [?] (see also Isbell [?], Nachbin [?], Kelley [?] and Khamsi [?]).

Proposition 2.8.3. Let M be a metric space. Then we have the follows:

(1) there exists an index set I and a natural isometric embedding from M to l^{∞}(I).

(2) If M is hyperconvex, the it is complete.

(3) M is hyperconvex if and only if for each metric space N which contains isometrically M, there exists a nonexpansive retract r : N → M; i.e., r is nonexpansive and r(x) = x for each x ∈ M. In particular if N is a normed space, for any non-empty finite points y_{1}, y_{2}, ⋯, y_{n} ∈ M, we have that r(conv{y_{1}, ⋯, y_{n}}) ⊂ co{y_{1}, y_{2}, ⋯, y_{n}}, where co{y_{1}, y_{2}, ⋯, y_{n}} is an admissible set defined by Definition 2.11.2 above.

(4) M is hyperconvex if and only if for each metric space N which is contained metrically in any space D, and any nonexpansive mapping T : N → M, there exists an extension T^{∗} : D → M which is nonexpansive, i.e., T (x) = T^{∗}(x) for each x ∈ N.

Let M be a metric space and consider the natural embedding into l^{∞}(I) given by statement (1) in Proposition 2.11.3. We denote by M_{∞} := co(M) ∈ A(l^{∞}(I)). Clearly M_{∞} is a hyperconvex subset of l^{∞}(I) and M_{∞} is also a convex subset of the linear space l^{∞}(I).

Let X be a non-empty set. We shall denote by F(X) and 2^{X} the family of all non-empty finite subsets of X and the family of all subsets of X, respectively. If A is a subset of a linear space E, the notion ‘conv(A)’ always means the convex hull of A.

It follows by Proposition 2.8.3 that a hyperconvex space is a nonexpansive retract of any metric space in which it is embedded. Since any metric space can be embedded in a Banach space, it follows that any hyperconvex metric space is contractible. Note that if {B(y_{i}, r_{i})}_{i∈I} is any collection of closed balls in a hyperconvex space, then ∩_{i∈I}B(y_{i}, r_{i}) itself is a hyperconvex space. Therefore for a hyperconvex space (Y, d), the set ∩_{i∈I}B(y_{i}, r_{i}) is contractible, or empty. For any non-empty finite subset A ∈ F(Y), the admissible set of A (also called ball hull of A by Sine [?]) defined by

F (A) := ∩{B(y, r) : A ⊂ B(y, r)}

defines an H-structure in Y and thus each hyperconvex metric space (Y, d) is an H-space with the H-structure given by ball hull above.

Each hyperconvex metric space, indeed, is a complete locally metrically convex H-space as shown by Lemma 2.8.4 below. This fact was first given by Horvath [?].

Lemma 2.8.4. Each hyperconvex metric space (Y, d) is a complete locally metrically convex H-space (of course, (Y, d) is a locally convex H-spaces).

Proof. By following Horvath in [?], for each y ∈ Y, F ((y)) = {y} which, of course, is an F -set. Weshow that for any H- convex set E and any r > 0, the set {y ∈ Y : d(y, e) < r} is an H-set. Let y_{0}, ⋯, y_{n} ∈ Y such that d(y_{i}, E) < r and y_{0} ∈ F ((y_{1}, ⋯, y_{n})), i.e., y_{0} belongs to any closed ball containing {y_{1}, ⋯, y_{n}}. We want to show that d(y_{0}, E) < r. Take any fixed points ŷ_{1}, ⋯, ŷ_{n} ∈ E such that d(y_{i}, ŷ_{i}) < r for i = 1, ⋯, n. Since F ({ŷ_{1}, ⋯, ŷ_{n}}) ⊂ E, it suffices to show that d(y_{0}, F ({ŷ_{1}, ⋯, ŷ_{n}})) < r. By the definition of F ({ŷ_{1}, ⋯, ŷ_{n}}), F ({ŷ_{1}, ⋯, ŷ_{n}}) = ∩_{i∈I}B(u_{i}, r_{i}), where (u_{i}, r_{i}) ∈ Y × R_{+}. Choose r′ ∈ (0, r) such that d(y_{i}, ŷ_{i}) < r′ for i = 1, ⋯, n. As {ŷ_{1}, ⋯, ŷ_{n}} ⊂ B_{i}(u_{i}, r_{i}) for i ∈ I, it follows that {y_{1}, ⋯, y_{n}} ⊂ B(u_{i}, r_{i} + r) for i ∈ I and y_{0} ∈ ∩_{i∈I}B(u_{i}, r_{i} + r′). By Sine’s Lemma in [?, p.864], there exists a retraction R : ∩_{i∈I}B(u_{i}, r_{i} + r′) → ∩_{i∈I}B(u_{i}, r_{i}) such that d(y, R(y)) ≤ r′ for any y ∈ ∩_{i∈I}B(u_{i}, r_{i} + r′). Then R(y_{0}) ∈ F ({ŷ_{1} ⋯, ŷ_{n}}) and d(y_{0}, R(y_{0})) ≤ r′ < r. This completes the proof. □

By the fact that each hyperconvex metric space is a locally convex H-space (thus, an H-space), we have the following hyperconvex version of Fan-Glicksberg fixed point theorem for set-valued upper semicontinuous with non-empty closed acyclic values as an application of Theorem 2.7.9.

Theorem 2.8.5. Let X ∈ A(M) be a compact subset of a hyperconvex metric space M. Suppose F : X → 2^{X} is a upper semicontinuous set-valued mappings with non-empty closed admissible values. Then F has at least one fixed point, i.e., there exists x ∈ X such that x ∈ F (x).

Proof. Note that each hyperconvex metric space is complete. It follows by Theorem 8.3.5 of Engelking [?] M has a uniform structure. Thus X is a locally convex H-space by Lemma 2.8.4. Then the conclusion follows by Theorem 2.7.9 and this completes the proof. □

We would like to note that Theorem 2.8.5 improves the corresponding fixed point theorms given by Horvath [?], Kirk and Shin [?], and Khamsi [?] as special cases.

2.8.2 KKM Theory in Hyperconvex Metric Spaces

In this section, our purpose is to establish the general KKM theory in hyperconvex metric spaces. As applications, we give hyperconvex versions of Fan’s celebrated minimax principle and Fan’s best approximation theorem for set-valued mappings.

Let (M, d) be a metric space. Following Khamsi [?], a subset S ⊂ M is said to be finitely metrically closed if for each F ∈ F(M), the set co(F) ∩ S is closed. Note that co(F) is always defined and belongs to A(M). Thus if S is closed in M, it is obviously finitely metrically closed. We also recall that a family {A_{α}}_{α∈D} of M has the finite intersection property if the intersection of each non-empty finite sub-family is not empty.

Definition 2.8.6. Let X be any non-empty set and let M be a metric space. A set-valued mapping G : X → 2^{M} \{∅} is said to be a generalized metric KKM mapping (in short, GMKKM) if for each non-empty finite set {x_{1}, ⋯, x_{n}} ⊂ X, there exists a sequence of points of M, y_{1}, ⋯, y_{n}, not necessarily all different, such that for each subsequenct {y_{i_{1}}, ⋯, y_{i_{k}}}, we have

co{y_{i_{j}} : j = 1, ⋯, k} ⊂ ∪_{j=1}^{k}G(x_{i_{j}}).

In partiuclar, the mapping G : X → 2^{M} is said to be a metric KKM (in short, MKKM) mapping if for each finite subset F ∈ F(X), we have co(F) ⊂ ∪_{x∈F} G(x).

As shown by Lemma 2.8.4, each hyperconvex metric space is an H-space and thus each GMKKM mapping is a GHKKM mapping, as a direct application of Theorem 2.2.15, the following characterization of the generalized in hyperconvex metric spaces which was first given by Kirk et al. [?].

Theorem 2.8.7. Let X be a non-empty set and let M be a hyperconvex metric space. Suppose G : X → 2^{M} \ {∅} is a set-valued mapping with finitely metrically closed values. Then the family {G(x) : x ∈ X} has the finite intersection property if and only if the mapping G is a generalized metric KKM mapping.

As an application of Theorem 2.8.7, we have the following characterization for the existence of a non-empty intersection for the values of a set-valued mapping in an hyperconvex metric space.

Theorem 2.8.8. Let X be a non-empty set and M be a hyperconvex metric space. Suppose G : X → 2^{M} \ {∅} is a set-valued mapping with non-empty closed values and there exists x_{0} ∈ X such that G(x_{0}) is compact. Then ∩_{x∈X}G(x) ≠ ∅ if and only if the mapping G is a generalized metric KKM mapping.

Proof. \underline{Necessity}: Since ∩_{x∈X}G(x) ≠ ∅, it follows that the family \{G(x) : x ∈ X\} has the finite intersection property. Since G(x) is closed for each x ∈ X, it is finitely metrically closed. By Theorem 2.8.7, G is a generalized metric KKM mapping.

\underline{Sufficiency}: Since G is a generalized metric KKM mapping, it follows by Theorem 2.8.7 that the family {G(x) : x ∈ X} has the finite intersection property. Rewriting this to {G(x)∩G(x_{0}) : x ∈ X} and noting G(x_{0}) is compact, we have

∅ ≠ ∩_{x∈X}G(x) ∩ G(x_{0}) = G(x_{0}) ∩_{x∈X} G(x) = ∩_{x∈X}G(x).

This completes the proof. □

As a special cases of Theorem 2.8.7, we also have the following result which includes Theorem 3 of Khamsi [?, p.303] as a special case.

Corollary 2.8.9. Let X be a non-empty subset of a hyperconvex metric space M. Suppose G : X → 2^{M} \ {∅} is a metric KKM mapping with finitely metrically closed values. Then the family {G(x) : x ∈ X} has the finite intersection property.

By Theorem 2.8.7 and Lemma 2.2.10, we have the following characterization of the finite intersection property of set-valued mappings.

Theorem 2.8.10. Let X be a non-empty set and let M be a hyperconvex metric space. Suppose G : X → 2^{M} \ {∅} is a set-valued mapping such that G is transfer closed valued and there exists a finite subset X_{0} of X such that the set ∩_{x∈X_{0}}clG(x) is non-empty and compact. Then the intersection ∩_{x∈X}G(x) is non-empty if and only if the mapping clG is a generalized metric KKM mapping.

Theorems 2.8.7, 2.8.8 and 2.8.10 tell us that in hyperconvex metric spaces the finite intersection property for a family of subsets is equivalent to the existence of a non-empty intersection for a set-valued mapping which is a generalized metric KKM mapping. In order to study in which situations such generalized metric KKM mappings can be derived, we now introduce the following definitions.

Definition 2.8.11. Let X ∈ F(M) be a non-empty subset of a hyperconvex space M. Then:

(1) the function f : X → [−∞, +∞] is said to be hyper quasi-convex (resp., concave) if the set {x ∈ X : f(x) ≤ λ} (resp., the set {x ∈ X : f(x) ≥ λ}) for each λ ∈ ℝ is an admissible set in M.

(2) the function ψ(x, y) : X × X → [−∞, +∞] is said to be hyper diagonal quasi-convex (resp., concave) in y if for each A ∈ F(X) and any y_{0} ∈ co(A), we have that ψ(y_{0}, y_{0}) ≤ max_{y_{i}∈A} ψ(y_{0}, y_{i}) (resp., ψ(y_{0}, y_{0}) ≥ min_{y_{i}∈A} ψ(y_{0}, y_{i})).

(3) ψ(x, y) is said to be hyper γ-diagonal quasi-convex (resp., concave) in y for some γ ∈ [−∞, +∞] if for any A ∈ F(X) and y_{0} ∈ co(A), we have that γ ≤ max_{y_{i}∈A} ψ(y_{0}, y_{i}) (resp., γ ≥ min_{y_{i}∈A} ψ(y_{0}, y_{i})).

Remark 2.8.12. We note that the inequality ‘ ≤’ (resp., ‘ ≥’) could be replaced equivalently by the strict inequality ‘ <’ (reps., ‘ >’) in the Definition 2.8.11(1). It is clear that if ψ(x, y) is hyper diagonal quasi-convex (resp., concave) in y, then ψ(x, y) must be hyper γ-diagonally quasi-convex (resp., concave) in y, where γ = inf_{x∈X} ψ(x, x) (resp, γ = sup_{x∈X} ψ(x, x)).

Definition 2.8.13. Let X ∈ F(M) be a non-empty set of a hyperconvex metric space M and γ ∈ (−∞, +∞]. Suppose ψ : M × X → (−∞, +∞] is a function. Then ψ is said to be hyper γ-generalized quasi-convex (resp., concave) in M if for each non-empty finite subset {x_{1}, ⋯, x_{n}} ⊂ X, there exists a sequence y_{1}, ⋯, y_{n} in M such that for each subsequenct y_{i_{1}}, ⋯, y_{i_{k}} and any x_{0} ∈ co{x_{i_{1}}, ⋯, x_{i_{k}}}, we have γ ≤ max_{1≤j≤k} ψ(x_{0}, y_{i_{j}}) (resp., γ ≥ min_{1≤j≤k} ψ(x_{0}, y_{i_{j}})).

By its definition, it is easy to see that following is true.

Lemma 2.8.14. Let X be a non-empty set, M a hyperconvex metric space and let γ ∈ ℝ be a given real number. Suppose ψ : M × X → (−∞, +∞] is a mapping. Then the following two statements are equivalent:

(1) the mapping G : X → 2^{M} \ {∅} defined by G(x) := {y ∈ M : ψ(y, x) ≤ γ} (resp., G(x) := {y ∈ M : ψ(y, x) ≥ γ}) for each x ∈ X is a generalized metric KKM mapping.

(2) the function ψ is hyper γ-generalized quasi-concave (resp., convex) in M.

We now give the following minimax inequality which is a hyperconvex version of Fan’s celebrated minimax inequality principle.

Theorem 2.8.15. Let X ∈ A(M) be a compact subset of a hyperconvex space M. Suppose ψ : X × X → (−∞, +∞] is a mapping such that

(1) for each fixed x ∈ X, y ↦ ψ(x, y) is hyper 0-generalized quasi-concave; and

(2) for each fixed y ∈ X, x ↦ ψ(x, y) is lower semicontinous.

Then there exists x_{0} ∈ X such that sup_{y∈X} ψ(x_{0}, y) ≤ 0.

Proof. We define a mapping G : X → 2^{X} by

G(x) := {y ∈ X : ψ(y, x) ≤ 0}

for each x ∈ X. Then it is clear that G is a GMKKM mapping by Lemma 2.8.14. Now by Theorem 2.8.8, it follows that ∩_{x∈X}G(x) ≠ ∅. Take any point x_{0} ∈ ∩_{x∈X}G(x), then we have that sup_{y∈X} ψ(x_{0}, y) ≤ 0 and the proof is completed. □

As an application of the generalized metric MKKM principle established above, we give the following hyperconvex version of Fan’s best approximation in hyperconvex spaces for set-valued mappings.

Theorem 2.8.16. Let M be a hyperconvex metric space and X ∈ A(M) be a non-empty compact subset of M. Suppose F : X → A(M) is a set-valued continuous mapping. Then there exists x_{0} ∈ X such that

d(x_{0}, F (x_{0})) = inf_{x∈X} d(x, F (x_{0})).

In particular if F (x_{0}) is compact and x_{0} ∉ F (x_{0}), x_{0} must be a boundary point of X (i.e., x_{0} ∈ Bd(X)).

Proof. Define a mapping G : X → 2^{M} \ {∅} by

G(x) := {y ∈ X : d(y, F (y)) ≤ d(x, F (y))}

for each x ∈ X. As F is continuous, G(x) is closed and non-empty for each x ∈ X. Now we claim that G is a metric KKM mapping. Suppose it were not. Then there exists a non-empty and finite subset {x_{1}, ⋯, x_{n}} and y ∈ co({x_{i} : i = 1, ⋯, n}) such that

d(x_{i}, F (y)) < d(y, F (y))

for i = 1, ⋯, n. Let ϵ > 0 such that d(x_{i}, F (y)) ≤ d(y, F (y)) − ϵ for i = 1, ⋯, n. Let r = dist(y, F (y)) − ϵ. Then for i = 1, ⋯, n, x_{i} ∈ F (y) + r, where F (y) + r := ∪{B(a; r) : a ∈ F (y)} and B(a, r) is a closed ball centered at a with the radius r. Note that F (y) ∈ A(M), it follows that F (y) + r ∈ A(M) (e.g., see Sine [?, p.864]). Thus co{x_{1}, x_{2}, ⋯, x_{n}} ⊂ F (y) + r. This in turn implies y ∈ F (y) + r, that is, dist(y, F (y)) ≤ r = dist(y, F (y)) − ϵ, which is impossible. Therefore G must be a metric KKM-mapping.

Note that X is compact, we have that ∩_{x∈X}G(x) ≠ ∅. Take any point x_{0} ∈ ∩_{x∈X}G(x), then it is clear that d(x_{0}, F (x_{0})) ≤ d(x, F (x_{0})) for all x ∈ X, which implies that d(x_{0}, F (x_{0})) = inf_{x∈X} d(x, F (x_{0})).

If F (x_{0}) is compact and x_{0} ∉ F (x_{0}), then there exists u_{0} ∈ F (x_{0}) such that d(x_{0}, u_{0}) = d(x_{0}, F (x_{0})) > 0. We now claim that x_{0} is a boundary point of X. Suppose it were not, i.e., x_{0} ∈ intX, where intX is the set of all relatively interior points of X in M. Then there exists a postive number r > 0 small enough such that B(x_{0}, r) ⊂ intX ⊂ X and

0 < r < d(x_{0}, F (x_{0})) ≤ d(z, F (x_{0}))

for all z ∈ B(x_{0}, r). Note that d(x_{0}, u_{0}) = d(x_{0}, F (x_{0})) = r+d(x_{0}, F (x_{0}))−r and d(x_{0}, F (x_{0}))−r > 0, it follows by the hyperconvexity of M that

B(x_{0}, r) ∩ B(u_{0}, d(x_{0}, F (x_{0})) − r) ≠ ∅.

Since u_{0} ∈ F (x_{0}) and thus B(u_{0}, d(x_{0}, F (x_{0})) − r) ⊂ B(F (x_{0}), d(x_{0}, F (x_{0})) − r), there exists y_{0} ∈ B(x_{0}, r) ∩ B(F (x_{0}), d(x_{0}, F (x_{0})) − r) such that d(y_{0}, F (x_{0})) ≤ d(x_{0}, F (x_{0})) − r. However, we do have that d(y_{0}, F (x_{0})) ≥ d(x_{0}, F (x_{0})) > 0, which is a contraction. Therefore x_{0} must be a boundary point of X, i.e., x_{0} ∈ Bd(X) and completes the proof. □

As a special case of our best approximation theorem, we have the following best approximation result for single-valued mappings, first given by Khamsi in [?].

Corollary 2.8.17. Let M be a hyperconvex metric space and X ∈ A(M) be a non-empty compact subset of M. Suppose F : X → M is a single-valued continuous mapping. Then there exists x_{0} ∈ X such that

d(x_{0}, F (x_{0})) = inf_{x∈X} d(x, F (x_{0})).

2.8.3 Intersection Theorems

In what follows, we shall give a dual form of above KKM principle in hyperconvex space in the sense that the family {G(x) : x ∈ X} of Theorem 2.8.7 still has a finite intersection property if G(x) takes finitely metrically open values instead of taking finitely metrically closed values for each x ∈ X.

Theorem 2.8.18 (The dual form of FKKM principle in hyperconvex space.) Let X be a non-empty subset of a hyperconvex metric space M. Suppose G : X → 2^{M} \ {∅} is a MKKM mapping with finitely metrically open values. Then the family {G(x) : x ∈ X} has the finite intersection property.

Proof. As each hyperconvex metric space is an H-space by Lemma 2.8.4, the conclusion follows by Proposition 2.2.6 and the proof is complete. □

As applications of both Theorems 2.8.7 and 2.8.18, we have the following Ky Fan matching theorem for both closed and open covers in hyperconvex spaces.

Theorem 2.8.19. Let X ∈ A(M) be a non-empty subset of a hyperconvx metric space M. Suppose {A_{i}}_{i=1}^{n} is a family of closed subsets of X such that ∪_{i=1}^{n}A_{i} = X and x_{1}, ⋯, x_{n} are (not necessarily distinct) n points of X. Then there exist k indices i_{1} < i_{2} < ⋯ < i_{k} between 1 and n such that co{x_{i_{1}}, x_{i_{2}}, ⋯, x_{i_{k}}} ∩ (∩_{j=1}^{k}A_{i_{j}}) ≠ ∅.

Proof. Let X_{0} = {x_{1}, x_{2}, ⋯, x_{n}} and we define a set-valued mapping F : X_{0} → 2^{X} by F (x_{i}) := X \A_{i} for each i = 1, 2, ⋯, n. Suppose the conclusion is false. Then F (x) is open for each x ∈ X_{0} and for each non-empty finite subset B ⊂ X_{0}, we have co{x : x ∈ B} ⊂ ∪{F (x) : x ∈ B}. By Theorem 2.8.18, it follows that ∩_{i=1}^{n}F (x_{i}) ≠ ∅. But this means that ∪_{i=1}^{n}A_{i} ≠ X, which contradicts our hypothesis and this contradiction completes our proof. □

As an application of Theorem 2.8.7, we also have the following Ky Fan matching theorem for open covers in hyperconvex spaces.

Theorem 2.8.20. Let X ∈ A(M) be a non-empty subset of a hyperconvx metric space M. Suppose {A_{i}}_{i=1}^{n} is a family of open subsets of X such that ∪_{i=1}^{n}A_{i} = X and x_{1}, ⋯, x_{n} are (not necessarily distinct) n points of X. Then there exist k indices i_{1} < i_{2} < ⋯ < i_{k} between 1 and n such that co{x_{i_{1}}, x_{i_{2}}, ⋯, x_{i_{k}}} ∩ (∩_{j=1}^{k}A_{i_{j}}) ≠ ∅.

Proof. Let X_{0} = {x_{1}, x_{2}, ⋯, x_{n}} and we define a set-valued mapping F : X_{0} → 2^{X} by F (x_{i}) := X \ A_{i} for each i = 1, 2, ⋯, n. Suppose the conclusion is false. Then F (x) is closed for each x ∈ X_{0} and for each non-empty finite subset B ⊂ X_{0}, we have co{x : x ∈ B} ⊂ ∪{F (x) : x ∈ B}. By Theorem 2.8.7, it follows that ∩_{i=1}^{n}F (x_{i}) ≠ ∅. But this means that ∪_{i=1}^{n}A_{i} ≠ X, which contradicts our hypothesis and this contradiction completes our proof. □

We note that Theorem 2.8.20 need not to assume that X is compact as required by Theorem 2.8.19.

In this part, as applications of Ky Fan matching theorems for closed covers, we shall establish some fixed point theorems for set-valued mappings and some existence of intersection theorems which are hyperconvex versions of the corresponding results given by Alexandroff-Pasynkoff and others.

First we have the following fixed point theorem for set-valued mappings in hyperconvex spaces.

Theorem 2.8.21. Let X ∈ A(M) be a non-empty subset of a hyperconvex metric space M. Suppose A : X → 2^{X} is a set-valued mapping with closed values (maybe taking empty values for some point x ∈ X) and there exists n points x_{1}, ⋯, x_{n} (not necessarily distinct) of X such that ∪_{i=1}^{n}A(x_{i}) = X and the set A^{−1}(y) := {x ∈ X : y ∈ A(x)} for each y ∈ X is admissible. Then A has a fixed point, i.e., there exists w ∈ X such that w ∈ A(w).

Proof. By Theorem 2.8.19, there exist k indices i_{1} < i_{2} < ⋯ < i_{k} between 1 and n such that co{x_{i_{1}}, x_{i_{2}}, ⋯, x_{i_{k}}} ∩ (∩_{j=1}^{k}A(x_{i_{j}})) ≠ ∅. Take any w ∈ co{x_{i_{1}}, x_{i_{2}}, ⋯, x_{i_{k}}} ∩ ∩_{j=1}^{k}A(x_{i_{j}}).

Then x_{i_{j}} ∈ A^{−1}(w) for j = 1, ⋯, k. By the fact that A^{−1}(w) is admissible, it follows that w ∈ co{x_{i_{1}}, ⋯, x_{i_{k}}} ⊂ A(w). Therefore w is a fixed point of A and the proof is complete. □

As an application of Theorem 2.8.20, we also have the following Browder-Fan type fixed point theorm for set-valued mappings in hyperconvex spaces.

Theorem 2.8.22. Let X ∈ A(M) be a non-empty subset of a hyperconvex metric space M. Suppose A : X → 2^{X} is a set-valued mapping with open values (maybe taking empty values for some point x ∈ X) and there exist n points x_{1}, ⋯, x_{n} (not necessarily distinct) of X such that ∪_{i=1}^{n}A(x_{i}) = X and the set A^{−1}(y) := {x ∈ X : y ∈ A(x)} for each y ∈ X is admissible. Then A has a fixed point, i.e., there exists w ∈ X such that w ∈ A(w).

Proof. By Theorem 2.8.20, there exist k indices i_{1} < i_{2} < ⋯ < i_{k} between 1 and n such that co{x_{i_{1}}, x_{i_{2}}, ⋯, x_{i_{k}}} ∩ (∩_{j=1}^{k}A(x_{i_{j}})) ≠ ∅. Take any w ∈ co{x_{i_{1}}, x_{i_{2}}, ⋯, x_{i_{k}}} ∩ (∩_{j=1}^{k}A(x_{i_{j}})). Then x_{i_{j}} ∈ A^{−1}(w) for j = 1, ⋯, k. By the fact that A^{−1}(w) is admissible, it follows that w ∈ co{x_{i_{1}}, ⋯, x_{i_{k}}} ⊂ A(w). Therefore w is a fixed point of A and the proof is complete. □

As equivalent forms of Theorems 2.8.21 and 2.8.22, we have the following fixed point theorems for set-valued mappings in hyperconvex spaces.

Theorem 2.8.23. Let X ∈ A(M) be a subset of a hyperconvex space M. Suppose F : X → 2^{X} is a set-valued mapping with non-empty admissible values. There exists a non-empty finite subset C of X such that F (x) ∩ C ≠ ∅ for each x ∈ X and F^{−1}(y) is closed for each y ∈ X. Then F has a fixed point, i.e., there exists w ∈ X such that w ∈ F (w).

Proof. Define A : X → 2^{X} by A(x) := F^{−1}(x) for each x ∈ X. Then it is easy to verify that A satisfies all hypotheses of Theorem 2.8.21. Thus the conclusion follows by Theorem 2.8.21 and the proof is complete. □

Theorem 2.8.24. Let X ∈ A(M) be a subset of a hyperconvex space M. Suppose F : X → 2^{X} is a set-valued mapping with non-empty admissible values. There exists a non-empty finite subset C of X such that F (x) ∩ C ≠ ∅ for each x ∈ X and F^{−1}(y) is open for each y ∈ X. Then F has a fixed point, i.e., there exists w ∈ X such that w ∈ F (w).

Proof. Define A : X → 2^{X} by A(x) := F^{−1}(x) for each x ∈ X. Then it is easy to verify that A satisfies all hypotheses of Theorem 2.8.22. Thus the conclusion follows by Theorem 2.8.22 and the proof is complete. □

The following are consequences of Theorems 2.8.23 and 2.8.24.

Theorem 2.8.25. Let X ∈ A(M) be a non-empty subset of a hyperconvex space M. Suppose G : X → 2^{X} is a upper semicontinuous set-valued mapping with non-empty closed admissible valuesand there exists a non-empty finite subset C of X such that G(x) ∩ C ≠ ∅ for each x ∈ X. Then G has a fixed point, i.e., there exists w ∈ X such that w ∈ G(w).

Proof. As G is upper semicontinuous with non-empty closed values, it follows that the graph of G is closed and thus the set G^{−1}(y) is closed for each y ∈ X. Therefore all hypotheses of Theorem 2.8.23 are satisfied. The conclusion follows from Theorem 2.9.23 and the proof is complete. □

Theorem 2.8.26. Let X ∈ A(M) be a compact subset of a hyperconvex space M. Suppose F : X → 2^{X} is a set-valued mapping with non-empty admissible values and F^{−1}(y) is open for each y ∈ X. Then F has a fixed point, i.e., there exists w ∈ X such that w ∈ F (w).

Proof. Note that F^{−1}(y) is open for each y ∈ X and F (x) ≠ ∅ for all x ∈ X, it follows that ∪_{y∈X}F^{−1}(y) = X. As X is compact, there exists a finite subset C of X such that X = ∪_{y∈C}F^{−1}(y). Therefore for each x ∈ X, we have that F (x) ∩ C ≠ ∅. Thus F satisfes all hypotheses of Theorem 2.8.24. The conclusion follows by Theorem 2.8.24 and completing the proof. □

Remark 2.8.27. Theorem 2.8.26 is a hyperconvex version of the famous Browder-Fan fixed point theorm. In next section, we will establish a more general form for non-compact hyperconvex spaces. We also note that both Theorems 2.8.21 and 2.8.22 are hyperconvex versions of the corresponding fixed points given by Kim [?] and Lassonde [?].

In the remainder of this section, as applications of both Theorems 2.8.18 and 2.8.20, we will establish some intersection theorems which are hyperconvex versions of the corresponding results due to Alexandroff-Pasynkoff [?] and others.

Theorem 2.8.28. Let X ∈ A(M) be a non-empty subset of a hyperconvex metric space M. Suppose {A_{i} : i = 1, 2, ⋯, n} is a closed cover of X (i.e., A_{i} is a closed subset of X for i = 1, ⋯, n and ∪_{i=1}^{n}A_{i} = X) and there exist n points {x_{i}, i ∈ I = (1, ⋯, n)} (not necessarily distinct) of X such that

co{x_{j} : i ∈ I and j ≠ i} ⊂ A_{i}

for each i ∈ I. Then we have ∩_{i=1}^{n}A_{i} ≠ ∅.

Proof. By Theorem 2.8.19, it follows that there exists a non-empty subset J ⊂ I such that co{x_{j} : j ∈ J} ∩ (∩_{j∈J}A_{j}) ≠ ∅. On the other hand, our assumption implies that co{x_{j} : j ∈ J} ⊂ ∩{A_{i} : i ∈ I \ J}. Therefore, it follows that there exists x_{0} ∈ co{x_{j} : j ∈ J} such that x_{0} ∈ ∩_{i∈I}A_{i}, i.e., ∩_{i=1}^{n}A_{i} ≠ ∅ and the proof is complete. □

Theorem 2.8.29. Let X ∈ A(M) be a non-empty subset of a hyperconvex metric space M. Suppose {A_{i} : i = 1, 2, ⋯, n} is an open cover of X (i.e., A_{i} is an open subset of X for i = 1, ⋯, n and ∪_{i=1}^{n}A_{i} = X) and there exist n points {x_{i}, i ∈ I = (1, ⋯, n)} (not necessarily distinct) of X such that

co{x_{j} : i ∈ I and j ≠ i} ⊂ A_{i}

for each i ∈ I. Then we have ∩_{i=1}^{n}A_{i} ≠ ∅.

Proof. By Theorem 2.8.20, it follows that there exists a non-empty subset J ⊂ I such that co{x_{j} : j ∈ J} ∩ ∩_{j∈J}A_{j} ≠ ∅. On the other hand, our assumption implies that co{x_{j} : j ∈ J} ⊂ ∩{A_{i} : i ∈ I \ J}. Therefore, it follows that there exists x_{0} ∈ co{x_{j} : j ∈ J} such that x_{0} ∈ ∩_{i∈I}A_{i}, i.e., ∩_{i=1}^{n}A_{i} ≠ ∅ and the proof is complete. □

Theorem 2.8.30. Let X ∈ A(M) be a non-empty subset of a hyperconvex metric space M. Suppose {A_{i} : i = 1, 2, ⋯, n} is a closed cover of X such that ∩_{j∈J}A_{j} ≠ ∅ for each J ⊂ I with J ≠ I, where I = {1, 2, ⋯, n}. Then ∩_{i=1}^{n}A_{i} ≠ ∅.

Proof. For each i ∈ I = {1, ⋯, n}. Take any x_{i} ∈ ∩{A_{j} : j ∈ I and j ≠ i}. Then by Theorem 2.8.19, it follows that there exists a non-empty subset J ⊂ I such that co{x_{j} : j ∈ J}∩(∩_{j∈J}A_{j}) ≠ ∅. On the other hand, our assumption implies that {x_{j} : j ∈ J} ⊂ ∩{A_{i} : i ∈ I \ J}. Therefore, it follows that there exists x_{0} ∈ {x_{j} : j ∈ J} such that x_{0} ∈ ∩_{i∈I}A_{i}, i.e., ∩_{i=1}^{n}A_{i} ≠ ∅ and the proof is complete. □

Theorem 2.8.31. Let X ∈ A(M) be a non-empty subset of a hyperconvex metric space M. Suppose {A_{i} : i = 1, 2, ⋯, n} is an open cover of X such that ∩_{j∈J}A_{j} ≠ ∅ for each J ⊂ I with J ≠ I, where I = {1, 2, ⋯, n}. Then ∩_{i=1}^{n}A_{i} ≠ ∅.

Proof. For each i ∈ I = {1, ⋯, n}. Take any x_{i} ∈ ∩{A_{j} : j ∈ I and j ≠ i}. Then by Theorem 2.8.20, it follows that there exists a non-empty subset J ⊂ I such that co{x_{j} : j ∈ J}∩∩_{j∈J}A_{j} ≠ ∅. On the other hand, our assumption implies that {x_{j} : j ∈ J} ⊂ ∩{A_{i} : i ∈ I \ J}. Therefore, it follows that there exists x_{0} ∈ {x_{j} : j ∈ J} such that x_{0} ∈ ∩_{i∈I}A_{i}, i.e., ∩_{i=1}^{n}A_{i} ≠ ∅ and the proof is complete. □

2.8.4 Fixed Point Theorems

In this part, we shall use the generalized metric KKM principle and Fan’s best approximation theorem in hyperconvex metric space established above, to derive hyperconvex versions of the Browder-Fan fixed point theorem and Schauder-Tychonoff fixed point theorms for set-valued mappings.

Theorem 2.8.32. Let X ∈ A(M) be a compact subset of a hyperconvex metric space M. Suppose that F : X → 2^{X} is a set-valued mapping with non-empty values such that:

(1) F is transfer open inversed valued; and

(2) for each x ∈ X, F (x) is admissible. Then there exists x_{0} ∈ X such that x_{0} ∈ F (x_{0}).

Proof. Define a set-valued mapping G : X → 2^{X} by

G(x) = X \ F^{−1}(x)

for each x ∈ X. Since F (x) is non-empty and transfer closed valued, X = ∪_{y∈X}F^{−1}(y). Therefore we have ∩_{x∈X}G(x) = ∅. By Theorem 2.8.10, G must not be a metric KKM mapping, thus there exists a subset {x_{1}, ⋯, x_{n}} ⊂ X and a point x_{0} ∈ co{x_{1}, ⋯, x_{n}} such that x_{0} ∉ G(x_{i}) for i = 1, 2, ⋯, n. It follows that x_{i} ∈ F (x_{0}) for all i = 1, ⋯, n. Note that F (x_{0}) is admissible, so it follows that x_{0} ∈ co{x_{i}, i = 1, ⋯, n} ⊂ F (x_{0}). Therefore x_{0} is a fixed point of F and thus the proof is completed. □

We note that Theorem 2.8.32 was also derived by Park [?]. As an application of Theorem 2.8.32, we have the following result which is Fan’s geometric lemma [?] in hyperconvex metric space.

Theorem 2.8.33. Let X ∈ A(M) be a non-empty compact hyperconvex subset of a hyperconvex metric space M. Suppose C is a non-empty subset of X × X such that:

(1) for each fixed x ∈ X, the set {y ∈ X : (x, y) ∉ C} is empty or admissible;

(2) for each fixed y ∈ X, the set {x ∈ X : (x, y) ∈ C} is closed; and

(3) for each x ∈ X, (x, x) ∈ C. Then there exists x_{0} ∈ X such that {x_{0}} × X ⊂ C.

Proof. In order to apply Theorem 2.8.32, we define a set-valued mapping F : X → 2^{X} by

F (x) := {y ∈ X : (x, y) ∉ C}

for each x ∈ X. Then we can verify that F satisfies all the hypotheses of Theorem 2.8.32. Since F has no fixed point, by our condition (3), it follows that there must exist x_{0} ∈ X such that F (x_{0}) = ∅. Thus we must have {x_{0}} × X ⊂ C by the definition of F and the proof is completed. □

As another application of Theorem 2.8.33, we have the following result.

Theorem 2.8.34. Let X ∈ A(M) be a non-empty compact subset of a hyperconvex metric space M. Suppose F : X → 2^{X} is a set-valued mapping such that:

(1) for each y ∈ X, X \ F^{−1}(y) is admissible;

(2) for each x ∈ X, the set F (x) is closed; and

(3) for each x ∈ X, x ∈ F (x). Then there exists x_{0} ∈ X such that x_{0} ∈ ∩_{x∈X}F (x).

Proof. In order to apply Theorem 2.8.32, let C := {(x, y) ∈ X × X : x ∈ F (y)}. Then we have that

(1) for each x ∈ X, the set {y ∈ X : (x, y) ∉ C} = {y ∈ X : x ∉ F (y)} = X \ F^{−1}(x) which is admissible by condition (1).

(2) for each fixed y ∈ X, the set {x ∈ X : (x, y) ∈ C} = F (y) is closed in X by the hypothesis (2).

(3) for each x ∈ X, it is clear that (x, x) ∈ X.

Thus all hypotheses of Theorem 2.8.33 are satisfied and it follows that there exists x_{0} ∈ X with {x_{0}} × X ⊂ C. This implies that x_{0} ∈ ∩_{x∈X}F (x) and the proof is completed. □

Remark 2.8.35. Both Theorems 2.8.33 and 2.8.34 are equivalent to each other. To see this, it suffices to prove Theorem 2.8.33 from Theorem 2.8.34. Define F : X → 2^{X} by F (y) = {x ∈ X : (x, y) ∈ C} for each y ∈ X. Then the hypotheses in Theorem 2.8.32.5.2 ensure that F satisfies all the hypotheses of Theorem 2.8.34. By Theorem 2.8.34, ∩_{x∈X}F (x) ≠ ∅. Take any x_{0} ∈ ∩_{x∈X}F (x), and we have {x_{0}} × X ⊂ C, as required.

As an equivalent statement of Theorem 2.8.32, we have the following existence theorem for maximal elements in hyperconvex spaces.

Theorem 2.8.36. Let X ∈ A(M) be a compact hyperconvex subset of a hyperconvex metric space M. Assume F : X → 2^{X} is a set-valued mapping such that:

(1) F is transfer open inversed valued; and

(2) for each x ∈ X, x ∉ F (x) and F (x) is admissible. Then there exists x_{0} ∈ X such that F (x_{0}) = ∅.

As an application of Theorem 2.8.16, we have the following Schauder-Tychonoff fixed point theorem for set-valued mappings.

Theorem 2.8.37. Let X ∈ F(M) be a compact subset of a hyperconvex metric space M. Suppose F : X → A(M) is a set-valued continuous mapping with non-empty closed values such that for each x ∈ X with x ∉ F (x), there exists z ∈ X such that d(z, F (x)) < d(x, F (x)). Then F has a fixed point in X. In particular when F takes compact values, the conclusion of Theorem 2.8.37 still holds if for all x ∈ Bd(X) with x ∉ F (x), there exists z ∈ X such that d(z, F (x)) < d(x, F (x)).

Proof. By Theorem 2.8.16, there exists x_{0} ∈ X such that

d(x_{0}, F (x_{0})) = inf_{x∈X} d(x, F (x_{0})).

We claim that x_{0} is a fixed point of F. Indeed, assume it were not true, i.e., x_{0} ∉ F (x_{0}), it follows that d(x_{0}, F (x_{0})) > 0. Then by assumption, there exists z_{0} ∈ X such that d(z_{0}, F (x_{0})) < d(x_{0}, F (x_{0})). On the other hand, note that d(z_{0}, F (x_{0})) ≥ d(x_{0}, F (x_{0})) > 0. This implies that 0 < d(z_{0}, F (x_{0})) < d(z_{0}, F (x_{0})), which is impossible and thus x_{0} must be a fixed point of F. When F takes compact values and x_{0} ∉ F (x_{0}), it follows by the particular case of Theorem 2.8.16, x_{0} ∈ Bd(X) and thus the conclusion follows by the same argument above and we complete the proof. □

As an immediate consequence of Theorem 2.8.37, we have the following fixed point theorem which is Corollary 3.5 of Kirk and Shin [?, p.180].

Corollary 2.8.38. Let X ∈ F(M) be a compact subset of a hyperconvex metric space M. Suppose F : X → A(M) is a set-valued continuous mapping with non-empty closed values for which F (x) ∩ X ≠ ∅ for all x ∈ X. Then F has a fixed point in X.

Proof. Note that F (x) ∩ X ≠ ∅ for each x ∈ X. It follows that for any x ∈ X with x ∉ F (x), if we take z ∈ F (x) ∩ X, we have d(z, F (x)) = 0 < d(x, F (x)). Thus all hypotheses of Theorem 2.8.37 are satisfied and the conclusion follows. □

Remark 2.8.39. We note that a spacial case of Theorem 2.8.37 was given by Khamsi [?] for a single-vaued mapping.

Before closing this part, we will prove one generalization of Browder-Fan fixed point theorem for set-valued mappings in hyperconvex. In order to do so, we first need the following Ky Fan matching theorem for open covers in hyperconvex spaces.

Theorem 2.8.40. Let X ∈ A(M) be a non-empty compact subset of a hyperconvex space M and A : X → 2^{X} be a set-valued mapping such that:

(1) A(x) is open for each x ∈ X; and

(2) A(X) := ∪_{x∈X}A(x) = X. Then there exists a non-empty finite subset {x_{1}, ⋯, x_{n}} ⊂ X and an x_{0} ∈ co{x_{1}, ⋯, x_{n}} such that x_{0} ∈ ∩_{i=1}^{n}A(x_{i}).

Proof. Define a set-valued mapping F : X → 2^{X} by F (x) := X \ A(x) for each x ∈ X. Suppose the conclusion is false. Then for each finite subset {x_{1}, ⋯, x_{n}} ⊂ X, it follows that co{x_{1}, ⋯, x_{n}} ⊂ ∪_{i=1}^{n}F (x_{i}). Therefore F is a MKKM mapping with closed values. As X is compact, by Theorem 2.8.7, we have that ∩_{x∈X}F (x) ≠ ∅. But this implies ∪_{x∈X}A(x) = A(X) ≠ X, which is against our assumption and this completes our proof. □

As an application of Theorem 2.8.40, we have the following corollary which is Theorem 1 of Park [?].

Corollary 2.8.41. Let X ∈ A(M) be a non-empty compact subset of a hyperconvex space M and Y be any given topological space. Suppose A : X → 2^{Y} is a set-valued mapping such that:

(1) A(x) is open for each x ∈ X; and

(2) A(X) := ∪_{x∈X}A(x) = Y. Then for each (single-valued) continuous mapping f : X → Y, there exists a non-empty finite subset {x_{1}, ⋯, x_{n}} ⊂ X and an x_{0} ∈ co{x_{1}, ⋯, x_{n}} such that f(x_{0}) ∈ ∩_{i=1}^{n}A(x_{i}).

Proof. Define a set-valued mapping A : X → 2^{X} by F (x) = f^{−1}(A(x)) for each x ∈ X. This definition is well defined as A(X) = Y. Then the mapping F satisfies all hypotheses of Theorem 2.8.40. By Theorem 2.8.40, there exists a finite subset {x_{1}, ⋯, x_{n}} ⊂ X and an x_{0} ∈ co{x_{1}, ⋯, x_{n}} such that x_{0} ∈ ∩_{i=1}^{n}f^{−1}(A(x_{i})), which implies that f(x_{0}) ∈ ∩_{i=1}^{n}A(x_{i}) and thus completing the proof. □

As an application of Theorem 2.8.40, we have the following Browder-Fan fixed point theorem for set-valued mappings in hyperconvex spaces.

Theorem 2.8.42. Let X ∈ A(M) be a compact subset of a hyperconvex metric space M. Suppose there exist two set-valued mappings F_{1}, F_{2} : X → 2^{X} \ {∅} such that:

(1) F_{1}(x) ⊂ F_{2}(x) for each x ∈ X;

(2) X = ∪{intF_{1}^{−1}(y) : y ∈ X} (i.e., F_{1} is transfer open inversed valued; and

(3) F_{2}(x) is admissible for each x ∈ X. Then there exists x_{0} ∈ X such that x_{0} ∈ F_{2}(x_{0}).

Proof. Define a set-valued mapping A : X → 2^{X} by A(x) := intF_{1}^{−1}(x) for each x ∈ X. Then we have that (1): A(x) is open for each x ∈ X; (2): ∪_{x∈X}A(x) = A(X) = X by condition (2); and (3): A^{−1}(x) ⊂ F_{2}(x) for each x ∈ X by condition (1). By Theorem 2.8.40, there exists a non-empty finite subset {x_{1}, ⋯, x_{n}} ⊂ X and an x_{0} ∈ co{x_{1}, ⋯, x_{n}} such that x_{0} ∈ ∩_{i=1}^{n}A(x_{i}). Therefore x_{i} ∈ A^{−1}(x_{0}) ⊂ F_{2}(x_{0}) for i = 1, ⋯, n. By the condition F (x_{0}) is admissible, it follows that x_{0} ∈ co{x_{1}, ⋯, x_{n}} ⊂ F_{2}(x_{0}), which is a fixed point of F_{2} and we complete the proof. □

Theorem 2.8.42 unifies Theorem 3 of Park [?] and Theorem 3.1 of Kirk et al. [?] for the Browder-Fan fixed point theorems in hyperconvex spaces. We also like to metion that by using the notion of noncompactness measure, and non-compact version of the KKM principle in hyperconvex metric space can be established. For more details, see Kirk et al. [?] and thus we omit their details here.

2.8.5 Existence of Saddle Points and Nash Equilibria

In this section, we use our generalized metric KKM principle in hyperconvex spaces to give some applications to the existence for saddle points, intersection theorems, and the existence of Nash equilibria in game theory. We first have the following result.

Theorem 2.8.43. Let X ∈ A(M) be a non-empty compact subset of a hyperconvex metric space M. Suppose ψ : X × X → (−∞, +∞) is a function such that:

(1) for each fixed y ∈ X, ψ(x, y) is lower semicontinuous in x; and for each fixed x ∈ X, ψ(x, y) is hyper 0-generalized quasi-concave in y; and

(2) for each fixed x ∈ X, ψ(x, y) is upper semicontinuous in y; and for each fixed x ∈ X, ψ(x, y) is hyper 0-generalized quasi-convex in x. Then ψ has at least one saddle point in X × X, i.e., there exists (x_{0}, y_{0}) ∈ X × X such that

max_{y∈X} inf_{x∈X} ψ(x, y) = ψ(x_{0}, y_{0}) = min_{x∈X} sup_{y∈X} ψ(x, y) = 0.

Proof. By our assumptions, ψ satisfies all the hypotheses of Theorem 2.8.15, it follows that there exists x_{0} ∈ X such that

sup_{y∈X} ψ(x_{0}, y) ≤ 0.  (2.8.4)

Let λ(x, y) = −ψ(x, y) for each (x, y) ∈ X × X, then λ : X × X → (−∞, +∞) is a mapping which also satisfies all hypotheses of Theorem 2.8.15. By Theorem 2.8.15, it follows that there exists y_{0} ∈ X such that

sup_{x∈X} λ(x, y_{0}) ≤ 0.  (2.8.5)

By combining (2.8.4) and (2.8.5), it follows that

max_{y∈X} min_{x∈X} ψ(x, y) = ψ(x_{0}, y_{0}) = min_{x∈X} sup_{y∈X} ψ(x, y) = 0,

thus completing the proof. □

In what follows, we shall establish one intersection theorem and the existence of Nash equilibria in hyperconvex metric spaces. We first recall the following notion: Given a Cartesian product X = Π_{i=1}^{n}X_{i} of topological spaces, let X^{j} = Π_{i≠j}X_{i} and let p_{i} : X → X_{i} and p^{i} : X → X^{i} denote the natural projections; write p_{i}(x) = x_{i} and p^{i}(x) = x^{i}. Given x, y ∈ X, we let (y_{i}, x^{i}) = (x_{1}, ⋯, x_{i−1}, y_{i}, x_{i+1}, ⋯, x_{n}).

The following intersection theorem is a hyperconvex version of Fan’s intersection theorem.

Theorem 2.8.44. Let X_{1}, ⋯, X_{n} be non-empty compact sets in hyperconvex metric spaces M_{i} such that X_{i} ∈ A(M_{i}) for i = 1, ⋯, n and let A_{1}, A_{2}, ⋯, A_{n} be n subsets of X (where, X = Π_{i=1}^{n}X_{i}) such that:

(1) for each fixed x ∈ X and for each i = 1, ⋯, n, the set

A_{i}(x) := {y ∈ X : (y_{i}, x^{i}) ∈ A_{i}}

is a non-empty and admissible set; and (2) for each fixed y ∈ X and each i = 1, ⋯, n, the set

A^{i}(y) := {x ∈ X : (y_{i}, x^{i}) ∈ A_{i}}

is open.

Then the intersection ∩_{i=1}^{n}A_{i} ≠ ∅.

Proof. Define a set-valued mapping G : X → 2^{X} by

G(x) := X \ ∩_{i=1}^{n}A^{i}(x)

for each x ∈ X. Then we can show that G is not a metric KKM mapping. Indeed, by condition (1), for each x ∈ X, there exists y ∈ X such that (y_{i}, x^{i}) ∈ A_{i} for all i = 1, 2, ⋯, n. This implies that x ∈ ∩_{i=1}^{n}A^{i}(y), which means that X ⊂ ∪_{y∈X}(∩_{i=1}^{n}A^{i}(y)). Therefore ∩_{y∈X}G(y) = ∅ and also by Corollary 2.8.9, G must not be a metric KKM mapping. Thus there exists a non-empty finite subset {x(1), x(2), ⋯, x(m)} ⊂ X for which there exists w ∈ co{x(1), ⋯, x(m)} with the property that w ∉ G(x(j)), i.e., w ∈ ∩_{i=1}^{n}A^{i}(x(j)) for all j = 1, ⋯, m. Note that the set A_{i}(w) is admissible and x(j) ∈ A_{i}(w) for all j = 1, 2, ⋯, m. It follows that w ∈ co{x(1), x(2), ⋯, x(m)} ⊂ A_{i}(w). Thus we have that w ∈ A_{i}(w) for all i = 1, 2, ⋯, n, i.e., w ∈ ∩_{i=1}^{n}A_{i} ≠ ∅ and the proof is completed. □.

As an immediate corollary of Theorem 2.8.44, we have the following existence of Nash equilibria in hyperconvex spaces.

Theorem 2.8.45. Let X_{1}, ⋯, X_{n} ∈ A(M) be non-empty compact sets in a hyperconvex metric space M. Let f_{1}, ⋯, f_{n} be n real-valued continuous function defined on X = Π_{i=1}^{n}X_{i} such that for each y ∈ X and for each i = 1, ⋯, n, the function x_{i} ↦ f_{i}(x_{i}, y^{i}) is hyper quasi-concave on X_{i}. Then there exists a point y_{0} ∈ X such that f_{i}(y_{0}) = max_{x_{i}∈X_{i}} f_{i}(x_{i}, y_{0}^{i}) for i = 1, ⋯, n.

Proof. For any given ϵ > 0, define for each i = 1, 2, ⋯, n,

A_{i}^{ϵ} = {z ∈ X : f_{i}(z) > max_{x_{i}∈X_{i}} f_{i}(x_{i}, y^{i}) − ϵ}.

It follows that for each fixed x ∈ X, there exists y ∈ X such that (y_{i}, x^{i}) ∈ A_{i}^{ϵ} for all i = 1, ⋯, n and the set A_{i}^{ϵ} is non-empty and admissible, by the hyper quasi-concavity of the mapping x_{i} ↦ f_{i}(x_{i}, y^{i}) for each y ∈ X. Secondly, for each y ∈ X and for each i = 1, ⋯, n, we note that x ∈ A_{i}^{ϵ}(y) if and only if (y_{i}, x^{i}) ∈ A_{i}^{ϵ}. Further A_{i}^{ϵ} is open by the upper semicontinuity of the mapping y^{i} ↦ f_{i}(x_{i}, y^{i}) for each fixed x ∈ X. Thus all the hypotheses of Theorem 2.8.44 are satisfied and so there exists x(ϵ) ∈ X such that x(ϵ) ∈ ∩_{i=1}^{n}A_{i}^{ϵ}. As X is compact, without loss of the generality, we may assume that x(ϵ) ↦ y_{0} ∈ X as ϵ ↦ 0^{+}. The conclusion then follows by the continuity of f_{i} for i = 1, 2, ⋯, n, and the proof is completed. □

Remark 2.8.46. By following a similar methodology and arguments to those used by Fan [?] (see also Park [?] and references therein for more recent study), many variations and generalizations of the results given in this section can be established. Since the ideas are basically the same as those used in Fan [?], we will not include further discussion here. Finally we point out that by using the notion of noncompactness measure, non-compact versions of results in this section can be derived. As their arguments are not particularly difficult we will omit their details.

2.8.6 Generic Stability of Fixed Points in Hyperconvex Metric Spaces

The aim of this section is to estabilish generic stability of fixed points for upper semicontinuous set-valued mappings in hyperconvex metric spaces. Our results show that almost all of upper semicontinous set-valued mappings defined in compact hyperconvex metric spaces have stable fixed points from viewpoint of Baire category theory.

Let (X, d) be a metric space and K(X) be the space of all non-empty compact subsets of X equipped with the Hausdorff metric h which is induced by the metric d. For any ϵ > 0, x_{0} ∈ X and A ∈ K(X), let U(ϵ, A) = {x ∈ X : d(u, x) < ϵ for some u ∈ A} and O(x_{0}, ϵ) = {x ∈ X : d(x_{0}, y) < ϵ}. Let Y be a topological space. Recall that a subset Q ⊂ Y is called a residual set if it is a countable intersection of open dense subsets of Y.

In order to study generic stability of fixed points for upper semicontinuous set-valued mappings in hyperconvex metric spaces, throughout the rest part of this section, we denote by X a hyperconvex metric space (and thus it is complete) and A(X) denotes the collection of all ball intersections (i.e., admissible sets) in X. By Theorem 15 of Sine [?] (see also Lemma 2 of Kirk [?]), we have the following fact.

Lemma 2.8.47. Let X be a hyperconvex metric space. Then the space (A(X), h) is a hyperconvex metric space with the Hausdorff metric h (thus (A(X), h) is a complete metric space).

Let X be a hyperconvex metric space and KA(X) the collection of all compact admissible sets of X. By Theorem 4.3.9 of Klein and Thompson [?], it follows that (KA(X), h) is still a complete metric space.

Let C := {f : X → KA(X) : f is upper semicontinuous with non-empty compact and admissible values on X} and for each f, f′ ∈ C, define ρ(f, f′) := sup_{x∈X} h(f(x), f′(x)). Clearly, ρ is a metric on C and we also have the following result.

Lemma 2.8.48. The space (C, ρ) is complete.

Proof. Let {f_{n}}_{n=1}^{∞} be each Cauchy sequence in C, then for any ϵ > 0, there is a positive integer N(ϵ) such that ρ(f_{n}, f_{m}) = sup_{x∈X} h(f_{n}(x), f_{m}(x)) < ϵ for any n, m ≥ N(ϵ). It follows that for each x ∈ X, {f_{n}(x)}_{n=1}^{∞} is a Cauchy sequence in A(X). Note that KA(X) is complete by Lemma 2.8.47 and Theorem 4.3.9 of Klein and Thompson [?], it follows that there is f : X → A(X) such that lim_{m→∞} f_{m}(x) = f(x). for all x ∈ X. Now for each given x ∈ X, each ϵ > 0 and each n ≥ N(ϵ), the inequality above implies that f_{n}(x) ⊂ U(2ϵ, f(x)) and f(x) ⊂ U(2ϵ, f_{n}(x)). Fix n ≥ N(ϵ); since f_{n} ∈ C, there is δ > 0 such that f_{n}(x′) ⊂ U(ϵ, f_{n}(x)) whenever d(x, x′) < δ. Thus, f(x′) ⊂ U(2ϵ, f_{n}(x′)) ⊂ U(3ϵ, f_{n}(x)) ⊂ U(5ϵ, f(x)) whenever d(x, x′) < δ. Therefore f is upper semicontinuous on X, so that f ∈ C and that f_{n} → f. Hence C is complete. □

Throughout this section, we set Y := C×KA(X) and for each y = (f, A) ∈ Y, y′ = (f′, A′) ∈ Y, define a metric D on Y by

D(y, y′) = ρ(f, f′) + h(A, A′)

Then it follows by Lemma 2.8.48, Lemma 2.8.47 and Theorem 4.3.9 of Klein and Thompson [?] again that Y is a complete metric space.

Define M := {y = (f, A) ∈ Y: there is x ∈ A such that x ∈ f(x)}. Then M is, indeed, a subset of Y and we do have the following result.

Lemma 2.8.49. The metric space (M, D) is complete.

Proof. Since M ⊂ Y and Y is complete, it is sufficient to prove that M is closed in Y. Let {y_{n}}_{n=1}^{∞} be a sequence in M and y_{n} → y ∈ Y. Set y_{n} = (f_{n}, A_{n}), n = 1, 2, ⋯ and y = (f, A); then f_{n} → f and A_{n} → A. For each n = 1, 2, ⋯, since y_{n} ∈ M, there is x_{n} ∈ A_{n} such that x_{n} ∈ f_{n}(x_{n}). Since A_{n} and A are compact and A_{n} → A, by A.5.1 (ii) of Mas-Colell [?, p.10], ∪_{n=1}^{∞}A_{n} ∪ A is compact. Since x_{n} ∈ A_{n} ⊂ ∪_{n=1}^{∞}A_{n} ∪ A, we may assume without loss of generality that x_{n} → x ∈ ∪_{n=1}^{∞}A_{n} ∪ A. If x ∉ A, since A is compact, there is a > 0 such that U(a, A) ∩ O(x, a) = ∅. Since A_{n} → A and x_{n} → x, there is N_{1} such that A_{n} ⊂ U(a, A) and x_{n} ∈ O(x, a) for all n ≥ N_{1}, which contradicts the assumption that x_{n} ∈ A_{n}. Hence we must have x ∈ A. If x ∉ f(x), since f(x) is compact, there is b > 0 such that U(b, f(x)) ∩ U(b, x) = ∅. Since f_{n} → f, there is N_{2} such that f_{n}(u) ⊂ U(b/2, f(u)) for all n ≥ N_{2} and for all u ∈ X. Since f is upper semicontinuous at x and x_{n} → x, there is N_{3} ≥ N_{2} such that f(x_{n}) ⊂ U(b/2, f(x)) and x_{n} ⊂ U(b/2, x) for all n ≥ N_{3}. Thus for all n ≥ N_{3}, f_{n}(x_{n}) ⊂ U(b/2, f(x_{n})) ⊂ U(b, f(x)) and x_{n} ⊂ U(b, x) which contradicts the assumption that f_{n}(x_{n}) ∩ x_{n} ≠ ∅. Hence we must also have x ∈ f(x). Therefore y = (f, A) ∈ M so that M is closed in Y. □

Now for each y = (f, A) ∈ M, define a fixed point mapping S : M → K(X) by S(y) := {x ∈ A : x ∈ f(x)}. Then S(y) ≠ ∅ and following fact is true.

Lemma 2.8.50. The fixed point set S(y) ∈ K(X) for each y ∈ M.

Proof. Let y = (f, A) ∈ M be given. Since that x_{n} → x ∈ ∪_{n=1}^{∞}A_{n} ∪ A. If x ∉ A, since A is compact, there is a > 0 such that U(a, A) ∩ O(x, a) = ∅. Since A_{n} → A and x_{n} → x, there is N_{1} such that A_{n} ⊂ U(a, A) and x_{n} ∈ O(x, a) for all n ≥ N_{1}, which contradicts the assumption that x_{n} ∈ A_{n}. Hence we must have x ∈ A. If x ∉ f(x), since f(x) is compact, there is b > 0 such that U(b, f(x)) ∩ U(b, x) = ∅. Since f_{n} → f and g_{n} → g, there is N_{2} such that f_{n}(u) ⊂ U(b/2, f(u)) all n ≥ N_{2} and for all u ∈ X. Since f is upper semicontinuous at x and x_{n} → x, there is N_{3} ≥ N_{2} such that f(x_{n}) ⊂ U(b/2, f(x)) and x_{n} ⊂ U(b/2, x) for all n ≥ N_{3}. Thus for all n ≥ N_{3}, f_{n}(x_{n}) ⊂ U(b/2, f(x_{n})) ⊂ U(b, f(x)) and x_{n} ⊂ U(b, gx) which contradicts the assumption that x_{n} ∈ f_{n}(x_{n}). Hence we must also have x ∈ f(x). Therefore y = (f, A) ∈ M so that M is closed in Y. □

By Lemma 2.8.50, the fixed point mapping y ↦ S(y) defines a set-valued correspindence from M to K(X); and moreove S is upper semicontinuous.

Lemma 2.8.51. The mapping S is upper semicontinuous on M.

Proof. Suppose S is not upper semicontinuous at y ∈ M, then there exist ϵ_{0} > 0 and a sequence {y_{n}}_{n=1}^{∞} in M with y_{n} → y such that for each n = 1, 2, ⋯, there exists x_{n} ∈ S(y_{n}) with x_{n} ∉ U(ϵ_{0}, S(y)). Let y_{n} = (f_{n}, A_{n}) and y = (f, A), then f_{n} → f and A_{n} → A. Since x_{n} ∈ A_{n} ⊂ ∪_{n=1}^{∞}A_{n} ∪ A and ∪_{n=1}^{∞}A_{n} ∪ A is compact, we may assume without loss of generality that x_{n} → x ∈ ∪_{n=1}^{∞}A_{n} ∪ A. Note that we must have x ∉ U(ϵ_{0}, S(y)). Now the same argument as in the proof of Lemma 2.12.26 shows that x ∈ A and x ∈ f(x) so that x ∈ S(y); this contradicts that x ∉ U(ϵ_{0}, S(y)). Therefore S must be upper semicontinuous. □

In order to study the stability of fixed points in hyperconvex spaces, we need the following notion.

Definition 2.8.52. Let M_{1} be a non-empty closed subset of M (since M is complete, so is M_{1}). For any given point y_{1} ∈ M_{1}, a point x in S(y) is said to be an essential fixed point of y_{1} with respect to M_{1} provided that for any ϵ > 0, there is δ > 0 such that for any y′ ∈ M_{1} with D(y_{1}, y′) < δ, there exists x′ ∈ S(y′) with d(x, x′) < ϵ. The point y is said to be essential with respect to M_{1} if every x ∈ S(y) is an essential fixed point of y with respect to M_{1}.

Remark 2.8.53. Definition 2.8.52 says that if x is an essential fixed point of the mapping f ∈ C. Then for any ϵ > 0, there exists a δ > 0 such that every mapping f′ ∈ C with sup_{x∈X}{d(f(x), f′(x)) : x ∈ X} < δ has at least on fixed point in the ϵ-neighborhood of the point y; in another word, the fixed point x of f is continuous (or say, stable) respect to the mapping f.

Though, in general the fixed point set of any given mapping f ∈ C is not stable, we do have the following characteristic of stability for fixed points of upper semicontinuous set-valued mappings in C.

Theorem 2.8.54. The mapping S is lower semicontinuous at y ∈ M_{1} if and only if y is essential with respect to M_{1}.

Proof. Suppose S is lower semicontinuous at y ∈ M_{1}. Then for any ϵ > 0, there is δ > 0 such that for any y′ ∈ M_{1} with d(y, y′) < δ, we have S(y) ⊂ U(ϵ, S(y′)) so that for any x ∈ S(y), there is x′ ∈ S(y′) with d(x, x′) < ϵ. Thus every x ∈ S(y) is an essential fixed point of y with respect to M_{1} and hence y is essential with respect to M_{1}.

Conversely, suppose that y is essential with respect to M_{1}. If S were not lower semicontinuous at y ∈ M_{1}, then there exist ϵ_{0} > 0 and a sequence {y_{n}}_{n=1}^{∞} in M with y_{n} → y such that for each n = 1, 2, ⋯, there is x_{n} ∈ S(y) with x_{n} ∉ U(ϵ_{0}, S(y_{n})). Since S(y) is compact, we may assume that x_{n} → x ∈ S(y). Since x is an essential fixed point of y with respect to M_{1}, y_{n} → y and x_{n} → x, there is N such that d(x_{n}, x) < ϵ_{0}/2 and x ∈ U(ϵ_{0}/2, S(y_{n})) for all n ≥ N. Hence x_{n} ∈ O(x, ϵ_{0}/2) ⊂ U(ϵ_{0}, S(y_{n})) for all n ≥ N which contradicts the assumption that x_{n} ∉ U(ϵ_{0}, S(y_{n})) for all n = 1, 2, · · ·. Hence S must be lower semicontinuous at y. □

The following result says that if an upper semicontinuous set-valued mapping with compact and admissible values has fixed points defined in hyperconvex metric spaces, then it can be approximated arbitrarily by an upper semicontinuous set-valued mapping with compact and admissible values whose fixed point set is stable.

Theorem 2.8.55. The set of essential points with respect to M_{1} is a dense residual set in M_{1}. In particular, every point in M_{1} can be arbitrarily approximated by an essential points in M_{1}.

Proof. By Lemma 2.8.50 and Lemma 2.8.51, the mapping S : M → K(X) is an usco mapping. Since M_{1} is complete, by Theorem 2 of Fort [?], the set of points where S is lower semicontinuous is a dense residual set in M_{1}. By Theorem 2.8.54, the set of essential points in M_{1} is a dense residual set in M_{1}. □

By Lemma 2.8.51, Theorems 2.8.54 and 2.8.55, we have the following:

Theorem 2.8.56. The mapping S is continuous at y ∈ M_{1} if and only if y is essential with respect to M_{1}. Moreover, the set of points at which S is continuous is a dense residual set in M_{1}.

Remark 2.8.57. We remark that S is continuous at y ∈ M_{1}, if and only if for each ϵ > 0, there is δ > 0 such that h(S(y), S(y′)) < ϵ for each y′ ∈ M with D(y, y′) < δ. Theorem 2.8.56 implies that if y = (f, A) ∈ M_{1}, then y is essential with respect to M_{1} if and only if its set S(y) of fixed points is stable in the sense that: S(y′) is close to S(y) whenever y′ is close to y.

We shall now give a sufficient condition that y ∈ M_{1} is essential with respect to M_{1}:

Theorem 2.8.58. If y ∈ M_{1} is such that S(y) is a singleton set, then y is essential with respect to M_{1}.

Proof. Suppose S(y) = {x}. It follows by Lemma 2.8.51 that S is upper semicontinuous at y. Thus for any ϵ > 0, there is δ > 0 such that for each y′ ∈ M_{1}, D(y, y′) < δ implies S(y′) ⊂ U(ϵ, S(y)) = O(x, ϵ) so that S(y) = {x} ⊂ U(ϵ, S(y′)). This shows that S is also lower semicontinuous at y. By Theorem 2.8.54, y is essential with respect to M_{1}. □

Finally as applications of Theorem 2.8.56, we have the following stability result of fixed points in hyperconvex metric spaces for upper semicontinuous set-valued mappings.

Theorem 2.8.59. Let X be a compact hyperconvex metric space and C_{1} be the space consisting of all upper semicontinuous set-valued mappings from X to itself with non-empty admissible values. Then there exists a dense residual subset C_{1} of C such that:

(1) the fixed point set of each mapping in C_{1} is essential; and

(2) any given f ∈ C and ϵ > 0, there exists δ > 0 such that for each f′ ∈ C_{1} with sup_{x∈X} h(f(x), f′(x)) < δ, we have that h(S(f), S(f′)) < ϵ.

Proof. By Theorem 2.8.5, it follows that C_{1} × {X} is a non-empty closed subset of M. Thus the conclusion follows by Theorem 2.8.56. This completes the proof. □

Remark 2.8.60. Theorem 2.8.59 show that almost all of fixed points of upper semicontinuous set-valued mappings defined in compact hyperconvex metric spaces are stable in the sense that each upper semicontinuous set-valued mapping with non-empty compact and admissible values (and thus having fixed points) defined in hyperconvex metric spaces can be approximated arbitrarily by an essentially upper semicontinuous set-valued mapping with non-empty compact and admissible values whose fixed point set is stable.

Before we clsoe this chapter, we remark that by introducing so-called ‘locally generalzied convex spaces’- which is another kind of general abstract convexity due to Ben-El-Mechaiekh et al. [?], Himmelberg’s fixed point theorem [?] has been generalized by replacing the usual convexity in topological vector spaces with an abstract topological notion of convexity that generalizes classic convexity as well as several metric convexity structures found of existing literatures [?].