=0 Chapter 1 METRIC ENVIROMENT OF TOPOLOGICAL FIXED POINT THEOREMS

Kazimierz Goebel

Maria Curie-Skłodowska University 20-031 Lublin, Poland

goebel@golem.umcs.lublin.pl

1. Introduction

In metric fixed point theory the term the fixed point property is usually related to a certain class of mappings described by some metric conditions. In topological part of the theory however, we use this term with respect to the wide class of spaces and families of continuous transformations. Let us begin with recalling the classical definition and facts.

Definition 1.1 A topological space X is said to have the (topological) fixed point property (t.f.p.p. for short) if each continuous mapping f : X → X has a fixed point.

It is easy to see that the above property is a topological invariant. Suppose X and Y are two homeomorphic spaces with X having the fixed point property. Let h : X → Y be a homeomorphism with h (X) = Y. Suppose f : Y → Y is continuous. Then g = h^{−1} ∘ f ∘ h : X → X and g is continuous. Thus there exists x ∈ X such that gx = x, and consequently, for y = hx we have y = fy.

Other operations which preserve the fixed point property are retractions. Recall that a subspace Y of X is a retract of X if there exists a continuous mapping (a retraction) r : X → Y such that rx = x for all x ∈ Y. For any given retract Y of X there can be many retractions on it. Suppose Y is a retract of X and let f : Y → Y be continuous. Then, with the use of any retraction r : X → Y, such f may be extended to a continuous mapping g = f ∘ r : X → X. Any fixed point of g must be a fixed point of f. Thus if X has the t.f.p.p. so have it all of its retracts.

The most famous and important theorem on t.f.p.p., The Brouwer’s Fixed Point Theorem, was first proved in 1912 by L.E.J. Brouwer. In our terminology it can be formulated as

The closed unit ball B^{n} in ℝ^{n} has the t.f.p.p.

Brouwer’s Theorem has many applications in analysis, differential equations and generally in proving all kinds of so called existence theorems for many types of equations. Its discovery has had a tremendous influence in the development of several branches of mathematics, especially algebraic topology. All known proofs of the Theorem require analytical, topological, combinatorial, “nonelementary” and “nonmetric” (whatever it means) arguments. The Theorem also has many equivalent formulations. The attempts (sometimes naive) to find more general versions, equivalents or an elementary proof of Brouwer’s Theorem have given rise to interesting questions of “metric type”. We are not going to present here proofs of the Theorem but rather discuss the metric “environment” of it.

To follow let us define the standard simplex in ℝ^{n} as

T^{n−1} = {x ∈ ℝ^{n} : x_{i} ≥ 0 for i = 1, 2, ..., n and ∑_{i=1}^{n} x_{i} = 1}.  (1)

and formulate:

For each n = 1, 2, ... , T^{n−1} has the t.f.p.p.

In view of the initial remarks one can immediately formulate the “more general version”.

Any nonempty, convex, closed and bounded subset K of ℝ^{n} has the t.f.p.p.

All three formulations are equivalent. It can be shown using combinations of some of the following facts:

a) All closed balls in ℝ^{n} (of positive radius and arbitrary center) are homeomorphic.

b) T^{n} ⊂ ℝ^{n+1} is homeomorphic with B^{n}.

c) Any nonempty convex and closed subset K of ℝ^{n} is a retract of ℝ^{n}. Thus it is a retract of any ball and any set containing it.

d) Any nonempty convex, bounded and closed subset K of ℝ^{n} is homeomorphic to a ball B^{k} for some k ≤ n.

There are many other equivalents of Brouwer’s Theorem. Let us recall two the most important. First is known as The Non-retraction Theorem.

For n = 1, 2, ... the unit sphere S^{n−1} is not the retract of B^{n}.

The second concerns the contractibility of spheres. Let us recall that a topological space X is said to be contractible to a point z ∈ X if there exists a continuous function (a homotopy) H : [0, 1]×X → X such that for all x ∈ X, H (0, x) = x and H (1, x) = z. If X is contractible to a point z then it is contractible to any other point of X, so we can simply say X is contractible. In other (topological) terminology X is contractible if the identity mapping I on X is homotopic to a constant map.

The following is the so called No Contractibility Theorem.

For n = 1, 2, ... the unit spheres S^{n−1} are not contractible.

The equivalence of Theorems 1, 4 and 5 (1, 4, 5 for short) can be shown in several ways. We present here arguments which will be useful later in the discussion of infinitely dimensional balls and spheres.

Proof. Proof of the equivalence. The unit sphere S^{n−1} does not have the t.f.p.p. since the antipodal map −I : x → −x is continuous (even isometric) on S^{n−1} without fixed points. Thus, since retractions preserve the t.f.p.p., 1 implies 4.

To prove the contrary let us assume that 1 is false. Suppose for certain n there exists a continuous mapping T : B^{n} → B^{n} such that T x ≠ x for all x ∈ B^{n}. Extend this mapping to the doubled ball 2B^{n} by defining

T_{1}x = \{ \begin{matrix}T x & for ∥x∥ ≤ 1 \\ (2 − ∥x∥) T (\frac{x}{∥x∥}) & for1 < ∥x∥ ≤ 2 \end{matrix}  (2)

and the new mapping T_{2} : B^{n} → B^{n} as

T_{2}x = \frac{1}{2}T_{1} (2x) .

Observe that both mappings T_{1} and T_{2} are fixed point free and for all x ∈ S^{n−1} we have T_{2}x = 0. Now we can contradict 4 by defining a retraction R : B^{n} → S^{n−1} as

Rx = \frac{x − T_{2}x}{∥x − T_{2}x∥}.  (3)

Thus 4 implies 1 and the equivalence of 1 and 4 is proved.

Suppose now that 5 does not hold and let H : [0, 1] × S^{n−1} → S^{n−1} be a homotopy with H (0, x) = x and H (1, x) = z = const. Take any radius 0 ≤ r < 1 and define

Rx = \{ \begin{matrix}z & for ∥x∥ ≤ r, \\ H (\frac{1−∥x∥}{1−r}, \frac{x}{∥x∥}) & for r < ∥x∥ ≤ 1. \end{matrix}  (4)

The consistency of the above formula for r = 0 comes from the uniform continuity of H. Observe that for each r, R is a retraction of B^{n} onto S^{n−1}. In view of this 4 implies 5.

On the other hand, if R is a retraction onto S^{n−1} then S^{n−1} would be contractible to R (0) by the homotopy defined as

H (t, x) = R ((1 − t) x) .  (5)

This concludes the proof of equivalence of all formulations 1, 2, 3, 4, 5. ■

To end this section we turn the readers attention to the fact that all the facts presented here are nonmetric in the sense that they do not depend on the norm ∥ ∥ used in ℝ^{n}. Geometric properties of balls will be important however, in the infinitely dimensional case we are going to discuss.

2. Schauder’s Theorem

From now on let X denote an infinitely dimensional Banach space with norm ∥ ∥ the closed unit ball B_{X} and the unit sphere S_{X} (B and S for short) . The crucial difference between infinitely dimensional spaces and ℝ^{n} is caused by the fact that closed, bounded and convex sets do not need to be compact. Recall that X is of finite dimension if and only if B_{X} is compact.

An important generalization of Brouwer’s Theorem was discovered in 1930 by J. Schauder and is known as Schauder’s Fixed Point Theorem.

Any nonempty, compact, convex subset K of a Banach space has the t.f.p.p.

Proof. Let K ⊂ X be convex and compact. For any ε > 0 there exists a finite ε−net {a_{1}, a_{2}, ..., a_{p}} ⊂ K. Define a collection {α_{i}} of real valued nonnegative functions on K by taking

α_{i} (x) = \{ \begin{matrix}0 & if ∥x − a_{i}∥ ≥ ε, \\ ε − ∥x − a_{i}∥ & if ∥x − a_{i}∥ ≤ ε, \end{matrix} i = 1, 2, .., p.

Put

β_{i} (x) = \frac{α_{i} (x)}{∑_{i=1}^{p} α_{i} (x)}

and observe that ∑_{i=1}^{p} β_{i} (x) = 1. Now define the function Φ : K → K_{0} = K ∩ span {a_{1}, a_{2}, ..., a_{p}} by taking

Φ (x) = ∑_{i=1}^{p} β_{i} (x) a_{i}.

Obviously Φ is continuous and represents on K an ε−approximation of identity, which means that for x ∈ K we have ∥Φ (x) − x∥ ≤ ε. Let T : K → K be a continuous mapping. Let us modify it, defining T_{0} : K → K by T_{0} = Φ ∘ T. Thus T_{0} : K_{0} → K_{0} and since K_{0} is homeomrphic to a compact convex subset of finite dimensional space, T_{0} has a fixed point x ∈ K_{0}. From this we have

∥x − T x∥ ≤ ∥x − T_{0}x∥ + ∥T_{0}x − T x∥ = ∥Φ ∘ T x − T x∥ ≤ ε.

Therefore ,inf {∥x − T x∥ : x ∈ K} = 0, and since K is compact F ixT ≠ ∅. ■

It is in general not possible, even in limited ways, to extend Brouwer and Schauder theorems to noncompact settings. Let us recall, in more or less historical order, facts explaining the cause of the failure of Schauder theorem on noncompact sets. The most commonly known are Kakutani’s Examples (1943). Example 1. Consider l^{2} (ℕ) as a model of a Hilbert space. For the unit ball, the mapping T : B → B

T (x_{1}, x_{2}, ...) = ((\sqrt{1 − ∥x∥^{2}}) , x_{1}, x_{2}, ...)

is clearly continuous with F ixT = ∅. Moreover it is easy to notice that T is uniformly continuous on B and T (B) ⊂ S. Also for any 0 < ε < 1 similar mappings

T_{ε} (x_{1}, x_{2}, ...) = (ε (1 − ∥x∥) , x_{1}, x_{2}, ...)

map B into B (but not into S) are fixed point free and are even lipschitzian. Indeed we have

∥T_{ε}x − T_{ε}y∥ ≤ \sqrt{(1 + ε^{2})} ∥x − y∥ .

The next of Kakutani’s constructions shows even more. Consider another model on Hilbert space namely l^{2} (ℤ) and an the isometric shift operator

A : x = {x_{i}}_{i∈ℤ} → {x_{i+1}}_{i∈ℤ} .

Thus the mapping T : B → B defined by

T x = \frac{1}{2} (1 − ∥x∥) e_{0} + Ax

where e_{0} = {δ_{i,0}}, is a lipschitzian homeomorphism of B onto B with lipschitzian inverse, yet F ixT = ∅.

All Kakutani’s examples have very simple structure. Each one represents a one dimensional perturbation of an isometry. They fail to have fixed points but share a somewhat weaker property. For all the above mappings we have

inf_{x∈B} ∥x − T x∥ = 0.  (6)

Since it is known that all nonexpansive self-mappings of B in l^{2} must have fixed points, Kakutani’s constructions show just how fragile is this property.

Balls in less regular spaces than l^{2} can accept even nonexpansive mappings without fixed points. Example 2. Consider the space C [a, b] (e.g. C [−1, 1]) of continuous functions and its unit ball B. Let us recall that there exists a nonexpansive retraction Q : C [a, b] → B defined with the use of the function

α (t) = min \{1, max [−1, t]\} = \{ \begin{matrix}−1 & for t ≤ −1 \\ t & for − 1 ≤ t ≤ 1 \\ 1 & for t ≥ 1 \end{matrix}

as

Qf (t) = α ∘ f (t) = α (f (t)) .  (7)

Take any function g ∈ C [a, b] satisfying g (a) ≤ −2, g (b) ≥ 2 (for example g (t) = 2t for C [−1, 1]). Define the mapping T : B → B by the formula

T f (t) = Q (f (t) + g (t)) .

For each such mapping we have T f (a) = −1, T f (b) = 1 so T : B → S. Moreover T is nonexpansive, fixed point free and again satisfies (6). Example 3. There exist isometries mapping the unit ball in c_{0} into its unit sphere without fixed points. The simplest formula for such a mapping is:

T (x_{1}, x_{2}, ...) = (1, x_{1}, x_{2}, ...) .

Our next example shows another type of singularity. This one is connected not with the metric regularity of the mapping but with its structure. Example 4. One of the properties distinguishing infinitely dimensional Banach spaces from spaces of finite dimension is the fact that if dim X = ∞ then X is homeomorphic with the punctured space X⧹ {0}. More, there exists a homeomorphism h : X → X⧹ {0} such that hx = x for ∥x∥ ≥ 1. Now the mapping T : B → B,

T x = h^{−1} (−hx)

lacks fixed points but represents an involution which means T^{2} = I. In other words all point are fixed under T^{2} (T^{2}x = x).

The first general theorem on the extent of the topological fixed point property has been presented by V. Klee in 1950.

A closed bounded convex subset C of a Banach space has the t.f.p.p. if and only if it is compact.

Recall two commonly known facts. First states that if C is noncompact then there exist a number r > 0 and a bounded sequence of points {x_{n}} , n = 0, 1, 2, .. in C such that

dist (x_{n+1,} span (x_{1}, x_{2}, ..., x_{n})) ≥ r > 0  (8)

The second is the well known Tietze’s Extension Theorem.

Let (M, ρ) be a metric space and let Z be a closed subset of M. Then for any continuous function f : Z → R there exists a continuous function (extension) f̃: M → R such that F (x) = f (x) for all x ∈ Z. Moreover if f (Z) ⊂ [a, b] then f̃(M) ⊂ [a, b].

With the above we can prove Klee’s result.

Proof. Proof of Theorem 7. Suppose C ⊂ X is a closed bounded and convex. Let for {x_{n}} , n = 0, 1, 2, ... be a sequence in C satisfying (8). Without loss of generality we may assume that 0 ∈ C and that ∥x_{0}∥ ≥ r > 0. Let us construct a piecewise linear curve joining consecutive points x_{0}, x_{1}, x_{2}, ... by putting

Γ = ⋃_{n=0}^{∞} Γ_{n} where Γ_{n} are segments Γ_{n} = conv (x_{n}, x_{n+1}) = [x_{n}, x_{n+1}]  (9)

Thus Γ is a closed subset of C and can be parametrized by a function γ : [0, ∞) → Γ. For t ∈ [0, ∞), let t = n + s where n is the integer part of t and 0 ≤ s < 1. Set

γ (t) = γ (n + s) = (1 − s) x_{n} + sx_{n+1}.

The function γ^{−1} : Γ → [0, ∞) is continuous and hence (in view of Tietze’s Theorem) has a continuous extension g : C → [0, ∞). Now a fixed point free mapping T : C → Γ ⊂ C can be defined as

T x = γ (g (x) + 1) .  (10)

Together with Schauder’s Theorem this completes the proof. ■

The assumption of boundedness of C is not crucial to our theorem. Unbounded convex sets either contain a halfline and are invariant under translations along it or contain a closed homeomorphic image of the half line and our arguments can be repeated.

The method of the construction used by Klee has a certain disadvantage. The curve Γ is obviously the retract of C via the retraction r (x) = γ (g (x)). However Γ is of infinite length but C is bounded, thus none of continuous mappings satisfying T (C) = Γ can be uniformly continuous. In view of our initial examples it is natural to ask if there is another method of construction leading to uniformly continuous fixed point free mappings?

Before we pass to the discussion of this problem let us notice that formulas (2) , (3) , (4) , (5) used in the proof of equivalence in the introductory section allow us to formulate:

If X is an infinitely dimensional Banach space then:

a the unit ball B does not have the t.f.p.p.,

b the unit sphere S is the retract of B,

c the unit sphere S is contractible.

Since we used explicit formulas all items a, b, c are equivalent in the sense that each one can be derived from another. The search of uniformly continuous extension of Theorem 9 has been initiated by B. Nowak (1979) who proved that for some Banach spaces there exists a uniformly continuous (even lipschitzian) retraction of B onto S. The result has been extended to all Banach spaces by Y. Benyamini and Y. Sternfeld in (1983). And finally Y. Sternfeld and P. K. Lin presented the following result If C is a closed bounded convex but noncompact subset of a Banach space then for any k > 1 there exists mapping T : C → C satisfying the Lipschitz condition

∥T x − T y∥ ≤ k ∥x − y∥  (11)

and such that

d (T) = inf [∥x − T x∥ : x ∈ C] > 0.  (12)

The proof is based on an idea similar to those we used showing Theorem 7. We only need a modification of the extension theorem.

Let (M, ρ) be a metric space and let Z be a closed subset of M. Suppose f : Z → ℝ is a function satisfying the Lipschitz condition

|fx − fy| ≤ kρ (x, y)  for x, y ∈ Z.  (13)

Then f has an extension f̃ : M → ℝ (f̃|_{Z} = f ) which satisfies the same Lipschitz

condition. Moreover, if a ≤ fx ≤ b for all x ∈ Z, then f̃ can be chosen so that a ≤ f̃x ≤ b for all x ∈ M.

Proof. It suffices to observe that the function defined for x ∈ M by

f̃_{1}x = sup {fy − kϱ (x, y) : y ∈ Z} ,

has the first property. The second holds upon taking

f̃x = max {a, min {b, f̃_{1}x}} .
■

Corollary 2.1 Analogous theorem holds for mappings F = {f_{1}, f_{2}, ...} : Z → l^{∞} satisfying

∥F x − F y∥_{l^{∞}} ≤ kρ (x, y) .

Since l^{∞} consists of bounded sequences with the supremum norm, the function F is represented by a sequence of “coordinate” functions {f_{i}} each of which satisfies (13). Extending each one f_{i}, according to the above theorem we can obtain the extension F̃ of F. Moreover, if the “coordinate functions” f_{i} are bounded all their extensions have the same bounds.

We are now in the position to prove the main result of this section.

Proof. Proof of Theorem 10. Let C be a bounded, closed and convex but noncompact subset of a Banach space X. As in the proof of Theorem 7 assume that 0 ∈ C, ∥x_{0}∥ ≥ r and a sequence {x_{n}} , n = 0, 1, 2, ... satisfies (8). Construct as before the piecewise linear curve Γ = Γ_{C} and define in C a sequence of triangles

Δ_{n} (C) = co {0, x_{n}, x_{n+1}} for n = 0, 1, 2, ...

and a “fan”

Δ (C) = ⋃_{n=1}^{∞} Δ_{n} (C) .

Each element x ∈ Δ (C) is of the form x = λγ_{C} (t) where λ ∈ [0, 1] and γ_{C} represents the already used parametrization of Γ_{C} by t = n + s ∈ [0, ∞). We use here C in the index position only to indicate that all objects are in the set C.

Let us pass now to the space l^{∞} (ℕ_{0}). For each n ∈ ℕ_{0} let e_{n} denote the standard unit vector {δ_{in}} = (0, ..., 0, 1, 0, ...) (where 1 is in the nth position). Define the sequence {Δ_{n}} of standard triangles as

Δ_{n} = co {0, e_{n}, e_{n+1}}
= {(0, ..., 0, ξ_{n}, ξ_{n+1}, 0, ...) : ξ_{n} ≥ 0, ξ_{n+1} ≥ 0, ξ_{n} + ξ_{n+1} ≤ 1}

and the “standard fan” as

Δ = ⋃_{n=0}^{∞} Δ_{n}.

Obviously again each point x of Δ can be represented as x = λγ (t) where λ and t are natural, for Δ and corresponding “frame” curve Γ generated by unit vectors, parameters. Both “fans” Δ (C) and Δ are locally finite dimensional (two dimensional) except for the neighborhood of zero. We leave to the reader a justification of the fact that both are homeomorphic and that the natural invertible mapping H : Δ → Δ (C) defined by

H (ξ_{n}e_{n} + ξ_{n+1}e_{n+1}) = ξ_{n}x_{n} + ξ_{n+1}x_{n+1}  (14)

is onto and moreover both H and H^{−1} are lipschitzian.

The next crucial step is to show that there exists a lipschitzian retraction of l^{∞} (ℕ_{0}) onto Δ. Let e be a vector whose coordinates are all equal to one. For x ∈ l^{∞} (ℕ_{0}) , set

E (x) = {ε ≥ 0 : (x − εe) ∨ 0 = {max {x_{i} − ε, 0}} ∈ Δ},

observe that since ∥x∥ ∈ E (x) , E (x) ≠ ∅ and put

ε (x) = inf E (x) .

Observe that, in the natural coordinatewise order, for x, y ∈ l^{∞} we have

x ≤ y + ∥x − y∥ e.

Thus ε (y) + ∥x − y∥ ∈ E (x) implying ε (x) ≤ ε (y) + ∥x − y∥. By symmetry we obtain

|ε (x) − ε (y)| ≤ ∥x − y∥ .

Therefore the retraction R : l^{∞} (ℕ_{0}) → Δ defined as Rx = (x − ε (x)) ∨ 0 fulfils the Lipschitz condition

∥Rx − Ry∥ ≤ 2 ∥x − y∥ .

In consequence, Δ (C) is a lipschitzian retract of C. Indeed, let H^{−1} : Δ (C) → Δ be the inverse homeomorphism to H defined by (14). By Corollary H^{−1} has a lipschitzian extension H̃^{−1} : C → l^{∞}. The desired lipschitzian retraction R_{C} : C → Δ (C) can be now defined as R_{C} = H ∘ R ∘ H̃^{−1}. Only we do not know an exact Lipschitz constant for it.

To complete the proof observe that we can start the construction of a lipschitzian mapping T : C → C with positive minimal displacement,

d (T) = inf {∥x − T x∥ : x ∈ C} > 0,

building first, for our standard “fan” Δ, a lipschtzian mapping T_{Δ} : Δ → Δ satisfying d (T_{Δ}) = inf {∥x − T_{Δ}x∥ : x ∈ Δ} > 0.

Once we have it, we can define the next mapping T_{Δ(C)} : Δ (C) → Δ (C) , also satisfying d (T_{Δ(C)}) > 0 by putting T_{Δ(C)} = H ∘ T_{Δ} ∘ H^{−1}. And we can finish with taking T = T_{Δ(C)} ∘ R_{C}.

Here is a formula for T_{Δ} in terms of (λ, t) coordinates. For the initial triangle Δ_{0} of Δ (i.e.,for 0 ≤ t ≤ 1) set

T_{Δ}x = T_{Δ} (λγ (t)) = \{ \begin{matrix}(1 − 3λt) e_{0} \\ (1 − t) e_{0} + (3λ − 1) t (\frac{e_{0}+e_{1}}{2} ) \\ γ (\frac{t}{2} + (3λ − 2)) \end{matrix}  for λ ∈ [0, \frac{1}{3}] , for λ ∈ (\frac{1}{3}, \frac{2}{3}] , for λ ∈ (\frac{2}{3}, 1] .

The above mapping maps Δ_{0} into Δ_{0} ∪ Δ_{1} without having fixed points. In view of compactness of Δ_{0} we have inf {∥x − T_{Δ}x∥ : x ∈ Δ_{0}} > 0.

Now we can extend T_{Δ} to all of Δ by putting for x ∈ ⋃_{n=1}^{∞} Δ_{n}, x = λγ (t) (t ≥ 1) :

T_{Δ}x = T_{Δ} (λγ (t)) = \{ \begin{matrix}(1 − 3λ) e_{0} \\ (3λ − 1) γ (t − \frac{1}{2}) \\ γ ((t − \frac{1}{2}) + (3λ − 2)) \end{matrix}  for λ ∈ [0, \frac{1}{3}] , for λ ∈ (\frac{1}{3}, \frac{2}{3}] , for λ ∈ (\frac{2}{3}, 1] .

We leave to the reader checking that T_{Δ} is lipschitzian and that

d(T_{Δ}) = inf {∥x − T_{Δ}x∥ : x ∈ Δ} = inf {∥x − T_{Δ}x∥ : x ∈ Δ_{0} ∪ Δ_{1}} > 0.

So we proved that there exists a mapping T : C → C satisfying the Lipschitz condition (11) and having positive minimal displacement, which means satisfying (12). We neither evaluated its Lipschitz constant k nor its minimal displacement d (T).

To show that such mappings do exists for any Lipschitz constant k > 1 observe that if T satisfies (11) and (12) then for any α ∈ (0, 1] the mapping T_{α} = (1 − α) I + αT : C → C is lipschitzian with constant (1 − α) + αk and d (T_{α}) = αd (T). This remark completes the proof of Theorem 10. ■

Theorem 10 has strong consequences concerning balls. Let T : B → B be a lipschitzian mapping having d (T) > 0. Close analysis of all the reasonings used in the proof of the equivalence of 1, 4, 5 leads to the following result which is stronger than Theorem 9

Let X be an infinitely dimensional Banach space with the unit ball B and the unit sphere S. Then:

a For any k > 1 there exists a mapping T : B → B satisfying (11) such that d (T) > 0.

b There exists a retraction R : B → S satisfying (11) with certain (sufficiently large) k > 1.

c S is contractible to a point by a lipschitzian homotopy.

The last statement means that there is a homotopy H : [0, 1]×S → S with H (0, x) = x, H (1, x) = x_{0} for which there are two constants A > 0, B > 0 such that for any t, s ∈ [0, 1] and x, y ∈ S,

∥H (t, x) − H (s, y)∥ ≤ A |t − s| + B ∥x − y∥ .

All three statements are formally equivalent in the sense that each one can be derived from any other.

This theorem completes the qualitative part of our considerations. In the sections to follow we will discuss some quantitative problems connected with concrete estimates for d (T) or a Lipschitz constant for retractions of a ball onto a sphere.

3. Minimal displacement problem

Minimal displacement problem

As we already observed, all the fixed point free mappings defined by Kakutani have d (T) = 0. This property is natural for certain classes of selfmappings of convex and bounded sets. For example this is true for all nonexpansive mappings, their compact perturbations or the so called k-set contractions. On the other hand for k-lipschitzian mappings we have Theorem 10.

The term minimal displacement problem was probably first used by the present author in 1973. It came long before the Sternfeld and Lin result, from the observation