Classical Theory of Nonexpansive Mappings

Kazimierz Goebel and W. A. Kirk

1. INTRODUCTION

Mappings which are defined on metric spaces and which do not increase distances between pairs of points and their images are called nonexpansive. Thus an abstract metric space is all that is needed to define the concept. At the same time, the more interesting results seem to require some notion of topology; more specifically a topology which assures that closed metric balls are compact. This is not a serious limitation, however, because many spaces which arise naturally in functional analysis possess such topologies; most notably the weak and weak^{∗} topologies in Banach spaces.

Here we present a brief overview of the classical theory of nonexpansive mappings, concentrating on the topics listed below. We give only passing attention to aspects of the theory which are covered in more detail elsewhere in this work.

• Existence: Classical results.

• The structure of the fixed point sets.

• The approximation of fixed points.

• Abstract Metric Theory

Recognition of fixed point theory for nonexpansive mappings as a noteworthy avenue of research almost surely dates from the 1965 publication of likely the most widely known result in the theory.

Theorem 1.1. If K is a bounded closed and convex subset of a uniformly convex Banach space X and if T : K → K is nonexpansive (that is, ∥T (x) − T (y)∥ ≤ ∥x − y∥ for each x, y ∈ K), then T has a fixed point.

The above theorem was proved independently by F. Browder [?] and D. Göhde [?] in the form stated above, and by W. Kirk [?] in a more general form. (Slightly earlier Browder obtained the Hilbert space version of this result ([?]) as a byproduct of the fact that a mapping T in a Hilbert space is nonexpansive if and only if the mapping I − T is monotone.) Browder and Kirk used the same line of argument – indeed, one which in fact yields a more general result – while the proof of Göhde relies on properties essentially unique to uniformly convex spaces. As a result, Göhde’s proof reveals that fixed points in this setting may be obtained as weak limits of sequences of ‘approximate’ fixed points.

1991 Mathematics Subject Classification. Primary 54H10 .

Key words and phrases. Contraction mappings, fixed points, Picard iterates.

The proof of Theorem ?? is contained in the proof of Theorem ?? given in the next section. (The proofs included in this article were chosen to illustrate a variety of standard methods in the theory.)

2. CLASSICAL EXISTENCE RESULTS

2.1. Overview. We begin with a study of nonexpansive mappings in a Banach space setting and take up the purely metric aspects of the theory later. If X is a subset of a Banach space and D ⊆ X, then a mapping T : D → X is said to be nonexpansive if for each x, y ∈ D,

∥T (x) − T (y)∥ ≤ ∥x − y∥ .

The study of the existence of fixed points for nonexpansive has generally fallen into three categories. We shall say that a Banach space has FPP if each of its nonempty bounded closed convex subsets has the fixed point property (f.p.p.) for nonexpansive self-mappings, the wc-FPP if each of its weakly compact convex subsets has f.p.p., and B-FPP if its closed unit ball (hence any ball) has f.p.p. This latter category is primarily relevant to dual spaces where the unit ball is always compact in the weak^{∗} topology relative to any predual, and in these cases the f.p.p. seems invariably to extend to sets which are intersections of closed balls. The classical nonreflexive space ℓ_{1} provides an example of a space which has B-FPP but not FPP (Karlovitz [?], Lim [?]). Also, c_{0} provides an example of a space which has wc-FPP but neither FPP nor B-FPP (Maurey [?]).

It is only natural that one of the central goals in the theory should be to fully characterize those Banach spaces which have FPP. This goal remains elusive. It is known that essentially all classical reflexive spaces - in particular, all uniformly convex spaces - have FPP, hence wc-FPP, via the presence of a geometric property called normal structure. A Banach space X, or more generally, a closed convex subset K of X, has normal structure if any bounded convex subset H of K which contains more than one point contains a nondiametral point - i.e., there exists a point x_{0} ∈ H such that

sup{∥x_{0} − x∥ : x ∈ H} < diam(K) := sup{∥x − y∥ : x, y ∈ H}.
For D ⊂ X let
diam (D) = sup {∥u − v∥ : u, v ∈ D};
r_{x} (D) = sup {∥x − v∥ : v ∈ D};
r (D) = inf {r_{x} (D) : x ∈ D} .

If X is reflexive and if D is bounded closed and convex, then weak compactness of closed balls in X yields the fact that the set

C (D) = {z ∈ D : r_{z} (D) = r (D)}

is a nonempty closed convex subset of D. The number r (D) and the set C (D) are called the Chebyshev radius and Chebyshev center of D, respectively.

We state the following theorem for reflexive spaces although clearly weak compactness of K suffices.

Theorem 2.1. Let X be a reflexive Banach space and suppose K is a bounded closed convex subset of X which has normal structure. Then any nonexpansive mapping T : K → K has a fixed point. In particular, if X has normal structure then X has FPP.

Proof. Let

𝔉 = {D ⊂ K : D closed, convex; D ≠ ∅; T (D) ⊂ D} .

Since the members of 𝔉 are weakly compact each descending chain in 𝔉 has a lower bound (the intersection of its members); hence by Zorn’s Lemma 𝔉 has a minimal element, say D_{0}. On the other hand,

T (D_{0}) ⊂ D_{0} ⇒ T (\overline{conv} (T (D_{0}))) ⊂ T (D_{0}) ⊂ \overline{conv} (T (D_{0})) .

Thus \overline{conv} (T (D_{0})) ∈ 𝔉 and by minimality, \overline{conv} (T (D_{0})) = D_{0}. Let u ∈ C (D_{0}) ; thus r_{u} (D_{0}) = r (D_{0}) . But ∥T (u) − T (v)∥ ≤ ∥u − v∥ ≤ r (D_{0}) for all v ∈ D_{0} ⇒ T (D_{0}) ⊂ B (T (u) ; r (D_{0})) . Therefore

D_{0} = \overline{conv} (T (D_{0})) ⊂ B (T (u) ; r (D_{0}))

which gives r_{T (u)} (D_{0}) = r (D_{0}) , from which T (u) ∈ C (D_{0}) . In view of minimality of D_{0} this implies C (D_{0}) = D_{0}. Thus diam (D_{0}) ≤ r (D_{0}) , and since K has normal structure it must be the case that D_{0} consists of a single point which is fixed under T.

A major obstacle in identifying spaces which have FPP is the fact that it is not known whether FPP is stable under equivalent renormings. There are other hindrances as well. It has been known virtually from the outset that f.p.p. for a bounded closed convex subset K of a Banach space depends strongly on ‘nice’ geometrical properties of the space or on the set K itself. However, two closed convex subsets K_{1}, K_{2} ⊂ X may have f.p.p. yet K_{1} ∩ K_{2} may fail to have f.p.p.! Indeed, Goebel and Kuczumow [?] have shown how to construct a descending sequence {K_{n}} of nonempty bounded closed convex subsets of ℓ_{1} which has the property that if n is odd, K_{n} has f.p.p., if n is even K_{n} fails to have f.p.p., and in fact the sequence {K_{n}} may be constructed so that ∩K_{n} is nonempty and falls into either category. The space ℓ_{1} provides the setting for another interesting example. It is possible to construct a family {K_{ϵ}} (ϵ > 0) of bounded closed convex sets in ℓ_{1} each of which has f.p.p., but which converges in the Hausdorff metric as ϵ → 0 to a bounded closed convex K_{0} which fails to have f.p.p. ([?], Example 5.3).

1. It has been known for some time that even in reflexive spaces normal structure is not essential for FPP. An example is provided by the spaces X_{β}, β > 0, defined by

X_{β} = {x ∈ ℓ_{2} : ∥x∥_{β} = max{∥x∥_{ℓ_{2}} , β ∥x∥_{∞}}}.

Some fact about X_{β} :

(i) X_{β} is reflexive (since it is isomorphic to the Hilbert space ℓ_{2}).

(ii) X_{β} has normal structure ⇔ β <\sqrt{2}.

(iii) X_{β} has asymptotic normal structure ⇔ β < 2.

The concept of asymptotic normal structure was introduced by Baillon and Schöneberg in 1981. A Banach space X has asymptotic normal structure if each nonempty bounded closed and convex subset K of X which contains more than one point has the property: If {x_{n}} ⊂ K satisfies ∥x_{n} − x_{n+1}∥ → 0 then there exists x ∈ K such that

lim inf_{n→∞} ∥x_{n} − x∥ < diam(K).

In [?] Baillon and Schöneberg proved the following theorem.

Theorem 2.2. In a reflexive Banach space, asymptotic normal structure ⇒ FPP.

In the same paper they went on to show that in fact X_{2} also has FPP, thus proving that asymptotic normal structure is not a necessary condition for FPP. Surprisingly, P. K. Lin proved in 1985 ([?]) that X_{β} has FPP for all β > 0.

In 1971 it was observed by Day, James, and Swaminathan ([?]) that every separable space has an equivalent norm which has normal structure (also see van Dulst [?]) Thus every separable reflexive space has an equivalent norm which has FPP.

The question of whether reflexivity is essential for FPP remains open, but there is compelling evidence that it might be. First, it is known that some bounded closed convex sets in the classical nonreflexive spaces c_{0} and (as noted above) in ℓ_{1} fail to have FPP. Also, Bessaga and Pełczynski have shown that if X is any Banach space with an unconditional basis, then X is reflexive ⇔ X contains a subspace isomorphic to c_{0} or ℓ_{1}. Thus all classical nonreflexive can be renormed so that they fail to have FPP.

This raises an obvious question: Can either c_{0} or ℓ_{1} be renormed so that they have FPP? Recall ([?]) that any renorming of ℓ_{1} contains almost isometric copies of ℓ_{1} suggesting, at least for ℓ_{1}, that the answer should be no. If indeed the answer is no, then by the Bessaga-Pełczynski result, in any space with an unconditional basis, FPP⇒ reflexivity. These matters are discussed in much more detail elsewhere in this volume.

THE SPACE L_{1} : As we have noted ℓ_{1} (hence L_{1}) fails to have FPP. However, in 1981, Alspach [?] proved much more, namely that L_{1} fails to have wc-FPP. At the same time, Maurey [?] proved that all reflexive subspaces of L_{1} do have FPP (hence wc-FPP). There has been another quite recent development. Dowling, Lennard, and Turret (1993) have shown that nonreflexive subspaces of L_{1} fail to have FPP. Thus: A subspace of L_{1} has FPP ⇔ it is reflexive.

Although it remains unknown whether superreflexive spaces have FPP, Maurey proved (also in 1981) that superreflexive spaces have FPP for nonlinear isometries (see [?] or [?] for a proof). Recall that superreflexive spaces are ones which have the property that every space which is finitely representable in such a space must itself be reflexive. Such spaces spaces are characterized by that fact that they all have equivalent uniformly convex norms ([?]).

2.2. The Goebel-Karlovitz Lemma. The opening strategy in the proof of Theorem ?? was the use of Zorn’s Lemma in conjunction with weak compactness to obtain a minimal nonempty closed convex T -invariant set. Then the normal structure property of the space was used to obtain a contradiction if this minimal set has positive diameter. Such an approach is standard. On the other hand, the proof of Theorem ?? requires a refinement of that approach.

Any nonempty closed convex set K which of a Banach space which is minimal with respect to being invariant under any mapping T : K → K must satisfy \overline{conv}(T (K)) = K. This has nothing to do with properties of the mapping T. However if T is nonexpansive and if K is also bounded then more can be said. The first is a key observation in the proof of Theorem ??.

Lemma 2.1. Each point of K is a diametral point (i.e., sup {∥x − y∥ : y ∈ K} = diam (K) for each x ∈ K).

Now let K be bounded closed and convex, fix z ∈ K, and let T : K → K be nonexpansive. Then for t ∈ (0, 1) the mapping T_{t} : K → K defined by setting T_{t} (x) = (1 − t) z + tT (x) is a contraction mapping with a unique fixed point x_{t}. Since

lim_{t→1^{−}} ∥x_{t} − T (x_{t})∥ = lim_{t→1^{−}} (1 − t) ∥z − T (x_{t})∥ = 0

we have the following.

Lemma 2.2. If K is a bounded closed and convex subset of a Banach space and if T : K → K is nonexpansive, then there exists a sequence {x_{n}} ⊂ K such that lim_{n→∞} ∥x_{n} − T (x_{n})∥ = 0.

This brings us to a fundamental fact which in conjunction with the previous lemma has proved extremely useful in extending fixed point results for nonexpansive mappings beyond those spaces which possess normal structure. It appears to have been discovered independently and at about the same time by Goebel [?] and Karlovitz [?]. This observation is crucial to the proof of Theorem ??.

Lemma 2.3. (Goebel-Karlovitz) Let K be a subset of a Banach space X, and suppose K is minimal invariant with respect to being nonempty, weakly compact, convex, and T -invariant for some nonexpansive mapping T. Suppose {x_{n}} ⊂ K satisfies lim_{n→∞} ∥x_{n} − T (x_{n})∥ = 0. Then for each x ∈ K, lim_{n→∞} ∥x − x_{n}∥ = diam (K) .

Proof. As we have just seen, if diam(K) > 0 then K can have no diametral points. Now suppose there exists {x_{n}} ⊆ K for which lim_{n→∞} ∥x_{n} − T (x_{n})∥ = 0, but for which lim_{n→∞} ∥x − x_{n}∥ = r < diam(K) for some x ∈ K. Let

C = {z ∈ K : lim sup_{n→∞} ∥z − x_{n}∥ ≤ r}.

It is easy to see that C is convex. Also C is closed. Indeed, if {u_{i}} ⊆ C and if lim_{i→∞} u_{i} = u, then for each i = 1, 2, ⋯,

lim sup_{n→∞} ∥u − x_{n}∥ ≤ lim sup_{n→∞} ∥u − u_{i}∥ + lim sup_{n→∞} ∥u_{i} − x_{n}∥ ≤ ∥u − u_{i}∥ + r.

Letting i → ∞, lim sup_{n→∞} ∥u − x_{n}∥ ≤ r. Thus u ∈ C and C is closed. Also if u ∈ K,

lim sup_{n→∞} ∥T (u) − x_{n}∥ ≤ lim sup_{n→∞}[∥T (u) − T (x_{n})∥ + ∥T (x_{n}) − x_{n}∥]

≤ lim sup_{n→∞} ∥T (u) − T (x_{n})∥

≤ lim sup_{n→∞} ∥u − x_{n}∥ ≤ r.

This proves that T : C → C. Since any closed convex subset of a weakly compact set is itself weakly compact, we conclude that C = K by minimality of K. However if diam (K) > 0 and if ϵ > 0 is chosen so that r + ϵ < diam(K), then the family {B(x; r + ϵ) : x ∈ K} has the finite intersection property since each such ball contains all but a finite number of terms of the sequence {x_{n}}. Since these balls are weakly compact

∩_{x∈K}B(x; r + ϵ) ≠ ∅.

But this implies r(K) ≤ r + ϵ < diam(K), contradicting the fact that K cannot have any diametral points.

2.3. Further Applications of the G-K Lemma. The Goebel-Karlovitz Lemma has been applied extensively in the study of fixed point theory for nonexpansive mappings, both as formulated in Lemma ?? and as stated in the language of ultraproducts (see [?] and references cited therein). One of the more recent applications is due to A. Jiménez-Melado and E. Lloréns-Fuster in [?] in connection with their introduction and study of a generalization of uniform convexity called orthogonal convexity where, among other things, they used the Goebel-Karlovitz Lemma to prove that weakly compact convex subsets of orthogonally convex spaces have the wc-FPP.

Orthogonal convexity is defined as follows. For points x, y of a Banach space X and λ > 0, let

M_{λ}(x, y) = \{z ∈ X : max(∥z − x∥ , ∥z − y∥) ≤ \frac{1}{2}(1 + λ) ∥x − y∥\}.

If A is a bounded subset of X, let |A| = sup{∥x∥ : x ∈ A}, and for a bounded sequence {x_{n}} in X and λ > 0, let

D({x_{n}}) = lim_{i→∞} sup( lim_{j→∞} sup ∥x_{i} − x_{j}∥);

A_{λ}({x_{n}}) = lim_{i→∞} sup( lim_{j→∞} sup |M_{λ}(x_{i}, x_{j})|).

Definition 2.1. A Banach space is said to be orthogonally convex if for each sequence {x_{n}} in X which converges weakly to 0 and for which D ({x_{n}}) > 0, there exists λ > 0 such that A_{λ} ({x_{n}}) < D ({x_{n}}) .

In general it is difficult to test whether a space is OC, but it is noted in [?] that every uniformly convex Banach space is OC. For spaces having a Schauder basis the condition arises more naturally. Other examples given in [?] include Banach spaces with the Schur property (hence ℓ_{1}), c_{0}, c, and James’s classical space J. (Also see [?], [?], [?].)

Theorem 2.3. Let K be a nonempty weakly compact convex subset of a Banach space X, and suppose K is orthogonally convex. Then every nonexpansive mapping T : K → K has a fixed point.

The opening move in the proof of this theorem is to invoke Zorn’s lemma and assume K is minimal with respect to being nonempty and T-invariant. Next let {x_{n}} ⊂ K satisfy lim_{n→∞} ∥x_{n} − T (x_{n})∥ = 0 and pass to a weakly convergent subsequence. Moreover (by a shift) one may assume {x_{n}} converges weakly to 0 ∈ K. The strategy is to then show that if diam (K) > 0 then K cannot be orthogonally convex. It takes several steps to accomplish this.

Another recent development is due to S. Prus [?] who has introduced a class of spaces which he calls uniformly noncreasy.

To give a precise definition of the concept we need some more notation. We shall use B_{X} and S_{X} to denote, respectively, the unit ball and unit sphere of a Banach space X. For any f ∈ S_{X^{∗}} and δ ∈ [0, 1], define the slice S(f, δ) to be the set

S(f, δ) = {x ∈ B_{X} : f(x) ≥ 1 − δ}.

Now consider two functionals f, g ∈ S_{X^{∗}} and a scalar δ ∈ [0, 1], and set

S(f, g, δ) = S(f, δ) ∩ S(g, δ).

A Banach space is said to have a crease if there exist two distinct functionals f, g ∈ S_{X^{∗}} such that diamS(f, g, 0) > 0. Since f(x) ≤ 1 for each f ∈ S_{X} and x ∈ B_{X}, given f ∈ S_{X} the hyperplane {x ∈ X : f(x) = 1} supports the unit ball B_{X}. Thus to say that X has a crease means that the sphere S_{X} contains a segment of positive length that lies on two different hyperplanes which support the unit ball B_{X}.

Uniformly noncreasy spaces are characterized by the absence of creases in a relatively strong sense. A space X is called noncreasy if its unit sphere does not have a crease. Any Banach space X with dim(X) ≤ 2 is trivially noncreasy, as is any strictly convex space.

Definition 2.2. A Banach space is said to be uniformly noncreasy if given any ε > 0 there is a δ > 0 such that if f, g ∈ X^{∗} and ∥f − g∥ ≥ ε, then diam(S (f, g, δ)) ≤ ε.

Every finite dimensional space, every uniformly convex space, and every uniformly smooth space is UNC. Also UNC is a self-dual property in the sense that X is UNC if and only if X^{∗} is UNC. Prus also proves that every UNC space is superreflexive.

Regarding FPP, Prus first shows that uniformly noncreasy is a super property; that is, if a Banach space X is uniformly noncreasy then the Banach space ultrapower X̃ of X over any nontrivial ultra filter U is also uniformly noncreasy. Then he shows (via an ultrapower approach) that uniformly noncreasy spaces have the FPP. It is in this latter step that Prus invokes the Goebel-Karlovitz Lemma. Combined, these fact yield the following theorem.

Theorem 2.4. If a Banach space X is UNC, then both X and X̃ have FPP.

2.4. Uniformly lipschitzian mappings. In 1973 the class of uniformly lipschitzian mappings was introduced to the theory by the present authors in [?]. This class forms a natural extension of the family of nonexpansive maps.

Let M be a metric space with metric ϱ. A mapping T : M → M is said to be lipschitzian (satisfies Lipschitz condition, is a Lipschitz map) if there exists k ≥ 0 such that for all x, y ∈ M,

ϱ (T x, T y) ≤ kϱ (x, y) .

The smallest k for which the above holds is called the Lipschitz constant of T and is denoted by k (T). For mappings T, S : M → M,

k (T ∘ S) ≤ k (T) k (S)

and in particular

k (T^{n}) ≤ k^{n} (T) , n = 0, 1, 2, · · · .

Nonexpansive mappings are those with k (T) ≤ 1. Obviously all the iterates of a nonexpansive map are nonexpansive.

Two metrics ϱ, r on M are said to be equivalent (uniformly equivalent) if there are two constants a > 0, b > 0 such that for all x, y ∈ M

ar(x, y) ≤ ϱ (x, y) ≤ br (x, y) .

If ϱ and r are equivalent then any ϱ-lipschitzian mapping is r-lipschitzian. Moreover for the relevant Lipschitz constants k_{ϱ} (T) and k_{r} (T) we have

\frac{a}{b}k_{r} (T) ≤ k_{ϱ} (T) ≤ \frac{b}{a}k_{r} (T) .

It follows that if T is r-nonexpansive, K_{r} (T) ≤ 1 then

sup \{k_{ϱ} (T^{n}) : n = 0, 1, 2, ...\} ≤ \frac{b}{a} < +∞.

This leads to the following definition.

Definition 2.3. A mapping T : M → M is uniformly lipschitzian if there exists a constant k ≥ 0 such that for all x, y ∈ M and n = 0, 1, 2, ⋯,

ϱ (T^{n}x, T^{n}y) ≤ kϱ (x, y) .

In other words for uniformly lipschitzian mappings we have k_{ϱ} (T^{n}) ≤ k for n = 0, 1, 2, · · · . If k is given we say also that T is uniformly k-lipschitzian. Any uniformly lipschitzian mapping generates an equivalent metric r given by

r (x, y) = sup{ϱ (T^{n}x, T^{n}y) : n = 0, 1, 2, ...},

with respect to which T is nonexpansive.

Thus uniformly lipschitzian mappings on a metric space (M, ρ) are completely characterized as those which are nonexpansive with respect at least one metric equivalent to ϱ.

In particular all periodic lipschitzian mappings are uniformly lipschitzian. Also if K is a lipschitzian retract of M and R : M → K is a lipschitzian retraction satisfying k (R) = k then any nonexpansive mapping T : K → K admits a natural uniformly lipschitzian extension T̃ : M → M, T̃ = T ∘R with k \binom{}{T^{n}} = k (T^{n} ∘ R) ≤ k.

To state our first fixed point results for uniformly lipschitzian mappings we recall some basic facts. The modulus of convexity of a Banach space X is the function δ_{X} : [0, 2] → [0, 1] defined by

δ_{X} (ε) = inf \{1 − ∥\frac{x + y}{2} ∥ : ∥x∥ ≤ 1, ∥y∥ ≤ 1, ∥x − y∥ ≥ ε\} .

The characteristic of convexity of X is the number ε_{0} (X) = sup {ε : δ_{X} (ε) = 0} . The space X is said to be uniformly convex if ε_{0} (X) = 0. Uniformly convex spaces as well as all spaces satisfying ε_{0} (X) < 1 have normal structure. Spaces with ε_{0} (X) < 2 are superreflexive. For extensive discussion of the properties of the above notions and related topics see the section by S. Prus in this volume.

The first fixed point result for uniformly lipschitzian mappings was given in [?].

Theorem 2.5. Let C be a nonempty closed, bounded and convex subset of an uniformly convex Banach space X and let T : C → C be a uniformly k−lipschitzian mapping with k satisfying

k (1 − δ_{X} (k)) < 1.

Then T has a fixed point in C.

The above result was obtained by what is now called the asymptotic center technique (see Section ??). This technique can also be extended to the case of spaces with ε_{0} (X) < 1.

The seminal result in this direction, couched in a general metric space setting, was obtained by E. A. Lifschitz in 1975 ([?]).

Let (M, ρ) be a complete metric space. The balls in M are said to be cregular for c ≥ 1 if the following holds: For any k < c there are numbers μ, α ∈ (0, 1) such that for any x, y ∈ M and any r > 0 with ρ (x, y) ≥ (1 − μ)r, the set B (x, (1 + μ) r) ∩ B (y, k (1 + μ) r) is contained in a closed ball of radius αr.

The balls in any metric space are 1-regular, and if the balls are c-regular for some c > 1 then they are d-regular for all d ∈ [1, c] .

The number

κ (M) = sup{c ≥ 1 : the balls in M are c-regular}

is called the Lifschitz characteristic of M. Obviously κ (M) ≥ 1, and balls in M are κ (M)-regular.

For Banach space, the above definition has a simpler and more intuitive formulation: if X is a Banach space and if r (D) denotes the Chebyshev radius of D ⊂ X, then

κ (X) = sup {c ≥ 1 : r (B (0, 1) ∩ B (x, c)) < 1 for all ∥x∥ = 1} .

It is not difficult to verify that κ (H) = \sqrt{2} if H is a Hilbert space and the same holds for any closed convex subset D of H which has dimension at least 2. For other Banach spaces this is not the case. The value of κ (D) may vary with the set D. In [?] Downing and Turret discussed the following coefficient for an arbitrary Banach space:

κ_{0} (X) = inf {κ (D) : D is a nonempty convex, closed and bounded subset of X} .

The condition κ_{0} (X) > 1 implies reflexivity of X. Also κ_{0} (X) ≥ γ, where 1 ≤ γ is the unique solution of the equation

γ (1 − δ_{X} (\frac{1}{γ} )) = 1.

Consequently ε_{0} (X) < 1 implies that κ_{0} (X) > 1.

Theorem 2.6. Let (M, ϱ) be a bounded and complete metric space, and suppose T : M → M is uniformly lipschitzian, with

ϱ (T^{n}x, T^{n}y) ≤ kϱ (x, y) ,

for x, y ∈ M, n = 1, 2, ⋯,where k < κ (M) . Then T has a fixed point in M.

Proof. The case κ (M) = 1 is obvious. So suppose κ (M) > 1. Select x ∈ M and let r (x) be defined as

r (x) = inf {r > 0 : for some y ∈ M, ϱ (x, T^{n}y) ≤ r, n = 1, 2, ...} .

Now let μ be the positive number associated with k in the definition of κ (M)-regular balls. For selected x ∈ M there is an integer m such that ϱ (x, T^{m}x) ≥ (1 − μ) r (x) . Also there is a point y ∈ M such that ϱ (x, T^{n}y) ≤ (1 + μ) r (x) for n = 1, 2, · · ·.

Thus by κ (M)-regularity, the set

D = B (x, (1 + μ) r (x)) ∩ B (T^{m}x, k (1 + μ) r (x))

is contained in a closed ball centered say at z ∈ M, and having radius αr (x) where α < 1. Next observe that for n > m,

ϱ (T^{m}x, T^{n}y) ≤ kϱ (x, T^{n−m}y) ≤ k (1 + μ) r (x) .

This shows that the orbit {T^{n}y : n > m} is contained in D; hence in B (z, αr (x)) . Consequently r (z) ≤ αr (x) . Also, for any u ∈ D,

ϱ(z, x) ≤ ϱ (z, u) + ϱ (u, x) ≤ αr (x) + (1 + μ) r (x) = Ar (x)

where A = α + 1 + μ.

By setting x_{0} = x and z (x_{0}) = z, it is possible to construct a sequence {x_{n}} with x_{n+1} = z (x_{n}) , where z (x_{n}) is obtained via the above procedure. Thus r (x_{n}) ≤ α^{n}r (x_{0}) and ϱ (x_{n+1}, x_{n}) ≤ Ar (x_{n}) , and hence {x_{n}} converges to a fixed point.

The next result is due to E. Cassini and E. Maluta [?]. A Banach space X is said to have a uniform normal structure if for some h ∈ (0, 1) and every convex bounded subset D ⊂ X, the Chebyshev radius r (D) ≤ hdiam(D) . The normal structure coefficient N (X) of the space X is defined as

N (X) = inf \{\frac{diam (D)}{r (D)} : D ⊂ X is closed, bounded and convex\} .

Thus X has uniform normal structure if and only if N (X) > 1. In general the normal structure coefficient is difficult to calculate. However, for example, S. Prus [?] has shown thatN (L^{p}) = min \{2^{\frac{1}{p}} , 2^{\frac{1}{q}} \} , where q = p (p − 1)^{−1} .

Theorem 2.7. Let C be a nonempty closed, bounded and convex subset of an uniformly convex Banach space X with N (X) > 1, and let T : C → C be a uniformly k-lipschitzian mapping with k satisfying

k < N (X)^{\frac{1}{2}} .

Then T has a fixed point in C.

The preceding three theorems are the most widely known results on existence of fixed points for uniformly k-lipschitzian mappings. We can now formulate the general scheme. We say that a metric space M has the fixed point property for uniformly k-lipschitzian mappings, F P P (k) in short, if all uniformly k-lipschitzian mappings T : M → M have fixed points. We now define another metric space characteristic:

γ (M) = sup {k ≥ 1 : M has F P P (k)} .

Also, for a Banach space X, we set:

γ_{0} (X) = inf {γ (D) : D is a convex, closed and bounded subset of X}

Our theorems give some evaluations for γ (M) . First, in general, γ (M) ≥ κ (M) . For any convex closed and bounded subset D of a Banach space X we have the uniform evaluations γ (D) ≥ γ_{0} (X) ≥ κ_{0} (X) ; also γ (D) ≥ γ_{0} (X) ≥ N (X)^{\frac{1}{2}} , and γ (D) ≥ γ where γ satisfies γ (1 − δ_{X} (\frac{1}{γ} )) = 1. This implies that

for spaces with ε_{0} (X) < 1, γ (D) > (1 − δ_{X} (1))^{−1} > 1. Mutual relations between such estimations have been discussed by several authors. Also there have been attempts to evaluate γ_{0} (X) for concrete spaces. For example Teck-Cheong Lim [?] proved that in L^{p} spaces with 2 < p < ∞,

γ_{0} (X) ≥ (1 + \frac{1 + α^{p−1}}{(1 + α)^{p−1}} )^{\frac{1}{p}} ,

where α is a unique solution of the equation (p − 2) x^{p−1} + (p − 1) x^{p−2} = 1 in the interval (0, 1) . For the case 1 < p ≤ 2, R. Smarzewski [?] found the estimate γ_{0} (L^{p}) ≥ \sqrt{p}. Tomas Domínguez Benavídes [?] presented an interesting improved evaluation involving three constants. For any Banach space X

γ_{0} (X) ≥ \frac{1 + \sqrt{1 + 4N (X) (κ_{0} (X) − 1)}}{2}.

The best evaluation known for any set D in a Hilbert space H is γ (D) ≥ κ (D) = κ_{0} (H) = \sqrt{2}. However in this case we know also an evaluation from above. An example was mentioned in the original work of Lifschitz and also independently presented by Jean-Bernard Baillon [?].

Example 1. Let B^{+} be the positive part of the unit ball in the space l^{2}. Define the mapping T : B^{+} → B^{+} as

T x = (cos \frac{π ∥x∥}{2}) e_{1} + \frac{1}{∥x∥} sin \frac{π ∥x∥}{2}P x,

with the natural extension T 0 = e_{1}, where e_{1} = (1, 0, 0, ⋯) , and P x = (0, x_{1}, x_{2}, ⋯) is the shift operator. The mapping T is uniformly \frac{π}{2} - lipschitzian and fixed point free.

In view of the above example \sqrt{2} ≤ γ_{0} (H) ≤ γ (B^{+}) ≤ π/2. The exact value of γ_{0} (H) is unknown.

There is another direction in which the study of uniformly lipschitzian mappings has been aimed. Let X be a Banach space having wc-F P P. There is a question whether this property is stable under renorming. We formalize this as follows.

Let B be a Banach space with the original norm ∥ ∥ . Let N be the family of all norms equivalent to ∥ ∥ on B. For two norms ∥ ∥_{1} , ∥ ∥_{2} set

K (∥ ∥_{1} , ∥ ∥_{2}) = sup_{x≠0} (\begin{matrix}∥x∥ & 2 \\ ∥x∥ & 1 \end{matrix} ) sup_{x≠0} (\begin{matrix}∥x∥ & 1 \\ ∥x∥ & 2 \end{matrix} ) .

The function K is called the Banach-Mazur distance between the equivalent norms. This is not a metric but the family N may be metrized by taking

ϱ (∥ ∥_{1} , ∥ ∥_{2}) = ln K (∥ ∥_{1} , ∥ ∥_{2}) .

Suppose now D ⊂ X is convex, closed and bounded and T : D → D is nonexpansive with respect a norm | | ∈ N . Then obviously T is K (∥ ∥ , | |)-uniformly lipschitzian with respect to the original norm ∥ ∥ . However | |-nonexpansivness im-

plies more. We have inf_{x∈D} ∥x − T x∥ = 0, and moreover the mapping F = \frac{I + T}{2} is asymptotically regular (see subsection ??).

Suppose now that the space (X, ∥ ∥) has wc-F P P. Define the new characteristic

γ_{N} (X) = sup {k : if K (∥ ∥ , | |) < k then (X, | |) has wc − F P P } .

Obviously 1 ≤ γ_{0} (X) ≤ γ_{N} (X) . We refer to γ_{N} (X) as the stability constant. Apart from Schur spaces (spaces for which weak and strong compactness coincide), where γ_{N} (X) = ∞, an exact value of the stability constant is not known for any space. A crucial question for fixed theory for nonexpansive mappings whether the assertion that all superreflexive spaces have wc-F P P is equivalent to the assertion that γ_{N} (X) = ∞ for all uniformly convex spaces.

A considerable effort has been devoted to finding estimations of γ_{N} in particular spaces, especially the l^{p} spaces, 1 < p < ∞. For example Benavides [?] first proved

that γ_{N} (l^{p}) ≥ (1 + 2^{\frac{q}{p}} )^{\frac{1}{q}} where q = p (p − 1)^{−1} . This gives for Hilbert space

γ_{N} (H) ≥ \sqrt{3}. This result has been improved by A. Jiménez Melado and E. Lorens-Fuster [?] who showed that γ_{N} (l^{p}) ≥ \sqrt{C} where C is the smallest solution of

C (C − 1) = (C^{\frac{q}{p}} + (2C − 2)^{\frac{q}{p}} )^{\frac{p}{q}}

which is greater than 1. For Hilbert space, p = 2, which gives γ_{N} (H) ≥ \sqrt{2 + \sqrt{2}} ≈ 1.85. Finally, with the use of nonstandard (ultrafilter) methods, Pei-Kee Lin [?]

established the estimate γ_{N} (H) ≥ \sqrt{(5 + \sqrt{13}) /2} ≈ 2.07.

There are various results concerning other coefficients and their applications to the fixed point theory for uniformly lipschitzian mappings scattered in the literature. Interested readers should consult the section by S. Prus in this volume, as well as the book by J.M. Ayerbe Toledano, T. Domínguez Benavides and G. Lopez Acedo [?]. Also we call attention the expository article by N.M. Gulevich [?].

We conclude this section with the following observation. Given a Banach space X. The family of all spaces obtained by renorming X can be identified with the space N of norms on X. Coefficients such as ε_{0}, κ_{0}, N can be considered as functions of the norm in the space N: ε_{0} (∥ ∥) , κ_{0} (∥ ∥) , N (∥ ∥) . It is not difficult to observe that they are continuous. Suppose X is superreflexive. Then there exists a uniformly convex norm | | ∈ N on X. It is a technicality to prove that for any other norm ∥ ∥ ∈ N and for any λ > 0 the norm ∥ ∥_{λ} = ∥ ∥ + λ | | is uniformly convex. Thus uniformly convex norms form a dense set in N. In view of the continuous dependence of ε_{0} on the norms in N , the family of all norms satisfying ε_{0} (∥ ∥) < 1 is open and dense in N.

Similarly, the sets of norms satisfying κ_{0} (∥ ∥) > 1 and N (∥ ∥) > 1 are open and dense in N .

The above implies that at least in superreflexive spaces the family of equivalent norms for which X has F P P or more general F P P (k) for some k > 1 contains an open and dense subset. As mentioned before it is not known if F P P is stable under renorming, but at least the norms failing F P P are in a topological sense rare.

2.5. Subclasses of nonexpansive mappings. Various subclasses of nonexpansive mappings arise in natural ways. In our standard setting, where D ⊂ X is a convex closed san bounded set the family F of nonexpansive mappings T : D → D can be viewed as convex closed and bounded subset of the space C [D, X] of all continuous mappings of D into X furnished with the natural uniform norm

∥T ∥_{C[D,X]} = sup {∥T x∥_{X} : x ∈ D} .

The subfamily F_{0} ⊂ F of all contractions is dense in F in the topology generated by this norm. Obviously, in view of Banach Contraction Principle, we can say that D has the fixed point property with respect to F_{0} but may fail the F P P with respect to the whole family F. There are several other interesting subclasses of F.

The isometries are mappings T : D → D satisfying

∥T x − T y∥ = ∥x − y∥ .

The notion of normal structure, which is so useful in the fixed point theory for nonexpansive mappings, was actually introduced by M. S. Brodskii and D.P. Milman in 1948 [?] Milman to study the fixed points of isometries. They proved that the normal structure plus weak compactness of D implies existence of a common fixed point of all isometries of D. A very deep result for isometries has been obtained by B. Maurey [?].

Theorem 2.8. Let D be a weakly compact convex subset of a superreflexive Banach space. Then D has the fixed point property relative to the class of isometries.

Another interesting class consists of so called contractive mappings. These are mappings satisfying

∥T x − T y∥ < ∥x − y∥

for all x, y ∈ D with x ≠ y. The above condition implies two things. A contractive mapping T has at most one fixed point, and if T is fixed point free and D is weakly compact then T has exactly one minimal closed convex invariant set (see Section ??). It is easy to see that D has the fixed point property for contractive mappings if D is weakly compact and such that for any closed convex K ⊂ D, there is a point z ∈ D (not necessarily z ∈ K) such that

sup {∥z − x∥ : x ∈ K} < diam (K) .

Observe that this condition is a modification of normal structure.

In general it is not known if there are convex sets having fixed point property for contractive mappings yet failing F P P for the whole class F.

In 1979 R. Bruck explicitly introduced the following notion. (The concept apparently originates with J. B. Baillon’s 1978 Paris VI Thesis.) Let Γ denote the set of strictly increasing convex functions γ : ℝ^{+}→ ℝ^{+} with γ (0) = 0. A mapping T : D → D is said to be of type Γ if there exists γ ∈ Γ such that

γ (∥cT x + (1 − c) T y − T (cx + (1 − c) y∥) ≤ ∥x − y∥ − ∥T x − T y∥ .

Three facts about such mappings are easy to observe. Mappings of type Γ are nonexpansive, affine nonexpansive mappings are of type Γ and the fixed point sets of mappings of type Γ are convex. The main result of Bruck [?] was the observation that in uniformly convex spaces any nonexpansive mapping T : D → D is of type Γ.

The connection with nonexpansive mappings in uniformly convex spaces is closer than might at first appear. Indeed, Khamsi [?] has shown that for a Banach space X the following are equivalent: (a) X is uniformly convex. (b) There exists γ ∈ Γ such that every nonexpansive mapping T : K → K with diam (K) = 1 is a mapping of type Γ for γ. So in these spaces the class of mappings of type Γ coincides with the whole class F. In less regular spaces there are mappings with nonconvex sets of fixed points (see Section ??), and this shows that there are nonexpansive mappings which are not of type Γ for any γ. The class of mappings of type Γ has proved to be very useful in connection with weak approximation of fixed points (see Section ??).

The next class was also introduced by R. Bruck [?]. A mapping T : D → D is said to be firmly nonexpansive if for any x, y ∈ D the function

Φ_{x,y} (t) = ∥(1 − t) (x − y) + t (T x − T y)∥ , t ∈ [0, 1]

is nonincreasing on [0, 1] . Obviously any firmly nonexpansive mapping is nonexpansive. The converse is not true. For example if D is a subset of Hilbert space H, then T : D → D is firmly nonexpansive if and only if it is of the form T = \frac{1}{2} (I + G) where G : D → H is nonexpansive. However there is an interesting observation. For any x ∈ D and α ∈ [0, 1) consider the equation

z = (1 − α) x + αT z.

Since the right hand side of this equation is a contraction with respect to z, it has exactly one solution, say z_{α}. Since this solution depends of x and α we can define a family of mappings F_{α} : D → D, α ∈ [0, 1), by putting F_{α}x = z_{α}. It is minor technicality to prove that all mappings F_{α} are firmly nonexpansive. Moreover for any α ∈ [0, 1) the fixed point set F ixF_{α} of F_{α} coincides with F ixT. This shows that the fixed point property for firmly nonexpansive mappings coincides with F P P for the whole class F. Finally boundedness of D implies that

lim_{α→1} ∥F_{α}x − T F_{α}x∥ = 0.

This fact has been used to study the strong approximation techniques (see Section??).

Finally we turn our attention to the class of rotative mappings. If T : D → D is nonexpansive then for any n = 1, 2, ⋯ we have

∥x − T^{n}x∥ ≤ n ∥x − T x∥ .

The mapping T is said to be (n, a)-rotative with a < n if for all x ∈ D we have

∥x − T^{n}x∥ ≤ a ∥x − T x∥ .

T is rotative if it is (n, a)- rotative for some n, with a < n. All contractions are rotative, there are also rotative isometries. There are also nonexpansive mappings which are not (n, a)- rotative for any pair (n, a) .

In contrast to the general case, the fixed point property for rotative nonexpansive mappings does not depend on the geometry of the set D. Any closed convex set D, regardless of whether it is bounded or not, has this property.

For a more elaborate discussion of this topic, see the section by W. Kaczor and M. Koter elsewhere in this volume.

2.6. The Approximate Fixed Point Property. A convex subset K of a Banach space is said to have the approximate fixed point property (a.f.p.p.) if every nonexpansive T : K → K satisfies

inf {∥x − T (x)∥ : x ∈ K} = 0.

As we have seen, bounded convex sets always have this property. The first observation that some unbounded sets in Hilbert space have this property was made by K. Goebel and T. Kuczumow. They proved in [?] that if K ⊂ ℓ_{2} is a block, i.e., a set of the form K = {x ∈ ℓ_{2} : |⟨x, e_{i}⟩| ≤ M_{i}} where e_{i} is an orthonormal set in ℓ_{2}, then K has the a.f.p.p. This result was extended by W. Ray in [?] to include all linearly bounded subsets of ℓ_{p}, 1 < p < ∞. (A subset K of a normed space is linearly bounded if K has bounded intersection with all lines in X.) Ray went on to prove in [?] that a closed convex subset K of a real Hilbert space has the fixed point property for nonexpansive mappings if and only if it is bounded, from which it follows that K has the a.f.p.p if and only if it is linearly bounded. Subsequently in [?], S. Reich proved that every linearly bounded closed convex subset of a reflexive Banach space has the a.f.p.p. Finally, in order to characterize those closed convex sets which have the a.f.p.p. in general Banach spaces, I. Shafrir [?] introduced the concept of a directionally bounded set. A directional curve in a Banach space X is a curve γ : [0, ∞) → X for which there exists b > 0 such that for each t ≥ s ≥ 0,

t − s − b ≤ ∥γ (s) − γ (t)∥ ≤ t − s.

A convex subset K of X is said to be directionally bounded if it contains no directional curves. Note that a line is a directional curve with b = 0. Thus directionally bounded sets are always linearly bounded. Shafrir proved two noteworthy things in [?]. (1) A convex subset of a Banach space has the a.f.p.p if and only if it is directionally bounded. (2) For a Banach space X the following are equivalent: (i) X is reflexive. (ii) Every closed convex linearly bounded subset of X is directionally bounded. These facts may be summarized as follows.

Theorem 2.9. For a Banach space X the following are equivalent.

(i) X is reflexive. (ii) A closed convex subset K of X has the a.f.p.p if and only if it is linearly bounded.

3. PROPERTIES OF THE FIXED POINT SET

The investigations of the properties of the sets of fixed points for nonexpansive mappings have their origin in the following observations.

Let K be a convex closed subset of a Banach space X and let T : K → K be nonexpansive. If the norm of X is strictly convex, the fixed point set F (T) is closed and convex. If X = H is a Hilbert space then any nonempty closed convex subset K_{0} of K is the range of a nonexpansive retraction r : K → K_{0}. The nearest point projection is a standard example of such retraction.

In the general case the above is no longer true. There may exist closed convex subsets of K which are not of the form F (T) for any nonexpansive F (see the example in Section 4.4). Also the fixed point sets and nonexpansive retracts do not need to be convex.

The investigations of the properties of fixed point sets concentrated mostly on the question under which condition the set F (T) is necessarily the range of a nonexpansive retraction.

3.1. Nonexpansive Retracts. There are three principal theorems about the structure of the fixed point sets of nonexpansive mappings. The first is a classical Banach space result due to R. E. Bruck [?], and the remaining two are metric space results due, respectively, to J. B. Baillon [?] (for hyperconvex spaces, discussed in more detail elsewhere in this work) and M. A. Khamsi [?] (for arbitrary metric spaces, treated in Section ?? below).

We first take up Bruck’s result, beginning with the separable case. It is perhaps surprising that in any separable space the following simple assumption is enough to assure the desired result.

Definition 3.1. A bounded closed convex subset K of a Banach space is said to have the hereditary fixed point property (HFPP) for nonexpansive mappings if every nonexpansive mapping f : K → K has a fixed point in every nonempty bounded closed convex f-invariant subset of K.

If K is a bounded closed convex subset of a Banach space which has FPP the clearly K has (HFPP). Thus by what we have already seen any bounded closed convex subset of a reflexive Banach space which has normal structure has (HFPP) as do any weakly compact convex subsets of orthogonally convex and uniformly noncreasy spaces.

Theorem 3.1. Suppose K is a bounded closed convex subset of a separable Banach space X, suppose K has (HFPP), and let T : K → K be nonexpansive. then the fixed point set F (T) is a (nonempty) nonexpansive retract of K.

This result is an easy consequence of the following rather deep fact, which is also proved in [?].

Theorem 3.2. Let M be a separable complete metric space and let S be a semigroup of nonexpansive self-mappings of M. Then there exists in \overline{S} a retraction r of M onto F (S) if and only if one of the following two (equivalent) conditions holds.

(i) Each nonempty closed S-invariant subset of M contains a fixed point of S. (ii) Whenever x ∈ M then clS (x) ∩ F (S) ≠ ∅.

Proof of Theorem ??. Let K^{K} denote the family of all mappings of K → K and put

S = {s ∈ K^{K} : s is nonexpansive and F (s) ⊇ F (T)}.

Obviously S is a semigroup on K and F (S) ⊇ F (T). Since T ∈ S we conclude that F (S) = F (T). For each x ∈ K the set S(x) = {s(x) : s ∈ S} is clearly nonempty, convex, and T -invariant. Since T is continuous, cl S(x) is a nonempty bounded closed convex T -invariant subset of K, so by (HFPP) cl S(x) ∩ F (T) ≠ ∅. This shows that S satisfies (ii). The conclusion now follows from Theorem ??.

The following is a direct consequence of the preceding theorems.

Corollary 3.1. Suppose K is a bounded closed convex subset of a separable Banach space, suppose that K has (HFPP), and let 𝔉 be a finite family of commuting nonexpansive mappings of K → K. then the common fixed point set F (𝔉) of 𝔉 is a (nonempty) nonexpansive retract of K.

The above corollary actually extends to arbitrary commutative families 𝔉. Rather than pursue this further, however, we take up the question of what can be said in the nonseparable case. The central observation is that the separability assumption on the space can be replaced with reflexivity or, more generally, with the assumption that the domain is weakly compact. The basic ideas here are also due to Bruck [?].

Theorem 3.3. Let K be a nonempty weakly compact convex subset of a Banach space X and suppose 𝔗 is a family of nonexpansive mappings of K → K with a nonempty common fixed point set A. Suppose further that

(3.1) A intersects every nonempty 𝔗-invariant closed convex subset of K.

Then A is a nonexpansive retract of K.

Proof. As before let K^{K} denote the family of all mappings of K → K and let

N = {f ∈ K^{K} : f is nonexpansive and F (f) ⊇ A},

where F (f) denotes the fixed point set of f. Obviously 𝔗 ⊆ 𝔑 so N ≠ ∅. Notice also that by Tychonoff’s Theorem K^{K} is compact in the topology of weak pointwise convergence, since this is the product topology on the space Π_{x∈K}K induced by the weak topology on K.

Now for f, f′ ∈ N say f ∼ f′ if ∥p − f(x)∥ = ∥p − f′(x)∥ for each p ∈ A and x ∈ K, and for f ∈ N let

[f] = {f′ ∈ N : f′ ∼ f}.

Let [N] = {[f] : f ∈ N} and introduce the partial order ⪯ on [N] by saying [f] ⪯ [g] if and only if ∥p − f(x)∥ ≤ ∥p − g(x)∥ for each p ∈ A and x ∈ K. To see that ([N], ⪯) has a minimal element, let {[f_{α}], α ∈ I} be a descending chain in ([N], ⪯). Since K^{K} is compact in the topology of weak pointwise convergence the net {f_{α}} has a subnet {f_{α_{ξ}}} which converges in this topology to some f ∈ K^{K}. Thus for each x ∈ K,

weak − lim_{ξ} f_{α_{ξ}}(x) = f(x).

Now let x ∈ K and p ∈ A. Since {[f_{α}], α ∈ I} is descending in ⪯ it is the case that lim_{ξ} ∥p − f_{α_{ξ}}(x)∥ exists, and since norm-closed balls in X are weakly closed, if ϵ > 0 it must be the case that ∥p − f(x)∥ ≤ r + ϵ for any ϵ > 0; hence

∥p − f(x)∥ ≤ lim_{ξ} ∥p − f_{α_{ξ}}(x)∥ = lim_{α} ∥p − f_{α}(x)∥ .

This proves that [f] ⪯ [f_{α}] for each α ∈ I. Also, for x, y ∈ K, {f_{α_{ξ}}(x) − f_{α_{ξ}}(y)} converges weakly to f(x)−f(y) (algebraic operations are always weakly continuous) so

∥f(x) − f(y)∥ ≤ lim_{ξ} ∥f_{α_{ξ}}(x) − f_{α_{ξ}}(y)∥ ≤ ∥x − y∥ .

Finally if x ∈ A then f_{α}(x) = x for each α ∈ I so it follows that f(x) = x. Thus F (f) ⊇ A and this proves that {[f_{α}], α ∈ I} is bounded below by [f] in ([N], ⪯). It is now possible to invoke Zorn’s lemma to conclude that ([N], ⪯) contains a minimal element [r].

Now suppose there exists a point z ∈ K such that r(z) ∉ A. Since ∥p − r ∘ r(x)∥ ≤ ∥p − r(x)∥ for each p ∈ A and x ∈ K, [r ∘ r] ⪯ [r]. But since [r] is minimal, [r ∘ r] = [r]. In particular, if z_{0} = r(z) then for all p ∈ A,

∥p − r(z_{0})∥ = ∥p − z_{0}∥ > 0.

Let

C = {f ∘ r(z_{0}) : f ∈ N}.

Since f, g ∈ N implies λf +(1−λ)g ∈ N for any λ ∈ [0, 1], C is convex. To see that C is weakly compact it suffices to show that C is closed. Let {x_{α}} be a net in C for which lim_{α} x_{α} = x. Then for each α there exists f_{α} ∈ N such that x_{α} = f_{α} ∘ r(z_{0}). But since K^{K} is compact in the topology of weak pointwise convergence {f_{α}} has a subnet {f_{α_{ξ}}}which converges to some f ∈ K^{K}. Arguing as before, f is nonexpansive and F (f) ⊇ A. Hence f ∈ N, so f ∘ r(z_{0}) ∈ C. But

x = weak- lim_{ξ} f_{α_{ξ}} ∘ r(z_{0}) = f ∘ r(z_{0}).

This proves that x ∈ C; hence C is closed.

Let f ∈ N. Then F (f) ⊇ A so for each p ∈ A,

∥p − f ∘ r(z_{0})∥ = ∥f(p) − f ∘ r(z_{0})∥ ≤ ∥p − r(z_{0})∥ .

Thus [f∘r] ⪯ [r]. But since [r] is minimal this implies ∥p − f ∘ r(z_{0})∥ = ∥p − r(z_{0})∥ > 0 for each p ∈ A. Therefore A ∩ C = ∅. But as noted at the outset 𝔗 ⊆ 𝔑, so s : C → C for each s ∈ 𝔗. Thus A ∩ C ≠ ∅ by our basic hypothesis. This is a contradiction, so we conclude that r(z) ∈ A for each z ∈ K.

The following is an obvious special case of the preceding theorem.

Corollary 3.2. Suppose K is a weakly compact convex subset of a Banach space, suppose K has (HFPP), and let T : K → K be nonexpansive. Then the fixed point set F (T) of T is a nonempty nonexpansive retract of K.

REMARK. In connection with the foregoing, T. C. Lim [?] proved that X is reflexive and has normal structure, then every left reversible semigroup of nonexpansive self-mappings of a bounded closed convex subset K of X has a common fixed point.

It is interesting, and sometimes useful, to know that the fixed point set of a nonexpansive mapping is a nonexpansive retract of its domain. The following simple facts serve to illustrate this.

Theorem 3.4. Suppose M is a metric space which has the property that the fixed point set of every nonexpansive mapping of M into M is a nonempty nonexpansive retract of M, and suppose T and G are commuting nonexpansive mappings of M into M. Then F (T) ∩ F (G) ≠ ∅.

Proof. First observe that if x ∈ F (T) then T ∘ G(x) = G ∘ T (x) = G(x). This proves G : F (T) → F (T). By assumption there is a nonexpansive retraction r of M onto the nonempty set F (T). Thus G ∘ r : M → F (T) is nonexpansive and by assumption has a nonempty fixed point set. However, since r is a retraction onto F (T), it must be the case that F (G ∘ r) = F (G) ∩ F (T). This proves that F (T)∩F (G) ≠ ∅ (and at the same time that F (T)∩F (G) is a nonexpansive retract of M).

A mapping T : M → M is said to be eventually nonexpansive if there exists an integer N such that for each n ≥ N, T^{n} is nonexpansive.

Theorem 3.5. Suppose M is a metric space which has the property that the fixed point set of every nonexpansive mapping of M into M is a nonempty nonexpansive retract of M, and suppose T : M → M is eventually nonexpansive. Then F (T) ≠ ∅.

Proof. For N sufficiently large the mappings T^{N} and T^{N+1} are commuting nonexpansive mappings of M into M. By the previous theorem F (T^{N})∩F (T^{N+1}) ≠ ∅. But it is trivial to check that

F (T) = F (T^{N}) ∩ F (T^{N+1}).

The above results are not really about nonexpansive mappings since they only require that there is a retraction from the domain of the mapping onto its fixed point set which can be drawn from a class of mappings for which the original domain has the fixed point property. It just turns out that the class of nonexpansive mappings is amenable to this approach.

3.2. General Case. The above results give rise to the opposite question. How “irregular” F (K) can be? Also there is a question concerning the regularity of “approximate” fixed point sets. Let again K be a bounded closed and convex, and for nonexpansive T : K → K and ε > 0, let

F_{ε} (T) = {x ∈ K : ∥x − T x∥ ≤ ε} .

R.Bruck [?] has shown

Theorem 3.6. For any K and T as above the set F_{ε} (T) is pathwise connected.

This may suggests that some kind of regularity should be inherited by F (T) . However it is not the case.For given Banach space X consider the space X × c_{0} furnished with the max norm

∥(x, y)∥ = ∥(x, (y_{1}, y_{2}, ...))∥ = max {∥x∥_{X} , ∥y∥_{c_{0}}} .

Let D ⊂ X be a nonempty closed set. Then the mapping T : X × c_{0} → X × c_{0} defined by

T (x, y) = T (x, (y_{1}, y_{2}, ...)) = (x, (dist (x, D) , y_{1}, y_{2}, ...))

is nonexpansive (even isometric) and we have F (T) = D × {0} . This proves

Theorem 3.7. Any nonempty closed subset of X is isometric to a fixed point set of an isometry of X × c_{0} into itself

Consequently, since any complete, separable metric space (Polish space) is isometric to a closed subset of C [0, 1] , we observe that any such space is isometric to a fixed point set of an isometry of C [0, 1] × c_{0}.The above remarks come from [?]. There it is also shown that any nonempty compact subset fo c_{0} is the fixed point set for certain nonexpansive T : c_{0} → c_{0}. Thus in general case the fixed point sets can be “bizzare”. For example. totally disconnected.

Finally, the regularity properties has been studied not only for fixed point sets but also for families of minimal invariant sets. Let us begin with the already mentioned Alspach’s example.

Example. Let X = L_{1}[0, 1] and let C be the order segment given by

C = {f ∈ L_{1} : 0 ≤ f ≤ 1}.

The baker transformation

(T f) (t) = \{ \begin{matrix}min \{2f (2t) , 1\} & if 0 ≤ t ≤ \frac{1}{2} \\ max \{2f (2t − 1) − 1, 0\} & if \frac{1}{2} < t ≤ 1 \end{matrix}

isometrically transforms C into itself. Only the two constant functions, 0 and 1, are fixed points of T. The whole set C = ⋃_{a∈[0,1]} C_{a} where

C_{a} = {f ∈ C : ∫_{0}^{1} f = a}

are T −invariant convex, closed slices of C by parallel hyperplanes. Since C, is an order segment and therefore weakly compact, so are all the slices C_{a}. In view of these observations, each of the sets C_{a}, for 0 < a < 1, contains at least one nontrivial, that is of strictly positive diameter, minimal invariant subset. Further, since each C_{a} is T −invariant, these are the only minimal invariant sets for T. It is also known that except for a = 0 or 1 each of the C_{a} are not themselves minimal invariant and that the closed convex hull of an orbit need not be minimal invariant.

Now let K be a convex weakly compact subset of a Banach space X. For any nonexpansive mapping T : K → K there exists at least one minimal convex T - invariant subset K_{0} of K. Alspach’s example shows that the family M (T) of such sets may consists of more than one set. It was shown in [?] and [?] that any two sets K_{0}, K_{1} in M (T) are metrically parallel in the sense that for any x_{0} ∈ K_{0} and x_{0} ∈ K_{0}

dist (x_{0}, K_{1}) = dist (x_{1}, K_{0}) = H (K_{0}, K_{1})

where H (K_{0}, K_{1}) is The Hausdorff distance between sets. This has several consequences. First the family M (T) is metrically convex with respect to H meaning that for any t ∈ [0, 1] and anyK_{0}, K_{1} in M (T) there exists at least one set K_{t} ∈ M (T) with H (K_{0}, K_{t}) = t and H (K_{t}, K_{1}) = 1−t. Moreover the sets K_{t} can be selected to form a continuous path isometric to the interval [0, H (K_{0}, K_{1})] . Finally there exists a linear functional f ∈ X^{∗} being constant on each K_{t}. If f (K_{0}) = a and f (K_{1}) = b then f (K_{t}) = (1 − t) a + tb.

This shows that, in case of the weak compactness, the families of minimal invariant sets show more regular behaviour than fixed point sets in general. It also shows that the situation observed in Alspach’s example is typical for general case.

It was also observed that some geometric conditions imposed on the space X may have further influence on M (T) . If, for example, X is strictly convex then any minimal invariant set is a shifted copy of any other, if X is a KK-space then all sets in M (T) are of the same diameter.

The structure of fixed point sets of nonexpansive mappings naturally gives rise to questions about the structure of ‘approximate’ fixed point sets. Let K be a bounded closed and convex, and for T : K → K and ε > 0, let

F_{ε} (T) = {x ∈ K : ∥x − T (x)∥ ≤ ε} .

In [?] it is shown that F_{ε} (T) is always pathwise connected.

It seems that little else is known in the general case. Approximate fixed point sets as well a minimal invariant sets deserve more study.

4. APPROXIMATION

4.1. Asymptotic Regularity. The concept of asymptotic regularity was formally introduced by Browder and Petryshyn in [?].

Definition 4.1. A mapping f of a metric space (M, d) into itself is said to be asymptotically regular if lim_{n→∞} d(f^{n}(x), f^{n+1}(x)) = 0 for each x ∈ M.

In 1976 Ishikawa obtained a surprising result, a special case of which may be stated as follows: Let K be an arbitrary bounded closed convex subset of a Banach space X, T : K → K nonexpansive, and λ ∈ (0, 1). Set T_{λ} = (1 − λ)I + λT. Then for each x ∈ K :

∥T_{λ}^{n}(x) − T_{λ}^{n+1}(x)∥ → 0.

Thus by iterating the ‘averaged’ mapping T_{λ} one can always reach points which are approximately fixed (but on the other hand, these points may not be near fixed points - indeed, it need not be the case that T even have a fixed point).

In 1978, Edelstein and O’Brien [?] proved that {T_{λ}^{n}(x)−T_{λ}^{n+1}(x)} converges to 0 uniformly for x ∈ K, and, in 1983, Goebel and Kirk [?] proved that this convergence is even uniform for T ∈ ℑ, where ℑ denotes the collection of all nonexpansive selfmappings of K. Thus:

Theorem 4.1. Suppose K is a bounded closed convex subset of a Banach space. Then for each ϵ > 0 there exists a positive integer N such that if x ∈ K and T ∈ ℑ, n ≥ N ⇒

∥T_{λ}^{n}(x) − T_{λ}^{n+1}(x)∥ ≤ ϵ.

For further information about this topic we refer to the article by Baillon and Bruck elsewhere in this volume. Here we just touch on something new by showing how Ishikawa’s result can be obtained by embedding the problem in a larger space. The following special case of Theorem ?? suffices to illustrate the method.

Theorem 4.2. Let K be a bounded convex subset of a Banach space X and let T : K → K be nonexpansive. Fix α ∈ (0, 1) and define f_{α} : K → K by setting f_{α}(x) = (1 − α)x + αT (x), (x ∈ K). Then f_{α} is asymptotically regular on K.

The preceding result is quite significant. In particular, since the fixed point sets of f and T coincide, it effectively reduces the study of fixed point properties of nonexpansive mappings in convex sets to the corresponding study for asymptotically regular nonexpansive mappings.

We base the proof given here on the following simple metric space identity.

Lemma 4.1. Let (M, d) be a metric space and α ∈ (0, 1), and suppose {x_{0}, · · ·, x_{N}} and {y_{0}, ⋯, y_{N}} are finite subsets of M satisfying:

(i) d(y_{n}, x_{n}) ≡ r ≥ 0, (0 ≤ n ≤ N);

(ii) d(x_{n}x_{n+1}) = αr and d(x_{n+1}, y_{n}) = (1 − α)r (0 ≤ n ≤ N − 1);

(iii) d(y_{n+1}, y_{n}) ≤ d(x_{n+1}, x_{n}) (0 ≤ n ≤ N − 1); (iv) d(y_{n}, x_{i}) ≤ (1 − α)d(y_{n}, x_{i−1}) + αd(y_{n}, y_{i−1}) (0 ≤ i ≤ n ≤ N).

Then d(y_{N}, x_{0}) = (1 + αN)r.

Lemma ?? can be proved using a straightforward induction argument on N.

Let K be a bounded convex subset of a Banach space X and denote by ℓ_{∞}(X) and c_{0}(X) the Banach spaces obtained by substitution of X into ℓ_{∞}(ℕ) and c_{0}(ℕ). Let X̃ denote the quotient space ℓ_{∞}(X)/c_{0}(X). For x = (x_{1}, x_{2}, ⋯) ∈ ℓ_{∞}(X), let x̃ or [(x_{n})] denote the equivalence class x + c_{0}(X). The quotient norm is given by

\left∥{ \atop x̃}\right∥_{q} = lim sup_{n→∞} ∥x_{n}∥ .

Then

K̃ := ∏_{n=1}^{∞} K/c_{0}(X)

is a bounded closed convex subset of X̃.

Using this embedding, Lemma ?? gives a very quick proof of Theorem ??.

Proof. Define T̃ : D(T̃) ⊂ K̃ → K̃ by setting for x̃ = (x_{1}, x_{2}, ⋯) ∈ K̃

T̃(x̃) = [(T (x_{1}), T (x_{2}), ⋯)]

whenever {T (x_{n})} is bounded, and define f̃ analogously. Now let x ∈ K and set

x̃_{0} = [(f(x), f^{2}(x), ⋯, f^{n}(x), ⋯)].

Since {f^{n}(x)} is bounded, x̃_{0} ∈ K̃. Also, as we have seen in the previous argument, the sequence

{∥f^{n}(x) − T (f^{n}(x)∥}

is nonincreasing. Thus since {f^{n}(x)} is bounded {T (f^{n}(x))} is also bounded. Suppose

∥x̃_{0} − T̃(x̃_{0})∥_{q} = lim sup_{n→∞} ∥f^{n}(x) − T (f^{n}(x))∥ = r ≥ 0.

Then if x̃_{1} := (f^{2}(x), f^{3}(x), ⋯, f^{n+1}(x), ⋯)

∥x̃_{1} − T̃(x̃_{1})∥_{q} = lim sup_{n→∞} ∥f^{n+1}(x) − T (f^{n+1}(x))∥ = r.

Continuing, if x̃_{k} := (f^{k+1}(x), f^{k+2}(x), ⋯, f^{k+n}(x), ⋯), then

∥x̃_{k} − T̃(x̃_{k})∥_{q} = lim sup_{n→∞} ∥f^{n+k}(x) − T (f^{n+k}(x))∥ = r.

Also notice that for each n,

∥f^{n}(x) − T (f^{n}(x))∥ = α ∥f^{n}(x) − f^{n+1}(x)∥ = αr.

Thus if ỹ_{n} := T̃(x̃_{n}) we conclude∥x̃_{n} − ỹ_{n}∥_{q} = r, n = 0, 1, 2, · · ·. Also

x̃_{n+1} = (1 − α)x̃_{n} + αT̃(x̃_{n}), n = 0, 1, · · ·.

These observations give (i) and (ii) of Lemma ??, and (iii) follows from the fact that T̃ is nonexpansive and ỹ_{n+1} = T̃(x̃_{n+1}) with x̃_{n+1} ∈ seg(x̃_{n}, ỹ_{n}). Since (iv) holds in any normed space we conclude that for each n ∈ ℕ,

∥ỹ_{n} − x̃_{0}∥_{q} = (1 + αn)r.

Since K̃ is bounded, this can only happen if r = 0.

The approach just discussed is expanded on further in [?].

4.2. Strong Convergence. It has been known for some time that even in a uniformly convex setting the averaged sequence of the previous section need not actually converge to a fixed point [?]. On the other hand, in 1971 Kaniel [?] described a rather complicated discrete convergence procedure for approximating fixed points of nonexpansive mappings in such spaces. For a refinement of Kaniel’s method, see Molony [?].

The following theorem due to Reich [?] provides what is perhaps the nicest (strong) convergence result.

Theorem 4.3. Let X be a uniformly smooth Banach space, K a bounded closed convex subset of X, and T : K → K nonexpansive. For fixed y ∈ K and t ∈ (0, 1) , let y_{t} denote the unique fixed point of the contraction mapping

T_{t}(·) = (1 − t)y + tT (·).

Then lim_{t→1^{−}} y_{t} exists and is a fixed point of T.

Baillon [?] first proved that T always has a fixed point under the above assumptions. Later Turett [?] proved that uniformly smooth spaces are both superreflexive and have normal structure (and thus have FPP).

Below we give an elegant proof of Theorem ?? due to Reich which makes use of Banach limits.

Recall that a Banach space is uniformly smooth if and only if its norm is uniformly Fréchet differential. Thus if J : X → 2^{X^{∗}} is the normalized duality mapping defined by

J (x) = {j ∈ X^{∗} : ⟨x, j⟩ = ∥x∥^{2} , and ∥j∥ = ∥x∥}

(whose existence is always assured by the Hahn Banach Theorem) then J is single valued and uniformly continuous on any bounded subset of X which is bounded away from 0. Indeed, J (z) = ∥z∥ D_{z} where D_{z} is given by

D_{z} (x) = lim_{h→0} h^{−1} [∥z + hx∥ − ∥z∥]

where the limit exists uniformly for ∥x∥ = ∥z∥ = 1.

Recall also that a Banach limit is a continuous linear functional F defined on ℓ_{∞} which satisfies for all {ξ_{n}} ∈ ℓ_{∞} :

(i) lim inf_{n→∞}ξ_{n} ≤ F ({ξ_{n}}) ≤ lim sup_{n→∞}ξ_{n}.

(ii) F ({ξ_{n+1}}) = F ({ξ_{n}}) .

The existence of Banach limits is assured by the Hahn Banach Theorem.

Proof of Theorem ??. Since T always has at least one fixed point we may assume K is bounded. Fix a sequence {t_{n}} ⊂ (0, 1) with t_{n} → 1 and let y_{n} = y_{t_{n}}. Note that y_{n} − T y_{n} → 0. Now let F be a Banach limit and define f : X → ℝ^{+} by

f (x) = F ({∥y_{n} − x∥^{2}}) .

Since f is a convex and continuous function and since X is reflexive f attains its minimum on K. Let C ⊂ K denote the set of minimizers of f. Then if u ∈ K

f (T (u)) = F ({∥y_{n} − T (u)∥^{2}})
= F ({∥T (y_{n}) − T (u)∥^{2}})
≤ F ({∥y_{n} − u∥^{2}}) = f (u) .

Thus C is closed convex and invariant under T. Another application of Baillon’s theorem yields a fixed point v of T which lies in C. Since T is nonexpansive

⟨(I − T) (u) − (I − T) (v) , j (u − v)⟩ ≥ 0

from which

⟨u − T (u) , J (u − v)⟩ ≥ 0, u ∈ K.

Also

y_{n} − T (y_{n}) = (1 − t_{n}) (y − T (y_{n})) ;

hence

y_{n} − y = t_{n} (T (y_{n}) − y) = (t_{n}/ (1 − t_{n})) (T (y_{n}) − y_{n}) .

Thus if v is a fixed point of T,

⟨y_{n} − y, J (y_{n} − v)⟩ ≤ 0, n ≥ 1. (4.1)

Now we use the fact that J is the subgradient of the function φ (·) = \frac{1}{2} ∥·∥^{2} . Therefore for x, z ∈ X,

2⟨x, J (z)⟩ ≤ ∥x + z∥^{2} − ∥z∥^{2} .

It follows that for all x ∈ X and t > 0,

2t⟨x, J (y_{n} − v − tx)⟩ ≤ ∥y_{n} − v − tx + tx∥^{2} − ∥y_{n} − v − tx∥^{2} .

Let ε > 0 and x ∈ X. Uniform continuity of J implies that for all t > 0 sufficiently small

|⟨x, J (y_{n} − v − tx) − J (y_{n} − v)⟩| < ε

from which

⟨x, J (y_{n} − v)⟩ < ε + ⟨x, J (y_{n} − v − tx)⟩
≤ ε + (1/2t) [∥y_{n} − v∥^{2} − ∥y_{n} − v − tx∥^{2}] .

Therefore, since v is a minimizer of f,

F (⟨x, J (y_{n} − v)⟩) ≤ ε + (1/2t) (f (v) − f (v + tx))
≤ ε.

Since ε > 0 is arbitrary F (⟨x, J (y_{n} − v)⟩) ≤ 0 for all x ∈ X, and in particular

F (⟨y − v, J (y_{n} − v)⟩) ≤ 0.

Combined with (??) this gives F ({∥y_{n} − v∥^{2}}) = 0. This implies that some subsequence of {y_{n}} converges strongly to v, say y_{n_{k}} → v_{1} as k → ∞. Then (??) implies

⟨v_{1} − y, J (v_{1} − v_{2})⟩ ≤ 0 and ⟨v_{2} − y, J (v_{2} − v_{1})⟩ ≤ 0.

Adding gives ∥v_{1} − v_{2}∥^{2} = 0, so v_{1} = v_{2}. It follows that lim_{n→∞} y_{n} exists and, in turn, lim_{t→1^{−}} exists.

4.3. Weak Convergence. There are three classical weak convergence results. The first involves an aspect of Göhde’s original proof which inspired Browder to formulate the so-called demiclosedness principle.

Theorem 4.4. Let K be a closed and convex subset of a uniformly convex Banach space X, and let T : K → X be nonexpansive. Then the mapping (I − T) is demiclosed on K. Specifically, if {u_{j}} is a sequence in K which converges weakly to u and if {(I − T (u_{j}))} converges strongly to w ∈ X, then u ∈ K and (I − T (u)) = w.

Browder’s original proof of the above theorem (see [?]) actually reveals a little more. First it shows that in a uniformly convex space the following is true: if {v_{j}} and {w_{j}} are bounded sequences in a convex set K for which both {v_{j} − T (v_{j})} and {w_{j} − T (w_{j})} converge (strongly) to 0, and if z_{j} for each j is a point of the segment joining z_{j} and w_{j} then {z_{j} − T (z_{j})} also converges to 0. This fact makes crucial use of the uniform convexity of the space. However once this fact is established the remainder of Browder’s argument, which consists of a subtle thinning of the original sequence {u_{j}} , may be carried out in any Banach space using only continuity of T.

It is also trivial to see that the demiclosedness principle also holds in spaces which satisfy the so-called Opial’s condition (i.e., the condition x_{n} → x_{0} weakly ⇒ lim inf_{n→∞} ∥x_{j} − x_{0}∥ < lim inf_{n→∞} ∥x_{j} − x∥ for all x ≠ x_{0}.) Uniformly convex spaces which possess weakly sequentially continuous duality maps satisfy this condition, an observation due to Opial [?]. (For a complete proof see Browder [?]; also see, e.g., [?] p. 108.) In particular Opial’s condition is independent of uniform convexity since the ℓ_{p} spaces satisfy this condition for 1 < p < ∞, while it fails for the L_{p} spaces, p ≠ 2. In fact, spaces satisfying Opial’s condition need not even by isomorphic to uniformly convex spaces ([?]).

There have been a number of attempts to extend the demiclosedness principle. It is known ([?]) that the demiclosedness principle does not hold in the space (ℓ_{2} ⊕ ℝ^{1})_{∞} . Some extensions and related problems are also discussed in [?]. On the positive side, it is shown in [?] that if X is uniformly convex and if Y has the Schur property, then (X ⊕ Y)_{p} satisfies the demiclosedness principle for 0 < p < ∞.

In [?] it is shown that if C is an closed convex subset of an arbitrary Banach space and if T : C → X is norm continuous and α-almost convex for a continuous strictly increasing α : ℝ^{+} → ℝ^{+} with α (0) = 0, then (I − T) is demiclosed at 0. The mapping T is α-almost convex on C if for each x, y ∈ C and λ ∈ [0, 1]

J_{T} (λx + (1 − λ) y) ≤ α (max {J_{T} (x) , J_{T} (y)})

where J_{T} (x) = ∥x − T (x)∥ . Earlier Khamsi [?] had obtained the same result for nonexpansive mappings of ‘convex type’ ; that is, for mappings T : C → X which satisfy: if J_{T} (x_{n}) → 0 and J_{T} (y_{n}) → 0 then J_{T} (\frac{1}{2} (x_{n} + y_{n})) → 0. Clearly mappings of convex type are α-almost convex.

Several people have considered the demiclosedness principle (at 0) for asymptotically nonexpansive mappings. Perhaps the strongest such result is due to Xu [?] who proved that if C is a bounded closed convex subset of a uniformly convex Banach space and if T : C → C is asymptotically nonexpansive (thus there exists a sequence {k_{n}} with k_{n} → 1 for which ∥T^{n} (x) − T^{n} (y)∥ ≤ k_{n} ∥x − y∥ , x, y ∈ C), then (I − T) is demiclosed at 0. However the usefulness of the demiclosedness principle for asymptotically nonexpansive mappings is severely limited by the fact that it is not known in general whether an asymptotically nonexpansive mapping T : C → C always has an approximate fixed point sequence – see [?] for a discussion.

There is another classical weak convergence that makes use of the fact that nonexpansive mappings in uniformly convex spaces are of type Γ (see Section ??).

As we have seen, if K ⊂ X is bounded and if T : K → K is nonexpansive, then ∥f^{n} (x) − f^{n+1} (x)∥ → 0 for each x ∈ K, where f = \frac{1}{2} (I + T) . Therefore if X is uniformly convex, demiclosedness of (I − T) implies that any weak subsequential limit point of {f^{n} (x)} is a fixed point of T. The following result due to Reich [?] shows that, if X is also uniformly smooth, then in fact {f^{n} (x)} always converges weakly. The significance of this result is that it extends a fact known already for the ℓ_{p} spaces, 1 < p < ∞, ([?]) also to the corresponding L_{p} spaces.

Theorem 4.5. Let X be uniformly convex and uniformly smooth, and let K ⊂ X be bounded closed and convex. Suppose {c_{n}} ⊂ [0, 1] satisfies ∑_{n=1}^{∞} c_{n} (1 − c_{n}) = +∞. Then for any x_{1} ∈ K the sequence {x_{n}} defined by

x_{n+1} = (1 − c_{n}) x_{n} + c_{n}T (x_{n}) , n = 1, 2, ⋯

converges weakly to a fixed point of T.

Proof. We only consider the case c_{n} ≡ \frac{1}{2}. Thus let x ∈ K, and with f de-

fined as above let f^{n+1} (x) = \frac{1}{2} (f^{n} (x) + T (f^{n} (x))) , n = 1, 2, · · ·. Minor technical modifications yield the general result.

Let w_{1} and w_{2} be two weak limit points of {x_{n}}, and for each t ∈ [0, 1] , let

a_{n} = a_{n} (t) = ∥w_{1} − w_{2} + t (f^{n} (x) − w_{1})∥ − ∥w_{1} − w_{2}∥

and

b_{n,m} = ∥f^{m} (tf^{n} (x) + (1 − t) w_{1}) − (tf^{n+m} (x) + (1 − t) w_{1})∥ .

Then

∥w_{1} − w_{2} + t (f^{n+m} (x) − w_{1})∥

= ∥w_{1} − w_{2} + f^{m} (tf^{n} (x) + (1 − t) w_{1}) − f^{m} (tf^{n} (x) + (1 − t) w_{1}) + t (f^{n+m} (x) − w_{1})∥

≤ ∥w_{1} − f^{m} (tf^{n} (x) + (1 − t) w_{1}) + t (f^{n+m} (x) − w_{1})∥

+ ∥f^{m} (tf^{n} (x) + (1 − t) w_{1}) − f^{m} (w_{2})∥

≤ ∥f^{m} (tf^{n} (x) + (1 − t) w_{1}) − (tf^{n+m} (x) + (1 − t) w_{1})∥+∥w_{1} − w_{2} + t (f^{n} (x) − w_{1})∥ . Thus

a_{n+m} ≤ b_{n,m} + a_{m}, m, n = 1, 2, · · ·. (4.2)

Let M = ∥x − w_{1}∥ . Since f^{m} is of type Γ there exists γ ∈ Γ such that

γ (b_{n,m}) = γ (∥f^{m} (tf^{n} (x) + (1 − t) w_{1}) − (tf^{m} (f^{n} (x)) + (1 − t) f^{m} (w_{1}))∥)
≤ ∥f^{n} (x) − w_{1}∥ − ∥f^{n+m} (x) − w_{1}∥ .

Since {∥f^{n} (x) − w_{1}∥} is monotone decreasing, lim_{n→∞} γ (b_{n,m}) = 0 uniformly in m. In view of (??),

lim sup_{n→∞}a_{n} ≤ lim inf_{n→∞}a_{n}

so lim_{n→∞} a_{n} (t) = a (t) exists for all t ∈ [0, 1).

Now assume w_{1} ≠ w_{2} and let

d_{n} = ⟨f^{n} (x) − w_{1}, J (w_{1} − w_{2}) / ∥w_{1} − w_{2}∥⟩.

Since X is uniformly smooth, if ε > 0 there exists t = t (ε) such that

|a_{n} (t) /t − d_{n}| < ε.

From this

lim sup_{n→∞}d_{n} ≤ a (t) /t + ε, lim inf_{n→∞}d_{n} ≥ a (t) /t − ε.

Therefore lim_{n→∞} d_{n} exists, and since some subsequence of {f^{n} (x)} converges weakly to w_{1} this limit must be 0. Since the corresponding limit with w_{1} and w_{2} interchanged must also be 0,

∥w_{1} − w_{2}∥^{2} = ⟨w_{1} − w_{2}, J (w_{1} − w_{2})⟩
= ⟨w_{1} − f^{n} (x) , J (w_{1} − w_{2})⟩ + ⟨w_{2} − f^{n} (x) , J (w_{2} − w_{1})⟩;

hence w_{1} = w_{2}.

The third classical weak convergence result is the mean ergodic theorem for nonlinear nonexpansive mappings. This fact asserts that if K is a bounded closed convex nonempty subset of a Banach space X which is both uniformly convex and has a Fréchet differentiable norm, and if T : K → K is nonexpansive, then the Cesàro means of {T^{n} (x)} always converge weakly to a fixed point of T. This fact was first established by Baillon for the L_{p} spaces, 1 < p < ∞, and a modification of his original proof establishes the more general fact. A simple direct proof of this fact can be obtained via the following result of Bruck [?].

Theorem 4.6. Suppose K is weakly compact, X has a Fréchet differentiable norm, and T : K → K is a nonexpansive mapping for which T^{n} is of type γ ∈ Γ for each n. Then for each x ∈ K, the sequence

\{\frac{1}{n} ∑_{i=0}^{n−1} T^{i+k} (x)\}

converges weakly as n → ∞ to a fixed point of T.

4.4. Asymptotic center technique. The sequences of iterates for nonexpansive mappings as well as asymptotic fixed point sequences may determine or localize the fixed points in other way then being convergent. A useful tool has been introduced in 1972 by M. Edelstein (see [?]).

For x ∈ X and a bounded {x_{n}} ⊂ X define the asymptotic radius of {x_{n}} at x as the number

r (x, {x_{n}}) = lim sup_{n→∞} ∥x − x_{n}∥ .

If {x_{n}} is fixed, r (x, {x_{n}}) is a nonnegative, continuous and convex function of x. Moreover r (x, {x_{n}}) = 0 if and only if x = lim_{n→∞} x_{n}.

For a subset K of X, the number

r_{K} ({x_{n}}) = inf_{x∈K} r (x, {x_{n}})

and the set

A_{K} ({x_{n}}) = {y ∈ K : r (y, {x_{n}}) = r_{K} ({x_{n}})}

are called respectively, the asymptotic radius and asymptotic center of {x_{n}} relative to K. Asymptotic center A_{K} ({x_{n}}) in general case may be empty. The standard assumption that guarantees that all bounded sequences have nonempty asymptotic centers is weak compactness of K. Obviously the convexity of K implies convexity of A_{K} ({x_{n}}) for all bounded sequences {x_{n}} .

Let T : K → K be nonexpansive. For x_{0} ∈ K put x_{n} = T^{n}x_{0} and let K_{1} = A_{K} ({x_{n}}) . In view of the nonexpansivness of T , for any z ∈ K_{1} we have

r (T z, {x_{n}}) = lim sup_{n→∞} ∥T z − T^{n}x_{0}∥ ≤ lim sup_{n→∞} ∥z − T^{n−1}x_{0}∥ = r (z, {x_{n}}) .

implying T z ∈ K_{1}. Thus the asymptotic center relative to K of any sequence of iterates of T is T invariant.

We observe the same effect for any asymptotic fixed point sequence. Indeed if lim_{n→∞} ∥y_{n} − T y_{n}∥ = 0 then for any z ∈ A_{K} ({y_{n}})

r (T z, {y_{n}}) = lim sup_{n→∞} ∥T z − y_{n}∥ ≤ lim sup_{n→∞} ∥T z − T y_{n}∥ + lim_{n→∞} ∥T y_{n} − y_{n}∥ ≤

≤ lim sup_{n→∞} ∥z − y_{n}∥ = r (z, {y_{n}}) .

This implies that A_{K} ({y_{n}}) is T invariant.

Using both facts one can construct a sequence K ⊃ K_{1} ⊃ K_{2} ⊃ K_{3}... of T invariant sets. Each K_{n+1} can be taken to be the asymptotic center relative to K_{n} either of a sequence of iterates or an asymptotic fixed point sequence. Obviously if K is convex and weakly compact so are all K_{i}.

We can continue this procedure using the intersections and transfinite induction. If the procedure terminates on a set consisting of one point, this must be a fixed point of T.

Some geometrical regularity conditions of the space X or the set K itself force the above scheme to stabilize ”quickly”. For example if the space X is uniformly convex, or uniformly convex in every direction, then the asymptotic center of any bounded sequence is a singleton. Thus K_{1} consists of exactly one point, a fixed point of T. Further, for spaces with the characteristic of convexity ε_{0} (X) = sup {ε : δ_{X} (ε) = 0} < 1 we have (see [?])

diamA_{K} ({u_{n}}) ≤ ε_{0} (X) r (K, {u_{n}})

for any sequence {u_{n}} ⊂ K. This implies that lim_{n→∞} diam (K_{n}) = 0 and consequently ∩_{n=1}^{∞}K_{n} is a singleton, a fixed point of T. Also k-uniformly rotund spaces of Sullivan [?] show nice properties. Asymptotic centers of bounded sequences are in this spaces compact in norm topology. So in this case we can stop at K_{1} and then, for example, search for fixed points as norm cluster points of asymptotic fixed point sequences.

Asymptotic center technique has some other applications. It proved to be useful for investigations of fixed points of nonexpansive multivalued mappings. This will be discussed in the next section. We end with an observation connected to Section 3. It was mentioned that in Hilbert space case all convex subsets of K ⊂ H are of the form F (T) for certain nonexpansive T : K → K. This is no longer true in general case even for uniformly convex spaces and even for spaces isomorphic to H.

Consider the space l_{3}^{p} ( the space ℝ^{3} furnished with the norm ∥(x_{1}, x_{2}, x_{3})∥ = (|x_{1}|^{p} + |x_{2}|^{p} + |x_{3}|^{p})^{\frac{1}{p}}) for 1 < p < ∞, and let E denote the triangle \{x = (x_{1}, x_{2}, x_{3}) : x_{i} ≥ 0, i = 1, 2, 3, x_{1} + x_{2} + x_{3} = 1}. The norm is uniformly convex. All spaces l_{3}^{p} are isomorphic to l_{3}^{2}. However for p ≠ 2 the set E is not the nonexpansive retract of l_{3}^{p} nor is of the form of F (T) for certain nonexpansive T : l_{3}^{p} → l_{3}^{p}. Indeed, suppose that E = F (T) . Consider the periodic sequence of unit vectors {e^{1}, e^{2}, e^{3}, e^{1}, e^{2}, e^{3}, ...} . Let z be the unique point of the asymptotic center of this sequence relative to the whole space. It can be calculated that z = (c, c, c) with

c = (¯1 + 2^{\frac{1}{p−1}} )^{−1} . Since our sequence consists of fixed points of T ( thus it is an asymptotic fixed point sequence) we should have z = T z and z ∈ E (T) . However it is not true unless p = 2 and we have a contradiction. The example carries over to the case of T being a nonexpansive self mapping of any closed convex set K containing E in its interior.

5. SET-VALUED NONEXPANSIVE MAPPINGS

The principal result in this direction is due to T. C. Lim [?].

Theorem 5.1. Let X be a uniformly convex Banach space (or, more generally, a reflexive space which is uniformly convex in every direction), let K be a bounded closed convex subset of X, and let T : K → ℭ (K) , where ℭ (K) denotes the collection of all nonempty compact subsets of K. Suppose T is nonexpansive relative to the Hausdorff metric on ℭ (K) . Then there exists a point x ∈ K such that x ∈ T (x) .

The proof of Theorem ?? uses the so-called asymptotic center method which was discussed in the previous section. Here we make use of the fact that if X is uniformly convex in every direction then A_{K} ({x_{n}}) is a singleton for each bounded sequence {x_{n}} in X. Also, such a sequence is said to be regular (relative to K) if r_{K} ({u_{n}}) = r_{K} ({x_{n}}) for each subsequence {u_{n}} of {x_{n}} . The following fact was noticed independently by Goebel [?] and Lim [?]. Since K is fixed throughout, we drop the subscripts.

Lemma 5.1. Every bounded sequence {x_{n}} in X has a regular subsequence.

Proof. For any bounded sequence {u_{n}} let

r̄({u_{n}}) = inf {r ({z_{n}}) : {z_{n}} is a subsequence of {u_{n}}} .

Now let {y_{n}^{1}} = {x_{n}} and, given {y_{n}^{i}} , let {y_{n}^{i+1}} be a subsequence of {y_{n}^{i}} for which

r ({y_{n}^{i+1}}) ≤ r̄({y_{n}^{i}}) + 1/i.

Then if {y_{k}} is a subsequence of the diagonal sequence {y_{n}^{n}}

r (\{y_{n}^{n}\}) − \frac{1}{i} ≤ r (\{y_{n}^{i+1}\}) − \frac{1}{i} ≤ r̄(\{y_{n}^{i}\}) ≤ r̄(\{y_{k}\}) ≤ r (\{y_{k}\}) ≤ r (\{y_{n}^{n}\}) ,

proving r ({y_{k}}) = r ({y_{n}^{n}}) .

Proof of Theorem ??. By choosing fixed points of set-valued contraction mappings which uniformly approximate T and passing to a subsequence, it is possible to obtain a regular sequence {x_{n}} in K for which dist (x_{n}, T (x_{n})) → 0. Let r = r ({x_{n}}) and let {v} = A ({x_{n}}) . For each n select y_{n} ∈ T (x_{n}) such that ∥y_{n} − x_{n}∥ → 0, and select z_{n} ∈ T (v) such that

∥y_{n} − z_{n}∥ ≤ H (T (x_{n}) , T (v)) ≤ ∥x_{n} − v∥ ,

where H denotes the Hausdorff distance on ℭ (K) . Since T (v) is compact some subsequence {z_{n_{k}}} of {z_{n}} converges to an element w ∈ T (v). Therefore

∥x_{n_{k}} − w∥ ≤ ∥x_{n_{k}} − y_{n_{k}}∥ + ∥y_{n_{k}} − z_{n_{k}}∥ + ∥z_{n_{k}} − w∥

and

∥y_{n_{k}} − z_{n_{k}}∥ ≤ ∥x_{n_{k}} − v∥ .

It follows that lim_{k} sup ∥x_{n_{k}} − w∥ ≤ r, proving w ∈ A ({x_{n}}) = {v} . Therefore v = w ∈ T (v) .

Earlier versions of Theorem ?? were obtained by Markin [?] in Hilbert spaces, by Browder [?] for spaces possessing weakly continuous duality mappings. In each these instances the mapping is assumed to have compact convex values. It has since been shown in [?] that under this additional assumption about the values of T Theorem ?? holds in an even wider class of spaces.

Theorem 5.2. Let X be a Banach space and let K be a bounded closed convex subset of X. Suppose K has the property that the asymptotic center of each sequence in K (relative to K) is nonempty and compact. Suppose T : K → ℭℭ (K) is nonexpansive, where ℭℭ (K) denotes the collection of all nonempty compact convex subsets of K endowed with the Hausdorff metric. Then there exists a point x ∈ K such that x ∈ T (x) .

Spaces which satisfy the assumptions of the above theorem include, for example, all the k-uniformly rotund spaces of Sullivan [?]. The initial step in the proof involves showing that every sequence in K has a subsequence with the property that each of its subsequences has the same asymptotic radius and asymptotic center. The proof also has a topological ingredient in that it invokes the Bohnenblust-Karlin extension [?] of a well known fixed point theorem of Kakutani [?] for upper semicontinuous set-valued mappings.

As with the demiclosedness principle, Theorem ?? also holds for spaces satisfying Opial’s condition. This fact is due to E. Lami Dozo [?].

Theorem 5.3. Let K be a weakly compact convex subset of a Banach space X which satisfies Opial’s condition and let T : K → ℭ (K) be nonexpansive. Then there exists x ∈ K such that x ∈ T (x) .

6. ABSTRACT THEORY

6.1. Introduction. The axiomatic approach in modern mathematics which flourished in the early part of the twentieth century has had a lasting impact. The study of nonexpansive mappings in a ‘purely’ metric space context (indeed, in an even wider context) appears well suited to this approach via the use of convexity structures. The advantages accrue from the elegance and generality achieved, and the generality leads in turn to a unification of applications within more concrete settings.

General set-theoretic convexity in the study of fixed point theory of nonexpansive mappings made its first explicit appearance in the work of J-P. Penot [?], although it is implicit in the much earlier studies of hyperconvex metric spaces [?]. The idea involved is to proceed only with those essential properties of a metric structure needed for the development of a well grounded theory. In addition to the usual assumptions involved in the study of abstract convexity structures, it turns out that in this study an additional assumption is indispensable; namely the the closed metric balls of the underlying space must be included in the structure. Indeed, the presence of the closed balls and their intersections in the structure lies at the heart of the approach.

It is only fair to acknowledge, however, that there is one major disadvantage to this more abstract approach. As seen in the previous section, a standard tool in many concrete results is the use of the so-called ‘asymptotic center’ technique. Unfortunately this technique involves assumptions which are not appealing in a more abstract framework.

The convexity structure approach described below has been the subject of a vigorous study in its own right (see, e.g., van de Vel [?]). However one needs to be cautious because the terminology is not always consistent. For example the term ‘normal’ as it is used below is different from its topologically motivated usage in [?].

6.2. Convexity Structures.

6.2.1. Metric convexity structures. If one were to describe the behavior of the collection C of all convex sets in a Banach space X the following three properties stand out:

(A-1) Both X and ∅ are in C.

(A-2) C is closed under intersections, that is, if {K_{α}} is any subcollection of C then ∩_{α}K_{α} ∈ C.

(A-3) If {K_{α}} is any chain (relative to set inclusion) of members of C, then ∪_{α}K_{α} ∈ C.

If only the behavior of closed sets is under consideration it is necessary to drop (A-3), or at the very least modify (A-3) so that the closure of ∪_{α}K_{α} is a member of C. Since this involves the explicit introduction of a topological concept, we drop (A-3).

Definition 6.1. Let X be a set and let C be a family of subsets of X. The pair (X, C) is called a closed convexity structure if

(A-1) Both X and ∅ are in C.

(A-2) C is closed under intersections, that is, if {K_{α}} is any subcollection of C then ∩_{α}K_{α} ∈ C.

If the metric properties of the space X are of primary concern an additional axiom is needed. The correct one seems to be the one given below.

Definition 6.2. Let X be a metric space and let C be a family of subsets of X. The pair (X, C) is called a metric convexity structure if

(A-1) Both X and ∅ are in C.

(A-2) C is closed under intersections, that is, if {K_{α}} is any subcollection of C then ∩_{α}K_{α} ∈ C.

(A-4) C contains the closed balls of X.

In the presence of (A-2), (A-4) leads to another fact. For each x ∈ X, {x} = ∩B(x; 1/n) ∈ C.

Given a metric convexity structure (X, C) there is a subcollection of C that is of special interest. These are the so-called admissible subsets of X. A set D ∈ C is said to be an admissible set if D = cov(D) where

cov(D) := ∩{B : B is a closed ball in X and B ⊇ D.

Notice that if A(X) denotes the family of all admissible subsets of X, then (X, A(X)) is itself a metric convexity structure.

We also denote

co(D) := ∩{F : F ∈ C and F ⊇ D}.

To summarize, D ∈ A(X) iff D = cov(D) and D ∈ C iff D = co(D). In general co (D) ⊆ cov (D) , but in considering only the structure (X, A(X)) there is no need for a distinction.

We note the following fact for later reference.

Proposition 6.1. Let (X, C) be a metric convexity structure and suppose D ∈ C is bounded. Then diam (co (D)) = diam(cov(D)) = diam(D).

Proof. Let d = diam(D). Obviously diam(cov(D)) ≥ diam (co (D)) ≥ d. On the other hand, if x ∈ D and y ∈ cov(D) then since

D ⊆ cov(D) ⊆ B(x; r_{x}(D)),

d(x, y) ≤ r_{x}(D) ≤ d. This implies r_{y}(D) ≤ d and thus

D ⊆ cov(D) ⊆ B(y; d).

Therefore if z ∈ cov(D), d(z, y) ≤ d.

Definition 6.3. Let X be a metric space and let A(X) denote the collection of all admissible subsets of X. Then the pair (X, A(X)) is called an admissible convexity structure.

We now formalize the basic definition of the mappings considered.

Definition 6.4. A mapping f : M → N from a metric space (M, d) into a metric space (N, ρ) is said to be nonexpansive if ρ(f(x), f(y)) ≤ d(x, y) for all x, y ∈ M.

The central problems treated in the ensuing sections are the following. Let (X, C) be a metric convexity structure, suppose D ∈ C is nonempty, and suppose f : D → D is a nonexpansive mapping.

Problem 1. What properties on (X, C) insure that f has at least one fixed point?

Problem 2. What properties on (X, C) insure that fix(f) := {x ∈ D : f(x) = x} ∈ C?

A note of caution. If (X, A(X)) is an admissible convexity structure and if D ∈ A(X), then (D, C) is a metric convexity structure where

C := {D ∩ A : A ∈ A(X)}.

However in general, (D, C) ≠ (D, A(D)). On the other hand, it may well be that (D, C) inherits nice properties of (X, A(X)).

6.2.2. Compact convexity structures. If (X, C) is a convexity structure, the family C is always a subbase for a topology on X, and by Alexander’s subbase theorem ( [?]) X is compact in this topology if (X, C) is compact in the following sense.

Definition 6.5. A convexity structure (X, C) is said to be compact [resp., countably compact] if every family [resp., countable family] of C which has the finite intersection property has nonempty intersection.

Examples of compact metric convexity structures are abundant. Suppose the underlying space X is taken to be a bounded closed convex subset of a Banach space E. For reflexive E, the convexity structure (X, C) where C is the family of all closed convex subsets of X is compact. If E is a conjugate space and if X is a weak^{∗} compact convex subset of E, then (X, A(X)) is an admissible compact convexity structure. The same is true if X is a bounded hyperconvex metric space.

Compact convexity structures (often countable compactness suffices) permits immediate passage to a minimal invariant set in the following sense.

Proposition 6.2. Let (X, C) be a compact convexity structure and let f : X → X. Then there exists D ∈ C such that D is minimal with respect to being nonempty and invariant under f. For such a minimal D it is the case that D = co(f(D)).

Proof. The existence of a minimal such D is a direct consequence of Zorn’s Lemma. The last statement follows from the fact that f(co(f(D))) ⊆ f(D) ⊆ co(f(D)) ⊆ D. Since co(f(D)) ∈ C, D = co(f(D)) by minimality.

6.2.3. Normal convexity structures. For a bounded subset A of a metric space M we denote

diam(A) = sup{d(x, y) : x, y ∈ A};
r_{x}(A) = sup{d(x, y) : y ∈ A} (x ∈ A);
r(A) = inf{r_{x}(A) : x ∈ A}.

Also set

C(A) := {x ∈ A : r_{x}(A) = r(A).

The number r(A) and set C(A) (which may be empty) are called, respectively, the Chebyshev radius and Chebyshev center of A.

In view of Proposition ?? a standard tactic in addressing Problem ?? becomes this. Find conditions on C which insure that any minimal nonempty invariant D ∈ C must be a singleton. The following condition is in some sense canonical.

Definition 6.6. A metric convexity structure (X, C) is said to be normal if r(D) < diam(D) for every D ∈ C with diam(D) > 0.

This brings us to the fundamental theorem for nonexpansive mappings.

Theorem 6.1. Let X be a bounded metric space and suppose (X, C) is a normal compact metric convexity structure. Then every nonexpansive T : X → X has at least one fixed point.

Proof. By Proposition ?? there exists D ∈ C which is minimal with respect to being nonempty and invariant under T. Let r = r(D). By the definition of r each of the sets

C_{n}(D) := ∩\{B(x; r + \frac{1}{n}) : x ∈ D\} ∩ D

is nonempty, and clearly C_{n}(D) ∈ C, n = 1, 2, · · ·. Therefore

C(D) = ∩_{n=1}^{∞}C_{n}(D)

is nonempty and in C. Now let x ∈ C(D). Then D ⊆ B(x; r). Since T is nonexpansive

T (D) ⊆ B(T (x); r).

This implies B(T (x); r) ⊇ co(T (D)) = D. Hence r_{T (x)}(D) ≤ r, from which T (x) ∈ C(D). Therefore T : C(D) → C(D) and by minimality, C(D) = D. But if r > 0, diam(C(D)) ≤ r < diam(D). It follows that r = 0 and this in turn implies that diam(D) = 0.

By strengthening the normality condition it is possible to replace explicit reference to compactness with completeness.

Definition 6.7. A metric convexity structure (X, C) is said to be uniformly normal if there exists c ∈ (0, 1) such that r(D) ≤ c diam(D) for every D ∈ C.

Theorem 6.2. Let X be a bounded complete metric space and suppose (X, C) is a uniformly normal metric convexity structure. Then every nonexpansive T : X → X has at least one fixed point.

The proof of this theorem requires two technical lemmas. The first can be proved by translating its Banach space counterpart (see [?], p. 41) into a metric space framework.

Lemma 6.1. Let X be a bounded metric space and suppose (X, C) is a metric convexity structure. Suppose T : X → X is nonexpansive, and let

𝔉 := {D ∈ C : D ≠ ∅ and T : D → D}.

Then for each D ∈ 𝔉 there exists Ď ∈ 𝔉 such that

diam(Ď) ≤ \frac{1}{2}(diam(D) + r(D)).

Compactness has not really been entirely eliminated in Theorem ?? because of the following fact. This observation is due to Khamsi [?]. Another proof is given by replacing ∥·∥ with d and ‘\overline{conv}’ with ‘co’ in the proof of Theorem 4.4 of [?].

Lemma 6.2. . Let X be a bounded complete metric space and suppose (X, C) is a uniformly normal metric convexity structure. Then (X, C) is countably compact.

In view of Lemma ??, Theorem ?? is a consequence of the following extension of Theorem ??.

Theorem 6.3. Let X be a bounded complete metric space and suppose (X, C) is a normal countably compact metric convexity structure. Then every nonexpansive T : X → X has at least one fixed point.

Proof. Let

𝔉 := {D ∈ C : D ≠ ∅ and T : D → D}

and define δ : 𝔉 → ℝ^{+} by

δ(D) = inf{diam(F) : F ∈ 𝔉 and F ⊆ D}.

Select D_{1} ∈ 𝔉, and with D_{1}, ⋯, D_{n} given, select D_{n+1} ∈ 𝔉 with D_{n+1} ⊆ D_{n} such that

diam(D_{n+1}) ≤ δ(D_{n}) + \frac{1}{n}.

Now let C := ∩_{n=1}^{∞}D_{n}. Then C ∈ C, and by countable compactness C ≠ ∅. Since T : C → C, C ∈ 𝔉. Also for each n ∈ ℕ,

diam(C) − \frac{1}{n} ≤ diam(D_{n+1}) − \frac{1}{n} ≤ δ(D_{n}) ≤ δ(Č) ≤ \frac{1}{2}(diam(C) + r(C)).

Letting n → ∞ gives diam(C) = r(C); hence C is a singleton.

REMARK. Kulesza and Lim [?] have recently made a very interesting observation in connection with Lemma ??; namely:

Theorem 6.4. Let M be a bounded complete metric space and suppose A (M) is countably compact and normal. Then A (M) is compact.

6.3. Khamsi’s Structure Theorem. The following concept is the key to the structure of fixed points sets of nonexpansive mappings in metric spaces M for which A (M) is compact and normal. It also paves the way to the proof that in such a setting commuting families of nonexpansive mappings have a nonempty common fixed point set.

Definition 6.8. A subset A of M is said to be a 1-local retract of M if for each family {B_{i}}_{i∈I} of closed balls centered at points of A and for which ∩_{i∈I}B_{i} ≠ ∅ it is the case that A ∩ (∩_{i∈I}B_{i}) ≠ ∅.

It is easy to check that every nonexpansive retract of M is a 1-local retract of M.

We now describe several fundamental properties of 1-local retracts.

Proposition 6.3. If M is a metric space for which A (M) is compact, then M is complete.

Now we collect several additional properties.

Proposition 6.4. Let M be a metric space and let A be a nonempty subset of M. Then:

(1) cov (A) = ∩ {B (x; r_{x} (A)) : x ∈ M};

(2) r_{x} (A) = r_{x} (cov (A)) for every x ∈ M;

(3) r (cov (A)) ≤ r (A) ;

(4) diam (cov (A)) = diam (A) .

Proof. (1) This is immediate since B (x; r_{x} (A)) is the smallest ball centered at x which contains A.

(2) Since A ⊂ cov (A) it follows from the definition of r_{x} that

r_{x} (A) ≤ r_{x} (cov (A)) .

On the other hand, (1) implies r_{x} (cov (A)) ≤ r_{x} (A) .

(3) This is immediate from the definition of r and (2).

(4) It obviously suffices to show that diam(cov (A)) ≤ diam (A) . Let z ∈ cov (A) . Then z ∈ B (x; r_{x} (A)) for each x ∈ M. In particular, d (x, z) ≤ r_{x} (A) ≤ diam (A) for each x ∈ A; thus A ⊂ B (z; diam (A)) . Hence

cov (A) ⊂ B (z; diam (A)) .

It follows that

diam (cov (A)) ≤ diam (A) .

In the case of 1-local retracts, normal structure is hereditary.

Proposition 6.5. Let M be a metric space and suppose A (M) is compact and normal. Suppose N is a given subset of M which is a 1-local retract of M. Then A (N) is compact and normal.

The key step in proving the above is the following lemma.

Lemma 6.3. Under the assumptions of Proposition ??, r (cov (A)) = r (A) for each A ∈ A (N) .

Proof. Suppose diam (A) > 0. Proposition ?? implies r (cov (A)) ≤ r (A) so we only need to establish the reverse inequality. By assumption A ∈ A (N) so A is of the form

A = N ∩ (∩_{i}B (x_{i}; r_{i}))

for some family {x_{i}} ⊂ N. Also

cov (A) ⊂ ∩_{i}B (x_{i}; r_{i}) .

Choose z ∈ cov (A) and let r = r_{z}. Then

z ∈ S := (∩_{x∈A}B (x; r)) ∩ (∩_{i}B (x_{i}; r_{i})) .

Thus S is nonempty and S ∈ A (M) . Since N is a 1-local retract of M and S is the intersection of balls centered in N, S ∩ N ≠ ∅. Let w ∈ S ∩ N. Then

w ∈ (∩_{i}B (x_{i}; r_{i})) ∩ N,

that is, w ∈ A. On the other hand, w ∈ ∩_{x∈A}B (x; r) , so r_{w} ≤ r. This implies

r (A) ≤ r ≤ r_{z} (cov (A)) .

Since z was an arbitrary element of cov (A) the proof is complete.

Proof of Proposition ??. The definition of a 1-local retract assures that A (N) is compact. To see that A (N) is normal, let A ∈ A (N) and note that by Proposition ?? diam (cov (A)) = diam (A) . By Lemma ??, r (cov (A)) = r (A) . Since A (M) is normal, if diam (A) > 0, r (A) = r (cov (A)) < r (diam (A)) .

It is now possible to prove the following.

Theorem 6.5. Let M be a bounded metric space for which A (M) is compact and normal, and let T : M → M be nonexpansive. Then the fixed point set F (T) of T is a nonempty 1-local retract of M. Moreover, A (F (T)) is compact and normal.

Proof. The fact that F (T) ≠ ∅ follows from previous results. To see that F (T) is a 1-local retract of M let {B_{i}} be a family of closed balls centered at points of F (T) for which S := ∩_{i}B_{i} ≠ ∅. Then since T is nonexpansive, T : S → S. Also, since S ∈ A (M) , A (S) is compact and normal. Therefore T has a fixed point in S; i.e., F (T) ∩ S ≠ ∅. The final assertion follows from Proposition ??.

The above result extends as well to arbitrary commutative families.

Theorem 6.6. Let M be a bounded metric space for which A (M) is compact and normal. Then every family Ψ of commuting nonexpansive mappings of M → M has a nonempty common fixed point set F (Ψ) . Moreover, F (Ψ) is a 1-local retract of M.

If Ψ is finite, say Ψ = {T_{1}, ⋯, T_{n}} the proof is easy. By Theorem ?? A (F (T_{i})) is compact and normal for each i. Since T_{1} and T_{2} commute, T_{2} : F (T_{1}) → F (T_{1}) . Thus F (T_{1}) ∩ F (T_{2}) ≠ ∅. Also F (T_{1}) ∩ F (T_{2}) is a 1-local retract of F (T_{1}) , hence of M. The result for finite Ψ now follows by induction. Full details in the infinite case may be found in [?].

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Maira Curie-Skłodowska University, 20-031 Lublin, Poland

    E-mail address: goebel@golem.umcs.lublin.pl

Department of Mathematics, The University of Iowa, Iowa City, IA 52242-1419 USA

    E-mail address: kirk@math.uiowa.edu