Applications of uniform asymptotic regularity to fixed point theorems

In this paper, we show that there are no nontrivial surjective uniformly asymptotically regular mappings acting on a metric space, and we derive some consequences of this fact. In particular, we prove that a jointly continuous left amenable or left reversible semigroup generated by firmly nonexpansive mappings on a bounded τ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tau}$$\end{document}-compact subset of a Banach space has a common fixed point, and we give a qualitative complement to the Markov–Kakutani theorem.


Introduction
The notion of asymptotic regularity was introduced in 1966 by Browder and Petryshyn [6] in connection with the study of fixed points of nonexpansive mappings. Recall that a mapping T : M → M acting on a metric space (M, d) is asymptotically regular if for all x ∈ M . Krasnoselskii [16] proved that if K is a compact convex subset of a uniformly convex Banach space and if T : K → K is nonexpansive (i.e., ∥T x − T y∥ ≤ ∥x − y∥ for x, y ∈ K), then for any x ∈ K, the sequence {( 1 2 I + 1 2 T ) n x} converges to a fixed point of T . He used the fact that the averaged mapping 1 2 (I + T ) is asymptotically regular. Subsequently, Ishikawa [15] proved that if C is a bounded closed convex subset of a Banach space E and T : C → C is nonexpansive, then the mapping T α = (1 − α)I + αT is asymptotically regular for each α ∈ (0, 1). Independently, Edelstein and O'Brien [9] showed that T α is uniformly asymptotically regular over x ∈ C, and Goebel S. Borzdyński and A. Wiśnicki and Kirk [10] proved that the convergence is uniform with respect to all nonexpansive mappings from C into C. Other examples of asymptotically regular mappings are given by the result of Anzai and Ishikawa [1]-if T is an affine mapping acting on a bounded closed convex subset of a locally convex space X, then T α = (1−α)I +αT is uniformly asymptotically regular.
Another important class of nonlinear operators are firmly nonexpansive mappings, i.e., mappings with the property for all x, y ∈ C and α ∈ (0, 1). Their origin is associated with the study of maximal monotone operators in Hilbert spaces. Reich and Shafrir [22] proved that every firmly nonexpansive mapping T : C → C is asymptotically regular provided C is bounded. For a thorough treatment of firmly nonexpansive mappings we refer the reader to [2].
Quite recently, Bader, Gelander and Monod [3] proved a remarkable fixed point theorem for affine isometries in L-embedded Banach spaces preserving a bounded subset A and showed its several applications. In particular, they simplified the Losert's proof and simultaneously obtained the optimal solution to the "derivation problem" studied since the 1960s. The present paper is partly motivated by the Bader-Gelander-Monod theorem. The paper [8] is also relevant at this point. Our basic observation is Proposition 2.2-there are no nontrivial surjective uniformly asymptotically regular mappings. It appears to be quite useful in fixed point theory of nonexpansive and affine mappings.
The well-known Day generalization of the Markov-Kakutani fixed point theorem asserts that a semigroup S is left amenable if and only if, whenever S acts as an affine continuous mapping on a nonempty compact convex subset C of a Hausdorff locally convex space, S has a common fixed point in C. In 1976, T.-M. Lau [17] posed the problem of characterizing left amenability of a semigroup S in terms of the fixed point property for nonexpansive mappings. Consider the following fixed point property: It is not difficult to show (see, e.g., [19, p. 528]) that property (F * ) implies that S is left amenable. Whether the converse is true is still an open problem though a few partial results are known. Lau and Takahashi [18,Theorem 5.3] proved that the answer is affirmative if C is separable. Recently, the authors of the present paper were able to prove that commutative semigroups have the (F * ) property (see [5,Theorem 3.6]). We show in Section 3 that a jointly continuous left amenable or left reversible semigroup generated by firmly nonexpansive mappings acting on a bounded τ -compact subset of a Vol. 18 (2016) Uniform asymptotic regularity 857 Uniform asymptotic regularity 3 Banach space has a common fixed point. This is a partial result but the techniques developed in this paper may lead to a solution of the original problem too.
In Section 4 we apply our techniques to the case of commutative semigroups of affine mappings and give the qualitative complement to the Markov-Kakutani theorem. In spite of its simplicity, the result seems to be generally new even for the weak topology, see Remark 4.6.

Preliminaries
Let (M, d) be a metric space. In this paper, we focus on uniformly asymptotically regular (UAR) self-mappings on M , i.e., mappings T : M → M which satisfy the condition The following observation is our basis for using uniform asymptotic regularity to generate fixed points.
is the identity mapping.
Proof. Take ε > 0. The assured property gives us a positive integer n such that Take arbitrary y ∈ M . From the surjectivity of T , there exists x ∈ M such that T n x = y. Then the above inequality turns out to be d(y, T y) < ε. Since ε > 0 is arbitrary, we have T y = y and consequently, T = Id.
Proposition 2.2. There are no nontrivial surjective UAR mappings.
Notice that uniformity assumption is necessary as T x = x 2 defined on [0, 1] meets the demands of the above lemma, except it is merely asymptotically regular and obviously not the identity.
Let C be a convex subset of a Banach space X and α ∈ (0, 1). For brevity and symmetry, we denote 1 − α byᾱ. A mapping of the form is called (α-)averaged nonexpansive (resp., affine) provided that S : C → C is nonexpansive (resp., affine). The term "averaged mapping" was coined in [4]. Whenever a τ -topology on C is mentioned it is assumed to be Hausdorff.

Applications to nonexpansive mappings
We start by giving a few statements which result in the uniform asymptotic regularity. The first one was proved by Edelstein and O'Brien [9, Theorem 1] (see also [10,Theorem 2]).
Lemma 3.1. An averaged nonexpansive mapping T : C → C defined on a convex and bounded subset C of a Banach space is UAR.
Recall that in a Hilbert space a mapping T : C → C is firmly nonexpansive if and only if T = 1 2 (I + S) for a nonexpansive mapping S. In general, there is no such relation. However, a counterpart of Lemma 3.1 can be drawn for firmly nonexpansive mappings by combining a method of [11, Theorem 9.4] with [22,Theorem 1].
Proof. Let C be a bounded subset of E and put Also, C ⋆ inherits boundedness from C: If T is firmly nonexpansive, then and, by taking suprema, we conclude that Ψ T is also firmly nonexpansive.
where the last equality follows from boundedness of C ⋆ . Taking S as the identity on C, and unwinding definitions of ∥ · ∥ and Ψ T , we get which is the desired conclusion.
Now from the ingredients above we have instantly the following result.
Vol. 18 (2016) Uniform asymptotic regularity 859 Uniform asymptotic regularity 5 Theorem 3.3. If for a firmly nonexpansive (resp., averaged nonexpansive) mapping T : C → C defined on a subset (resp., convex subset) of a Banach space there exists a nonempty bounded set D ⊂ C such T (D) = D, then D ⊂ Fix T . In particular, Fix T is nonempty.
Proof. It follows from the fact that F is firmly nonexpansive (see [11, p. 122]) and Fix F = Fix T .
Below we use our technique to obtain fixed point theorems for semigroups of uniformly asymptotically regular mappings. But first let us introduce some notions.
Since, due to Lemma 2.1, surjectivity is intertwined into all results of this paper, the following definition seems natural.
There is a fundamental example of such representations.
is jointly continuous, then we say that the semigroup is (τ -)properly represented.
Since it is known that, notably, commuting semigroups are left amenable, we can now conveniently state a generalization of Lemma 3.7.  Proof. It follows from subsurjectivity that there exists D ⊂ M such that T s (D) = D for s ∈ S. Now, Proposition 2.2 yields that every generator of S is the identity on D. It follows that T s x = x for every x ∈ D and s ∈ S.
Combining the above theorem with Lemma 3.2 we obtain the following corollary.
Corollary 3.11. Let S be a subsurjective semigroup on a bounded subset C of a Banach space whose generators consist of firmly nonexpansive mappings. Then S has a common fixed point.

Vol. 18 (2016)
Uniform asymptotic regularity 861 Uniform asymptotic regularity 7 Note that a special case of the above corollary gives a partial answer to the problem of T.-M. Lau [20, Question 1] (as described in the Introduction) for semigroups generated by firmly nonexpansive mappings. Corollary 3.12. Let S = {T s : s ∈ S} be a representation of a left amenable semigroup on w * -compact subset C of a Banach space generated by firmly nonexpansive mappings such the mapping S × C ∋ (s, x) → T s x is jointly continuous (where C is equipped with the weak * topology). Then S has a common fixed point.
Assuming proper representability instead of just subsurjectivity, we get a result stronger than Theorem 3.10.
The famous Markov-Kakutani theorem states that if C is a nonempty compact and convex subset of a (Hausdorff) linear topological space X and {T i } i∈I a commutative family of continuous affine mappings of C into C, then there exists a common fixed point x ∈ C: T i x = x for every i ∈ I. Furthermore, ∩ i∈I Fix T i is compact and convex, and thus is an absolute retract provided C is a metrizable subset of a locally convex space (see, e.g., [12,Theorem 7.7.5]). In particular, then, there exists a (continuous) retraction A natural question arises whether we can also obtain an affine retraction onto ∩ i∈I Fix T i . To the best of our knowledge, only very partial results regarding almost periodic actions are in the literature (see [21]). In this section we prove a general result of this kind. To this end we need the following lemma, see [1,Lemma 1] and [23,Theorem 3]. Then is nonnegative up to m = ⌊ᾱ(n + 1)⌋. Using the identity and having in mind boundary cases, we get by telescoping, Also, since our coefficients are negative above m, we similarly get Since C is convex, it follows that for each x ∈ C, If α = 1 2 and n is even, we have from the Stirling's approximation of the central binomial coefficient, The odd case, n = 2k − 1 gives Vol. 18 (2016) Uniform asymptotic regularity 863 Uniform asymptotic regularity 9 This proves that a n vanishes at infinity, and since C is bounded and a n does not depend on x, we conclude that T is UAR for α = 1 2 (which turns out to be sufficient for our applications). The case α > 1 2 can also be easily proved: define U : C → C as λ-averaged mapping of S with λ = 2α. Notice that U is an affine mapping and T is its 1/2-averaged mapping, so once again T is UAR. As for the case α > 1 2 , we recommend the reader to compare [1, Lemma 1] or [23,Theorem 3].
Notice that in Lemma 3.13 D need not be a convex set. This helps us to prove our final theorem. But first recall the following variant of Bruck's theorem (see [7,Theorem 3]). The following corollary will be sufficient for us. Proof. LetS denote the closure of S in C C in the τ -product topology (i.e., topology of pointwise convergence with respect to the τ -topology). Notice that if S, T ∈S, then there exists nets {S α } α∈A , {T β } β∈B of mappings in S such that Sx = τ -lim α S α x and T x = τ -lim β T β x for x ∈ C. Notice that for every α, {S α T β } β ⊂ S and from τ -continuity of S α , where the above limits are to be understood in the topology of τ -pointwise convergence. Hence {S α T } α ⊂S and again, which follows from the pointwise definition of the above limit. ThusS is a semigroup. Since it is also compact as the closed subset of the compact set C C (in the τ -product topology) and consists of affine mappings, we can apply Bruck's theorem to get inS an affine idempotent r from C onto FixS which is our desired mapping since FixS = Fix S. Assume now that mappings from S are locally τ -equicontinuous. Pick x 0 ∈ C, a net {T α } α∈A ⊂ S and let T be the limit of this net in the τ -product topology. As earlier, letĀ denote the closure of A in this topology. Pick any B, U ∈ τ 0 such thatB ⊂ U , where τ 0 denotes τ -neighbourhoods of the origin. Then we conclude from local τ -equicontinuity that where τ x0 denotes τ -neighbourhoods of x 0 . With V satisfying the above condition, we have T x − T y ∈B ⊂ U . Thus which means that each T ∈S is τ -continuous at any x 0 ∈ C. In particular, it relates to r.
We are now ready to state the following qualitative version of the Markov-Kakutani theorem. Proof. Construct from S a semigroupŜ generated by averaged mappings { 1 2 I + 1 2 T : T ∈ S} which is obviously commutative and consists of affine τcontinuous mappings. Recall that a commutative semigroup is subsurjective on compact sets. Now it is enough to notice thatŜ is locally τ -equicontinuous whenever S is, and apply Theorem 4.4.
Remark 4.6. Note that in the case of w-topology the assumption about local w-equicontinuity can be changed to (strong) local equicontinuity because every strongly continuous affine mapping is weakly continuous. Even in this case the result seems to have been known only in strictly convex Banach spaces (see [21,Theorem 5.8]).
Remark 4.7. The results of this section are formulated for subsets of Banach spaces only, but similar arguments apply to the case of locally convex spaces, as well.
Note, on the margin, that starting again from Lemma 3.13 and basically repeating the scheme presented in this section, we get a "firmly nonexpansive" counterpart of Theorem 4.4.
Vol. 18 (2016) Uniform asymptotic regularity 865 Uniform asymptotic regularity 11 Theorem 4.8. Suppose that S is a τ -properly represented semigroup generated by firmly nonexpansive mappings on C, where C is a τ -compact bounded subset of a Banach space. Then there exists inS an idempotent mapping r from C onto Fix S. If, moreover, the norm is τ -lower semicontinuous, then Fix S is a nonexpansive retract of C.
Furthermore, we have the following theorem.