Applications of uniform asymptotic regularity to fixed point theorems

We show that there are no nontrivial surjective uniformly asymptotically regular mappings acting on a metric space and 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 $\tau$-compact subset of a Banach space has a common fixed point, and give a qualitative complement to the Markov-Kakutani theorem.


Introduction
The notion of asymptotic regularity was introduced by Browder and Petryshyn in [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 lim n→∞ d(T n x, T n+1 x) = 0 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 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 affine continuous mappings 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] (see also [20,Question 1]) 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: (F * ): Whenever S = {T s : s ∈ S} is a representation of S as normnonexpansive mappings on a non-empty weak * compact convex set C of a dual Banach space E and the mapping (s, x) → T s (x) from S × C to C is jointly continuous, where C is equipped with the weak * topology of E, then there is a common fixed point for S in C.
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 (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 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 self-mappings on M (abbr. UAR), 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. Proof. Take ε > 0. The assured property gives us a positive integer n such that ∀ x∈M d(T n+1 x, T n x) < ε. 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 corollary, 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]). Recall that in a Hilbert space a mapping T : C → C is firmly nonexpansive iff 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].

Lemma 3.2. A firmly nonexpansive mapping T : C → C defined on a bounded subset of a Banach space E is UAR.
Proof. Let C be a bounded subset of E and put Also, C ⋆ inherits boundedness from C: and, by taking suprema, we conclude that Ψ T is also firmly nonexpansive. Now, from [22, Theorem 1] we have where the last equality follows from boundedness of C ⋆ . Taking S as the identity on C, and unwiding definitions of · and Ψ T , we get which is the desired conclusion.
Now from the ingredients above we have instantly the following result.

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 ⊂ F ix 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. Definition 3.5. We say that S = {T s : s ∈ S} is a representation of S on a topological space (C, τ ) if for each s ∈ S, T s is a mapping from C into C and T st x = T s (T t x) for every s, t ∈ S and x ∈ C.
Since, due to Lemma 2.1, surjectivity is intertwined into all results of this paper, the following definition seems natural: Definition 3.6. We say that the representation S on the space C is subsurjective if there is a (nonempty) set D ⊂ C such that every T ∈ S is surjective on D.
There is a fundamental example of such representations. Proof. Using the Kuratowski-Zorn lemma, we get a minimal compact Sinvariant set D. Now, picking any T ∈ S, we notice that for every P ∈ S, It is also compact so, by the minimality of D, T (D) = D.
Let S be a semitopological semigroup, i.e., a semigroup with a Hausdorff topology such that the mappings S ∋ s → ts and S ∋ s → st are continuous for each t ∈ S. Notice that every semigroup can be equipped with the discrete topology and then it is called a discrete semigroup. The semigroup S is said to be left reversible if any two closed right ideals of S have a non-void intersection. In this case (S, ≤) is a directed set with the relation Let ℓ ∞ (S) be the Banach space of bounded real-valued functions on S with the supremum norm. For s ∈ S and f ∈ ℓ ∞ (S), we define the element l s f in ℓ ∞ (S) by l s f (t) = f (st) for every t ∈ S. Denote by C b (S) the closed subalgebra of ℓ ∞ (S) consisting of continuous functions and let LUC(S) be the space of bounded left uniformly continuous functions on S, i.e., all f ∈ C b (S) such that the mapping S ∋ s → l s f from S to C b (S) is continuous when C b (S) has the sup norm topology. Note that if S is a topological group, then LUC(S) is equivalently the space of bounded right uniformly continuous functions on S (see [13]). A semigroup S is called left amenable if there exists a left invariant mean on LUC(S), i.e., a functional µ ∈ LUC(S) * such that µ = µ(I S ) = 1 and µ(l s f ) = µ(f ) for each s ∈ S and f ∈ LUC(S). If S is discrete and left amenable then S is left reversible. In general there is no such relation (see [14, p. 335]).
Let us introduce one more temporary definition. 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.
Note that a special case of the above corollary gives a partial answer to the problem of A. T.-M. Lau [20, Question 1] (as described in the Introduction) for semigroups generated by firmly nonexpansive mappings. Assuming proper representability instead of just subsurjectivity, we get the 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 r : C → i∈I Fix T i . 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], [23,Theorem 3]. Proof. Let T : C → C be an α-averaged of an affine mapping S. Notice that from the affinity we get a quite compact binominal expansion: T n = (αI +ᾱS) n = n k=0 n k α (n−k)ᾱk S k .
With the notation n n+1 = n −1 = 0, we have Then We get that n k is nonnegative up to m = ⌊ᾱ(n + 1)⌋. Using the identity n k and having in mind boundary cases, we get by telescoping, m k=0 n k = n m α (n−m)ᾱm+1 = a n ≥ 0.
Since C is convex, it follows that for each x ∈ C, m k=0 n k a n S k x = x 1 ∈ C, n+1 k=m+1 n k a n S k x = x 2 ∈ C.

Notice that
T n x − T n+1 x = a n x 1 − x 2 ≤ a n diam C.
If α = 1 2 and n is even, we have from the Stirling's approximation of the central binomial coefficent a n = 1 2 n+1 The odd case, n = 2k − 1 gives This proves that a n vanishes in 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 the α > 1 2 can also be easily proved: define U : C → C as λ-averaged of S with λ = 2α. Notice that U is affine mapping and T is its 1/2-averaged, 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. Theorem 4.4. Suppose S is a τ -properly represented semigroup generated by averaged affine mappings on C, where C is a convex and τ -compact bounded subset of a Banach space. Then there exists an affine, surjective idempotent r : C → Fix S. If, moreover, elements of S are locally τ -equicontinuous, then we can pick r to be also τ -continuous.
Proof. Since from Lemma 4.1 generators of S are UAR, the existence of an affine, surjective idempotent r follows instantly from Lemma 3.13 and Corollary 4.3.
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 τproduct 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 x 0 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 τ -continuous mappings. Recall that a commutative semigroup is subsurjective on compact sets. Now it is enough to notice thatŜ is locally τequicontinuous 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.
Theorem 4.8. Suppose 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.
Even more generally,