CHARACTERIZATIONS OF FIXED POINTS OF NONEXPANSIVE MAPPINGS TOMONARI SUZUKI

Using the notion of Banach limits, we discuss the characterization of fixed points of nonexpansive mappings in Banach spaces. Indeed, we prove that the two sets of fixed points of a nonexpansive mapping and some mapping generated by a Banach limit coincide. In our discussion, we may not assume the strict convexity of the Banach space.


Introduction
Let C be a closed convex subset of a Banach space E. A mapping T on C is called a nonexpansive mapping if Tx − T y ≤ x − y for all x, y ∈ C. We denote by F(T) the set of fixed points of T. Kirk [17] proved that F(T) is nonempty in the case that C is weakly compact and has normal structure.See also [2,3,5,6,11] and others.
Convergence theorems to fixed points are also proved by many authors; see [1,7,8,9,10,13,15,18,23,30] and others.Very recently, the author proved the convergence theorems for two nonexpansive mappings without the assumption of the strict convexity of the Banach space.To prove this, the author proved the following theorem, which plays an extremely important role in [26].
Theorem 1.1 (see [26]).Let C be a compact convex subset of a Banach space E and let T be a nonexpansive mapping on C. Then z ∈ C is a fixed point of T if and only if holds.
The author also proved the following theorem.Using it, we give one nonexpansive retraction onto the set of all fixed points.Theorem 1.2 (see [27]).Let E be a Banach space with the Opial property and let C be a weakly compact convex subset of E. Let T be a nonexpansive mapping on C. Put for n ∈ N and x ∈ C. Then for z ∈ C, the following are equivalent: (i) z is a fixed point of T; (ii) {M(n, z)} converges weakly to z; (iii) there exists a subnet {M(ν β ,z) : β ∈ D} of a sequence {M(n, z)} in C converging weakly to z.
In this paper, using the notion of Banach limits, we generalize Theorems 1.1 and 1.2.We remark that the proofs of our results are simpler than the proofs of Theorems 1.1 and 1.2.In our discussion, we may not assume the strict convexity of the Banach space.

Preliminaries
Throughout this paper, we denote by N the set of all positive integers and by R the set of all real numbers.For a subset A of N, we define a function I A from N into R by Let E be a Banach space.We denote by E * the dual of E. We recall that E is said to have the Opial property [21] if for each weakly convergent sequence {x n } in E with weak limit x 0 , liminf n x n − x 0 < liminf n x n − x for all x ∈ E with x = x 0 .All Hilbert spaces, all finite-dimensional Banach spaces, and p (1 ≤ p < ∞) have the Opial property.A Banach space with a duality mapping which is weakly sequentially continuous also has the Opial property; see [12].We know that every separable Banach space can be equivalently renormed so that it has the Opial property; see [31].Gossez and Lami Dozo [12] prove that every weakly compact convex subset of a Banach space with the Opial property has normal structure.See also [19,20,22,25] and others.
We denote by ∞ the Banach space consisting of all bounded functions from N into R (i.e., all bounded real sequences) with supremum norm.We recall that µ Tomonari Suzuki 1725 for a Banach limit µ, a ∈ ∞ , and k ∈ N. We know that Banach limits exist; see [4].We also know that Let T be a nonexpansive mapping on a weakly compact convex subset C of a Banach space E. Let µ be a Banach limit.Then we know that for x ∈ C, there exists a unique element x 0 of C satisfying for all f ∈ E * ; see [14,19].Following Rodé [24], we denote such x 0 by T µ x.We also know that T µ is a nonexpansive mapping on C. We now prove the following lemmas, which are used in Section 3.
Lemma 2.1.Let a,b ∈ ∞ and let µ be a Banach limit.Then the following hold.
Proof.We first show (i).We note that a(n 0 + n) ≤ b(n 0 + n) for all n ∈ N. Since µ is a Banach limit, we have It is obvious that (ii) follows from (i).This completes the proof.
Lemma 2.2.Let A 1 ,A 2 ,A 3 ,...,A k be subsets of N and let µ be a Banach limit.Put Suppose that α > 0.Then, holds and holds for all n 0 ∈ N.
Proof.It is obvious that n ∈ A if and only if and n ∈ N \ A if and only if Therefore we obtain for all n ∈ N. Hence, (2.13) We suppose that {n ∈ N : This is a contradiction.This completes the proof.

Main results
In this section, we prove our main results.We first prove the following theorem, which plays an important role in this paper.
Theorem 3.1.Let E be a Banach space and let C be a weakly compact convex subset of E.
Let T be a nonexpansive mapping on C. Let µ be a Banach limit.Suppose that T µ z = z for some z ∈ C. Then there exist sequences {p n } in N and { f n } in E * such that for all n ∈ N, where Tomonari Suzuki 1727 Before proving this theorem, we need some preliminaries.In the following lemmas and the proof of Theorem 3.1, we put for f ∈ E * and ε > 0, and for ε > 0.
Lemma 3.2.For every n ∈ N, Proof.Since µ is a Banach limit, we have µ n ( T n z − z ) ≤ λ.Fix m ∈ N. By the Hahn-Banach theorem, there exists f ∈ E * such that For n ∈ N, we have Hence This completes the proof.
Proof.For n > m, by Lemma 3.2, we have (3.11) On the other hand, by the definition of A( f ,ε), for all n ∈ N \ A( f ,ε).Therefore, for n ∈ N with n > m, we have (3.13)By Lemma 2.1, we have Hence, we obtain This completes the proof.
Proof.We fix ε > 0 and η ∈ R with 1/2 < η < 1 and put We note that 0 < δ < ε/2.By the definition of λ, there exists m ∈ N such that So, using Lemma 3.3, we have For n ∈ N with m + n ∈ A( f ,ε/2), we have and hence n ∈ B(ε).Therefore for all n ∈ N.So we obtain Since η is arbitrary, we obtain the desired result.
Proof of Theorem 3.1.By the definition of λ, there exists p 1 ∈ N such that By Lemma 3.3, we have We now define inductively sequences {p n } in N and { f n } in E * .Suppose that p k ∈ N and f k ∈ E * are known.Since we have by Lemma 2.2.So we can choose p k+1 ∈ N such that p k+1 > p k , Proof.We first assume that z ∈ C is a fixed point of T.Then, we have for all f ∈ E * , and hence T µ z = z.Conversely, we assume that T µ z = z.By Theorem 3.1, there exist sequences {p n } in N and { f n } in E * satisfying the conclusion of Theorem 3.1.
We put λ = limsup n T n z − z .Since C is weakly compact, there exists a subsequence for all ∈ N. Since and hence Tz = u.Therefore Tz = z.This completes the proof.
Now, we prove our main results.Theorem 3.5.Let C be a weakly compact convex subset of a Banach space E with the Opial property.Let T be a nonexpansive mapping on C. Let µ be a Banach limit.Then z ∈ C is a fixed point of T if and only if T µ z = z.
{p nk } of {p n } such that {T pn k z} converges weakly to some pointu ∈ C. If n k > , then f T pn k z − z ≤