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International Journal of Mathematics and Mathematical Sciences
Volume 2008, Article ID 687815, 10 pages
http://dx.doi.org/10.1155/2008/687815
Research Article

Convergence to Common Fixed Point for Generalized Asymptotically Nonexpansive Semigroup in Banach Spaces

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

Received 13 May 2008; Accepted 14 August 2008

Academic Editor: Nils Ackermann

Copyright © 2008 Yali Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let be a nonempty closed convex subset of a reflexive and strictly convex Banach space with a uniformly Gâteaux differentiable norm, a generalized asymptotically nonexpansive self-mapping semigroup of , and a fixed contractive mapping with contractive coefficient . We prove that the following implicit and modified implicit viscosity iterative schemes defined by and strongly converge to as and is the unique solution to the following variational inequality: for all .

1. Introduction

Let be a closed convex subset of a Hilbert space and a nonexpansive mapping from into itself. We denote by the set of fixed points of . Let be nonempty and an element of . For each with , let   be the unique fixed point of the contraction Browder [1] showed that defined by converges strongly to the element of which is nearest to in as

In 2004, for a contraction and a nonexpansive mapping , Xu [2] proposed the following viscosity approximation method in Banach space:and Song and Xu [3] studied the convergence of the following implicit viscosity iterative scheme:where

On the other hand, for a fixed Lipschitz strongly pseudocontractive mapping and a continuous pseudocontractive mapping , Song and Chen [4] proposed the following motivated implicit viscosity iterative scheme:

In this paper, we will still study the implicit viscosity iterative scheme (1.2) and propose the following iterative scheme:where is a generalized asymptotically nonexpansive self-mappings semigroup and a fixed contractive mapping with contractive coefficient .

2. Preliminaries

Throughout this paper, we assume that is a Banach space and a nonempty closed convex subset of . Let be a dual space of , the normalized duality mapping defined bywhere denotes the generalized duality pairing.

Definition 2.1 (see {[5]}). A mapping is said to be total asymptotically nonexpansive if there exist nonnegative real sequences and , with and as , and strictly increasing and continuous functions with such that

Remark 2.2. If , the total asymptotically nonexpansive mapping coincides with generalized asymptotically nonexpansive mapping. In addition, for all , if , then generalized asymptotically nonexpansive mapping coincide with asymptotically nonexpansive mapping; if , where , then generalized asymptotically nonexpansive mapping coincide with asymptotically nonexpansive mapping in the intermediate sense; if and , then we obtain from (2.2) the class of nonexpansive mapping.

Remark 2.3. In [5], for the total asymptotically nonexpansive mapping, the authors assume that there exist such that for all , so for , for all , then the total asymptotically nonexpansive mapping studied by [5] coincides with generalized asymptotically nonexpansive mapping.

A (one-parameter) generalized asymptotically nonexpansive semigroup is a family of self-mapping of such that

(i) for (ii) for and (iii) for (iv)for each is generalized asymptotically nonexpansive, that is,

We will denote by the common fixed point set of , that is,

Definition 2.4. A Banach space is said to be strictly convex if for and

Definition 2.5. Let , the norm of is said to be uniformly differentiable, if for each , exists uniformly for .

Definition 2.6. Let be a continuous liner functional on and let . One writes instead of One calls a Banach limit when satisfies and for each .

For a Banach limit , one knows that for every . So if and as , one has

Definition 2.7. Let be a nonempty closed convex subset of a Banach space , a continuous operator semigroup on . Then is said to be uniformly asymptotically regular (in short, u.a.r.) on if for all and any bounded subset of ,

Lemma 2.8 (see {[6]}). Let be a Banach space with a uniformly Gâteaux differentiable norm, then the normalized duality mapping defined by (2.1) is single-valued and uniformly continuous from the norm topology of to the weak topology of on each bounded subset of .

The single-valued normalized duality mapping is denoted by .

Lemma 2.9. Let be a reflexive and strictly convex Banach space with a uniformly Gâteaux differentiable norm, a nonempty closed convex subset of . Suppose that is a bounded sequence in , a continuous generalized asymptotically nonexpansive semigroup from into itself such that for all Define the setIf , then .

Proof. Set , then is a convex and continuous function, and as . Using [7, Theorem 1.3.11], there exists such that by the reflexivity of , that is, is nonempty. Clearly, is closed convex by the convexity and continuity of .
Since and is continuous for all we haveHence
Let . Since is closed convex set, there exists a unique such that
Since and ,Therefore, Since for all , then we havefor all . Therefore and the proof is complete.

Lemma 2.10 (see {[8]}). Let be a nonempty convex subset of a Banach space with a uniformly Gâteaux differentiable norm, and a bounded sequence of . If thenif and only if

Lemma 2.11 (see {[9]}). Let be a sequence of nonnegative real numbers satisfying the following conditions:where is some nonnegative integer, with and . Then as .

3. Implicit Iteration Scheme

Theorem 3.1. Let be a real reflexive and strictly convex Banach space with a uniformly Gâteaux differentiable norm, a nonempty closed convex subset of , a u.a.r generalized asymptotically nonexpansive semigroup from into itself with sequences such that , and a fixed contractive mapping with contractive coefficient . If is given by (1.2), where and then converges strongly to some common fixed point of such that is the unique solution in to variational inequality:

Proof. For any fixed Let Since for all , there exists such that for all
Furthermore,for all . That is, for all . Thus is bounded, so are and . This imply thatSince is u.a.r and then for all ,where is any bounded subset of containing . Since is continuous, henceThat is, for all We claim that the set is sequentially compact. Indeed, define the setBy Lemma 2.9, we can found . Using Lemma 2.10, we get thatIt follows from (3.3) that
Then we have
Hence, there exists a subsequence of which strongly converges to as
Next we show that is a solution in to the variational inequality (3.1). In fact, for any fixed , there exists a constant such that thenTherefore,Taking limit as in two sides of (3.11), by Lemma 2.8 and as we obtainThat is, is a solution of variational inequality (3.1).
Suppose that satisfy (3.1), we have Combining (3.13) and (3.14), it follows thatHence , that is, is the unique solution of variational inequality (3.1), so each cluster point of sequence is equal to . Therefore, converges to and the proof is complete.

Remark 3.2. Let and be as in Theorem 3.1, a u.a.r nonexpansive semigroup from into itself, then our result coincides with Theorem 3.2 in [3].

4. Modified Implicit Iteration Scheme

Theorem 4.1. Let be a real reflexive and strictly convex Banach space with a uniformly Gâteaux differentiable norm, a nonempty closed convex subset of , a u.a.r generalized asymptotically nonexpansive semigroup from into itself with sequences , such that , and a fixed contractive mapping with contractive coefficient . If is given by (1.4), where , , , , and then converges strongly to some common fixed point of such that is the unique solution in to variational inequality (3.1).

Proof. For any fixed Let Hence,By induction, we get thatSince we know from Abel–Dini theorem that there exists such that . Thus is bounded, so are , and . This imply thatSince is u.a.r. and for all where is any bounded subset of containing . Since is continuous, henceThat is, for all From Theorem 3.1, there exists the unique solution to the variational inequality (3.1). Since for all we haveThereforeThat is,where and Since , is bounded, we have So we only need to show that that is,Let , where and satisfy the condition of Theorem 3.1. Then it follows from Theorem 3.1 that
SinceFurthermore,where is a constant such that . Hence, taking upper limit as firstly, and then as in (4.13), we haveOn the other hand, since and by Lemma 2.8, we haveThus given there exists such that if for all we haveHence, taking upper limit as firstly, and then as in two sides of (4.16), we get thatFor the arbitrariness of , (4.11) holds. By Lemma 2.11, and the proof is complete.

Acknowledgments

This work was supported by National Natural Science Foundation of China (10771173) and Natural Science Foundation Project of Henan (2008B110012).

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