Journal of Applied MathematicsVolume 2012 (2012), Article ID 787419, 13 pagesdoi:10.1155/2012/787419

Research Article

Iterative Algorithm for Common Fixed Points of Infinite Family of Nonexpansive Mappings in Banach Spaces

Songnian He and Jun Guo

College of Science, Civil Aviation University of China, Tianjin 300300, China

Received 9 January 2012; Accepted 18 January 2012

Academic Editor: Yonghong Yao

Copyright © 2012 Songnian He and Jun Guo. 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 real uniformly smooth Banach space , an infinite family of nonexpansive mappings with the nonempty set of common fixed points , and a contraction. We introduce an explicit iterative algorithm , where and with . Under certain appropriate conditions on , we prove that converges strongly to a common fixed point of , which solves the following variational inequality: , where is the (normalized) duality mapping of . This algorithm is brief and needs less computational work, since it does not involve -mapping.

1. Introduction

Let be a real Banach space, a nonempty closed convex subset of , and the dual space of . The (normalized) duality mapping is defined by If is a Hilbert space, then , where is the identity mapping. It is well known that if is smooth, then is single valued.

Recall that a mapping is a contraction, if there exists a constant such that We use to denote the collection of all contractions on , that is, A mapping is said to be nonexpansive, if We use to denote the set of fixed points of , namely, . One classical way to study nonexpansive mappings is to use contractions to approximate a nonexpansive mapping ([1–11]). Browder [1] first considered the following approximation in a Hilbert space. Fix and define a contraction from into itself by where . Banach contraction mapping principle guarantees that has a unique fixed point in . Denote by the unique fixed point of , that is, In the case of having fixed points, Browder [1] proved the following.

Theorem 1.1. In a Hilbert space, as , defined in (1.6) converges strongly to a fixed point of that is closest to , that is, the nearest point projection of onto .

Halpern [3] introduced an iteration process (discretization of (1.6)) in a Hilbert as follows: where are arbitrary (but fixed) and is a sequence in . Lions [4] proved the following.

Theorem 1.2. In a Hilbert space, if satisfies the following conditions: (K1);(K2);(K3). Then converges strongly to the nearest point projection of onto .

The Banach space versions of Theorems 1.1 and 1.2 were obtained by Reich [5]. He proved the following.

Theorem 1.3. In a uniformly smooth Banach space , both defined in (1.6) and defined in (1.7) converge strongly to a same fixed point of T. If one defines by then is the sunny nonexpansive retraction from onto . Namely, Q satisfies the property: where is the duality mapping of .

Moudafi [6] introduced a viscosity approximation method and proved the strong convergence of both the implicit and explicit methods in Hilbert spaces. Xu [7] extended Moudafi's results in Hilbert spaces. Given a real number and a contraction , define a contraction by Let be the unique fixed point of . Thus, Corresponding explicit iterative process is defined by where is arbitrary (but fixed) and is a sequence in . It was proved by Xu [7] that under certain appropriate conditions on , both defined in (1.11) and defined in (1.12) converged strongly to , which is the unique solution of the variational inequality: Xu [7] also extended Moudafi's results to the setting of Banach spaces and proved the strong convergence of both the implicit method (1.11) and explicit method (1.12) in uniformly smooth Banach spaces.

In order to deal with some problems involving the common fixed points of infinite family of nonexpansive mappings, -mapping is often used, see [12–20]. Let be an infinite family of nonexpansive mappings and let be a real number sequence such that for every . For any , we define a mapping of into itself as follows: Such is called the -mapping generated by and , see [12, 13].

Yao et al. [10] introduced the following iterative algorithm for infinite family of nonexpansive mappings. Let be a uniformly convex Banach space with a uniformly Gâteaux differentiable norm and a nonempty closed convex subset of . Sequence is defined by where are arbitrary (but fixed) and . It was proved that under certain appropriate conditions on , the sequence generated by (1.15) converges strongly to a common fixed point of [13].

Since -mapping contains many composite operations of , it is complicated and needs large computational work. In this paper, we introduce a new iterative algorithm for solving the common fixed point problem of infinite family of nonexpansive mappings. Let be a real uniformly smooth Banach space, a nonempty closed convex subset of , an infinite family of nonexpansive mappings with the nonempty set of common fixed points , and . Given any , define a sequence by where , , and with . Under certain appropriate conditions on , we prove that converges strongly to , which solves the following variational inequality: where is the duality mapping of . Because doesn't contain many composite operations of , this algorithm is brief and needs less computational wok.

We will use to denote a constant, which may be different in different places.

2. Preliminaries

Let denotes the unit sphere of . A Banach space is said to be strictly convex, if holds for all , . A Banach space is said to be uniformly convex if for each , there exists a constant such that for any , implies . It is known that a uniformly convex Banach space is reflexive and strictly convex, see [21].

The norm of is said to be Gâteaux differentiable if exists for each and in this case is said to be smooth. The norm of is said to be uniformly Gâteaux differentiable if for each , the limit (2.1) is attained uniformly for . The norm of is said to be Frêchet differentiable, if for each , the limit (2.1) is attained uniformly for . The norm of is said to be uniformly Frêchet differentiable, if the limit (2.1) is attained uniformly for and in this case is said to be uniformly smooth.

Let be a nonempty subset of . A mapping is said to be sunny [22] if whenever . A mapping is called a retraction if for all . Furthermore, is sunny nonexpansive retraction from onto if is a retraction from onto which is also sunny and nonexpansive.

A subset of is called a sunny nonexpansive retraction of if there exits a sunny nonexpansive retraction from onto .

Lemma 2.1 (see [22]). Let be a closed convex subset of a smooth Banach space . Let be a nonempty subset of and be a retraction. Then the following are equivalent. (a) is sunny and nonexpansive.(b), for all .(c), for all .

Lemma 2.2 (see [23]). Let be a sequence of nonnegative real numbers satisfying where , and satisfy the following conditions: (A1);(A2);(A3) (), . Then .

Lemma 2.3 (see [24]). In a Banach space , the following inequality holds: where .

Lemma 2.4 (see [25]). Let be a closed convex subset of a strictly convex Banach space . Let be a sequence of nonexpansive mappings on . Suppose is nonempty. Let be a sequence of positive numbers with . Then a mapping on defined by for is well defined, nonexpansive and holds.

Lemma 2.5 (see [7]). Let be a uniformly smooth Banach space, a closed convex subset of , a nonexpansive mapping with , and . Then defined by converges strongly to a point in . If we define a mapping by then solves the variational inequality:

Lemma 2.6. Let be a Banach space, a bounded sequence of , and a sequence of positive numbers with . Then is convergent in .

Lemma 2.7. Let X be Banach space, a sequence of nonexpansive mappings on X with , and a sequence of positive numbers with . Let , , and . Then uniformly converges to in each bounded subset of .

Proof. , we observe that where . Taking in above last inequality, we have that holds uniformly for and this completes the proof.

3. Main Results

Theorem 3.1. Let be a nonempty closed convex subset of a real uniformly smooth Banach space , an infinite family of nonexpansive mappings with , and a sequence of positive numbers with . Let , , and with coefficient . Given any , let be a sequence generated by where satisfies the following conditions: (A1);(A2);(A3)either or . Then converges strongly to , which solves the following variational inequality:

Proof. Step 1. We show that is bounded.Noticing nonexpansiveness of , take a to derive that By induction, we obtain and is bounded, so are , , and .Step 2. We prove that .By (3.1), We have where . At the same time, we observe that where . Applying Lemma 2.6 and compatibility test of series, we have Put It follows that It is easily seen from (A2), (A3), and (3.7) that Applying Lemma 2.2 to (3.9), we obtain Step 3. We show that .Indeed we observe that Hence, by (3.11), (A1), and Lemma 2.7, we have Step 4. We prove that where with being the fixed point of the contraction From Lemma 2.5, we have and By Lemma 2.4, we have and By , we have It follows from Lemma 2.3 that where (). It follows from above last inequality that Taking in (3.20) yields where for all and . Taking in (3.21), we have Noticing the fact that two limits are interchangeable due to the fact the duality mapping is norm-to-norm uniformly continuous on bounded sets, it follows from (3.22), we have Hence (3.14) holds.Step 5. Finally, we prove that ().Indeed we observe that By view of (3.14) and condition (A2), it follows from Lemma 2.2 that . This completes the proof.

Corollary 3.2. Let be a nonempty closed convex subset of a real uniformly smooth Banach space , an infinite family of nonexpansive mappings with , a sequence of positive numbers with . Let , , and . Given any , let be a sequence generated by where satisfies the following conditions: (A1);(A2);(A3)either or . Then converges strongly to , which solves the following variational inequality:

Acknowledgment

This paper is supported by the Fundamental Research Funds for the Central Universities (ZXH2011D005).

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