Abstract and Applied Analysis

Volume 2010 (2010), Article ID 301305, 18 pages

http://dx.doi.org/10.1155/2010/301305

## Iterative Schemes for Fixed Points of Relatively Nonexpansive Mappings and Their Applications

^{1}Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand^{2}Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand

Received 19 September 2010; Accepted 28 October 2010

Academic Editor: W. A. Kirk

Copyright © 2010 Somyot Plubtieng and Wanna Sriprad. 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

We present two iterative schemes with errors which are proved to be strongly convergent to a common element of the set of fixed points of a countable family of relatively nonexpansive mappings and the set of fixed points of nonexpansive mappings in the sense of Lyapunov functional in a real uniformly smooth and uniformly convex Banach space. Using the result we consider strong convergence theorems for variational inequalities and equilibrium problems in a real Hilbert space and strong convergence theorems for maximal monotone operators in a real uniformly smooth and uniformly convex Banach space.

#### 1. Introduction

Let be a real Banach space, and the dual space of . The function is denoted by for all , where is the normalized duality mapping from to . Let be a closed convex subset of , and let be a mapping from into itself. We denote by the set of fixed points of . A point in is said to be an asymptotic fixed point of [1] if contains a sequence which converges weakly to such that the strong equals 0. The set of asymptotic fixed points of will be denoted by . A mapping from into itself is called nonexpansive if for all and nonexpansive with respect to the Lyapunov functional [2] if for all and it is called relatively nonexpansive [3–6] if and for all and . The asymptotic behavior of relatively nonexpansive mapping was studied in [3–6].

There are many methods for approximating fixed points of a nonexpansive mapping. In 1953, Mann [7] introduced the iteration as follows: a sequence is defined by where the initial guess element is arbitrary and is a real sequence in . Mann iteration has been extensively investigated for nonexpansive mappings. One of the fundamental convergence results was proved by Reich [1]. In an infinite-dimensional Hilbert space, Mann iteration can yield only weak convergence (see [8, 9]). Attempts to modify the Mann iteration method (1.2) so that strong convergence is guaranteed have recently been made. Nakajo and Takahashi [10] proposed the following modification of Mann iteration method (1.2) for nonexpansive mapping in a Hilbert space: in particular, they studied the strong convergence of the sequence generated by where and is the metric projection from onto .

Recently, Takahashi et al. [11] extended iteration (1.6) to obtain strong convergence to a common fixed point of a countable family of nonexpansive mappings; let be a nonempty closed convex subset of a Hilbert space . Let and be families of nonexpansive mappings of into itself such that and let . Suppose that satisfies the NST-condition (I) with ; that is, for each bounded sequence , implies that for all . For , define a sequence of as follows: where . Then, converges strongly to .

On the other hand, Halpern [12] introduced the following iterative scheme for approximating a fixed point of : for all , where and is a sequence of . Strong convergence of this type of iterative sequence has been widely studied: for instance, see [13, 14] and the references therein. In 2006, Martinez-Yanes and Xu [15] have adapted Nakajo and Takahashi's [10] idea to modify the process (1.5) for a nonexpansive mapping in a Hilbert space: where is the metric projection from onto . They proved that if and , then the sequence generated by (1.6) converges strongly to .

The ideas to generalize the process (1.3) and (1.6) from Hilbert space to Banach space have been studied by many authors. Matsushita and Takahashi [6], Qin and Su [16], and Plubtieng and Ungchittrakool [17] generalized the process (1.3) and (1.6) and proved the strong convergence theorems for relatively nonexpansive mappings in a uniformly convex and uniformly smooth Banach spacey; see, for instance, [2, 6, 16, 18–22] and the references therein.

Recently, Nakajo et al. [18] introduced the following condition. Let be a nonempty closed convex subset of a Hilbert space , let be a family of mappings of into itself with , and let denote the set of all weak subsequential limits of a bounded sequence in . is said to satisfy the NST-condition (II) if, for every bounded sequence in , Very recently, Nakajo et al. [19] introduced the more general condition so-called the NST-condition, is said to satisfy the -condition if, for every bounded sequence in , It follows directly from the definitions above that if satisfies the NST-condition (I), then satisfies the -condition.

Motivated and inspired by Wei and Cho [2], in this paper, we introduce two iterative schemes (3.1) and (3.14) and use the -condition for a countable family of relatively nonexpansive mappings to obtain the strong convergence theorems for finding a common element of the set of fixed points of a countable family of relatively nonexpansive mappings and the set of fixed points of nonexpansive mappings in the sense of Lyapunov functional in a real uniformly smooth and uniformly convex Banach space. Using this result, we also discuss the problem of strong convergence concerning variational inequality, equilibrium, and nonexpansive mappings in Hilbert spaces. Moreover, we also apply our convergence theorems to the maximal monotone operators in Banach spaces. The results obtained in this paper improve and extend the corresponding result of Matsushita and Takahashi [6], Qin and Su [16], Wei and Cho [2], Wei and Zhou [22], and many others.

#### 2. Preliminaries

Let be a real Banach space with norm and let be the dual of . For all and , we denote the value of at by . We denote the strong convergence and the weak convergence of a sequence to in by and , respectively. We also denote the convergence of a sequence to in by . An operator is said to be *monotone* if whenever . We denote the set by . A monotone is said to be *maximal* if its graph is not properly contained in the graph of any other monotone operator. If is maximal monotone, then the solution set is closed and convex.

The normalized duality mapping from to is defined by for . By Hahn-Banach theorem, is nonempty; see [23] for more details. A Banach space is said to be strictly convex if for all with and . It is also said to be uniformly convex if, for each , there exists such that for with and . Let be the unit sphere of . Then the Banach space is said to be smooth provided that exists for each . It is also said to be uniformly smooth if the limit is attained uniformly for . It is well known that if is smooth, strictly convex, and reflexive, then the duality mapping is single valued, one-to-one, and onto.

Let be a smooth, strictly convex, and reflexive Banach space and let be a nonempty closed convex subset of . Throughout this paper, we denote by the function defined by It is obvious from the definition of the function that for all .

Following Alber [24], the generalized projection from onto is defined by If is a Hilbert space, then and is the metric projection of onto .

We need the following lemmas for the proof of our main results.

Lemma 2.1 (Kamimura and Takahashi [25]). *Let be a uniformly convex and smooth Banach space and let and be sequences in such that either or is bounded. If , then .*

Lemma 2.2 (Alber [24], Kamimura and Takahashi [25]). *Let be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space . Then
*

Lemma 2.3 (Alber [24], Kamimura and Takahashi [25]). *Let be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space , let , and let . Then
*

Lemma 2.4 (Matsushita and Takahashi [6]). *Let be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space , and let be a relatively nonexpansive mapping from into itself. Then is closed and convex.*

Lemma 2.5 (see [2]). *Let be a real smooth and uniformly convex Banach space. If is a mapping which is nonexpansive with respect to the Lyapunov functional, then is convex and closed subset of .*

#### 3. Main Results

In this section, by using the -condition, we proved two strong convergence theorems for finding a common element of the set of fixed points of a countable family of relatively nonexpansive mappings and the set of fixed points of nonexpansive mappings in the sense of Lyapunov functional in a real uniformly smooth and uniformly convex Banach space.

Theorem 3.1. *Let be a real uniformly smooth and uniformly convex Banach space, let be a nonempty closed convex subset of , and let be nonexpansive with respect to the Lyapunov functional and weakly sequentially continuous. Let be a family of relatively nonexpansive mappings of into itself such that and satisfy the -condition. Then the sequence generated by
**
converges strongly to provided that*(i)* with for some ;*(ii)*the error sequence is such that as .*

*Proof. *We split the proof into five steps.*Step 1 (Both and are closed and convex subset of ). *Noting the facts that
we can easily know that is closed and convex subset of . It is obvious that is also a closed and convex subset of .*Step 2 ( for all ). *To observe this, take . Then it follows from the convexity of that
for all . Hence for all . On the other hand, it is clear that . Then and is well defined. Suppose that for some . Then and is well defined. It follows from Lemma 2.3 that
which implies that . Therefore , and hence is well defined. Then by induction, the sequence generated by (3.1) is well defined, for each . Moreover for each nonnegative integer .*Step 3 ( is bounded sequence of ). *In fact, for all , it follows from Lemma 2.2 that
By the definition of and Lemma 2.2, we note that and hence
Therefore is bounded.*Step 4 (). *From the facts , , and Lemma 2.2, we have
Therefore, exists. Then , which implies from Lemma 2.1 that as . Since , we have
Note that
Since is uniform continuous on each bounded subset of and , we have as . This implies that and as . Using Lemma 2.1, we obtain , , and . Since both and are uniformly continuous on bounded subsets, it follows that .

Put for all . Since is bounded and , we have is bounded. Note that
for all . It implies that . Therefore, we have since satisfies -condition. Note that as . Hence, we also have .

On the other hand, from Step 3, we know that . Then, for all , there exists a subsequence of such that as . Therefore, and as . Since is weakly continuous and , we have . Hence .*Step 5 (, as ). *Let be any subsequence of which weakly converges to . Since and , we have . Then,
which yields
Hence as . It follows from Lemma 2.1 that as . Therefore, as . This completes the proof.

Corollary 3.2 (see Matsushita and Takahashi [6]). *Let be a real uniformly smooth and uniformly convex Banach space, let be a nonempty closed convex subset of , let be a relatively nonexpansive mappings of into itself, and let be a sequence of real numbers such that and . Suppose that is given by
**
where is the duality mapping on . If is nonempty, then converges strongly to , where is the generalized projection from onto .*

*Proof. *Suppose that and put , , and for all . Let be a bounded sequence in with and let . Then there exists subsequence of such that . It follows directly from the definition of that . Hence satisfies NST-condition; by Theorem 3.1, converges strongly to .

Theorem 3.3. *Let be a real uniformly smooth and uniformly convex Banach space, let be a nonempty closed convex subset of , and let be nonexpansive with respect to the Lyapunov functional and weakly sequentially continuous. Let be a family of relatively nonexpansive mappings of into itself such that and satisfy the -condition. Then the sequence generated by
**
converges strongly to provided that*(i)* is a sequence such that as ;*(ii)*the error sequence is such that as .*

*Proof. *By slightly modifying the corresponding proof in Theorem 3.1, we can easily obtain the result.

Setting , , and for all in Theorem 3.3, we obtain the following corollary.

Corollary 3.4 (see Qin and Su [16]). *Let be a real uniformly smooth and uniformly convex Banach space, and let be a nonempty closed convex subset of . Let be a relatively nonexpansive mappings of into itself, and let be a sequence in and . Define a sequence in by
**
where is the duality mapping on . If is nonempty, then converges strongly to , where is the generalized projection from onto .*

#### 4. Applications to the Variational Inequality Problem, Equilibrium Problem, and Fixed Points Problem of Nonexpansive Mappings in a Real Hilbert Space

In this section, using Theorems 3.1 and 3.3, we prove the strong convergence theorems for finding a common element of the set of fixed points of nonexpansive mapping, the solution set of the variational inequality problems, and the solution set of an equilibrium problems in a Hilbert space.

##### 4.1. Common Solutions of a Fixed Point Problem and a Variational Inequality Problem

Let be a nonempty closed convex subset of a real Hilbert space and let be a mapping of into . A classical variational inequality problem, denoted by , is to find an element such that for all . It is known that for , is a solution of the variational inequality of if and only if , where is the metric projection from onto . We denote by the normal cone for at a point , that is, for all . Define by
Then is maximal monotone and (see [26]). A mapping is called *α*-inverse strongly monotone if there exists an such that
It is well known that if , then is nonexpansive of into itself.

Lemma 4.1 (see [27]). *Let be a nonempty closed convex subset of a real Hilbert space and let be α-inverse strongly monotone mapping of into . Let and let be a metric projection from onto . Let , for all . Then is a sequence of nonexpansive mappings and satisfies NST-condition.*

By using Lemma 4.1 and Theorem 3.1 with , for all , we obtain the following theorem.

Theorem 4.2. *Let be a nonempty closed convex subset of a real Hilbert space and let be α-inverse strongly monotone mapping of into . Let be a nonexpansive mapping of into itself such that . Let be a metric projection from onto . Then the sequence generated by
*

*converges strongly to provided that*(i)

*with for some ;*(ii)

*for some with .*

Next, by using Lemma 4.1 and Theorem 3.3 with , for all , we obtain the following theorem.

Theorem 4.3. *Let be a nonempty closed convex subset of a real Hilbert space and let be α-inverse strongly monotone mapping of into . Let be a nonexpansive mapping of into itself such that . Let be a metric projection from onto . Then the sequence generated by
*

*converges strongly to provided that*(i)

*is a sequence such that as ;*(ii)

*for some with .*

##### 4.2. Common Solutions of a Fixed Point Problem and an Equilibrium Problem

Let be a nonempty closed convex subset of and let be a bifunction of into , where is the set of real numbers. The equilibrium problem for is to find such that The set of solutions of (4.5) is denoted by . Numerous problems in physics, optimization, and economics reduce to find a solution of (4.5). In 1997, Combettes and Hirstoaga [15] introduced an iterative scheme of finding the best approximation to the initial data when is nonempty and proved a strong convergence theorem.

For solving the equilibrium problem for a bifunction , let us assume that satisfies the following conditions:(A1) for all ;(A2) is monotone, that is, forall ;(A3) for each , (A4) for each , is convex and lower semicontinuous.

The following lemma appears implicitly in [15, 28].

Lemma 4.4 (see [15, 28]). *Let be a nonempty closed convex subset of and let be a bifunction of in to satisfying (A1)–(A4). Let and . Then, there exists such that
**
Define a mapping as follows:
**
for all . Then, the following holds:*(1)* is single-valued; *(2)* is firmly nonexpansive, that is, for any ,
*(3)*;
*(4)* is closed and convex. *

Lemma 4.5 (see [27]). *Let be a nonempty closed convex subset of a real Hilbert space . Let be a bifunction from into satisfying (A1)–(A4) and . If is a sequence in satisfying , then is a family of firmly nonexpansive mappings of into and satisfies NST-condition.*

The following theorem follows directly from Lemma 4.5 and Theorem 3.1.

Theorem 4.6. *Let be a nonempty closed convex subset of a real Hilbert space and let be a bifunction from into satisfying (A1)–(A4). Let be a nonexpansive mapping of into itself such that . Then the sequence generated by
**
converges strongly to provided that*(i)* with for some ;*(ii)* such that ,*

*Proof. *Put and for all . Then, it follows from Lemma 4.5 and Theorem 3.1 that converges strongly to .

Next, by using Lemma 4.5 and Theorem 3.3 with , for all , we obtain the following result.

Theorem 4.7. *Let be a nonempty closed convex subset of a real Hilbert space and let be a bifunction from into satisfying (A1)–(A4). Let be a nonexpansive mapping of into itself such that . Then the sequence generated by
**
converges strongly to provided that*(i)* is a sequence such that as ;*(ii)* such that ,*

#### 5. Applications to Maximal Monotone Operators in Banach Space

In this section, we discuss the problem of strong convergence concerning maximal monotone operators in a real uniformly smooth and uniformly convex Banach space.

Let be a smooth, strictly convex, and reflexive Banach space and let be a maximal monotone operator. Then for each and , there corresponds a unique element satisfying
see [23]. We define the *resolvent* of by . In other words, for all . We know that is relatively nonexpansive and for all (see [6, 23]), where denotes the set of all fixed points of . We can also define, for each , the *Yosida approximation* of by . We know that for all .

Lemma 5.1. *Let be a real uniformly smooth and uniformly convex Banach space and let be a maximal monotone operator with and for all . Let be a sequence of relatively nonexpansive mappings of into itself defined by for all , where is a sequence in such that . Then, satisfies the NST-condition.*

*Proof. *It easy to see that . Let be a bounded sequence in such that and let . Then, there exists a subsequence of such that . By the uniform smoothness of , we have . Since , we have
Let . Then it holds from the monotonicity of that
for all . Letting , we get . Then, the maximality of implies . Hence . Therefore, satisfies the NST-condition.

Using Theorem 3.1 and Lemma 5.1, we first obtain the result of [2].

Theorem 5.2 (see [2]). *Let be a real uniformly smooth and uniformly convex Banach space and let be nonexpansive with respect to the Lyapunov functional and weakly sequentially continuous. Let be a maximal monotone operator with and for all . Then, sequence generated by the following scheme
**
converges strongly to provided that*(i)* with for some ;*(ii)* is a sequence such that ;*(iii)*the error sequence is such that as .*

*Proof. *Put for all . Hence by using Lemma 5.1 and Theorem 3.1, we obtain the result.

Putting in Theorem 5.2, we obtain the following corollary.

Corollary 5.3 (see [22]). *Let be a real uniformly smooth and uniformly convex Banach space and let be a maximal monotone operator with . and for all . Then, sequence generated by the following scheme
**
converges strongly to provided that*(i)* with for some ;*(ii)* is a sequence such that ;*(iii)*the error sequence is such that as .*

The following theorem follows directly from Lemma 5.1 and Theorem 3.3

Theorem 5.4 (see [2]). *Let be a real uniformly smooth and uniformly convex Banach space and let be nonexpansive with respect to the Lyapunov functional and weakly sequentially continuous. Let be a maximal monotone operator with . and for all . Then, sequence generated by the following scheme
**
converges strongly to provided that*(i)* is a sequence such that as ;*(ii)* is a sequence such that ;*(iii)*the error sequence is such that as .*

Putting in Theorem 5.4, we obtain the following corollary.

Corollary 5.5 (see [22]). *Let be a real uniformly smooth and uniformly convex Banach space, let be a nonempty closed convex subset of , and let be a maximal monotone operator with and for all . Then, sequence generated by the following scheme
**
converges strongly to provided that*(i)* is a sequence such that as ;*(ii)* is a sequence such that ;*(iii)*the error sequence is such that as .*

#### Acknowledgment

The authors thank the National Centre of Excellence in Mathematics, PERDO, for the Financial support. Moreover, the second author would like to thank the National Centre of Excellence in Mathematics, PERDO, under the Commission on Higher Education, Ministry of Education, Thailand

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