Journal of Applied Mathematics

Volume 2012 (2012), Article ID 902437, 12 pages

doi:10.1155/2012/902437

Research Article

Approximation of Common Fixed Points of a Sequence of Nearly Nonexpansive Mappings and Solutions of Variational Inequality Problems

D. R. Sahu,¹ Shin Min Kang,² and Vidya Sagar¹

¹Department of Mathematics, Banaras Hindu University, Varanasi 221005, India

²Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea

Received 10 April 2012; Accepted 14 May 2012

Academic Editor: Yonghong Yao

Copyright © 2012 D. R. Sahu 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.

We introduce an explicit iterative scheme for computing a common fixed point of a sequence of nearly nonexpansive mappings defined on a closed convex subset of a real Hilbert space which is also a solution of a variational inequality problem. We prove a strong convergence theorem for a sequence generated by the considered iterative scheme under suitable conditions. Our strong convergence theorem extends and improves several corresponding results in the context of nearly nonexpansive mappings.

1. Introduction

Let be a nonempty subset of a real Hilbert space with inner product and norm , respectively. A mapping is called the following:

(1)monotone if

(2)-strongly monotone if there exists a positive real number such that

(3)-Lipschitzian if there exists a constant such that

(4)nonexpansive if

(5)nearly nonexpansive [1, 2] with respect to a fixed sequence in with if

In [3], Moudafi proposed viscosity approximation methods of selecting a particular fixed point of a given nonexpansive mapping in Hilbert spaces (see [4] for further developments in both Hilbert and Banach spaces). Let be a contraction on . Starting with an arbitrary initial , define a sequence recursively by

where is a sequence in . It is proved in [4] that under appropriate conditions imposed on , the sequence generated by (1.6) strongly converges to the unique solution of the variational inequality

where , the set of fixed points of .

In 2006, Marino and Xu [5] introduced the viscosity iterative method for nonexpansive mappings. Starting with an arbitrary initial , define a sequence recursively by

They proved that the sequence generated by (1.8) converges strongly to the unique solution of the variational inequality

which is the optimality condition for the minimization problem

where is a potential function for (i.e., for all ), and is a strongly positive bounded linear operator on ; that is, there is a constant such that

The applications of the iterative method (1.8) have been studied by some researchers (see [6, 7]).

Also, Wang [8, 9] and Wang and Hu [10] introduced the iterative method for nonexpansive mappings.

Recently, Tian [11] proposed an implicit and an explicit schemes on combining the iterative methods of Marino and Xu [5] and Yamada [12]. He also proved the strong convergence of these two schemes to a fixed point of a nonexpansive mapping defined on a real Hilbert space under suitable conditions.

More recently, Ceng et al. [13] introduced an implicit and an explicit schemes using the properties of projection for finding the fixed points of a nonexpansive mapping defined on the closed convex subset of a real Hilbert space. They also proved the strong convergence of the sequences generated by the proposed schemes to a fixed point of a nonexpansive mapping which is also a solution of a variational inequality defined on the set of fixed points.

Aoyama et al. [14] proved strong convergence of an iterative scheme for a sequence of nonexpansive mappings as follows.

Theorem 1.1. Let be a uniformly convex Banach space whose norm is uniformly Gâteaux differentiable and C be a nonempty closed convex subset of . Let be a sequence of nonexpansive mappings from into itself such that . Let be a mapping from into itself defined by for all and . Let be a sequence in generated by the following iterative process:

where is a sequence in satisfying the following conditions:

(a) and ;(b) either or and ;(c) for any bounded subset of .

Then, the sequence converges strongly to , where is the sunny nonexpansive retraction from onto .

Let be a nonempty subset of a real Hilbert space . Let be a sequence of mappings from into itself. We denote by the set of common fixed points of the sequence , that is, . Fix a sequence in with , and let be a sequence of mappings from into . Then, the sequence is called a sequence of nearly nonexpansive mappings [15] with respect to a sequence if

It is obvious that the sequence of nearly nonexpansive mappings is a wider class of sequence of nonexpansive mappings.

In this paper, inspired by Aoyama et al. [14], Ceng et al. [13], and Sahu et al. [15], we introduce an explicit iterative scheme and prove a strong convergence theorem for computing an element of , the set of common fixed points of a sequence of nearly nonexpansive mappings which is also a solution of a variational inequality over . Our result generalizes and improves the results of Ceng et al. [13], Tian [11], and many other related works.

2. Preliminaries

Throughout this paper, we denote by the identity operator of . Also, we denote by and the strong convergence and weak convergence, respectively. The symbol stands for the set of all natural numbers.

Let be a nonempty closed convex subset of a real Hilbert space . Then, for any , there exists a unique nearest point in , denoted by , such that

The mapping is called the metric projection from onto (see [1]).

Let be a nonempty subset of a real Hilbert space and be two mappings. We denote , the collection of all bounded subsets of . The deviation between and on , denoted by , is defined by

The following lemmas will be needed to prove our main result.

Lemma 2.1 (see [16]). The metric projection mapping is characterized by the following properties:

(a) for all ;(b) for all and ;(c) for all and ;(d) for all .

Lemma 2.2 (see [13]). Let be an -Lipschitzian mapping and be a -Lipschitzian and -strongly monotone operator. Then, for ,

That is, is strongly monotone with coefficient .

Lemma 2.3 (see [12]). Let be a nonempty subset of a real Hilbert space . Suppose that and . Let be a -Lipschitzian and -strongly monotone operator on . Define the mapping by

Then is a contraction that provided . More precisely, for ,

where .

Lemma 2.4 (see [1]). Let be a nonempty closed convex subset of a real Hilbert space . Let be a nonexpansive mapping. Then is demiclosed at zero; that is, if is a sequence in weakly converging to some and the sequence converges strongly to , then .

Lemma 2.5 (see [17]). Assume that is a sequence of nonnegative real numbers such that

where and are sequences of nonnegative real numbers which satisfy the following conditions:

(a) and ;(b), or(b’) is convergent.

Then .

Lemma 2.6 (see [18]). Let be a nonempty closed convex subset of a real Hilbert space and . such that . Let be nonexpansive mappings such that , and let . Then is nonexpansive from into itself and .

3. Main Result

Theorem 3.1. Let be a nonempty closed convex subset of a real Hilbert space . Let be a -Lipschitzian and -strongly monotone operator and be an -Lipschitzian mapping. Let be a sequence of nearly nonexpansive mappings from into itself with respect to a sequence such that and be a mapping from into itself defined by for all . Suppose that and , where . For an arbitrary , consider the sequence in generated by the following iterative process:

where is a sequence in satisfying the conditions:

(a) and ;(b)either or ;(c)either or for each ;(d).

Then, the sequence converges strongly to , where is the unique solution of variational inequality

Proof. Let be a mapping from into itself defined by for all . It is clear that is a nonexpansive mapping. So, we have . Now, we proceed with the following steps.

Step 1. ( is bounded). Let . From (3.1), we have

Note that , so there exists a constant such that

Thus, we have

Hence, is bounded. So and are bounded.

Step 2. ( as ). From (3.1), we have

for some constant . Thus, using Lemma 2.5, we get as .

Step 3. We have ( as ). Note that

Since

it follows that .

Step 4. We have (). Let us choose a subsequence of such that

Without loss of generality, we may assume that . By using Lemma 2.4, we get that . Note that , it follows that . Hence from (3.2), we get the following:

Step 5. We have ( as ). Set and . Noticing that . From (3.1), we have

Hence, we have

where

Noticing that , it follows from Lemma 2.5 that . This completes the proof.

Now, we derive the main result of Ceng et al. ([13], Theorem  3.2) as the following corollary.

Corollary 3.2. Let be a nonempty closed convex subset of a real Hilbert space . Let be a -Lipschitzian and -strongly monotone operator and be an -Lipschitzian mapping. Let be a nonexpansive mapping such that . Suppose that and , where . For an arbitrary , consider the sequence generated by the following iterative process:

where is a sequence in satisfying the conditions (a) and (b) of Theorem 3.1.

Then, the sequence converges strongly to , where is the unique solution of the following variational inequality:

Again, we derive the result of Tian ([11], Theorem  3.2) as the following corollary.

Corollary 3.3. Let be a real Hilbert space. Let be an -contraction on and be a -Lipschitzian and -strongly monotone operator. Let be a nonexpansive mapping such that . Suppose that and , where . For an arbitrary , consider the sequence generated by the following iterative process:

where is a sequence in satisfying the conditions and of Theorem 3.1.

Then, the sequence converges strongly to , where is the unique solution of the following variational inequality:

The following result obtains immediately from Theorem 3.1.

Corollary 3.4. Let be a nonempty closed convex subset of a real Hilbert space . Let be a -Lipschitzian and -strongly monotone operator and be an -Lipschitzian mapping. Let be a sequence of nonexpansive mappings from into itself such that and be a mapping from into itself defined by for all . Suppose that and , where . For an arbitrary , consider the sequence in generated by the following iterative process:

where is a sequence in satisfying the conditions (a)–(c) of Theorem 3.1.

Then, the sequence converges strongly to , where is the unique solution of the following variational inequality:

4. Application

Recall that the so-called problem of image recovery is essentially to find a common element of finitely many nonexpansive retracts of with . It is easy to see that every nonexpansive retraction of onto is a nonexpansive mapping of C into itself. There is no doubt that the problem of image recovery is equivalent to finding a common fixed point of finitely many nonexpansive mappings of into itself. Applying our main result, we obtain the following result which improves a number of results connected to the problem of image recovery.

Theorem 4.1. Let be a nonempty closed convex subset of a real Hilbert space . Let be a -Lipschitzian and -strongly monotone operator and be an -Lipschitzian mapping. Let such that and be nonexpansive mappings such that . Suppose that and , where . For an arbitrary , consider the sequence in generated by the following iterative process:

where is a sequence in satisfying the conditions and of Theorem 3.1.

Then, the sequence converges strongly to , where is the unique solution of the following variational inequality:

Proof. Define . Then is nonexpansive mapping from into itself. Thus, using Lemma 2.6, we get . Therefore, the proof follows from Corollary 3.2.

5. Numerical Example

For showing the effectiveness and convergence of the sequence generated by the considered iterative scheme, we discuss the following example.

Example 5.1. Let and . Let be a self-mapping defined by for all . Let be two mappings defined by and for all , where is a -Lipschitzian and -strongly monotone, and is an -Lipschitzian mapping. We take and , and we have , and . Define in by . Without loss of generality, we may assume that for all . For each , define by

In [15], it is proved that is a sequence of nearly nonexpansive mappings from into itself such that and for all , where is nonexpansive mapping.

It can be observed that all the assumptions of Theorem 3.1 are satisfied and the sequence generated by (3.1) converges to a unique solution of variational inequality (3.2) over . The graphical presentation of the convergence of the sequence generated by the iterative scheme (3.1) is given in Figure 1.

Figure 1

Acknowledgments

The authors would like to thank the referees for useful comments and suggestions.

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