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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 376421, 18 pages
http://dx.doi.org/10.1155/2012/376421
Research Article

The Meir-Keeler Type for Solving Variational Inequalities and Fixed Points of Nonexpansive Semigroups in Banach Spaces

1Department of Mathematics and Statistics, Faculty of Science and Agricultural Technology, Rajamangala University of Technology Lanna Tak, Tak 63000, Thailand
2Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bang Mod, Thrung Khru, Bangkok 10140, Thailand

Received 7 June 2012; Revised 19 August 2012; Accepted 27 August 2012

Academic Editor: Juan Torregrosa

Copyright © 2012 Phayap Katchang and Poom Kumam. 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

The aim of this paper is to introduce a new iterative scheme for finding common solutions of the variational inequalities for an inverse strongly accretive mapping and the solutions of fixed point problems for nonexpansive semigroups by using the modified viscosity approximation method associate with Meir-Keeler type mappings and obtain some strong convergence theorem in a Banach spaces under some parameters controlling conditions. Our results extend and improve the recent results of Li and Gu (2010), Wangkeeree and Preechasilp (2012), Yao and Maruster (2011), and many others.

1. Introduction

The theory of variational inequalities and variational inclusions are among the most interesting and important mathematical problems and have been studied intensively in the past years since they have wide applications in the optimization and control, economics, engineering science, physical sciences, and applied sciences. For these reasons, many existence result and iterative algorithms for various variational inclusion have been studied extensively by many authors (see, e.g., [18]). The important generalization of variational inequalities has been extensively studied and generalized in different directions to study a wide class of problems arising in mechanics, optimization, nonlinear programming, finance, and applied sciences (see, e.g., [2, 8]).

Let be a nonempty closed convex subset of a real Banach space and be the dual space of with norm and pairing between and . For , the generalized duality mapping is defined by for all . In particular, if , the mapping is called the normalized duality mapping and, usually, written . Further, we have the following properties of the generalized duality mapping : (i) for all with ; (ii) for all and ; (iii) for all .

Recall that a mapping is said to be (i) Lipschitzian with Lipschitz constant if , for all ; (ii) contraction if there exists a constant such that , for all ; (iii) nonexpansive if , for all . An operator is said to be(i)accretive if there exists such that (ii)-strongly accretive if there exists a constant such that (iii)-inverse strongly accretive if, for any ,

Let be a subset of and . Then is said to sunny if , whenever for and . A subset of is said to be a sunny nonexpansive retract of if there exists a sunny nonexpansive retraction of onto . A mapping is called a retraction if . If a mapping is a retraction, then for all is in the range of .

A family of mappings of into itself is called a nonexpansive semigroup on if it satisfies the following conditions: (i) for all ; (ii) for all ;(iii) for all and ;(iv)for all is continuous.

We denote by the set of all common fixed points of ,. It is known that is closed and convex (see also [9, 10]).

A mapping is said to be an L function if for each , and for every there exists such that for all . As a consequence, every -function satisfies for each .

Definition 1.1. Let be a matric space. A mapping is said to be:(i)-contraction if is an -function and for all with ;(ii)Meir-Keeler type mapping if for each there exists such that for each with we have .

Remark 1.2. From Definition 1.1, if , , then we get the usual contraction mapping with coefficient .

At the same time, we are also interesting in the variational inequality problems for an inverse strongly accretive mappings in Banach spaces. In 2006, Aoyama et al. [11] introduced the following iteration scheme for an inverse strongly accretive operator in Banach spaces : for all , where and is a sunny nonexpansive retraction from onto . They proved a weak convergence theorem in a Banach spaces. Moreover, the sequence in (1.5) solved the generalized variational inequality problem for finding a point such that for all . The set of solutions of (1.6) is denoted by .

An interesting is the proof by using a nonexpansive semigroup and Meir-Keeler type mapping, in 2010, Li and Gu [12] defined the following sequence: Wangkeeree and Preechasilp [13] introduced the following iterative scheme:

In 2011, Yao and Maruster [8] proved some strong convergence theorems for finding a solution of variational inequality problem (1.6) in Banach spaces. They defined a sequence iteratively by given arbitrarily and where is a sunny nonexpansive retraction from a uniformly convex and 2-uniformly smooth Banach space , and is an -inverse strongly accretive operator of into .

Motivated and inspired by the idea of Li and Gu [12], Wangkeeree and Preechasilp [13], and Yao and Maruster [8], in this paper, we introduce a new iterative scheme for finding common solutions of the variational inequalities for an inverse strongly accretive mapping and the solutions of fixed point problems for a nonexpansive semigroup by using the modified viscosity approximation method associated with Meir-Keeler type mapping. We will prove the strong convergence theorem under some parameters controlling conditions. Our results extend and improve the recent results of Li and Gu [12], Wangkeeree and Preechasilp [13], Yao and Maruster [8], and many others.

2. Preliminaries

Let . A Banach space is said to uniformly convex if, for any , there exists such that, for any , implies . It is known that a uniformly convex Banach space is reflexive and strictly convex. A Banach space is said to be smooth if the limit exists for all . It is also said to be uniformly smooth if the limit is attained uniformly for . The modulus of smoothness of is defined by where is a function. It is known that is uniformly smooth if and only if . Let be a fixed real number with . A Banach space is said to be q-uniformly smooth if there exists a constant such that for all .

A Banach space is said to satisfy Opial’s condition if for any sequence in , implies that By [14, Theorem 1], it is well known that if admits a weakly sequentially continuous duality mapping, then satisfies Opial’s condition, and is smooth.

The following result describes a characterization of sunny nonexpansive retractions on a smooth Banach space.

Proposition 2.1 (see [15]). Let be a smooth Banach space and let be a nonempty subset of . Let be a retraction, and let be the normalized duality mapping on . Then the following are equivalent:(i) is sunny and nonexpansive;(ii);(iii).

Proposition 2.2 (see [16]). Let be a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space , and let be a nonexpansive mapping of into itself with . Then the set is a sunny nonexpansive retract of .

Lemma 2.3 (see [17]). Let be a nonempty bounded closed convex subset of a uniformly convex Banach space and be a nonexpansive mapping. If is a sequence of such that and , then is a fixed point of .

We need the following lemmas for proving our main results.

Lemma 2.4 (see [18]). Let , and let be a uniformly convex Banach space. Then, there exists a continuous, strictly increasing, and convex function with such that for all and .

Lemma 2.5 (see [19]). Let be a real smooth and uniformly convex Banach space, and let . Then there exists a strictly increasing, continuous, and convex function such that and

Lemma 2.6 (see [18]). Let be a real 2-uniformly smooth Banach space with the best smooth constant . Then the following inquality holds:

Lemma 2.7 (see [20]). Let be a real Banach space and be the normalized duality mapping. Then, for any , one has for all with .

Lemma 2.8 (see [21]). Let and be bounded sequences in a Banach space , and let be a sequence in with . Suppose that for all integers and . Then, .

Lemma 2.9 (see [22]). Assume that is a sequence of nonnegative real numbers such that where is a sequence in , and is a sequence in such that (1)(2) or . Then .

Theorem 2.10 (see [23]). Let be a complete metric space and a Meir-Keeler type mapping. Then has a unique fixed point.

Theorem 2.11 (see [24]). Let be a metric space and a mapping. Then the following assertions are equivalent:(i)f is a Meir-Keeler type mapping;(ii)there exists an L function such that f is a contraction.

Proposition 2.12 (see [21]). Let be a convex subset of a Banach space . Let be a Meir-Keeler type mapping. Then for each there exists such that for each with , one has

Proposition 2.13 (see [21]). Let be a convex subset of a Banach space . Let be a nonexpansive mapping on , and let be a Meir-Keeler-type mapping. Then the following holds:(i) is a Meir-Keeler type mapping on ;(ii)for each the mapping is a Meir-Keeler-type mapping on .

The following lemma is characterized by the set of solutions of variational inequality by using sunny nonexpansive retractions.

Lemma 2.14 (see [11]). Let be a nonempty closed convex subset of a smooth Banach space . Let be a sunny nonexpansive retraction from onto , and let be an accretive operator of into . Then, for all , where .

3. Strong Convergence Theorems

In this section, we suppose that the function from the definition of the contraction is continuous and strictly increasing and , where , . In consequence, we have that is a bijection on and the function satisfies the assumption in Remark 1.2.

Suppose that , and satisfy the following conditions: (C1) and ;(C2) and ; (C3); (C4); (C5), bounded subset of .

Next, we stat the main result.

Theorem 3.1. Let be a uniformly convex and 2-uniformly smooth Banach space with the best smooth constant and a nonempty closed convex subset of . Let be a sunny nonexpansive retraction from onto and be an -inverse-strongly accretive operator. Let be a nonexpansive semigroup from into itself and be a Meir-Keeler contraction of into itself. Suppose that and the conditions (C1)–(C5). For arbitrary given , the sequences are generated by Then converges strongly to which also solves the following variational inequality:

Proof. First we prove that bounded. Let , we have So, we get , for all . It follows that
By induction, we conclude that This implies that bounded, so are , , , , and .
Next, we show that , we observe that It follows that By (C1), (C2), and (C4), they imply that Applying Lemma 2.8, we obtain Therefore, we have On the other hand, we consider Then, we obtain that By (C1), (C2), (C3), and (3.10), we get From Proposition 2.1 (ii) and Lemma 2.5, we also have So, we get, Therefore, using (3.11), we obtain Then we get By (C1), (3.10), and (3.13), we have It follows from the property of that Again, we consider It follows that By using (3.11), we obtain Therefore, we have By (C1) and (3.10), we have From the property of , we get According (3.19) and (3.25), we also have Since from (3.9) and (3.26), we get
Now, we show that . We can choose a sequence of such that is bounded, and there exists a subsequence of which converges weakly to . Without loss of generality, we can assume that .
(I) We show that . From the assumption, we see that control sequence is bounded. So, there exists a subsequence that converges to . We may assume, without loss of generality, that . Observe that where is as appropriate constant such that . It follows from (3.26) and that We know that is nonexpansive, and it follows from Lemma 2.3 that . By using Lemma 2.14, we can obtain that .
(II) Next, we show that . Let such that Fix . Notice that For all , we have Since a Banach space with a weakly sequentially continuous duality mapping satisfies the Opial’s condition, this implies that . Therefore , so .
Next, we show that , where , is a sunny nonexpansive retraction of onto . Since we have (3.26) and , then we have such that
Finally, we show that converges strongly to . Suppose that does not converge strongly to , and then there exist and a subsequence of such that for all . By Proposition 2.12, for this there exists such that So, by Lemma 2.7, we have It follows from (3.11) that Now, from (C1), (3.34) and applying Lemma 2.9 to (3.37), we get as . This is a contradiction, and hence the sequence converges strongly to . The proof is completed.

Corollary 3.2. Let be a uniformly convex and 2-uniformly smooth Banach space with the best smooth constant and a nonempty closed convex subset of . Let be a sunny nonexpansive retraction from onto and be an -inverse strongly accretive operator. Let be a nonexpansive semigroup from into it self and be a Meir-Keeler contraction of into itself. Suppose that . For arbitrary given , the sequences are generated by where , and satisfy the conditions (C1)–(C3) in Theorem 3.1 and assume that , bounded subset of , and . Then converges strongly to , which also solves the following variational inequality:

Corollary 3.3. Let be a uniformly convex and 2-uniformly smooth Banach space with the best smooth constant and a nonempty closed convex subset of . Let be a sunny nonexpansive retraction from onto . Let be a nonexpansive semigroup from into itself and be a Meir-Keeler contraction of into itself. Suppose that , , and satisfy the conditions (C1), (C2), (C4), and (C5) in Theorem 3.1. For arbitrary given , the sequences are generated by Then converges strongly to , which also solves the following variational inequality:

Proof. Taking in Theorem 3.1, we can conclude the desired conclusion easily. This completes the proof.

Corollary 3.4. Let be a uniformly convex and 2-uniformly smooth Banach space with the best smooth constant and a nonempty closed convex subset of . Let be a sunny nonexpansive retraction from onto and be an -inverse strongly accretive operator. Let be a Meir-Keeler contraction of into itself. Suppose that , and satisfy the conditions (C1)–(C3) in Theorem 3.1. For arbitrary given , the sequences are generated by Then converges strongly to , which also solves the following variational inequality:

Proof. Taking for all in Theorem 3.1, we can conclude the desired conclusion easily. This completes the proof.

Corollary 3.5. Let be a uniformly convex and 2-uniformly smooth Banach space with the best smooth constant and a nonempty closed convex subset of . Let be a sunny nonexpansive retraction from onto and be an strongly accretive and -Lipschitz continuous operator. Let be a nonexpansive semigroup from into it self and be a Meir-Keeler contraction of into itself. Suppose that , and satisfy the conditions (C1) and (C3)–(C5) in Theorem 3.1. If the sequence is generated by and (3.1) and and , then the sequence converges strongly to , which also solves the following variational inequality:

Proof. Since be an strongly accretive and -Lipschitz continuous operator of into , we have Therefore, is -inverse strongly accretive. Using Theorem 3.1, we can obtain that converges strongly to . This completes the proof.

The following corollary is defined in a real Hilbert space. Let be a closed convex subset of a real Hilbert space . Let be a mapping. The classical variational inequality problems are to find such that for all . For every point , there exists a unique nearest point in , denoted by , such that is called the metric projection of onto . It is well known that is a nonexpansive mapping of onto and satisfies for every . Moreover, is characterized by the following properties: and for all .

It is well known in Hilbert spaces the smooth constant and (identity mapping). From Theorem 3.1, we can obtain the following result immediately.

Corollary 3.6. Let be a nonempty compact convex subset of a real Hilbert space . Let be a metric projection of onto and be an -inverse strongly accretive operator. Let be a nonexpansive semigroup from into itself and be a Meir-Keeler contraction of into itself. Suppose that , and satisfy the conditions (C1)–(C5) in Theorem 3.1. For arbitrary given , the sequences are generated by Then converges strongly to which also solves the following variational inequality:

Remark 3.7. Question and Open problems. Can we extend Theorem 3.1 to more general variational inequalities in the sense of Noor [1] on Banach spaces?

Acknowledgments

The first author gratefully acknowledges support provided by King Mongkuts University of Technology Thonburi (KMUTT) during the first authors stay at King Mongkuts University of Technology Thonburi (KMUTT) as a postdoctoral fellow. Moreover, the authors would like to thank the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission (NRU-CSEC no. 55000613) for financial support. Both authors thank the referees for their comments which improved the presentation of this paper.

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