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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 328740, 18 pages
http://dx.doi.org/10.1155/2013/328740
Research Article

Mann-Type Viscosity Approximation Methods for Multivalued Variational Inclusions with Finitely Many Variational Inequality Constraints in Banach Spaces

1Department of Mathematics, Shanghai Normal University, Scientific Computing Key Laboratory of Shanghai Universities, Shanghai 200234, China
2Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
3Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received 13 September 2013; Accepted 23 October 2013

Academic Editor: Chi-Ming Chen

Copyright © 2013 Lu-Chuan Ceng 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 Mann-type viscosity approximation methods for finding solutions of a multivalued variational inclusion (MVVI) which are also common ones of finitely many variational inequality problems and common fixed points of a countable family of nonexpansive mappings in real smooth Banach spaces. Here the Mann-type viscosity approximation methods are based on the Mann iteration method and viscosity approximation method. We consider and analyze Mann-type viscosity iterative algorithms not only in the setting of uniformly convex and 2-uniformly smooth Banach space but also in a uniformly convex Banach space having a uniformly Gáteaux differentiable norm. Under suitable assumptions, we derive some strong convergence theorems. In addition, we also give some applications of these theorems; for instance, we prove strong convergence theorems for finding a common fixed point of a finite family of strictly pseudocontractive mappings and a countable family of nonexpansive mappings in uniformly convex and 2-uniformly smooth Banach spaces. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literature.

1. Introduction

Let be a real Banach space whose dual space is denoted by . The normalized duality mapping is defined by where denotes the generalized duality pairing. It is an immediate consequence of the Hahn-Banach theorem that is nonempty for each . Let denote the unite sphere of . A Banach space is said to be uniformly convex if, for each , there exists such that, for all ,

It is known that a uniformly convex Banach space is reflexive and strict convex. A Banach space is said to be smooth if the limit exists for all ; in this case, is also said to have a Gáteaux differentiable norm. is said to have a uniformly Gáteaux differentiable norm if, for each , the limit is attained uniformly for . Moreover, it is said to be uniformly smooth if this limit is attained uniformly for . The norm of is said to be the Fréchet differential if, for each , this limit is attained uniformly for . In addition, we define a function called the modulus of smoothness of as follows:

It is known that is uniformly smooth if and only if . Let be a fixed real number with . Then a Banach space is said to be -uniformly smooth if there exists a constant such that for all . It is well-known that no Banach space is -uniformly smooth for . In addition, it is also known that is single-valued if and only if is smooth, whereas if is uniformly smooth, then the mapping is norm-to-norm uniformly continuous on bounded subsets of . If has a uniformly Gáteaux differentiable norm then the duality mapping is norm-to-weak* uniformly continuous on bounded subsets of .

Let be a nonempty closed convex subset of a real Banach space . A mapping is called nonexpansive if

The set of fixed points of is denoted by . We use the notation to indicate the weak convergence and the one to indicate the strong convergence.

Definition 1. Let be a mapping of into . Then is said to be(i)accretive if for each there exists such that where is the normalized duality mapping;(ii)-strongly accretive if for each there exists such that for some ;(iii)-inverse strongly accretive if for each there exists such that for some ;(iv)-strictly pseudocontractive if for each there exists such that for some .

Let be a real smooth Banach space. Let be a nonempty closed convex subset of and let be a nonlinear mapping. The so-called variational inequality problem (VIP) is the problem of finding such that which was considered by Aoyama et al. [1]. Note that VIP (10) is connected with the fixed point problem for nonlinear mapping (see e.g., [2]), the problem of finding a zero point of a nonlinear operator (see e.g., [3]), and so on. In particular, whenever a Hilbert space, the VIP (10) reduces to the classical VIP of finding such that

whose solution set is denoted by . Recently, in order to find a solution of VIP (10), Aoyama et al. [1] introduced Mann-type iterative scheme for an accretive operator as follows: where is a sunny nonexpansive retraction from onto . Then they proved a weak convergence theorem.

Definition 2. Let be a nonempty convex subset of a real Banach space . Let be a finite family of nonexpansive mappings of into itself and let be real numbers such that for every . Define a mapping as follows: Such a mapping is called the -mapping generated by and .

Lemma 3 (see [4]). Let be a nonempty closed convex subset of a strictly convex Banach space. Let be a finite family of nonexpansive mappings of into itself with and let be real numbers such that for every and . Let be the -mapping generated by and . Then .

From Lemma 3, it is easy to see that the -mapping is a nonexpansive mapping.

On the other hand, let be the family of all nonempty, closed, and bounded subsets of a real smooth Banach space . Also, we denote by the Hausdorff metric on defined by

Let be two multivalued mappings, let be an -accretive mapping, let be a single-valued mapping, and let be a nonlinear mapping. Then for any given , , Chidume et al. [5] introduced and studied the multivalued variational inclusion (MVVI) of finding such that is a solution of the following:

If and , then the MVVI (15) reduces to the problem of finding such that is a solution of the following:

We denote by the set of such solutions for MVVI (16).

The authors [5] established an existence theorem for MVVI (15) in a smooth Banach space and then proved that the sequence generated by their iterative algorithm converges strongly to a solution of MVVI (16).

Theorem 4 (see [5, Theorem ]). Let be a real smooth Banach space. Let , and be three multivalued mappings, let be a single-valued mapping, and let be a single-valued continuous mapping satisfying the following conditions:(C1) is -accretive and -uniformly continuous;(C2) is -uniformly continuous;(C3) is -uniformly continuous;(C4)the mapping is -strongly accretive and --Lipschitz with respect to the mapping , where is a strictly increasing function with ;(C5)the mapping is accretive and --Lipschitz with respect to the mapping .
For arbitrary , define the sequence iteratively by where is defined by for any , , and some , where is a positive real sequence such that . Then, there exists such that, for and for all , converges strongly to , and, for any and , is a solution of the MVVI (16).

Let be a nonempty closed convex subset of a real smooth Banach space and let be a sunny nonexpansive retraction from onto . Let be a contraction with coefficient . Motivated and inspired by the research going on this area, we introduce Mann-type viscosity approximation methods for finding solutions of the MVVI (16) which are also common ones of finitely many variational inequality problems and common fixed points of a countable family of nonexpansive mappings. Here, the Mann-type viscosity approximation methods are based on the Mann iteration method and viscosity approximation method. We consider and analyze Mann-type viscosity iterative algorithms not only in the setting of uniformly convex and 2-uniformly smooth Banach space but also in a uniformly convex Banach space having a uniformly Gáteaux differentiable norm. Under suitable assumptions, we derive some strong convergence theorems. In addition, we also give some applications of these theorems; for instance, we prove strong convergence theorems for finding a common fixed point of a finite family of -strictly pseudocontractive mappings () and a countable family of nonexpansive mappings in uniformly convex and 2-uniformly smooth Banach spaces. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literature; see, for example, [611].

2. Preliminaries

Let be a real Banach space with dual . We denote by the normalized duality mapping from to defined by where denotes the generalized duality pairing. Throughout this paper, the single-valued normalized duality map is still denoted by . Unless otherwise stated, we assume that is a smooth Banach space with dual .

A multivalued mapping is said to be(i)accretive, if (ii)-accretive, if is accretive and , for all , where is the identity mapping;(iii)-inverse strongly accretive, if there exists a constant such that (iv)-strongly accretive, if there exists a strictly increasing continuous function with such that (v)-expansive, if

It is easy to see that if is -strongly accretive, then is -expansive.

A mapping is said to be -uniformly continuous, if for any given , there exists a such that whenever then .

A mapping is -strongly accretive, with respect to , in the first argument if

A mapping is called lower semicontinuous, if is open in whenever is open.

We list some propositions and lemmas that will be used in the sequel.

Proposition 5 (see [12]). Let and be sequences of nonnegative numbers and a sequence satisfying the conditions that is bounded, , and , as . Let the recursive inequality be given where is a strictly increasing function such that it is positive on and . Then , as .

Proposition 6 (see [13]). Let be a real smooth Banach space. Let , and be two multivalued mappings, and let be a nonlinear mapping satisfying the following conditions:(i)the mapping  is  -strongly accretive with respect to the mapping  ;(ii)the mapping  2009  is accretive with respect to the mapping  .

Then the mapping defined by is -strongly accretive.

Proposition 7 (see [14]). Let be a real Banach space and let be a lower semicontinuous and -strongly accretive mapping; then, for any is a one-point set; that is, is a single-valued mapping.

Lemma 8 can be found in [15]. Lemma 9 is an immediate consequence of the subdifferential inequality of the function .

Lemma 8. Let be a sequence of nonnegative real numbers satisfying where , and satisfy the following conditions:(i) and ;(ii);(iii), for all , and .Then .

Lemma 9. In a smooth Banach space , there holds the inequality

Lemma 10 (see [1]). 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 ,

Let be a subset of and let be a mapping of into . Then is said to be sunny if whenever for and . A mapping of into itself is called a retraction if . If a mapping of into itself is a retraction, then for every where is the range of . A subset of is called a sunny nonexpansive retract of if there exists a sunny nonexpansive retraction from onto . The following lemma concerns the sunny nonexpansive retraction.

Lemma 11 (see [16]). Let be a nonempty closed convex subset of a real smooth Banach space . Let be a nonempty subset of . Let be a retraction of onto . Then the following are equivalent:(i) is sunny and nonexpansive;(ii), for all ;(iii), for all .

It is well known that if a Hilbert space, then a sunny nonexpansive retraction is coincident with the metric projection from onto ; that is, . If is a nonempty closed convex subset of a strictly convex and uniformly smooth Banach space and if is a nonexpansive mapping with the fixed point set , then the set is a sunny nonexpansive retract of .

Lemma 12 (see [17]). Let be a uniformly convex Banach space and , . Then there exists a continuous, strictly increasing, and convex function , such that for all and all with .

Lemma 13 (see [18]). Let be a nonempty closed convex subset of a Banach space . Let be a sequence of mappings of into itself. Suppose that . Then for each , converges strongly to some point of . Moreover, let be a mapping of into itself defined by for all . Then .

Let be a nonempty closed convex subset of a Banach space and let be a nonexpansive mapping with . As previous, let be the set of all contractions on . For and , let be the unique fixed point of the contraction on ; that is,

Lemma 14 (see [19]). Let be a uniformly smooth Banach space or a reflexive and strictly convex Banach space with a uniformly Gáteaux differentiable norm. Let be a nonempty closed convex subset of , let be a nonexpansive mapping with , and let . Then the net defined by converges strongly to a point in . If one defines a mapping by , for all , then solves the VIP as follows:

Lemma 15 (see [20]). Let be a nonempty 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 defined well and nonexpansive, and holds.

Lemma 16 (see [21]). Given a number . A real Banach space is uniformly convex if and only if there exists a continuous strictly increasing function , such that for all and such that and .

3. Mann-Type Viscosity Algorithms in Uniformly Convex and 2-Uniformly Smooth Banach Spaces

In this section, we introduce Mann-type viscosity iterative algorithms in uniformly convex and -uniformly smooth Banach spaces and show strong convergence theorems. We will use the following useful lemma.

Lemma 17. Let be a nonempty closed convex subset of a real -uniformly smooth Banach space . Let be an -inverse strongly accretive mapping. Then, one has where . In particular, if , then is nonexpansive.

Theorem 18. Let be a uniformly convex and -uniformly smooth Banach space and let be a nonempty closed convex subset of such that . Let be a sunny nonexpansive retraction from onto . Let , , and be three multivalued mappings, let be a single-valued mapping, and let be a single-valued continuous mapping satisfying conditions (C1)–(C5) in Theorem 4. Consider that
(C6) is -inverse strongly accretive with .

Let be an -inverse strongly accretive mapping for each . Define the mapping by for , where and is the -uniformly smooth constant of . Let be the -mapping generated by and , where , for all and . Let be a contraction with coefficient . Let be a countable family of nonexpansive mappings of into itself such that . Suppose that , and are the sequences in and satisfy the following conditions:(i);(ii) and ;(iii) for some ;(iv);(v);(vi).

For arbitrary , define the sequence iteratively by where is defined by for any , , and some . Assume that for any bounded subset of and let be a mapping of into itself defined by for all and suppose that . Then converges strongly to , which solves the following VIP: and, for any and , is a solution of the MVVI (16).

Proof. First of all, by Lemma 17 we know that is a nonexpansive mapping, where for each . Hence, from the nonexpansivity of , it follows that is a nonexpansive mapping for each . Since is the -mapping generated by and , by Lemma 3, we deduce that . Utilizing Lemma 10, and the definition of , we get for each . Thus, we have
Now, let us show that for any , , there exists a point such that is a solution of the MVVI (15), for any and . Indeed, following the argument idea in the proof of Chidume et al. [5, Theorem 3.1], we put for all . Then by Proposition 6, is -strongly accretive. Since and are -uniformly continuous and is continuous, is continuous and hence lower semicontinuous. Thus, by Proposition 7,   is single-valued. Moreover, since is -strongly accretive and by assumption is -accretive, we have that is an -accretive and -strongly accretive mapping, and hence by Cioranescu [22, page 184], for any , we have that is closed and bounded. Therefore, by Morales [23], is surjective. Hence, for any and , there exists such that , where and . In addition, in terms of Proposition 7, we know that is a single-valued mapping. Assume that is -inverse strongly accretive with . Then by Lemma 17, we conclude that the mapping is nonexpansive.
Without loss of generality, we may assume that and . Let and let be sufficiently large such that . Then such that for any and . Let , , , . Then as , and are -uniformly continuous on , for , , and , there exist , , such that for any and imply and , respectively.
Let us show that for all . We show this by induction. First, by construction. Assume that . We show that . If possible we assume that , then . Further from (35) it follows that and hence
which immediately yields
Since is -strongly accretive with respect to and is accretive, we deduce from (41) that
Again from (35), we have that
Also, from Proposition 7, is a single-valued mapping; that is, for any and , we have and . On the other hand, it follows from Nadler [24] that, for and , there exist and such that respectively. Therefore, from (42) and (36), we have
So, we get , a contradiction. Therefore, is bounded.
Let us show that and .
Indeed, we define by for all . Then, is a nonexpansive mapping and the iterative scheme (35) can be rewritten as follows:
Taking into account condition (iv), we may assume that for some . From (47), we can rewrite by where . Now, we have where for some . By simple calculation, we have
So, from (49), we get
Also, for convenience, we write
By simple calculation, we get
From (51) and (53), we deduce that and hence where for some . Utilizing Lemma 17, we conclude from (55), conditions (i), (ii), and (vi), and the assumption on that
Furthermore, utilizing Lemma 16, we obtain from (39) and (47) that