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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 928950, 12 pages
http://dx.doi.org/10.1155/2014/928950
Research Article

Convergence Theorems for Accretive Operators with Nonlinear Mappings in Banach Spaces

1Department of Mathematics, Shanghai Normal University, Shanghai 200234, China
2Department of Mathematics, Shanghai Normal University and Scientific Computing Key Laboratory of Shanghai Universities, Shanghai 200234, China

Received 4 December 2013; Accepted 17 March 2014; Published 29 April 2014

Academic Editor: Satit Saejung

Copyright © 2014 Yan-Lai Song and Lu-Chuan Ceng. 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 purpose of this paper is to present two new forward-backward splitting schemes with relaxations and errors for finding a common element of the set of solutions to the variational inclusion problem with two accretive operators and the set of fixed points of strict pseudocontractions in infinite-dimensional Banach spaces. Under mild conditions, some weak and strong convergence theorems for approximating these common elements are proved. The methods in the paper are novel and different from those in the early and recent literature. Further, we consider the problem of finding a common element of the set of solutions of a mathematical model related to equilibrium problems and the set of fixed points of a strict pseudocontractions.

1. Introduction

The theory of nonexpansive mappings is very important because it is applied to convex optimization, the theory of nonlinear evolution equations, and others. Browder and Petryshyn [1] introduced a class of nonlinear mappings, called strict pseudocontractions, which includes the class of nonexpansive mappings. For strict pseudocontractions, we are interested in finding fixed points of the mappings. We also know the class of inverse-strongly accretive operators which is related to nonexpansive mappings. For inverse-strongly accretive operators, we are interested in finding zero points of the mappings.

Let be a nonempty closed convex subset of a real Hilbert space . Let be a single-valued nonlinear mapping and let be a multivalued mapping. The so called quasi-variational inclusion problem is to find a such that The set of solutions of (1) is denoted by . A number of problems arising in structural analysis, mechanics, and economics can be studied in the framework of this kind of variational inclusions; see, for instance, [25]. The problem (1) includes many problems as special cases. (1)If , where is a proper convex lower semicontinuous function and is the subdifferential and if only erential of , then the variational inclusion problem (1) is equivalent to finding such that which is called the mixed quasi-variational inequality (see, Noor [6]).(2)If , where is a nonempty closed convex subset of and is the indicator function of , that is, then the variational inclusion problem (1) is equivalent to finding such that This problem is called Hartman-Stampacchia variational inequality (see, e.g., [7]).

In [8], Zhang et al. investigated the problem of finding a common element of the set of solutions to the inclusion problem and the set of fixed points of nonexpansive mappings by considering the following iterative algorithm: where is an -cocoercive mapping, is a maximal monotone mapping, is a nonexpansive mapping, and is a sequence in . Under mild conditions, they obtained a strong convergence theorem.

In [9], Manaka and Takahashi introduced the following iteration: where is a sequence in (0, 1), is a positive sequence, is a nonexpansive mapping, is an inverse-strongly monotone mapping, is a maximal monotone operator, and is the resolvent of . They showed that the sequence generated in (6) converges weakly to some provided that the control sequence satisfies some restrictions.

It is well known that the quasi-variational inclusion problem in the setting of Hilbert spaces has been extensively studied in the literature; see, for instance, [410]. However, there is little work in the existing literature on this problem in the setting of Banach spaces (though there was some work on finding a common zero of a finite family of accretive operators [1113]). The main difficulties are due to the fact that the inner product structure of a Hilbert space fails to be true in a Banach space. To overcome these difficulties, López et al. [14] use the new technique to carry out certain initiative investigations on splitting methods for accretive operators in Banach spaces. They considered the following algorithms with errors in Banach spaces: where ,, and is the resolvent of . Then they studied the weak and strong convergence of algorithms (7) and (8), respectively.

Motivated and inspired by Zhang et al. [8], Manaka and Takahashi [9], Takahashi et al. [10], Chen and Fan [13], López et al. [14], and Cho et al. [15], the purpose of this paper is to introduce two iterative forward-backward splitting methods for finding a common element of the set of solutions of the variational inclusion problem (1) with -accretive operators and inverse-strongly accretive operators and the set of fixed points of strict pseudocontractions in the setting of Banach spaces. Under suitable conditions, some weak and strong convergence theorems for approximating to these common elements are proved. The results presented in the paper improve and extend the corresponding results in [810, 1315].

2. Preliminaries

Throughout this paper, we denote by and a real Banach space and the dual space of , respectively. Let be a subset of and let be a mapping on . We use to denote the set of fixed points of . The expressions and denote the strong and weak convergence of the sequence , respectively, and stands for the set of weak limit points of the sequence . will denote the closed ball with center zero and radius .

Let be a real number. The (generalized) duality mapping is defined by for all , where denotes the generalized duality pairing between and . In particular, is called the normalized duality mapping and for . If is a Hilbert space, then where is the identity mapping. It is well known that if is smooth, then is single-valued, which is denoted by .

A Banach space is said to be 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.

The norm of a Banach space is said to be Gâteaux differentiable if the limit exists for all on the unit sphere . If, for each , limit (10) is uniformly attained for , then the norm of is said to be uniformly Gâteaux differentiable. The norm of is said to be Fréchet differentiable if, for each , limit (10) is attained uniformly for .

Let be the modulus of smoothness of defined by

A Banach space is said to be uniformly smooth if as . Let . A Banach space is said to be -uniformly smooth, if there exists a fixed constant such that . It is well known that is uniformly smooth if and only if the norm of is uniformly Fréchet differentiable. If is -uniformly smooth, then and is uniformly smooth, and hence the norm of is uniformly Fréchet differentiable; in particular, the norm of is Fréchet differentiable. Typical examples of both uniformly convex and uniformly smooth Banach spaces are , where . More precisely, is -uniformly smooth for every .

A Banach space is said to satisfy Opial’s condition if for any sequence in the condition that converges weakly to implies that the inequality holds for every with .

A Banach space is said to have the Kadec-Klee property if, for every sequence in , and together imply . As we know the duals of reflexive Banach spaces with Fréchet differentiable norms have the Kadec-Klee property.

Definition 1. A mapping is said to be (1)nonexpansive if (2)-Lipschitz if there exists such that in particular, if , then is called contractive and if , then reduces to a nonexpansive mapping; (3)-strict pseudocontractive in the terminology of Browder and Petryshyn if for all , there exists and such that (4)accretive if for all , there exists such that (5)-strongly accretive if for all , there exists and such that (6)-inverse-strongly accretive if for all , there exists and such that

Remark 2. The conception of strict pseudocontractions was firstly introduced by Browder and Petryshyn [1] in a real Hilbert space. Let be a nonempty subset of a real Hilbert space , and let be a mapping. In light of [1], is said to be a -strict pseudocontraction, if there exists a such that

Remark 3. The class of strictly pseudocontractive mappings has been studied by several authors (see, e.g., [1, 16, 17]). However, their iterative methods are far less developed though Browder and Petryshyn [1] initiated their work in 1967. As a matter of fact, strictly pseudocontractive mappings have more powerful applications in solving inverse problems (see, e.g., [18]). Therefore it is interesting to develop the theory of iterative methods for strictly pseudocontractive mappings.

Remark 4. If is a real Hilbert space, then accretive and strongly accretive operators coincide with monotone and strongly monotone operators, respectively.

Definition 5. A set-valued mapping is said to be(1)accretive if for any , there exists , such that for all and , (2)-accretive if is accretive and for every (equivalently, for some) , where is the identity mapping. In real Hilbert spaces, -accretive operators coincide with maximal monotone operators.
Let be -accretive. The mapping defined by is called the resolvent operator associated with , where is any positive number and is the identity mapping. It is well known that is single-valued and nonexpansive.

In order to prove our main results, we need the following lemmas.

Lemma 6 (see [19]). Let be a Banach space and let be a generalized duality mapping. Then for any given , the following inequality holds: In particular, we have, for any given ,

Lemma 7 (see [19]). Let ,, be given.(i)If is uniformly convex, then there exists a continuous, strictly increasing, and convex function with such that where ,.(ii)If is a real -uniformly smooth Banach space, then there exists a constant such that

Lemma 8 (see [20]). Let ,, and be sequences of nonnegative real numbers satisfying the inequality If and , then exists. In particular, whenever there exists a subsequence in which strongly converges to zero.

Lemma 9 (see [21]). Let be a sequence of nonnegative numbers satisfying the following property: where ,, satisfy the restrictions (i),(ii),,(iii).Then, .

Lemma 10 (see [16]). Let be a nonempty convex subset of a real -uniformly smooth Banach space and let be a -strict pseudocontraction. For , we define . Then, as , , is nonexpansive such that .

Lemma 11 (see [22]). Let be a uniformly convex Banach space, a closed convex subset of , and a nonexpansive mapping with . Then, is demiclosed at zero.

Lemma 12. Let be a nonempty closed convex subset of a real -uniformly smooth Banach space . Let the mapping be an -inverse-strongly accretive operator. Then the following inequality holds: In particular, if , then is nonexpansive.

Proof. Indeed, for all , it follows from Lemma 7 that It is clear that if , then is nonexpansive. This completes the proof.

Lemma 13 (see [23]). If is a uniformly convex Banach space and is a closed convex bounded subset of , there is a continuous strictly increasing function with such that for all , and nonexpansive mapping .

Lemma 14 (see [24]). Let be a real reflexive Banach space such that its dual has the Kadec-Klee property. Let be a bounded sequence in and ; here denotes the weak -limit set of . Suppose exists for all . Then .

Lemma 15. Assume that is a real uniformly convex and -uniformly smooth Banach space. Suppose that is -inverse-strongly accretive operator for some and is an -accretive operator. Moreover, denote by and by Then, it holds for all that .

Proof. From the definition of , we have This completes the proof.

Lemma 15 alludes to the fact that, in order to solve the inclusion problem (1), it suffices to find a fixed point of . Since is already split, an iterative algorithm for corresponds to a splitting algorithm for (1). However, to guarantee convergence (weak or strong) of an iterative algorithm for , we need good metric properties of such as nonexpansivity. To this end, some related geometric conditions on the underlying space are very necessary (see Lemmas 16 and 17 below).

Lemma 16 (see [14]). Assume that is a real uniformly convex and -uniformly smooth Banach space. Suppose that is -inverse-strongly accretive operator for some and is an -accretive operator. Then, the following relations hold.(i)Given and , (ii)Given , there exists a continuous, strictly increasing, and convex function with such that, for all ,

Lemma 17. Let be a real uniformly convex and -uniformly smooth Banach space. Suppose that is a nonexpansive mapping, is an -inverse-strongly accretive operator for some , and is an -accretive operator. Assume that . Then .

Proof. Suppose that ; it is sufficient to show that . Indeed, for , we have by Lemma 16 that The property of and the condition together imply that It turns out that which imply Noticing the assumption of , we can deduce . This means that .

Lemma 18 (see [25]). Let be a nonempty, closed, and convex subset of a real -uniformly smooth Banach space . Let be a -Lipschitz and -strongly accretive operator with constants . Let and . Then for , the mapping defined by is a contraction with a constant .

Next we give a weak convergence theorem in a Banach space .

3. Main Results

Theorem 19. Let be a uniformly convex and -uniformly smooth Banach space. Let be -inverse-strongly accretive, -accretive, and -strict pseudocontractive. Assume that . Define a mapping for all . For arbitrarily given and , where , let be the sequence generated iteratively by where ,,, and . Assume that (i)and(ii)(iii). Then converges weakly to some point .

Proof. We divide the proof into several steps.
Step  1. We prove exists for any point .
Putting , one has where Then the iterative formula (40) turns into the form Thus, by virtue of Lemmas 10 and 12 and nonexpansivity of , we have By (44) and condition (i), we have that Since , according to Lemmas 10 and 15, we can deduce . Lemma 16 and condition (iii) together imply is nonexpansive. Therefore, we get from (43) that In view of (45), (46), and Lemma 8, we get that exists. Therefore is bounded.
Step  2. We show .
Let be such that , for all and let . By (43), Lemmas 6, 10, and 16, we have Meanwhile, by the fact that , and (47), we get that Thanks to (45), existence of , (ii) and (iii), one has It turns out that Step  3. We prove .
Noticing (45) and Lemma 7, we have which implies where . From (45), (52), (ii), and existence of , it turns out that It follows from the property of and (45) that
Step  4. We prove .
Since , there exists such that for all . Then, by Lemma 16, we have It follows from (50), (54), and (55) that By Lemmas 10, 11, and 17 and (56), we get Step  5. We show converges weakly to a fixed point of .
Indeed, it suffices to show consists of exactly only one point. To this end, we suppose that two different points and are in . Then there exist two different subsequences and such that and as and . Define by Then can be written where Thanks to the nonexpansivity of , we have It follows from (45) that Let Applying Lemma 13 to the closed convex bounded subset , we obtain Since exists, (62), (64), and the property of together imply that Furthermore, we have After taking first and then in (66) and using (62) and (65), we get So exists for all . It follows from Lemma 14 that . This completes the proof.

Remark 20. Compared with the known results in the literature, our results are very different from those in the following aspects. (i)Theorem 19 improves and extends Theorem 3 of Kamimura and Takahashi [4] and Theorem 3.1 of Manaka and Takahashi [9] from Hilbert spaces to uniformly convex and -uniformly smooth Banach spaces.(ii)Theorem 19 also improves and extends Theorem 3.6 of López et al. [14] from the problem of finding an element of to the problem of finding an element of , where is -strictly pseudocontractive on .
In the following, we give a strong convergence theorem in a Banach space .

Theorem 21. Let be a uniformly convex and -uniformly smooth Banach space which admits a weakly sequentially continuous generalized duality mapping . Let be -inverse-strongly accretive, -accretive, -Lipschitz and -strongly accretive, -Lipschitz, and -strictly pseudocontractive. Define a mapping for all . For arbitrarily given and , where , let be the sequence generated iteratively by Assume that , and satisfying the following conditions: (i)and,(ii),and(iii)and. Suppose in addition that , and , where . Then converges strongly to some point which solves the variational inequality: for all.

Proof. Let be a sequence generated by where . We show .
It follows from Lemmas 10, 12, and 18 that By virtue of Lemma 8, (i), and (70), we have .
Hence, to show the desired result, it suffices to prove that .
Step  1. We prove that the sequence is bounded. Taking , it follows from Lemmas 10, 12, 15, and 16 and condition (iii) that By induction, we have Hence, is bounded, and so are and .
Step  2. We prove that Putting , it follows from Lemma 16 that where , , and . Hence from (69) and (74) we have where . It follows from Lemma 9, (ii), and (iii) that .
Again from Lemmas 6 and 16, we obtain