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Abstract and Applied Analysis
Volume 2014, Article ID 659870, 11 pages
http://dx.doi.org/10.1155/2014/659870
Research Article

Stable Perturbed Iterative Algorithms for Solving New General Systems of Nonlinear Generalized Variational Inclusion in Banach Spaces

1Department of Mathematics, Sichuan University of Science and Engineering, Zigong, Sichuan 643000, China
2Key Laboratory Higher Education of Sichuan Province for Enterprise Informationalization and Internet of Things, Zigong, Sichuan 643000, China

Received 19 June 2014; Accepted 26 July 2014; Published 14 October 2014

Academic Editor: Jong Kyu Kim

Copyright © 2014 Ting-jian Xiong and Heng-you Lan. 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 and study a new general system of nonlinear variational inclusions involving generalized -accretive mappings in Banach space. By using the resolvent operator technique associated with generalized -accretive mappings due to Huang and Fang, we prove the existence theorem of the solution for this variational inclusion system in uniformly smooth Banach space, and discuss convergence and stability of a class of new perturbed iterative algorithms for solving the inclusion system in Banach spaces. Our results presented in this paper may be viewed as an refinement and improvement of the previously known results.

1. Introduction

Let be a given positive integer, for any , a real Banach space with dual space . , all endowed with the norm , and the dual pair between and (as matter of convenience). Let denote the family of all the nonempty subsets of , , single-valued mappings, and generalized -accretive mapping for . In this paper, we consider the following new general system for nonlinear variational inclusion involving generalized -accretive mappings. Find such that for all . Some special cases of the problem (1) had been studied by many authors. See, for example, [134] and the reference therein. Here, we mention some of them as follows.

Case 1. The problem (1) with , the Hilbert spaces, was introduced and studied as general system of monotone nonlinear variational inclusions problems by Peng and Zhao [29].
If and , is proper, convex, and lower semi-continuous functional on , and denote the subdifferential operators of the for , then the problem (1) is equivalent to finding such that
When , 2-uniformly smooth Banach space with the smooth constant , is a nonempty closed convex subset of , , where and and for ; the problem (2) reduces to the following system of finding such that
Further, in the problem (3), when is the indicator function of a nonempty closed convex set , in defined by then the system (3) reduces to finding such that which was introduced and studied by Zhu et al. [34].

Case 2. If , then the system (3) is equivalent to finding such that It is easy to see that the mathematical model studied by Saewan and Kumam [31] is a variant of (6).

Case 3. If , then the problem (1) reduces to find such that Problem (7) is called a system of strongly nonlinear quasivariational inclusion involving generalized -accretive mappings, it is considered and studied by Lan [19]. There are many special cases of the problems (7) that can be found in [3, 7, 1214, 17, 20, 28, 30] and the references cited therein.

Case 4. If and , then the problem (1) reduces to finding such that which was introduced and studied by Fang and Huang [8]. We remark that for appropriate and suitable choices of positive integer , the mappings , , and , and the spaces for , one can know that the problem (1) includes a number of general class of variational character known problems, including minimization or maximization (whether constraint or not) of functions and minimax problems et al. as special cases. For more details, see [134] and the reference therein.
On the other hand, many authors discussed stability of the iterative sequence generated by the algorithm for solving the problems that they studied. Lan [19] introduced the notion of -stable or stable with respect to . Moreover, Agarwal et al. [1, 2], Jin [16], Kazmi and Bhat [18], and Lan and Kim [21] constructed some stability under suitable conditions, respectively.
Motivated and inspired by the above works, the main purpose of this paper is to introduce and study the new general system of nonlinear variational inclusions (1) involving generalized -accretive mapping in uniformly smooth Banach spaces. By using the resolvent operator technique for generalized -accretive, we prove the existence theorem of the solution for this kind of system of variational inclusions in Banach spaces and discuss the convergence and stability of a new perturbed iterative algorithm for solving this general system of nonlinear variational inclusions in Banach spaces.

2. Preliminaries

In order to get the main results of the paper, we need the following concepts and lemmas. Let be a real Banach space with dual space , the dual pair between and , and denote the family of all the nonempty subsets of . The generalized duality mapping is defined by where is a constant. In particular, is the usual normalized duality mapping. It is known that if is strictly convex or is a uniformly smooth Banach space, then is single-valued (see [33]), and if , the Hilbert space, then becomes the identity mapping on . We will denote the single-valued duality mapping by .

In order to construct convergence and stability for researching the problem (1), we need to be using the following definition and lemma.

Definition 1. Let be Banach spaces, and let be single mappings for . Then is said to be (i)-strongly accretive with respect to th argument if for any , , there exists , such that where is a constant;(ii)-Lipschitz continuous if there exists constants , , such that for all and .

Remark 2. When , is different or the same as Hilbert spaces, (i) and (ii) in Definition 1 reduce to strongly monotonicity with respect to th argument of and -Lipschitz continuity of , respectively (see [29]).

Definition 3. Let be single-valued mapping. Then set-valued mapping is said to be (i)accretive if (ii)-accretive if (iii)-accretive if is accretive and for all , where denotes the identity operator on ;(iv)generalized -accretive if is -accretive and for all .

Remark 4. When , (i)–(iv) of Definition 3 reduce to the definitions of monotone operators, -monotone operators, classical maximal monotone operators, and maximal -monotone operators; if , then (ii) and (iv) of Definition 3 reduce to the definitions of accretive and -accretive of uniformly smooth Banach spaces (see [10, 11]).

Definition 5. The mapping is said to be (i)-strongly monotone if there exists a constant such that (ii)-Lipschitz continuous if there exists a constant such that

In [10], Huang and Fang show that for any , inverse mapping is single-valued, if is strict monotone and is generalized -accretive mapping, where is the identity mapping. Based on this fact, Huang and Fang [10] gave the following definition.

Definition 6. Let be strictly monotone mapping, and let be generalized -accretive mapping. Then the resolvent for is defined as follows: where is a constant and denotes the identity mapping on for .

Lemma 7 (see [10, 11]). Let be -Lipschitz continuous and -strongly monotone, and let be generalized -accretive mapping. Then for any , the resolvent operator for is -Lipschitz continuous; that is,

The modules of smoothness is a measure, it is depicted geometric structure of the underlying Banach space. The modules of smoothness of Banach space are the function defined by A Banach space is called uniformly smooth if . is called -uniformly smooth if there exists a constant such that , where is a real number.

Remark that is single-valued if is uniformly smooth, and Hilbert space and (or ) spaces are 2-uniformly smooth Banach spaces. In what follows, we will denote the single-valued generalized duality mapping by .

In the study of characteristic inequalities in -uniformly smooth Banach spaces, Xu [35] proved the following result.

Lemma 8. Let be a given real number and let be a real uniformly smooth Banach space. Then is -uniformly smooth if and only if there exists a constant such that for all , , there holds the following inequality:

Definition 9. Let be a self-map of , , and let define an iteration procedure which yields a sequence of points in . Suppose that and converges to a fixed point of . Let and let . If implies that , then the iteration procedure defined by is said to be -stable or stable with respect to .

Lemma 10 (see [36]). Let , , and be three nonnegative real sequences satisfying the following condition: there exists a natural number such that where , , , and . Then converges to as .

3. Existence Theorem

In this section, we will give the existence theorem of the problem (1). The solvability of the problem (1) depends on the equivalence between (1) and the problem of finding the fixed point of the associated generalized resolvent operator. It follows from the definition of generalized resolvent operator that we can obtain the following conclusion.

Lemma 11. Let , single-valued mappings, and generalized -accretive mapping for . Then the following statements are mutually equivalent. (i)An element is a solution to the problem (1).(ii)There is an such that where , and is constants for all .(iii)For any given constants , the map is defined by for all and , has a fixed point , where maps are defined by for and .

Proof. We first prove that (i) (ii). Let satisfy the relation in (ii). Then, the definition of resolvent operator implies that this equality holds if and only if for ; that is where . Thus is the solution of the problem (1).
Next, we show (ii) (iii). If satisfy following relation: then, for any , it follows from that Hence, is a fixed point of the mapping Conversely, if is a fixed point of the mapping , then for . Hence, from we have for . Therefore satisfy the relation of (ii).

Theorem 12. Let be a real -uniformly smooth Banach space with and let be -Lipschitz continuous and -strongly monotone for any . Suppose that is generalized -accretive mapping, and is -strongly accretive in the th argument and -Lipschitz continuous for . If where is the constants as in Lemma 8 for , then problem (1) has a unique solution .

Proof. For any given and , we first define as follows: for all . Now define on by It is easy to see that is a Banach space. In fact (i), the negative being satisfied;(ii)for all real number , homogeneity being satisfied;(iii)for all , the triangle inequality being satisfied;(iv)let ; that is, ; this implies that ; thus ; we get is a norm on the ;(v)let is Cauchy sequence; that is, for , there exists a positive integer ; let ; we have Thus, for all , we have ; that is, is also Cauchy sequence; thus for ; we get and is a cluster point on the ; we claim is a Banach space.
Now, by (34), for any given , define mapping by where for .
In the sequel, we prove that is a contractive mapping on the . In fact, for any and , it follows from (34) and Lemma 7 that By assumptions and Lemma 8, we have From (40)-(41), we obtain for . Equation (42) implies that where . By (33), we know that . It follows from (43) that This proves that is a contraction mapping. Hence, there exists a unique such that that is, for ; that is, By Lemma 11, is the unique solution of problem (1). This completes the proof.

Remark 13. If , then Theorem 12 reduces to Theorem 3.2 of Lan [19].

Corollary 14. Let be real Hilbert space and be -Lipschitz continuous and -strongly monotone for any . Suppose that is maximal -monotone mapping, is -strongly monotone in the th argument, and -Lipschitz continuous for . If then problem (1) has a unique solution .

Corollary 15. Let be real Hilbert space for any . Suppose that is proper, convex, and lower semicontinuous functional on and is -strongly monotone in the th argument and -Lipschitz continuous for . If then problem (2) has a unique solution .

4. Perturbed Iterative Algorithms

In this section, by using Definition 9 and Lemma 10, we construct a new perturbed iterative algorithm with mixed errors for solving problem (1) and prove the convergence and stability of the iterative sequence generated by the algorithm.

Algorithm 16. Let and be single-valued mappings and let be generalized -accretive mapping for . For any given initial point , the perturbed iterative sequence for problem (1) is defined by where , , is a sequence in , and are errors to take into account a possible inexact computation of the resolvent operator point satisfying the following conditions: (i);(ii);(iii), for .
Let be any sequence in and define by for .

Algorithm 17. Let and be single-valued mappings and let be maximal -monotone mapping for . For any given initial point , the perturbed iterative sequence for problem (1) is defined by where , , is a sequence in , and are errors to take into account a possible inexact computation of the resolvent operator point satisfying the following conditions: (i);(ii) for .
Let be any sequence in and define by for .

Algorithm 18. Let be single-valued mappings and is proper, convex, and lower semi-continuous functional on for . For any given initial point , the perturbed iterative sequence for problem (2) is defined by where , , is a sequence in , are errors to take into account a possible inexact computation of the resolvent operator point satisfying the condition for . Let be any sequence in and define by for .

Remark 19. If , then Algorithm 16 reduces to Algorithm 4.3 of Lan [19].
Next we will show the convergence and stability of Algorithm 16.

Theorem 20. Suppose that , , , and are the same as in Theorem 12. If and condition (33) holds, then the perturbed iterative sequence defined by Algorithm 16 converges strongly to the unique solution of the problem (1). Moreover, if there exists for all , then if and only if where is defined by (50).

Proof. From Theorem 12, we know that problem (1) has a unique solution It follows from (49) and the proof of (42) in Theorem 12 that, for , It follows from (58), we have where is the same as in (43). Letting , , and , then it follows from and (i)–(iii) of Algorithm 16 that
Setting , then (59) can be rewritten as It follows from Lemma 10 that ; that is, thus
Hence, we know that the sequence converges strongly to the unique solution of the problem (1).
Now we prove the second conclusion. By (50), now we know where . As the proof of inequality (59), we have