ISRN Applied MathematicsVolume 2011 (2011), Article ID 709715, 21 pagesdoi:10.5402/2011/709715

Research Article

--Accretive Mappings and a New System of Generalized Variational Inclusions with --Accretive Mappings in Banach Spaces

Sayyedeh Zahra Nazemi

Department of Mathematics, Yasouj University, Yasouj 75914, Iran

Received 5 June 2011; Accepted 29 June 2011

Academic Editors: Y.-K. Chang and Z. Hou

Copyright © 2011 Sayyedeh Zahra Nazemi. 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 a new class of generalized accretive mappings, named --accretive mappings, in Banach spaces. We define a resolvent operator associated with --accretive mappings and show its Lipschitz continuity. We also introduce and study a new system of generalized variational inclusions with --accretive mappings in Banach spaces. By using the resolvent operator technique associated with --accretive mappings, we construct a new iterative algorithm for solving this system of generalized variational inclusions in Banach spaces. We also prove the existence of solutions for the generalized variational inclusions and the convergence of iterative sequences generated by algorithm. Our results improve and generalize many known corresponding results.

1. Introduction

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 mechanics, physics, optimization and control, nonlinear programming, economics and transportation equilibrium, and engineering sciences, and so forth (see, e.g., [1–4]).

Recently Ding [5], Huang and Fang [6], Verma [7], Fang and Huang [8, 9], Huang and Fang [10], Fang et al. [11], Kazmi and Khan [12], and Lan et al. [13, 14] introduced the concepts of -subdifferential operators, maximal -monotone operators, A-monotone operators, and -monotone operators in Hilbert spaces, H-accretive operators, generalized m-accretive mappings, -accretive operators, P--accretive operators, and -accretive mappings in Banach spaces, and their resolvent operators, respectively. In [15], Luo and Huang introduced a new concept of --monotone mappings in Banach spaces and defined the proximal mapping associated with --monotone mappings.

Motivated and inspired by the research work going on this field, in this paper, we introduce a new concept of --accretive mappings and give the definition of its resolvent operator in Banach spaces. We also introduce and study a new system of generalized variational inclusions with --accretive mappings in Banach spaces, and we construct a new iterative algorithm for solving this system of generalized variational inclusions in Banach spaces. We also prove the existence of solutions for the generalized variational inclusions and the convergence of iterative sequences generated by algorithm. The results in this paper improve and extend some known results in the literature.

2. Preliminaries

Let be a real Banach space equipped with norm ; let be the topological dual space of ; let be the pair between and ; let be the power set of ; let be the Hausdorff metric on defined by

Definition 2.1 (see [16, 17]). For , a mapping is said to be generalized duality mapping if it is defined by In particular, is the usual normalized duality mapping on .It is well known that Note that if is a real Hilbert space, then becomes the identity mapping on .

Definition 2.2 (see [18]). A Banach space is called smooth, if for every with , there exists a unique such that . The modulus of is the function , defined by

Definition 2.3 (see [17]). The Banach space is said to be(i)uniformly smooth, if (ii)q-uniformly smooth, for , if there exists a constant such that It is well known (see [16]) thatNote that if is uniformly smooth, becomes single-valued, and is single-valued if is strictly convex. In the sequel, unless otherwise specified, we always suppose that is a real Banach space such that is single-valued.

Lemma 2.4 (see [19]). Let be a real number and let be a smooth Banach space. Then is q-uniformly smooth if and only if there exists a constant such that for every ,

Definition 2.5 (see [9, 20]). Let be a single-valued mapping. is said to be(i)accretive if (ii)strictly accretive if P is accretive and (iii)r-strongly accretive if there exists a constant such that (iv)m-relaxed accretive if there exists a constant such that (v)-relaxed cocoercive if there exist constants such that (vi)s-Lipschitz continuous if there exists a constant such that

Definition 2.6 (see [12]). A mapping is said to be -Lipschitz continuous if there exists a constant such that

Definition 2.7 (see [20]). Let and be single-valued mappings. Then a multi-valued mapping is said to be(i)-accretive if (ii)strictly -accretive if and equality holds if and only if ;(iii)-strongly -accretive if there exists a constant such that (iv)m-relaxed -accretive if there exists a constant such that (v)--accretive if is -accretive and holds for all ;(vi)--accretive if is -accretive and holds for all .

Definition 2.8. Let be real Banach spaces, and, for , let be a single-valued mapping. Then is said to be -Lipschitz continuous in the th argument, if there exists a constant such that

Definition 2.9. Let be a set-valued mapping. is said to be -Lipschitz continuous, if there exists a constant such that where denotes the Hausdorff metric on .

Definition 2.10. Let be a Hilbert space, and let be a single-valued mapping. is said to be(i)coercive if (ii)hemicontinuous if for any fixed , the function is continuous at .

3. --Accretive Mappings

In this section, we will introduce a new class of generalized accretive mappings, --accretive mappings, and discuss some properties of --accretive mappings.

Definition 3.1. Let be a Banach space and let , be single-valued mappings and a multi-valued mapping. The mapping is said to be a --accretive mapping, if is -accretive and .

Remark 3.2. (i) If is -accretive and , for all , , then --accretive mapping reduces to the --accretive mapping studied by Kazmi and Khan [12].(ii) If , for all , is -accretive, and , for all , , then --accretive mapping reduces to the -accretive mapping studied by Fang and Huang [9].(iii) If is -relaxed -accretive and , for all , , then --accretive mapping reduces to the -accretive mapping studied by Lan et al. [14].Similarly, we give the following definition.

Definition 3.3. Let be single-valued mappings and a multi-valued mapping. The mapping is said to be a -accretive mapping, if is accretive and .

Example 3.4. Let be a Hilbert space and for every , , , where is a constant. Let be a maximal monotone mapping and a bounded, coercive, hemicontinuous, and -strongly -accretive mapping. Then it follows from Corollary  32.26 of [21] that is --accretive mapping.

Theorem 3.5. Let , be single-valued mappings, a strictly -accretive mapping, and a --accretive mapping. Then is a single-valued mapping.

Proof. For any given , let . It follows that Then -accretivity of implies that This implies that and so is a single-valued mapping. This completes the proof.

By Theorem 3.5, we can define the resolvent operator associated with an -accretive mapping as follows.

Definition 3.6. Let , be single-valued mappings, let be a strictly -accretive mapping, and let be a --accretive mapping. A resolvent operator is defined by

Remark 3.7. (i) If is -accretive and , for all , , then the resolvent operator reduces to the --proximal point mapping introduced by Kazmi and Khan [12].(ii) If , for all , is -accretive, and , for all , , then the resolvent operator reduces to the proximal-point mapping introduced by Fang and Huang [9].(iii) If is -relaxed -accretive and , for all , , then the resolvent operator reduces to the resolvent operator introduced by Lan at el. [14].

Theorem 3.8. Let be a single-valued mapping, let be a -Lipschitz continuous mapping, let be a -strongly -accretive mapping, and let be a --accretive mapping. Then the resolvent operator is Lipschitz continuous with constant , that is,

Proof. Let be any given points in , it follows from Definition 3.6 that This implies that Since is --accretive, we have The inequality above implies that Since is Lipschitz continuous with a constant , we have It follows from (3.8) and (3.9) that Hence, we get This completes the proof.

4. A New System of Generalized Variational Inclusions

In this section, we will introduce a new system of generalized variational inclusions with --accretive mappings and construct a new iterative algorithm for solving this system of generalized variational inclusions. In what follows, for each , suppose that is a Banach space, , , are single-valued mappings, is a set-valued mapping, and is a --accretive mapping in the second argument. Assume that , for each . We consider the following system of generalized variational inclusions. Find such that for each , , , , , and The following are some special cases of problem (4.1).

(i) If , , and are three Hilbert spaces, and, for each , , , , for all , , for all , and , for all , where is a proper lower semicontinuous and -subdifferential function, is the -subdifferential of at , is the -subdifferential of at , and is the -subdifferential of at , then SGVIP (4.1) reduces to the following system of variational inequalities, which is to find such that

If , for all , , for all , , for all , and is the subdifferential of at , is the subdifferential of at , and is the subdifferential of at , then problem (4.2) reduces to the following system of variational inequalities, which is to find such that

If , , and , for all , , and , where , , and are three nonempty, closed, and convex subsets, , , and denote the indicator functions of , , and , respectively, then problem (4.3) reduces to the following system of variational inequalities, which is to find such that

If is a Hilbert space, is a nonempty, closed, and convex subset, , , and , for all , where are mappings on , are three numbers, then problem (4.4) reduces to the following system of variational inequalities, which is to find such that Problem (4.5) was introduced and studied by Cho and Qin [22].

(ii) If , , , and and for each , , , , for all , then SGVIP (4.1) reduces to the following system of variational inclusions, which is to find such that

If and are two Hilbert spaces and , for all , , for all , then the problem (4.6) reduces to the following system of variational inequalities, which is to find such that

If , for all , , for all , is the subdifferential of at , and is the subdifferential of at , then the problem (4.7) reduces to the following system of variational inequalities, which is to find such that Problem (4.8) was introduced and studied by Cho et al. [23].

If , , for all , , where , are two nonempty, closed, and convex subsets and and denote the indicator functions of , , respectively, then problem (4.8) reduces to the following system of variational inequalities, which is to find such that Problem (4.9) is just the problem [24] with and being single-valued.

If is a Hilbert space and is a nonempty, closed, and convex subset, and , for all , where is a mapping on , are two numbers, then problem (4.9) reduces to the following problem, which is to find such that Problem (4.10) was introduced and studied by Verma [25].

Lemma 4.1. Let, for , be a single-valued mapping satisfying and , a single-valued mapping, a strictly -accretive mapping and a --accretive mapping in the second argument. Then in which , , , is a solution of the problem (4.1) if and only if where .

Proof. The fact directly follows from Definition 3.6.

Algorithm 4.2. For any , take , , and . For , let Since , , , by Nodler's theorem [26], there exist , , , such that, for each , For , let Again by Nodler's theorem [26], there exist , , , such that, for each , By induction, we can compute the sequences , , , by the following iterative schemes such that, for each , for all .

Now we prove the existence of solution of the SGVIP (4.1) and the convergence of Algorithm 4.2.

Theorem 4.3. Let, for , be a q-uniformly smooth Banach space and let be a -Lipschitz continuous mapping satisfying and . Let be -Lipschitz continuous, let be a -strongly -accretive and -Lipschitz continuous mapping, be -relaxed cocoercive and -Lipschitz continuous, and let be a strongly accretive mapping with constant and Lipschitz continuous with constant . Suppose that is -Lipschitz continuous in the kth argument and be -Lipschitz continuous in the kth argument for , is a --accretive mapping in the second argument, and set-valued mappings , , are -Lipschitz continuous with constants , , , respectively. In addition if for all , one has Then the problem (4.1) admits a solution and sequences converge to , , , , , , , , , , respectively, where are sequences generated by Algorithm 4.2.

Proof. For , let By Algorithm 4.2 and (4.17), we have Since is -relaxed cocoercive and -Lipschitz continuous and is -strongly accretive and -Lipschitz continuous, we have From (4.19), we have Since is -Lipschitz continuous in the argument and by continuity of , , , we have It follows from the Lipschitz continuity of and the -Lipschitz continuity of that It follows from (4.20)–(4.24) that, for each , Therefore, where Let We know that as .Define on by It is easy to see that is a Banach space. Define . Then we have It follows from (4.18) that , and hence there exists an and such that , for all . Therefore, by (4.26) and (4.30), we have Hence, for any , it follows that Since , it follows from (4.32) that as , and hence is a Cauchy sequence in . By the same argument, we also have is a Cauchy sequence in . Thus, there exist , , such that , , as .Now we prove that , , . In fact, it follows from the Lipschitz continuity of , , that for, , From (4.33), we have that , , are also Cauchy sequences. Therefore, there exist , , such that , , as . Further, for , Since is closed, we have . Similarly , . By continuity of , , , , , , , , and Algorithm 4.2, we know that , , , , , , , , , satisfy the following relation: By Lemma 4.1, is a solution of the SGVIP (4.1). This completes the proof.

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