# A Hybrid Proximal Point Three-Step Algorithm for Nonlinear Set-Valued Quasi-Variational Inclusions System Involving -Accretive Mappings

- Hong-Gang Li
^{1}Email author, - AnJian Xu
^{1}and - MaoMing Jin
^{2}

**2010**:635382

**DOI: **10.1155/2010/635382

© Hong-Gang Li et al. 2010

**Received: **29 December 2009

**Accepted: **7 April 2010

**Published: **16 May 2010

## Abstract

The main purpose of this paper is to introduce and study a new class of generalized nonlinear set-valued quasi-variational inclusions system involving -accretive mappings in Banach spaces. By using the resolvent operator due to Lan-Cho-Verma associated with -accretive mappings and the matrix analysis method, we prove the convergence of a new hybrid proximal point three-step iterative algorithm for this system of set-valued variational inclusions and an existence theorem of solutions for this kind of the variational inclusions system. The results presented in this paper generalize, improve, and unify some recent results in this field.

## 1. Introduction

The variational inclusion, which was introduced and studied by Hassouni and Moudafi [1], is a useful and important extension of the variational inequality. It provides us with a unified, natural, novel, innovative, and general technique to study a wide class of problems arising in different branches of mathematical and engineering sciences. Various variational inclusions have been intensively studied in recent years. Ding and Luo [2], Verma [3, 4], Huang [5], Fang et al. [6], Fang and Huang [7], Fang et al. [8], Lan et al. [9], Zhang et al. [10] introduced the concepts of -subdifferential operators, maximal -monotone operators, -monotone operators, -monotone operators, -monotone operators, -accretive mappings, -monotone operators, and defined resolvent operators associated with them, respectively. Moreover, by using the resolvent operator technique, many authors constructed some approximation algorithms for some nonlinear variational inclusions in Hilbert spaces or Banach spaces. Verma has developed a hybrid version of the Eckstein-Bertsekas [11] proximal point algorithm, introduced the algorithm based on the -maximal monotonicity framework [12], and studied convergence of the algorithm. For the past few years, many existence results and iterative algorithms for various variational inequalities and variational inclusion problems have been studied. For details, please see [1–37] and the references therein.

On the other hand, some new and interesting problems for systems of variational inequalities were introduced and studied. Peng and Zhu [14], Cohen and Chaplais [15], Bianchi [16], and Ansari and Yao [17] considered a system of scalar variational inequalities. Ansari et al. [18] introduced and studied a system of vector equilibrium problems and a system of vector variational inequalities using a fixed point theorem. Allevi et al. [19] considered a system of generalized vector variational inequalities and established some existence results with relative pseudomonotonicity. Kassay and Kolumbán [20] introduced a system of variational inequalities and proved an existence theorem through the Ky Fan lemma. Kassay et al. [21] studied Minty and Stampacchia variational inequality systems with the help of the Kakutani-Fan-Glicksberg fixed point theorem. J. K. Kim and D. S. Kim [22] introduced a new system of generalized nonlinear quasi-variational inequalities and obtained some existence and uniqueness results on solutions for this system of generalized nonlinear quasi-variational inequalities in Hilbert spaces. Cho et al. [23] introduced and studied a new system of nonlinear variational inequalities in Hilbert spaces. They proved some existence and uniqueness theorems for solutions for the system of nonlinear variational inequalities. As generalizations of a system of variational inequalities, Agarwal et al. [24] introduced a system of generalized nonlinear mixed quasi-variational inclusions and investigated the sensitivity analysis of solutions for this system of generalized nonlinear mixed quasi-variational inclusions in Hilbert spaces. Kazmi and Bhat [25] introduced a system of nonlinear variational-like inclusions and gave an iterative algorithm for finding its approximate solution. Fang and Huang [26], Fang et al. [8] introduced and studied a new system of variational inclusions involving H-monotone operators and -monotone operators, respectively. Yan et al. [27] introduced and studied a system of set-valued variational inclusions which is more general than the model in [3].

Inspired and motivated by recent research work in this field, in this paper, a general set-valued quasi-variational inclusions system with -accretive mappings is studied in Banach spaces, which includes many variational inclusions (inequalities) as special cases. By using the resolvent operator associated with -accretive operator due to Lan, an existence theorem of solution for this class of variational inclusions is proved, and a new hybrid proximal point algorithm is established and suggested, and the convergence of iterative sequences generated by the algorithm is discussed in -uniformly smooth Banach spaces. The results presented in this paper generalize, and unify some recent results in this field.

## 2. Preliminaries

where is a constant.

Remark 2.1.

In particular, is the usual normalized duality mapping. It is known that, for all , is single-valued if is strictly convex [10], or is uniformly smooth (Hilbert space and space are uniformly Banach space), and if , the Hilbert space, then becomes the identity mapping on . In what follows we always denote the single-valued generalized duality mapping by in real uniformly smooth Banach space unless otherwise states.

Let us recall the following results and concepts.

Definition 2.2.

Definition 2.3.

A single-valued mapping is said to be

(ii)strictly accretive if is accretive and if and only if for all ;

Definition 2.4.

A set-valued mapping is said to be

where is the Hausdorff metric on

where .

Definition 2.5.

Let be a single-valued mapping and be a set-valued mapping . For , a single-valued mapping is said to be

In a similar way, we can define Lipschitz continuity and -relaxed cocoercive with respect to of in the second, or the three argument.

Definition 2.6.

- (i)accretive if(215)

(iv) -accretive if is accretive and for all ;

(v) -accretive if is -relaxed -accretive and for every .

Based on [9], we can define the resolvent operator as follows.

Definition 2.7 (see [9]).

where is a constant.

Remark 2.8.

The -accretive mappings are more general than -monotone mappings, -accretive mappings, -monotone operators, -subdifferential operators, and -accretive mappings in Banach space or Hilbert space, and the resolvent operators associated with -accretive mappings include as special cases the corresponding resolvent operators associated with -monotone operators, -accretive mappings, -accretive mappings, -monotone operators, -subdifferential operators [5, 6, 11, 14, 15, 26, 27, 35–37].

Lemma 2.9 (see [9]).

where , , and .

In the study of characteristic inequalities in -uniformly smooth Banach spaces , Xu [29] proved the following result.

Lemma 2.10 (see [29]).

Theorem 2.11.

Proof.

This completes the proof.

Corollary 2.12.

Proof.

The proof directly follows from the (i) in the Theorem 2.11.

Definition 2.13 (see [38]).

is called the -norm, and -norm, respectively.

Obviously, may be a Banach space on real field , which is called the real matrix-Banach space.

Definition 2.14 (see [38]).

then the matrix is called the limit matrix of matrix sequence , noted by , where is a real sequence, , and , .

Lemma 2.15 (see [38]).

Hence, if , then

In this paper, the matrix norm symbol is noted by .

Definition 2.16.

Let be real numbers, and and be two real vectors, then if and only if .

## 3. Quasi-Variational Inclusions System Problem and Hybrid Proximal Point Algorithm

where is a constant. Now, we consider the following generational nonlinear set-valued quasi-variational inclusions system problem with -accretive mappings (GNSVQVIS) problem.

Let , , and be single-valued mappings for . Let be a set-valued -accretive mapping and be set-valued mappings for .

where

Remark 3.1.

- (i)If , , , , and is a Hilbert space, then the problem (3.2) reduces to the problem associated with the system of variational inclusions with -monotone operators, which is finding such that , , , , , , , , , and(33)

where

- (ii)If , , (Hilbert space) and, , where is a proper, -subdifferentiable functional and denotes the -subdifferential operator of , then problem(3.3) changes to the problem associated with the following system of variational-like inequalities, which is finding such that(34)

- (iii)If , , , (Hilbert space) and , where is a proper, convex, lower semicontinuous functional and denotes the subdifferential operator of , then problem (3.3) changes to the problem associated with the following system of variational inequalities, which is finding such that(35)

- (iv)If , (Hilbert space), and , where is a nonempty, closed, and convex subsets and denotes the indicator of , then problem (3.5) reduces to the problem associated with the following system of variational inequalities, which is finding such that(36)

- (v)If , and is a Hilbert space, is a nonempty, closed and convex subset, for all , , where is a mapping on , is a constant, then problem (3.6) changes to the following problem: find such that(37)

where

Moreover, if , problem (3.7) becomes the problem introduced and studied by Verma [31].

We can see that problem (3.2) includes a number of known classes of system of variational inequalities and variational inclusions as special cases (see, e.g., [2–9, 11–27, 29, 32–37]). It is worth noting that problems (3.2)–(3.7) are all new mathematical models.

Theorem 3.2.

Let be a Banach space, be -Lipschtiz continuous mapping, be an -strongly -accretive mapping, and be a set-valued -accretive mapping for . Then the following statements are mutually equivalent.

(i)An element is a solution of problem (3.2), .

where is a constant and , respectively.

Proof.

This directly follows from definition of and the problem (3.2) for .

Algorithm 3.3.

*,*and be three nonnegative sequences such that

Step 1.

for .

Step 2.

for and .

Remark 3.4.

If we choose suitable some operators , and space , then Algorithm 3.3 can be degenerated to a number of known algorithms for solving the system of variational inequalities and variational inclusions (see, e.g., [2–9, 11–27, 29, 31–35, 38, 39]).

## 4. Existence and Convergence

In this section, we prove the existence of solutions for problem (3.2) and the convergence of iterative sequences generated by Algorithm 3.3.

Theorem 4.1.

where is the same as in Lemma 2.10, , and . Then the problem (3.2) has a solution .

Proof.

for , where is a constant. Thus, by Theorem (3.3), we know that is solution of problem (3.2). This completes the proof.

Corollary 4.2.

where is the same as in Lemma 2.10, , and . Then problem (3.2) has a solution .

Remark 4.3.

For a suitable choice of the mappings , we can obtain several known results in [2–5, 9, 11–27, 29, 32–37] as special cases of Theorem 4.1 and Corollary 4.2.

## Declarations

### Acknowledgment

The authors acknowledge the support of the Educational Science Foundation of Chongqing, Chongqing (KJ091315).

## Authors’ Affiliations

## References

- Hassouni A, Moudafi A:
**A perturbed algorithm for variational inclusions.***Journal of Mathematical Analysis and Applications*1994,**185**(3):706–712. 10.1006/jmaa.1994.1277MathSciNetView ArticleMATHGoogle Scholar - Ding XP, Luo CL:
**Perturbed proximal point algorithms for general quasi-variational-like inclusions.***Journal of Computational and Applied Mathematics*2000,**113**(1–2):153–165. 10.1016/S0377-0427(99)00250-2MathSciNetView ArticleMATHGoogle Scholar - Verma RU:
**-monotonicity and applications to nonlinear variational inclusion problems.***Journal of Applied Mathematics and Stochastic Analysis*2004, (2):193–195.Google Scholar - Verma RU:
**Projection methods, algorithms, and a new system of nonlinear variational inequalities.***Computers & Mathematics with Applications*2001,**41**(7–8):1025–1031. 10.1016/S0898-1221(00)00336-9MathSciNetView ArticleMATHGoogle Scholar - Huang N-J:
**Nonlinear implicit quasi-variational inclusions involving generalized -accretive mappings.***Archives of Inequalities and Applications*2004,**2**(4):413–425.MathSciNetMATHGoogle Scholar - Fang YP, Cho YJ, Kin JK:
**-accretive operators and approximating solutions for systems of variational inclusions in Banach spaces.**to appear in*Applied Mathematics Letters* - Fang Y-P, Huang N-J:
**-accretive operators and resolvent operator technique for solving variational inclusions in Banach spaces.***Applied Mathematics Letters*2004,**17**(6):647–653. 10.1016/S0893-9659(04)90099-7MathSciNetView ArticleMATHGoogle Scholar - Fang Y-P, Huang N-J, Thompson HB:
**A new system of variational inclusions with -monotone operators in Hilbert spaces.***Computers & Mathematics with Applications*2005,**49**(2–3):365–374. 10.1016/j.camwa.2004.04.037MathSciNetView ArticleMATHGoogle Scholar - Lan H-Y, Cho YJ, Verma RU:
**Nonlinear relaxed cocoercive variational inclusions involving -accretive mappings in Banach spaces.***Computers & Mathematics with Applications*2006,**51**(9–10):1529–1538. 10.1016/j.camwa.2005.11.036MathSciNetView ArticleMATHGoogle Scholar - Zhang Q-B, Ding X-P, Cheng C-Z:
**Resolvent operator technique for generalized implicit variational-like inclusion in Banach space.***Journal of Mathematical Analysis and Applications*2010,**361**(2):283–292. 10.1016/j.jmaa.2006.01.090MathSciNetView ArticleMATHGoogle Scholar - Eckstein J, Bertsekas DP:
**On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators.***Mathematical Programming*1992,**55**(1–3):293–318.MathSciNetView ArticleMATHGoogle Scholar - Verma RU:
**A hybrid proximal point algorithm based on the -maximal monotonicity framework.***Applied Mathematics Letters*2008,**21**(2):142–147. 10.1016/j.aml.2007.02.017MathSciNetView ArticleMATHGoogle Scholar - Shim SH, Kang SM, Huang NJ, Cho YJ:
**Perturbed iterative algorithms with errors for completely generalized strongly nonlinear implicit quasivariational inclusions.***Journal of Inequalities and Applications*2000,**5**(4):381–395. 10.1155/S1025583400000205MathSciNetMATHGoogle Scholar - Peng J-W, Zhu D-L:
**Three-step iterative algorithm for a system of set-valued variational inclusions with -monotone operators.***Nonlinear Analysis: Theory, Methods & Applications*2008,**68**(1):139–153. 10.1016/j.na.2006.10.037MathSciNetView ArticleMATHGoogle Scholar - Cohen G, Chaplais F:
**Nested monotony for variational inequalities over product of spaces and convergence of iterative algorithms.***Journal of Optimization Theory and Applications*1988,**59**(3):369–390. 10.1007/BF00940305MathSciNetView ArticleMATHGoogle Scholar - Bianchi M:
*Pseudo P-monotone operators and variational inequalities.*Istituto di Econometria e Matematica per le Decisioni Economiche, Universita Cattolica del Sacro Cuore, Milan, Italy; 1993.Google Scholar - Ansari QH, Yao J-C:
**A fixed point theorem and its applications to a system of variational inequalities.***Bulletin of the Australian Mathematical Society*1999,**59**(3):433–442. 10.1017/S0004972700033116MathSciNetView ArticleMATHGoogle Scholar - Ansari QH, Schaible S, Yao JC:
**System of vector equilibrium problems and its applications.***Journal of Optimization Theory and Applications*2000,**107**(3):547–557. 10.1023/A:1026495115191MathSciNetView ArticleMATHGoogle Scholar - Allevi E, Gnudi A, Konnov IV:
**Generalized vector variational inequalities over product sets.***Nonlinear Analysis: Theory, Methods & Applications*2001,**47**(1):573–582. 10.1016/S0362-546X(01)00202-4MathSciNetView ArticleMATHGoogle Scholar - Kassay G, Kolumbán J:
**System of multi-valued variational inequalities.***Publicationes Mathematicae Debrecen*2000,**56**(1–2):185–195.MathSciNetMATHGoogle Scholar - Kassay G, Kolumbán J, Páles Z:
**Factorization of Minty and Stampacchia variational inequality systems.***European Journal of Operational Research*2002,**143**(2):377–389. 10.1016/S0377-2217(02)00290-4MathSciNetView ArticleMATHGoogle Scholar - Kim JK, Kim DS:
**A new system of generalized nonlinear mixed variational inequalities in Hilbert spaces.***Journal of Convex Analysis*2004,**11**(1):235–243.MathSciNetMATHGoogle Scholar - Cho YJ, Fang YP, Huang NJ, Hwang HJ:
**Algorithms for systems of nonlinear variational inequalities.***Journal of the Korean Mathematical Society*2004,**41**(3):489–499.MathSciNetView ArticleMATHGoogle Scholar - Agarwal RP, Cho YJ, Huang NJ:
**Sensitivity analysis for strongly nonlinear quasi-variational inclusions.***Applied Mathematics Letters*2000,**13**(6):19–24. 10.1016/S0893-9659(00)00048-3MathSciNetView ArticleMATHGoogle Scholar - Kazmi KR, Bhat MI:
**Iterative algorithm for a system of nonlinear variational-like inclusions.***Computers& Mathematics with Applications*2004,**48**(12):1929–1935. 10.1016/j.camwa.2004.02.009MathSciNetView ArticleMATHGoogle Scholar - Fang YP, Huang NJ:
**-monotone operators and system of variational inclusions.***Communications on Applied Nonlinear Analysis*2004,**11**(1):93–101.MathSciNetMATHGoogle Scholar - Yan W-Y, Fang Y-P, Huang N-J:
**A new system of set-valued variational inclusions with -monotone operators.***Mathematical Inequalities & Applications*2005,**8**(3):537–546.MathSciNetView ArticleMATHGoogle Scholar - Zou Y-Z, Huang N-J:
**-accretive operator with an application for solving variational inclusions in Banach spaces.***Applied Mathematics and Computation*2008,**204**(2):809–816. 10.1016/j.amc.2008.07.024MathSciNetView ArticleMATHGoogle Scholar - Xu HK:
**Inequalities in Banach spaces with applications.***Nonlinear Analysis: Theory, Methods & Applications*1991,**16**(12):1127–1138. 10.1016/0362-546X(91)90200-KMathSciNetView ArticleMATHGoogle Scholar - Zou Y-Z, Huang N-J:
**A new system of variational inclusions involving -accretive operator in Banach spaces.***Applied Mathematics and Computation*2009,**212**(1):135–144. 10.1016/j.amc.2009.02.007MathSciNetView ArticleMATHGoogle Scholar - Verma RU:
**Generalized system for relaxed cocoercive variational inequalities and projection methods.***Journal of Optimization Theory and Applications*2004,**121**(1):203–210.MathSciNetView ArticleMATHGoogle Scholar - Chang S-S, Cho YJ, Zhou H:
*Iterative Methods for Nonlinear Operator Equations in Banach Spaces*. Nova Science, Huntington, NY, USA; 2002:xiv+459.MATHGoogle Scholar - Weng X:
**Fixed point iteration for local strictly pseudo-contractive mapping.***Proceedings of the American Mathematical Society*1991,**113**(3):727–731. 10.1090/S0002-9939-1991-1086345-8MathSciNetView ArticleMATHGoogle Scholar - Agarwal RP, Huang N-J, Tan M-Y:
**Sensitivity analysis for a new system of generalized nonlinear mixed quasi-variational inclusions.***Applied Mathematics Letters*2004,**17**(3):345–352. 10.1016/S0893-9659(04)90073-0MathSciNetView ArticleMATHGoogle Scholar - Huang N-J, Fang Y-P:
**A new class of general variational inclusions involving maximal -monotone mappings.***Publicationes Mathematicae Debrecen*2003,**62**(1–2):83–98.MathSciNetMATHGoogle Scholar - Jin M-M:
**Perturbed algorithm and stability for strongly nonlinear quasi-variational inclusion involving -accretive operators.***Mathematical Inequalities & Applications*2006,**9**(4):771–779.MathSciNetView ArticleMATHGoogle Scholar - Peng J, Yang X:
**On existence of a solution for the system of generalized vector quasi-equilibrium problems with upper semicontinuous set-valued maps.***International Journal of Mathematics and Mathematical Sciences*2005, (15):2409–2420. - Horn RA, Johnson CR:
*Matrix Analysis*. Cambridge University Press, Cambridge, UK; 1985:xiii+561.View ArticleMATHGoogle Scholar - Nadler SB Jr.:
**Multi-valued contraction mappings.***Pacific Journal of Mathematics*1969,**30:**475–488.MathSciNetView ArticleMATHGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. 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.