# An Ishikawa-Hybrid Proximal Point Algorithm for Nonlinear Set-Valued Inclusions Problem Based on -Accretive Framework

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

**2010**:501293

**DOI: **10.1155/2010/501293

© Hong Gang Li et al. 2010

**Received: **30 April 2010

**Accepted: **8 June 2010

**Published: **28 June 2010

## Abstract

A general nonlinear framework for an Ishikawa-hybrid proximal point algorithm using the notion of -accretive is developed. Convergence analysis for the algorithm of solving a nonlinear set-valued inclusions problem and existence analysis of solution for the nonlinear set-valued inclusions problem are explored along with some results on the resolvent operator corresponding to -accretive mapping due to Lan-Cho-Verma in Banach space. The result that sequence generated by the algorithm converges linearly to a solution of the nonlinear set-valued inclusions problem with the convergence rate is proved.

## 1. Introduction

The set-valued inclusions problem, which was introduced and studied by Di Bella [1], Huang et al. [2], and Jeong [3], is a useful extension of the mathematics analysis. And the variational inclusion(inequality) is an important context in the set-valued inclusions problem. 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[4], Verma [5], Huang [6], Fang and Huang [7], Lan et al. [8], Fang et al. [9], and 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. Recently, Verma has developed a hybrid version of the Eckstein and Bertsekas [11] proximal point algorithm, introduced the algorithm based on the -maximal monotonicity framework [12], and studied convergence of the algorithm.

On the other hand, in 2008, Li [13] studied the existence of solutions and the stability of perturbed Ishikawa iterative algorithm for nonlinear mixed quasivariational inclusions involving -accretive mappings in Banach spaces by using the resolvent operator technique in [14].

Inspired and motivated by recent research work in this field, in this paper, a general nonlinear framework for a Ishikawa-hybrid proximal point algorithm using the notion of -accretive is developed. Convergence analysis for the algorithm of solving a nonlinear set-valued inclusions problem and existence analysis of solution for the nonlinear set-valued inclusions problem are explored along with some results on the resolvent operator corresponding to -accretive mapping due to Lan et al. in Banach space. The result that sequence generated by the algorithm converges linearly to a solution of the nonlinear set-valued inclusions problem as the convergence rate is proved.

## 2. Preliminaries

Remark 2.1.

In particular, is the usual normalized duality mapping, and (for all ). If is strictly convex [15], or is uniformly smooth Banach space, then is single valued. In what follows we always denote the single-valued generalized duality mapping by in real uniformly smooth Banach space unless otherwise stated.

Let
;
be single-valued mappings. Let
be a set-valued
-accretive mapping. We consider *nonlinear set-valued mixed variational inclusions problem with*
*-accretive mappings (NSVMVIP).*

Remark 2.2.

A special case of problem (2.5) is the following.

(i)If is a Hilbert space, is the zero operator in , is the identity operator in , and , then problem (2.5) becomes the parametric usual variational inclusion with a -maximal monotone mapping , which was studied by Verma [12].

(ii)If is a real Banach space, is the identity operator in , and , then problem (2.5) becomes the parametric usual variational inclusion with a -accretive mapping, which was studied by Li [13].

It is easy to see that a number of known special classes of variational inclusions and variational inequalities in the problem (2.5) are studied (see [2, 7, 12–14]).

Let us recall the following results and concepts.

Definition 2.3.

Definition 2.4.

A single-valued mapping is said to be

(ii)strictly accretive, if is accretive and if and only if ,

(iii) strongly -accretive if there exists a constant such that

Definition 2.5.

A single-valued mapping is said to be

(i) -Lipschitz continuous if there exist constants such that

where are single-valued mappings.

Definition 2.6.

Let , and let be single-valued mappings. A set-valued mapping is said to be

(iii) -strongly accretive if there exists a constant such that

(iv) -relaxed -accretiveif there exists a constant such that

(v) -accretive, if is accretive and for all

(vi) -accretive if is -relaxed -accretive and for all .

Based on the literature [8], we can define the resolvent operator as follows.

Definition 2.7 (see [8]).

Remark 2.8.

The -accretive mappings are more general than -monotone mappings 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, -monotone operators, -subdifferential operators [3–14, 16, 17].

Lemma 2.9 (see [8]).

In the study of characteristic inequalities in -uniformly smooth Banach spaces, Xu [18] proved the following result.

Lemma 2.10 (see [18]).

## 3. The Existence of Solutions

Now, we are studing the existence for solutions of problem (2.5).

Lemma 3.1.

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

Proof.

This directly follows from the definition of .

Theorem 3.2.

where is the same as in Lemma 2.10, and , then the problem (2.5) has a solution .

Proof.

## 4. Ishikawa-Hybrid Proximal Point Algorithm

Based on Lemma 3.1, we develop an Ishikawa-hybrid proximal point algorithm for finding an iterative sequence solving problem (2.5) as follows.

Algorithm 4.1.

Step 1.

Step 2.

Remark 4.2.

For a suitable choice of the mappings , space , and nonnegative sequences , , Algorithm 4.1 can be degenerated to a number of algorithms involving many known algorithms which are due to classes of variational inequalities and variational inclusions [12–14].

Theorem 4.3.

Proof.

and the convergence rate is .By (4.4), if , then it follows that and . Therefor, the sequence generated hybrid proximal point Algorithm 4.1 converges linearly to a solution of problem (2.5) with convergence rate . This completes the proof.

Remark 4.4.

For a suitable choice of the mappings , we can obtain several known results [12–14, 17] as special cases of Theorem 3.2 and Theorem 4.3.

## Authors’ Affiliations

## References

- Di Bella B:
**An existence theorem for a class of inclusions.***Applied Mathematics Letters*2000,**13**(3):15–19. 10.1016/S0893-9659(99)00179-2MathSciNetView ArticleMATHGoogle Scholar - Huang NJ, Tang YY, Liu YP:
**Some new existence theorems for nonlinear inclusion with an application.***Nonlinear Functional Analysis and Applications*2001,**6**(3):341–350.MathSciNetMATHGoogle Scholar - Jeong JU:
**Generalized set-valued variational inclusions and resolvent equations in Banach spaces.***Computers & Mathematics with Applications*2004,**47**(8–9):1241–1247. 10.1016/S0898-1221(04)90118-6MathSciNetView 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:
**Approximation-solvability of a class of A-monotone variational inclusion problems.***Journal KSIAM*2004,**8**(1):55–66.Google Scholar - Huang NJ:
**Nonlinear implicit quasi-variational inclusions involving generalized**m**-accretive mappings.***Archives of Inequalities and Applications*2004,**2**(4):413–425.MathSciNetMATHGoogle Scholar - Fang YP, Huang NJ: H
**-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 - Lan HY, 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 - Fang YP, Huang NJ, Thompson HB:
**A new system of variational inclusions with****-monotone operators in Hilbert spaces.***Computers & Mathematics with Applications. An International Journal*2005,**49**(2–3):365–374. 10.1016/j.camwa.2004.04.037MathSciNetView ArticleMATHGoogle Scholar - Zhang QB, Zhanging XP, Cheng CZ: Resolvent operator technique for solving generalized implicit variational-like inclusions in Banach space. Journal of Mathematical Analysis and Applications 2007, (20):216–221.Google Scholar
- Eckstein J, Bertsekas DP:
**On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators.***Mathematical Programming*1992,**55**(3):293–318. 10.1007/BF01581204MathSciNetView 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 - Li HG:
**Perturbed Ishikawa iterative algorithm and stability for nonlinear mixed quasi-variational inclusions involving****-accretive mappings.***Advances in Nonlinear Variational Inequalities*2008,**11**(1):41–50.MathSciNetMATHGoogle Scholar - Huang NJ, Fang YP:
**Generalized**m**-accretive mappings in Banach spaces.***Journal of Sichuan University*2001,**38**(4):591–592.MATHGoogle Scholar - Weng XL:
**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 - 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 - 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 - 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

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