- Research Article
- Open Access

# 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

https://doi.org/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.

## Keywords

- Banach Space
- Variational Inequality
- Monotone Operator
- Maximal Monotone
- Real Banach Space

## 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

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