- Research Article
- Open Access

# The Over-Relaxed -Proximal Point Algorithm for General Nonlinear Mixed Set-Valued Inclusion Framework

- XianBing Pan
^{1}Email author, - HongGang Li
^{2}and - AnJian Xu
^{3}

**2011**:840978

https://doi.org/10.1155/2011/840978

© Xian Bing Pan et al. 2011

**Received:**16 November 2010**Accepted:**10 January 2011**Published:**6 February 2011

## Abstract

The purpose of this paper is (1) a general nonlinear mixed set-valued inclusion framework for the over-relaxed -proximal point algorithm based on the ( , )-accretive mapping is introduced, and (2) it is applied to the approximation solvability of a general class of inclusions problems using the generalized resolvent operator technique due to Lan-Cho-Verma, and the convergence of iterative sequences generated by the algorithm is discussed in -uniformly smooth Banach spaces. The results presented in the paper improve and extend some known results in the literature.

## Keywords

- Nonexpansive Mapping
- Monotone Operator
- Smooth Banach Space
- Inclusion Problem
- Resolvent Operator

## 1. Introduction

In recent years, various set-valued variational inclusion frameworks, which have wide applications to many fields including, for example, mechanics, physics, optimization and control, nonlinear programming, economics, and engineering sciences have been intensively studied by Ding and Luo [1], Verma [2], Huang [3], Fang and Huang [4], Fang et al. [5], Lan et al. [6], Zhang et al. [7], respectively. Recently, Verma [8] has intended to develop a general inclusion framework for the over-relaxed -proximal point algorithm [9] based on the -maximal monotonicity. In 2007-2008, Li [10, 11] has studied the algorithm for a new class of generalized nonlinear fuzzy set-valued variational inclusions involving -monotone mappings and an existence theorem of solutions for the variational inclusions, and a new iterative algorithm [12] for a new class of general nonlinear fuzzy mulitvalued quasivariational inclusions involving -monotone mappings in Hilbert spaces, and discussed a new perturbed Ishikawa iterative algorithm for nonlinear mixed set-valued quasivariational inclusions involving -accretive mappings, the stability [13] and the convergence of the iterative sequences in -uniformly smooth Banach spaces by using the resolvent operator technique due to Lan et al. [6].

Inspired and motivated by recent research work in this field, in this paper, a general nonlinear mixed set-valued inclusion framework for the over-relaxed -proximal point algorithm based on the -accretive mapping is introduced, which is applied to the approximation solvability of a general class of inclusions problems by the generalized resolvent operator technique, and the convergence of iterative sequences generated by the algorithm is discussed in -uniformly smooth Banach spaces. For more literature, we recommend to the reader [1–17].

## 2. Preliminaries

Let
be a real Banach space with dual space
, and let
be the dual pair between
and
, let
denote the family of all the nonempty subsets of
, and let
denote the family of all nonempty closed bounded subsets of
. The generalized duality mapping
is single-valued if
is strictly convex [14], or
is uniformly smooth space. In what follows we always denote the single-valued generalized duality mapping by
in real uniformly smooth Banach space
unless otherwise stated. We consider the following *general nonlinear mixed set-valued inclusion problem with*
*-accretive mappings (GNMSVIP)*.

where , be single-valued mappings; be an -accretive set-valued mapping. A special case of problem (2.1) is the following:

if is a Hilbert space, is the zero operator in , and , then problem (2.1) becomes the inclusion problem with a -maximal monotone mapping , which was studied by Verma [8].

Definition 2.1.

Let be a real Banach space with dual space , and let be the dual pair between and . Let and be single-valued mappings. A set-valued mapping is said to be

(iv) -accretive, if is -relaxed -accretive and for every .

Based on the literature [6], we can define the resolvent operator as follows.

Definition 2.2.

Remark 2.3.

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 [1–7, 11–13].

Lemma 2.4 (see [6]).

In the study of characteristic inequalities in -uniformly smooth Banach spaces, Xu [14] proved the following result.

Lemma 2.5.

## 3. The Over-Relaxed -Proximal Point Algorithm

This section deals with an introduction of a generalized version of the over-relaxed proximal point algorithm and its applications to approximation solvability of the inclusion problem of the form (2.1) based on the -accretive set-valued mapping.

Let be a set-valued mapping, the set be the graph of , which is denoted by for simplicity, This is equivalent to stating that a mapping is any subset of , and . If is single-valued, we shall still use to represent the unique y such that rather than the singleton set . This interpretation will depend greatly on the context. The inverse of is .

Definition 3.1.

holds.

Lemma 3.2.

Proof.

Therefore, (3.3) holds.

Lemma 3.3.

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

(i)An element is a solution of problem (2.1).

Proof.

This directly follows from definitions of and .

Lemma 3.4.

where is the same as in Lemma 2.5, and . Then the problem (2.1) has a solution .

Proof.

This completes the proof.

Based on Lemma 3.3, we can develop a general over-relaxed -proximal point algorithm to approximating solution of problem (2.1) as follows.

Algorithm 3.5.

where , and each satisfies condition (3.7).

Step 1.

Step 2.

Remark 3.6.

For a suitable choice of the mappings , , , , , and space , then the Algorithm 3.5 can be degenerated to the hybrid proximal point algorithm [16, 17] and the over-relaxed -proximal point algorithm [8].

Theorem 3.7.

Let be a -uniformly smooth Banach space. Let and be single-valued mappings, and let be a set-valued mapping and be the inverse mapping of the mapping satisfying the following conditions:

(ii) be an -strongly -accretive mapping and nonexpansive;

(iii) be an -Lipschtiz continuous and -strongly -accretive mapping;

(iv) be an -accretive set-valued mapping;

(v)the be -Lipschitz continuous at 0 ;

where , and each satisfies condition (3.7),

Proof.

By (3.22), it follows that from the condition (vi), and the sequence generated by the hybrid proximal point Algorithm 3.5 converges linearly to a solution of problem (2.1) with convergence rate . This completes the proof.

Corollary 3.8.

This is Theorem 3.2 in [8], and if, in addition, , then we can have the Proposition 2 in [9].

## Authors’ Affiliations

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