Open Access

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

Fixed Point Theory and Applications20102010:501293

DOI: 10.1155/2010/501293

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

Let be a real Banach space with dual space and and let the dual pair between and , denote the family of all the nonempty subsets of and the family of all nonempty closed bounded subsets of . The generalized duality mapping is defined by
(2.1)

where is a constant.

The modulus of smoothness of is the function defined by
(2.2)
A Banach space is called uniformly smooth if
(2.3)
is called -uniformly smooth if there exists a constant such that
(2.4)

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).

For any , finding , such that
(2.5)

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, 1214]).

Let us recall the following results and concepts.

Definition 2.3.

A single-valued mapping is said to be -Lipschitz continuous if there exists a constant such that
(2.6)

Definition 2.4.

A single-valued mapping is said to be

(i)accretive if
(2.7)

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

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

(2.8)
(iv) -Lipschitz continuous if there exists a constant such that
(2.9)

Definition 2.5.

A single-valued mapping is said to be

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

(2.10)
(ii) - -relaxed cocoercive with respect to in the first argument if there exist constants , and such that
(2.11)

where are single-valued mappings.

Definition 2.6.

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

(i)accretive if
(2.12)

(ii) -accretive if

(2.13)

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

(2.14)

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

(2.15)

(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]).

Let be a single-valued mapping, a strictly -accretive single-valued mapping and a -accretive mapping. The resolvent operator is defined by
(2.16)

where is a constant.

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 [314, 16, 17].

Lemma 2.9 (see [8]).

Let be -Lipschtiz continuous mapping, be an -strongly -accretive mapping, and an -accretive mapping. Then the generalized resolvent operator is -Lipschitz continuous, that is,
(2.17)

where

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

Lemma 2.10 (see [18]).

Let be a real uniformly smooth Banach space. Then is -uniformly smooth if and only if there exists a constant such that for all ,
(2.18)

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.

(i)An element is a solution of problem (2.5).
  1. (ii)
    For a and any , there exists such that
    (3.1)
     

where is a constant.

Proof.

This directly follows from the definition of .

Theorem 3.2.

Let be a -uniformly smooth Banach space. Let ; be single-valued mappings, and be a -Lipschtiz continuous mapping, a -strongly -accretive and -Lipschitz continuous mapping, be a -strongly accretive and -Lipschitz continuous mapping, and a -Lipschitz continuous mapping, respectively. Let be -Lipschitz continuous, and - -relaxed cocoercive with respect to in the first argument. Let be a set-valued -accretive mapping. If the following condition holds:
(3.2)

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

Proof.

Define a mapping as follows:
(3.3)
For elements , if we let and
(3.4)
then by (3.1), (3.3), and Lemma 2.10, we have
(3.5)
Since is - -relaxed cocoercive with respect to in the first argument and is a -Lipschitz continuous mapping so we obtain
(3.6)
(3.7)
By -strongly accretivity of , we have
(3.8)
Combining(3.5), (3.6), (3.7), and (3.8), we can get
(3.9)
where
(3.10)
It follows from (3.2) and (3.9) that has a fixed point in , that is, there exists a point such that , and
(3.11)

where . This completes the 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.

Let be a solution of problem ( 2.5 ). Let , , , and , be five nonnegative sequences such that
(4.1)

Step 1.

For an arbitrarily initial point , we choose suitable , letting
(4.2)

Step 2.

The sequences and are generated by an iterative procedure
(4.3)

where

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 [1214].

Theorem 4.3.

Let be the same as in Theorem 3.2, then condition (3.2) holds. Let , , , and be the same as in Algorithm 4.1. Then the sequence generated by hybrid proximal point Algorithm 4.1 converges linearly to a solution of problem (2.5) as
(4.4)
where is the same as in Lemma 2.10, , and the convergence rate is
(4.5)

Proof.

Suppose that the sequence is the the sequence generated by the Ishikawa-hybrid proximal point Algorithm 4.1, and that is a solution of problem (2.5). From Lemma 3.1 and condition , we can get
(4.6)

where .

For all , and , setting
(4.7)
we find the estimation
(4.8)
By the conditions and Lemma 2.10, we have
(4.9)
(4.10)
It follows from (4.8)–(4.10) that
(4.11)
where
(4.12)
(4.13)
Since and (4.3), and
(4.14)
Next, we calculate
(4.15)
This implies that
(4.16)
letting
(4.17)
For all , set
(4.18)
For the same reason,
(4.19)
where
(4.20)
(4.21)
Furthermore,
(4.22)
Combining (4.16)-(4.22), then we have
(4.23)
By (4.4) and the condition , we can see that
(4.24)

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 [1214, 17] as special cases of Theorem 3.2 and Theorem 4.3.

Authors’ Affiliations

(1)
Institute of Applied Mathematics Research, Chongqing University of Posts and Telecommunications
(2)
Institute of Nonlinear Analysis Research, Changjiang Normal University

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© Hong Gang Li et al. 2010

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