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

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.

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).

Finding such that

(2.1)

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

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

(2.2)

(ii)-relaxed -accretive, if there exists a constant such that

(2.3)

(iii)-cocoercive, if there exists a constant such that

(2.4)

(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.

Let be a single-valued mapping, be a strictly -accretive single-valued mapping and be an -accretive set-valued mapping. The resolvent operator is defined by

(2.5)

where is a constant.

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]).

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

(2.6)

where .

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

Lemma 2.5.

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

(2.7)

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.

Let be a set-valued mapping. The map , the inverse of , is said to be general -Lipschitz continuous at 0 if, and only if there exist two constants for any , a solution of the inclusion exist and the such that

(3.1)

holds.

Lemma 3.2.

Let be a -uniformly smooth Banach space, be a -Lipschtiz continuous mapping, be an -strongly -accretive mapping, be a -Lipschtiz continuous mapping, and be an -accretive set-valued mapping. If , and for all , and

(3.2)

then

(3.3)

Proof.

Let be a -uniformly smooth Banach space, be a -Lipschtiz continuous mapping, be an -strongly -accretive mapping, and be an -accretive set-valued mapping. Let us set and , then by using Definition 2.2, Lemmas 2.4, 2.5, and (3.2), we can have

(3.4)

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).

(ii)For a , such that

(3.5)

(iii)For a , holds

(3.6)

where is a constant.

Proof.

This directly follows from definitions of and .

Lemma 3.4.

Let be a -uniformly smooth Banach space, be a -Lipschtiz continuous mapping, be an -strongly -accretive and nonexpansive mapping, be an -Lipschtiz continuous and -strongly -accretive mapping, and , and be an -accretive set-valued mapping. If the following conditions holds

(3.7)

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

Proof.

Define as follows:

(3.8)

For elements , if letting

(3.9)

then by (3.1) and (3.3), we have

(3.10)

By using -strongly -accretive of , -strongly -accretive of , and Lemma 2.5, we obtain

(3.11)

Combining (3.10)-(3.11), by using nonexpansivity of , we have

(3.12)

where

(3.13)

It follows from (3.7)–(3.12) that has a fixed point in , that is, there exist a point such that , and

(3.14)

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.

Let be a -uniformly smooth Banach space, be a -Lipschtiz continuous mapping, be an -strongly -accretive and nonexpansive mapping, be an -strongly -accretive mapping and -Lipschitz continuous, and , and be an -accretive set-valued mapping. Let , and be three nonnegative sequences such that

(3.15)

where , and each satisfies condition (3.7).

Step 1.

For an arbitrarily chosen initial point , set

(3.16)

where the satisfies

(3.17)

Step 2.

The sequence is generated by an iterative procedure

(3.18)

and satisfies

(3.19)

where .

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:

(i) is -Lipschtiz continuous;

(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;

(vi), and be three nonnegative sequences such that

(3.20)

where , and each satisfies condition (3.7),

(vii)let the sequence generated by the general over-relaxed-proximal point algorithm (3.6) be bounded and be a solution of problem (2.1), and the condition

(3.21)

(3.22)

hold. Then the sequence converges linearly to a solution of problem (2.1) with convergence rate , where

(3.23)

Proof.

Let the be a solution of the Framework (2.1) for the conditions (i)–(iv) and Lemma 3.4. Suppose that the sequence which generated by the hybrid proximal point Algorithm 3.5 is bounded, from Lemma 3.4, we have

(3.24)

We infer from Lemma 3.3 that any solution to (2.1) is a fixed point of . First, in the light of Lemma 3.2, we show

(3.25)

where and .

For , and under the assumptions (including the condition (vii) (3.21)), then since the is -Lipschitz continuous at 0. Indeed, it follows that from . Next, by using the condition (iv) and (3.1), and setting and , we have

(3.26)

Now applying Lemma 3.3, we get

(3.27)

Therefore,

(3.28)

where and .

Next we start the main part of the proof by using the expression

(3.29)

Let us set and for simple. We begin with estimating and later using (3.2), the nonexpansivity of , (3.21) and (3.28) as follows:

(3.30)

Thus, we have

(3.31)

where

(3.32)

and , , , and .

Since , we have . It follows that

(3.33)

Next, we can obtain

(3.34)

This implies from (3.38) and (3.39) that

(3.35)

Since is an -strongly -accretive mapping (and hence, ), this implies from (3.35) that the sequence converges strongly to for

(3.36)

where , , and .

Hence, we have

(3.37)

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.

Let be a Hilbert space , be an -strongly monotone and nonexpansive mapping , is a zero operator, be an -maximal set-valued monotone. , and the condition (3.21) hold, the be -Lipschitz continuous at 0 . Let , and be the same as in Algorithm 3.5. If

(3.38)

then the bounded sequence generated by the general over-relaxed -proximal point algorithm converges linearly to a solution of problem (2.1) with convergence rate , where

(3.39)

and , , .

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