A New System of Generalized Nonlinear Mixed Variational Inclusions in Banach Spaces

Abstract

We introduce and study a new system of generalized nonlinear mixed variational inclusions in real -uniformly smooth Banach spaces. We prove the existence and uniqueness of solution and the convergence of some new -step iterative algorithms with or without mixed errors for this system of generalized nonlinear mixed variational inclusions. The results in this paper unify, extend, and improve some known results in literature.

1. Introduction

Variational inclusion problems are among the most interesting and intensively studied classes of mathematical problems and have wide applications in the fields of optimization and control, economics and transportation equilibrium, as well as engineering science. For the past years, many existence results and iterative algorithms for various variational inequality and variational inclusion problems have been studied. For details, see [1–25] and the references therein.

Recently, some new and interesting problems, which are called to be system of variational inequality problems, were introduced and studied. Pang [1], Cohen and Chaplais [2], Bianchi [3], and Ansari and Yao [4] considered a system of scalar variational inequalities, and Pang showed that the traffic equilibrium problem, the spatial equilibrium problem, the Nash equilibrium, and the general equilibrium programming problem can be modeled as a system of variational inequalities. Ansari et al. [5] considered a system of vector variational inequalities and obtained its existence results. Allevi et al. [6] considered a system of generalized vector variational inequalities and established some existence results with relative pseudomonoyonicity. Kassay and Kolumbán [7] introduced a system of variational inequalities and proved an existence theorem by the Ky Fan lemma. Kassay et al. [8] studied Minty and Stampacchia variational inequality systems with the help of the Kakutani-Fan-Glicksberg fixed point theorem. Peng [9], Peng and Yang [10] introduced a system of quasivariational inequality problems and proved its existence theorem by maximal element theorems. Verma [11–15] introduced and studied some systems of variational inequalities and developed some iterative algorithms for approximating the solutions of system of variational inequalities in Hilbert spaces. J. K. Kim and D. S. Kim [16] introduced and studied a new system of generalized nonlinear quasivariational inequalities in Hilbert spaces. Cho et al. [17] introduced and studied a new system of nonlinear variational inequalities in Hilbert spaces. They proved some existence and uniqueness theorems of solutions for the system of nonlinear variational inequalities.

As generalizations of system of variational inequalities, Agarwal et al. [18] introduced a system of generalized nonlinear mixed quasivariational inclusions and investigated the sensitivity analysis of solutions for this system of generalized nonlinear mixed quasivariational inclusions in Hilbert spaces. Peng and Zhu [19] introduce a new system of generalized nonlinear mixed quasivariational inclusions in -uniformly smooth Banach spaces and prove the existence and uniqueness of solutions and the convergence of several new two-step iterative algorithms with or without errors for this system of generalized nonlinear mixed quasivariational inclusions. Kazmi and Bhat [20] introduced a system of nonlinear variational-like inclusions and proved the existence of solutions and the convergence of a new iterative algorithm for this system of nonlinear variational-like inclusions. Fang and Huang [21], Verma [22], and Fang et al. [23] introduced and studied a new system of variational inclusions involving -monotone operators, -monotone operators and -monotone operators, respectively. Yan et al. [24] introduced and studied a system of set-valued variational inclusions which is more general than the model in [21]. Peng and Zhu [25] introduced and studied a system of generalized mixed quasivariational inclusions involving -monotone operators which contains those mathematical models in [11–16, 21–24] as special cases.

Inspired and motivated by the results in [1–25], the purpose of this paper is to introduce and study a new system of generalized nonlinear mixed quasivariational inclusions which contains some classes of system of variational inclusions and systems of variational inequalities in the literature as special cases. Using the resolvent technique for the -accretive mappings, we prove the existence and uniqueness of solutions for this system of generalized nonlinear mixed quasivariational inclusions. We also prove the convergence of some new -step iterative sequences with or without mixed errors to approximation the solution for this system of generalized nonlinear mixed quasivariational inclusions. The results in this paper unifies, extends, and improves some results in [11–16, 19] in several aspects.

2. Preliminaries

Throughout this paper we suppose that is a real Banach space with dual space, norm and the generalized dual pair denoted by , and , respectively, is the family of all the nonempty subsets of , denotes the domain of the set-valued map and the generalized duality mapping is defined by

(2.1)

where is a constant. In particular, is the usual normalized duality mapping. It is known that, in general, , for all , and is single-valued if is strictly convex.

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)

Note that is single-valued if is uniformly smooth.

Xu [26] and Xu and Roach [27] proved the following result.

Lemma 2.1.

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

(2.5)

Definition 2.2 (see [28]).

Let be a multivalued mapping:

(i) is said to be accretive if, for any , , , there exists such that

(2.6)

(ii) is said to be -accretive if is accretive and holds for every (equivalently, for some) , where is the identity operator on .

Remark 2.3.

It is well known that, if is a Hilbert space, then is -accretive if and only if is maximal monotone (see, e.g., [29]).

We recall some definitions needed later.

Definition 2.4 (see [28]).

Let the multivalued mapping be -accretive, for a constant , the mapping which is defined by

(2.7)

is called the resolvent operator associated with and .

Remark 2.5.

It is well known that is single-valued and nonexpansive mapping (see [28]).

Definition 2.6.

Let be a real uniformly smooth Banach space, and let be a single-valued operator. However, is said to be

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

(2.8)

or equivalently,

(2.9)

(ii)-Lipschitz continuous if there exists a constant such that

(2.10)

Remark 2.7.

If is -strongly accretive, then is -expanding, that is,

(2.11)

Lemma 2.8 (see [30]).

Let be three real sequences, satisfying

(2.12)

where , , for all , , . Then .

3. System of Generalized Nonlinear Mixed Variational Inequalities

In this section, we will introduce a new system of generalized nonlinear mixed variational inclusions which contains some classes of system of variational inclusions and systems of variational inequalities in literature as special cases.

In what follows, unless other specified, we always suppose that is a zero element in , and for each , and are single-valued mappings, is an -accretive operator. We consider the following problem: find such that

(3.1)

which is called the system of generalized nonlinear mixed variational inclusions, where are constants.

In what follows, there are some special cases of the problem (3.1).

• (i) If , then problem (3.1) reduces to the system of nonlinear mixed quasivariational inclusions introduced and studied by Peng and Zhu [19].

  If is a Hilbert space and , then problem (3.1) reduces to the system of nonlinear mixed quasivariational inclusions introduced and studied by Agarwal et al. [18].

• (ii) If is a Hilbert space, and for each , for all , where is a proper, convex, lower semicontinuous functional, and denotes the subdifferential operator of , then problem (3.1) reduces to the following system of generalized nonlinear mixed variational inequalities, which is to find such that

  

  (3.2)

  where are constants.

• (iii) If , then (3.2) reduces to the problem of finding such that

  

  (3.3)

  Moreover, if , then problem (3.3) becomes the system of generalized nonlinear mixed variational inequalities introduced and studied by J. K. Kim and D. S. Kim in [16].

• (iv) For , if (the indicator function of a nonempty closed convex subset ) and , then (3.2) reduces to the problem of finding , such that

  

  (3.4)

Problem (3.4) is called the system of nonlinear variational inequalities. Moreover, if , then problem (3.4) reduces to the following system of nonlinear variational inequalities, which is to find such that

(3.5)

If and , then (3.5) reduces to the problem introduced and researched by Verma [11–13].

Lemma 3.1.

For any given , is a solution of the problem (3.1) if and only if

(3.6)

where is the resolvent operators of for .

Proof.

It is easy to know that Lemma 3.1 follows from Definition 2.4 and so the proof is omitted.

4. Existence and Uniqueness

In this section, we will show the existence and uniqueness of solution for problems (3.1).

Theorem 4.1.

Let be a real -uniformly smooth Banach spaces. For , let be strongly accretive and Lipschitz continuous with constants and , respectively, let be Lipschitz continuous with constant , and let be an -accretive mapping. If for each ,

(4.1)

then (3.1) has a unique solution .

Proof.

First, we prove the existence of the solution. Define a mapping as follows:

(4.2)

For , since is a nonexpansive mapping, is strongly accretive and Lipschitz continuous with constants and , respectively, and is Lipschitz continuous with constant , for any , we have

(4.3)

It follows from (4.1) that

(4.4)

Thus, (4.3) implies that is a contractive mapping and so there exists a point such that

(4.5)

Let

(4.6)

then by the definition of , we have

(4.7)

that is, is a solution of problem (3.1).

Then, we show the uniqueness of the solution. Let be another solution of problem (3.1). It follows from Lemma 3.1 that

(4.8)

As the proof of (4.3), we have

(4.9)

It follows from (4.1) that

(4.10)

Hence,

(4.11)

and so for , we have

(4.12)

This completes the proof.

Remark 4.2.

• (i) If is a 2-uniformly smooth space, and there exist () such that

  

  (4.13)

  Then (4.1) holds. We note that the Hilbert spaces and (or ) spaces () are 2-uniformly smooth.

• (ii) Let , by Theorem 4.1, we recover [19, Theorem  3.1]. So Theorem 4.1 unifies, extends, and improves [19, Theorem  3.1, Corollaries  3.2 and 3.3], [16, Theorems  2.1–2.4] and the main results in [13].

5. Algorithms and Convergence

This section deals with an introduction of some -step iterative sequences with or without mixed errors for problem (3.1) that can be applied to the convergence analysis of the iterative sequences generated by the algorithms.

Algorithm 5.1.

For any given point , define the generalized -step iterative sequences as follows:

(5.1)

where , is a sequence in , and , are the sequences satisfying the following conditions:

(5.2)

Theorem 5.2.

Let , , and be the same as in Theorem 4.1, and suppose that the sequences are generated by Algorithm 5.1. If the condition (4.1) holds, then converges strongly to the unique solution of the problem (3.1).

Proof.

By the Theorem 4.1, we know that problem (3.1) has a unique solution , it follows from Lemma 3.1 that

(5.3)

By (5.1) and (5.3), we have

(5.4)

For , since is strongly monotone and Lipschitz continuous with constants and , respectively, and is Lipschitz continuous with constant , we get for ,

(5.5)

where

It follows from (5.4) and (5.5) that

(5.6)

By (5.1), (5.3), and (5.5), we have

(5.7)

since is strongly accretive and Lipschitz continuous with constants and , respectively, and is Lipschitz continuous with constant , we get

(5.8)

where .

It follows from (5.6)–(5.8) that

(5.9)

Let

(5.10)

Then (5.9) can be written as follows:

(5.11)

From the assumption (5.2), we know that satisfy the conditions of Lemma 2.8.

Thus , that is, . It follows from (5.6)–(5.8) that So for . That is, converges strongly to the unique solution of (3.1).

For , let and , by Algorithm 5.1 and Theorem 5.2, it is easy to obtain the following Algorithm 5.3 and Theorem 5.4.

Algorithm 5.3.

For any given point , define the generalized -step iterative sequences as follows:

(5.12)

where , is a sequence in , satisfying

Theorem 5.4.

Let , , and be the same as in Theorem 4.1, and suppose that the sequences are generated by Algorithm 5.3. If (4.1) holds, then converges strongly to the unique solution of (3.1).

Remark 5.5.

Theorem 5.4 unifies and generalizes [19, Theorems  4.3 and 4.4] and the main results in [11, 12]. So Theorem 5.2 unifies, extends, and improves the corresponding results in [11–14, 16, 19].