# Auxiliary Principle for Generalized Strongly Nonlinear Mixed Variational-Like Inequalities

- Zeqing Liu
^{1}, - Lin Chen
^{1}, - Jeong Sheok Ume
^{2}Email author and - Shin Min Kang
^{3}

**2009**:758786

**DOI: **10.1155/2009/758786

© Zeqing Liu et al. 2009

**Received: **4 February 2009

**Accepted: **27 April 2009

**Published: **12 May 2009

## Abstract

We introduce and study a class of generalized strongly nonlinear mixed variational-like inequalities, which includes several classes of variational inequalities and variational-like inequalities as special cases. By applying the auxiliary principle technique and KKM theory, we suggest an iterative algorithm for solving the generalized strongly nonlinear mixed variational-like inequality. The existence of solutions and convergence of sequence generated by the algorithm for the generalized strongly nonlinear mixed variational-like inequalities are obtained. The results presented in this paper extend and unify some known results.

## 1. Introduction

It is well known that the auxiliary principle technique plays an efficient and important role in variational inequality theory. In 1988, Cohen [1] used the auxiliary principle technique to prove the existence of a unique solution for a variational inequality in reflexive Banach spaces, and suggested an innovative and novel iterative algorithm for computing the solution of the variational inequality. Afterwards, Ding [2], Huang and Deng [3], and Yao [4] obtained the existence of solutions for several kinds of variational-like inequalities. Fang and Huang [5] and Liu et al. [6] discussed some classes of variational inequalities involving various monotone mappings. Recently, Liu et al. [7, 8] extended the auxiliary principle technique to two new classes of variational-like inequalities and established the existence results for these variational-like inequalities.

Inspired and motivated by the results in [1–13], in this paper, we introduce and study a class of generalized strongly nonlinear mixed variational-like inequalities. Making use of the auxiliary principle technique, we construct an iterative algorithm for solving the generalized strongly nonlinear mixed variational-like inequality. Several existence results of solutions for the generalized strongly nonlinear mixed variational-like inequality involving strongly monotone, relaxed Lipschitz, cocoercive, relaxed cocoercive and generalized pseudocontractive mappings, and the convergence results of iterative sequence generated by the algorithm are given. The results presented in this paper extend and unify some known results in [9, 12, 13].

## 2. Preliminaries

where is a coercive continuous bilinear form, that is, there exist positive constants and such that

(C1)

(C2) Clearly,

Let satisfy the following conditions:

(C3) for each , is linear in the first argument;

(C4) is bounded, that is, there exists a constant such that

(C5) ;

(C6) for each , is convex in the second argument.

Remark 2.1.

It is easy to verify that

(m1)

(m2) ,

where (m2) implies that for each , is continuous in the second argument on .

Special Cases

which was introduced and studied by Ansari and Yao [9], Ding [11] and Zeng [13], respectively.

which was introduced and studied by Yao [12].

In brief, for suitable choices of the mappings and , one can obtain a number of known and new variational inequalities and variational-like inequalities as special cases of (2.1). Furthermore, there are a wide classes of problems arising in optimization, economics, structural analysis and fluid dynamics, which can be studied in the general framework of the generalized strongly nonlinear mixed variational-like inequality, which is the main motivation of this paper.

Definition 2.2.

Let and be mappings.

*relaxed Lipschitz*with constant if there exists a constant such that

*cocoercive*with constant with respect to in the first argument if there exists a constant such that

*-cocoercive*with constant with respect to in the first argument if there exists a constant such that

*relaxed*

*-cocoercive*with respect to in the first argument if there exist constants such that

*Lipschitz continuous*with constant if there exists a constant such that

*relaxed Lipschitz*with constant with respect to in the second argument if there exists a constant such that

*-relaxed Lipschitz*with constant with respect to in the second argument if there exists a constant such that

*-generalized pseudocontractive*with constant with respect to in the second argument if there exists a constant such that

*strongly monotone*with constant if there exists a constant such that

*relaxed Lipschitz*with constant if there exists a constant such that

*cocoercive*with constant if there exists a constant such that

*Lipschitz continuous*with constant if there exists a constant such that

*Lipschitz continuous*in the first argument if there exists a constant such that

Similarly, we can define the Lipschitz continuity of in the second argument.

Definition 2.3.

Let be a nonempty convex subset of and let be a functional.

*convex*if for any and any ,

(d2)
is said to be *concave* if
is convex;

(d3)
is said to be *lower semicontinuous* on
if for any
, the set
is closed in
;

(d4)
is said to be *upper semicontinuous* on
, if
is lower semicontinuous on
.

In order to gain our results, we need the following assumption.

Assumption 2.4.

The mappings satisfy the following conditions:

(d5)

(d6) for given the mapping is concave and upper semicontinuous on .

Remark 2.5.

It follows from (d5) and (d6) that

(m5)

(m6) for any given , the mapping is convex and lower semicontinuous on .

Proposition 2.6 (see [9]).

Let be a nonempty convex subset of . If is lower semicontinuous and convex, then is weakly lower semicontinuous.

Proposition 2.6 yields that if is upper semicontinuous and concave, then is weakly upper semicontinuous.

Lemma 2.7 (see [10]).

Let be a nonempty closed convex subset of a Hausdorff linear topological space , and let be mappings satisfying the following conditions:

- (b)
for each is upper semicontinuous on

- (c)
for each the set is a convex set

- (d)
there exists a nonempty compact set and such that

Then there exists such that

## 3. Auxiliary Problem and Algorithm

where
is a constant,
is a mapping. The problem is called a *auxiliary problem* for the generalized strongly nonlinear mixed variational-like inequality (2.1).

Theorem 3.1.

Let be a nonempty closed convex subset of the Hilbert space . Let be a coercive continuous bilinear form with (C1) and (C2), and let be a functional with (C3)–(C6). Let be Lipschitz continuous and relaxed Lipschitz with constants and , respectively. Let be Lipschitz continuous with constant , and let satisfy Assumption 2.4. Then the auxiliary problem (3.1) has a unique solution in .

Proof.

which means that is a solution of (3.1).

which implies that . That is, the auxiliary problem (3.1) has a unique solution in . This completes the proof.

Applying Theorem 3.1, we construct an iterative algorithm for solving the generalized strongly nonlinear mixed variational-like inequality (2.1).

- (i)
At step , start with the initial value .

- (ii)At step , solve the auxiliary problem (3.1) with . Let denote the solution of the auxiliary problem (3.1). That is,(3.13)

- (iii)
If, for given , stop. Otherwise, repeat (ii).

## 4. Existence of Solutions and Convergence Analysis

The goal of this section is to prove several existence of solutions and convergence of the sequence generated by Algorithm 3.2 for the generalized strongly nonlinear mixed variational-like inequality (2.1).

Theorem 4.1.

then the generalized strongly nonlinear mixed variational-like inequality (2.1) possesses a solution and the sequence defined by Algorithm 3.2 converges to

Proof.

This completes the proof.

Theorem 4.2.

and one of (4.3)–(4.6), then the generalized strongly nonlinear mixed variational-like inequality (2.1) possesses a solution and the sequence defined by Algorithm 3.2 converges to

Proof.

The rest of the argument is the same as in the proof of Theorem 4.1 and is omitted. This completes the proof.

Theorem 4.3.

If there exists a constant satisfying (4.2) and one of (4.3)–(4.6), then the generalized strongly nonlinear mixed variational-like inequality (2.1) possesses a solution and the sequence defined by Algorithm 3.2 converges to

Proof.

The rest of the proof is identical with the proof of Theorem 4.1 and is omitted. This completes the proof.

Theorem 4.4.

and one of (4.3), (4.4), and (4.6), then the generalized strongly nonlinear mixed variational-like inequality (2.1) possesses a solution and the sequence defined by Algorithm 3.2 converges to

Proof.

The rest of the argument follows as in the proof of Theorem 4.1 and is omitted. This completes the proof.

Theorem 4.5.

and one of (4.3)–(4.6), then the generalized strongly nonlinear mixed variational-like inequality (2.1) possesses a solution and the sequence defined by Algorithm 3.2 converges to

Proof.

The rest of the proof is similar to the proof of Theorem 4.1 and is omitted. This completes the proof.

Remark 4.6.

Theorems 4.1–4.5 extend, improve, and unify the corresponding results in [9, 12, 13].

## Declarations

### Acknowledgments

The authors thank the referees for useful comments and suggestions. This work was supported by the Science Research Foundation of Educational Department of Liaoning Province (2009A419) and the Korea Research Foundation (KRF) grant funded by the Korea government (MEST)(2009-0073655).

## Authors’ Affiliations

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