# On Solvability of a Generalized Nonlinear Variational-Like Inequality

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

**2009**:467512

https://doi.org/10.1155/2009/467512

© Zeqing Liu et al. 2009

**Received: **12 July 2009

**Accepted: **28 September 2009

**Published: **11 October 2009

## Abstract

A new generalized nonlinear variational-like inequality is introduced and studied. By applying the auxiliary principle technique and KKM theory, we construct a new iterative algorithm for solving the generalized nonlinear variational-like inequality. By means of the Banach fixed-point theorem, we establish the existence and uniqueness of solution for the generalized nonlinear variational-like inequality. The convergence of the sequence generated by the iterative algorithm is also discussed.

## 1. Introduction

It is well known that variational inequality theory has become a very effective and powerful tool for studying a wide range of problems arising in many diverse fields. One of the most interesting and important problems in the variational inequality theory is the development of an efficient iterative algorithm to compute approximate solutions. The researchers in [1–10] suggested a lot of iterative algorithms for solving various variational inequalities and variational-like inequalities. By using the auxiliary principle technique, Ding and Yao [3], Ding et al. [4], Huang and Deng [5], Liu et al. [8, 9], and others studied several classes of nonlinear variational inequalities and variational-like inequalities in reflexive Banach and Hilbert spaces, respectively, suggested some iterative algorithms to compute approximate solutions for these nonlinear variational inequalities and variational-like inequalities, and proved the existence of solutions for the nonlinear variational inequalities and variational-like inequalities involving different monotone mappings.

Motivated and inspired by the research work in [1–10], we introduce and study a generalized nonlinear variational-like inequality. By using the auxiliary principle technique and KKM theorem due to Zhang and Xiang [2], we suggest a new iterative scheme for solving the generalized nonlinear variational-like inequality. Utilizing the Banach fixed-point theorem, we prove the existence and uniqueness of solution for the generalized nonlinear variational-like inequality. Under certain conditions, we discuss the convergence of the iterative sequence generalized by the iterative algorithm.

## 2. Preliminaries

Throughout this paper, let be a real Hilbert space endowed with an inner product and a norm , respectively, and . Let be a nonempty closed convex subset of . Assume that is a coercive continuous bilinear form, that is, there exist positive constants such that

Remark 2.1.

It follows from (a1) and (a2) that

Let be nondifferentiable and satisfy the following conditions:

(b1) is linear in the first argument,

(b2) is convex in the second argument,

Remark 2.2.

which implies that is continuous in the second argument.

Let be mappings and . Now we consider the following generalized nonlinear variational-like inequality.

It is clear that for appropriate and suitable choices of the mappings and , the generalized nonlinear variational-like inequality (2.3) includes some variational inequalities and variational-like inequalities in [1–10] as special cases.

Recall the following concepts and results.

Definition 2.3.

- (3)

- (5)

- (8)

Lemma 2.4 (see [2]).

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

Lemma 2.5 (see [11]).

Assumption 2.6.

(2)for any the mapping is convex and lower semicontinuous in

## 3. Auxiliary Problem and Algorithm

Now we consider the following auxiliary problem with respect to the generalized nonlinear variational-like inequality (2.3). For each find such that

Theorem 3.1.

Then for each , the auxiliary problem (3.1) has a unique solution in

Proof.

Now we prove that the functionals and satisfy all the conditions of Lemma 2.4 in the weak topology. It is easy to see for all

That is, is a solution of the auxiliary problem (3.1).

Now we prove the uniqueness of solution for the auxiliary problem (3.1). Suppose that are two solutions of the auxiliary problem (3.1) with respect to . It follows that

which yields that by (3.2). That is, is the unique solution of the auxiliary problem (3.1). This completes the proof.

The proof of the below result is similar to that of Theorem 3.1 and is omitted.

Theorem 3.2.

Then for each , the auxiliary problem (3.1) has a unique solution in

Based on Theorems 3.1 and 3.2, we suggest the following iterative algorithm with errors for solving the generalized nonlinear variational-like inequality (2.3).

Algorithm 3.3.

## 4. Existence and Convergence

In this section, we prove the existence of solution for the generalized nonlinear variational-like inequality (2.3) and discuss the convergence of the sequence generated by Algorithm 3.3.

Theorem 4.1.

Then the generalized nonlinear variational-like inequality (2.3) possesses a unique solution and the iterative sequence generated by Algorithm 3.3 converges strongly to

Proof.

that is, is a solution of the generalized nonlinear variational-like inequality (2.3).

Now we prove the uniqueness. Suppose that the generalized nonlinear variational-like inequality (2.3) has two solutions . It follows that

which together with (4.1) implies that That is, the generalized nonlinear variational-like inequality (2.3) has a unique solution in .

Next we discuss the convergence of the iterative sequence generated by Algorithm 3.3. Taking in (4.7) and in (3.18), and adding these inequalities, we infer that

where is defined by (4.6). It follows from (3.19), (4.1), (4.14), and Lemma 2.5 that the iterative sequence generated by Algorithm 3.3 converges strongly to This completes the proof.

As in the proof of Theorem 4.1, we have the following theorem.

Theorem 4.2.

Then the generalized nonlinear variational-like inequality (2.3) possesses a unique solution and the iterative sequence generated by Algorithm 3.3 converges strongly to

Remark 4.3.

The conditions of Theorems 3.1, 3.2, 4.1, and 4.2 are different from the conditions of the results in [1–10]. In particular, the mappings and with respect to the second argument in Theorems 3.2 and 4.2 are Lipschitz continuous, but other mappings in Theorems 3.2 and 4.2 are not Lipschitz continuous.

## Declarations

### Acknowledgments

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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