- Research Article
- Open Access

# Generalized Strongly Nonlinear Implicit Quasivariational Inequalities

- Salahuddin
^{1}Email author and - M. K. Ahmad
^{1}

**2009**:124953

https://doi.org/10.1155/2009/124953

© Salahuddin and M. K. Ahmad. 2009

**Received:**11 February 2009**Accepted:**17 June 2009**Published:**29 July 2009

## Abstract

We prove an existence theorem for solution of generalized strongly nonlinear implicit quasivariational inequality problems and convergence of iterative sequences with errors, involving Lipschitz continuous, generalized pseudocontractive and generalized -pseudocontractive mappings in Hilbert spaces.

## Keywords

- Variational Inequality
- Iterative Process
- Nonempty Closed Convex Subset
- Real Sequence
- Quasivariational Inequality

## 1. Introduction

Variational inequality was initially studied by Stampacchia [1] in 1964. Since then, it has been extensively studied because of its crucial role in the study of mechanics, physics, economics, transportation and engineering sciences, and optimization and control. Thanks to its wide applications, the classical variational inequality has been well studied and generalized in various directions. For details, readers are referred to [2–5] and the references therein.

It is known that one of the most important and difficult problems in variational inequality theory is the development of an efficient and implementable approximation schemes for solving various classes of variational inequalities and variational inclusions. Recently, Huang [6–8] and Cho et al. [9] constructed some new perturbed iterative algorithms for approximation of solutions of some generalized nonlinear implicit quasi-variational inclusions (inequalities), which include many iterative algorithms for variational and quasi-variational inclusions (inequalities) as special cases. Inspired and motivated by recent research works [1, 9–19], we prove an existence theorem for solution of generalized strongly nonlinear implicit quasi-variational inequality problems and convergence of iterative sequences with errors, involving Lipschitzian, generalized pseudocontractivity and generalized -pseudocontractive mappings in Hilbert spaces.

## 2. Preliminaries

Let be a real Hilbert space with norm and inner product . For a nonempty closed convex subset , let be the projection of onto . Let be a set valued mapping with nonempty closed convex values, and be the mappings. We consider the following problem.

The problem (2.1) is called the *generalized strongly nonlinear implicit quasi-variational inequality problem*.

- (i)
If , for all , where is a nonempty closed convex subset of and is a mapping, then the problem (2.1) is equivalent to finding such that and

- (ii)
If we assume as identity mappings, then (2.1) reduces to the problem of finding such that and

- (iii)
If we assume , then (2.3) reduces to the following problem of finding such that and

- (iv)
If we assume , then (2.4) reduces to the following problem of finding such that and

- (v)
If , an identity mapping, then (2.5) is equivalent to finding such that

- (vi)
If , a nonempty closed convex subset of and for all , where a nonlinear mapping, then the problem (2.6) is equivalent to finding such that

- (vii)
If , for all , then (2.7) reduces to the following problem for finding such that

which is a classical variational inequality considered by [1, 4, 5].

- (1)

- (2)

- (3)

for , is called the Mann iterative process.

- (4)

- (5)
In particular, if for and . The sequence defined by

- (6)
In particular, if and for all . The sequence defined by

for , is called the Mann iterative process with errors, where is a summable sequence in and a sequence in [ 0,1] satisfying certain restrictions.

- (7)
Let be a nonempty convex subset of a Banach space and a mapping. For any given , the sequence defined by

(8) If for the sequence defined by

(9) If for , the sequence defined by

for , is called the Mann iterative process with errors.

For our main results, we need the following lemmas.

Lemma 2.1 (see [3]).

where is the projection of onto .

Lemma 2.2 (see [10]).

Lemma 2.3 (see [10]).

Lemma 2.4 (see [21]).

Then

has a unique fixed point, where is a constant.

## 3. Main Results

In this section, we establish an existence theorem for solution of generalized strongly nonlinear implicit quasi-variational inequality problems and convergence of the iterative sequences generated by (2.18). First, we give some definitions.

Definition 3.1.

It is easy to check that (3.1) is equivalent to

For in (3.1), we get the usual concept of pseudo-contractive of , introduced by Browder and Petryshyn [10], that is,

Definition 3.2.

Let and be the mappings. The mapping is said to be as follows.

In a similar way, we can define Lipschitz continuity of N with respect to the second and third arguments.

Definition 3.3.

Definition 3.4.

Remark 3.5.

where is a single-valued mapping and a nonempty closed convex subset of . If is Lipschitz continuous with constant , then from Lemma 2.3, is Lipschitz continuous with Lipschitz constant .

Now, we give the main result of this paper.

Theorem 3.6.

Let be a real Hilbert space and a set-valued mapping with nonempty closed convex values. Let be the Lipschitz continuous mappings with positive constants and respectively. Let be the mapping such that and are Lipschitz continuous with positive constants and respectively. A trimapping is generalized pseudo-contractive with respect to in the first argument of with constant and generalized -pseudo-contractive with respect to in the second argument of with constant , Lipschitz continuous with respect to the first, second, and third arguments with positive constants respectively. Suppose that is Lipschitz continuous with constant . Let , and be the three bounded sequences in and , , , , , , , and are sequences in satisfying the following conditions:

(1)

(2)

(3)

If the following conditions hold:

where and .

Then there exists a unique satisfying the generalized strongly nonlinear implicit quasi-variational inequality (2.1) and as , where is the three-step iteration process with errors defined as follows:

for .

Proof.

has a unique fixed point in .

Let be two arbitrary points in . From Lemma 2.2 and Lipschitz continuity of and , we have

Since is generalized pseudo-contractive with respect to in the first argument of and Lipschitz continuous with respect to first argument of and also is Lipschitz continuous, we have

Again since is generalized -pseudo-contractive with respect to in the second argument of and Lipschitz continuous with respect to second argument of and is Lipschitz continuous, we have

It follows from (3.13)–(3.16) that

From (3.10), we know that and so has a unique fixed point , which is a unique solution of the generalized strongly nonlinear implicit quasi-variational inequality (2.1).

Now we prove that converges to . In fact, it follows from (3.11) and that

From (3.17) and (3.19), it follows that

Similarly, we have

Again,

Let

Similarly, we deduce from (3.21) the following:

From the above inequalities, we get

Since , it follows from conditions (1) and (3) that

Therefore,

From (3.29)-(3.31) and Lemma 2.4, we know that converges to the solution . This completes the proof.

Remark 3.7.

We now deduce Theorem 3.6 in the direction of Ishikawa iteration.

Theorem 3.8.

Let be a real Hilbert space and a set-valued mapping with the nonempty closed convex values. Let and be the same as in Theorem 3.6. Suppose that is Lipschitz continuous with constant . Let and be the two bounded sequences in and , , , , and be six sequences in satisfying the following conditions:

(1)

(2)

(3)

If the following conditions holds:

Then there exists a unique satisfying the generalized strongly nonlinear implicit quasi-variational inequality (2.1) and as , where is the Ishikawa iteration process with errors defined as follows:

for .

Remark 3.9.

We can also deduce Theorem 3.6 in the direction of (2.16).

Theorem 3.10.

Let and be the same as in Theorem 3.6. Let be a bounded sequence in and , and be three sequences in satisfying the following conditions:

(1) for ,

(2) ,

(3) and

If the conditions of (3.10) hold, then there exists a unique satisfying the generalized strongly nonlinear implicit quasi-variational inequality (2.1) and as , where is the Mann iterative process with errors defined as follows:

for .

Our results can be further improved in the direction of (2.25).

Theorem 3.11.

for .

Now, we deduce Theorem 3.6 for three step iterative process in terms of (2.10).

Theorem 3.12.

Let and be the same as in Theorem 3.6. Let , , , , and be six sequences in satisfying conditions:

(1) for

(2) ,

(3)

If the conditions of (3.10) hold, then there exists satisfying (2.1) and as , where the three-step iteration process is defined by

for .

Next, we state the results in terms of iterations (2.10) and (2.25).

Theorem 3.13.

for .

Remark 3.14.

Theorem 3.13 can also be deduce for Ishikawa and Mann iterative process.

## Declarations

### Acknowledgment

The authors thank the editor Professor R. U. Verma and anonymous referees for their valuable useful suggestions that improved the paper.

## Authors’ Affiliations

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