Generalized Strongly Nonlinear Implicit Quasivariational Inequalities

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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 Open image in new window -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, 3, 4, 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, 7, 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, 10, 11, 12, 13, 14, 15, 16, 17, 18, 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 Open image in new window -pseudocontractive mappings in Hilbert spaces.

2. Preliminaries

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

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

Special Cases
  1. (i)

    If Open image in new window , for all Open image in new window , where Open image in new window is a nonempty closed convex subset of Open image in new window and Open image in new window is a mapping, then the problem (2.1) is equivalent to finding Open image in new window such that Open image in new window and

     
the problem (2.2) is called generalized nonlinear quasi-variational inequality problem.
  1. (ii)

    If we assume Open image in new window as identity mappings, then (2.1) reduces to the problem of finding Open image in new window such that Open image in new window and

     
which is known as general implicit nonlinear quasi-variational inequality problem.
  1. (iii)

    If we assume Open image in new window , then (2.3) reduces to the following problem of finding Open image in new window such that Open image in new window and

     
which is known as generalized implicit nonlinear quasi-variational inequality problem, a variant form as can be seen in [20, equation ( 2.6)].
  1. (iv)

    If we assume Open image in new window , then (2.4) reduces to the following problem of finding Open image in new window such that Open image in new window and

     
The problem (2.5) is called the generalized strongly nonlinear implicit quasi-variational inequality problem, considered and studied by Cho et al. [9].
  1. (v)

    If Open image in new window , Open image in new window an identity mapping, then (2.5) is equivalent to finding Open image in new window such that

     
Problem (2.6) is called generalized strongly nonlinear quasi-variational inequality problem, see special cases of Cho et al. [9].
  1. (vi)

    If Open image in new window , Open image in new window a nonempty closed convex subset of Open image in new window and Open image in new window for all Open image in new window , where Open image in new window a nonlinear mapping, then the problem (2.6) is equivalent to finding Open image in new window such that

     
which is a nonlinear variational inequality, considered by Verma [17].
  1. (vii)

    If Open image in new window , for all Open image in new window , then (2.7) reduces to the following problem for finding Open image in new window such that

     

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

Now, we recall the following iterative process due to Ishikawa [13], Mann [14], Noor [15] and Liu [21].
  1. (1)
    Let Open image in new window be a nonempty convex subset of Open image in new window and Open image in new window a mapping. The sequence Open image in new window , defined by
     
, is called the three-step iterative process, where Open image in new window , Open image in new window , and Open image in new window are three real sequences in [ 0,1] satisfying some conditions.
, is called the Ishikawa iterative process, where Open image in new window and Open image in new window are two real sequences in [ 0,1] satisfying some conditions.

for Open image in new window , is called the Mann iterative process.

Recently Liu [21] introduced the concept of three-step iterative process with errors which is the generalization of Ishikawa [13] and Mann [14] iterative process, for nonlinear strongly accretive mappings as follows.
  1. (4)
    For a nonempty subset Open image in new window of a Banach spaces Open image in new window and a mapping Open image in new window , the sequence Open image in new window , defined by
     
, is called the three-step iterative process with errors. Here Open image in new window , Open image in new window and Open image in new window are three summable sequences in Open image in new window (i.e., Open image in new window , Open image in new window and Open image in new window ), and Open image in new window , Open image in new window and Open image in new window are three sequences in [ 0,1] satisfying certain restrictions.
, is called the Ishikawa iterative process with errors. Here Open image in new window and Open image in new window are two summable sequences in Open image in new window (i.e., Open image in new window and Open image in new window ; Open image in new window and Open image in new window are two sequences in [ 0,1] satisfying certain restrictions.
  1. (6)
     

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

However, in a recent paper [19] Xu pointed out that the definitions of Liu [21] are against the randomness of the errors and revised the definitions of Liu [21] as follows.
  1. (7)

    Let Open image in new window be a nonempty convex subset of a Banach space Open image in new window and Open image in new window a mapping. For any given Open image in new window , the sequence Open image in new window defined by

     

 (8) If Open image in new window for Open image in new window the sequence Open image in new window defined by

for Open image in new window , is called the Ishikawa iterative process with errors, where Open image in new window and Open image in new window are two bounded sequences in Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window and Open image in new window are six sequences in [ 0,1] satisfying the conditions

 (9) If Open image in new window for Open image in new window , the sequence Open image in new window defined by

for Open image in new window , is called the Mann iterative process with errors.

For our main results, we need the following lemmas.

Lemma 2.1 (see [3]).

If Open image in new window is a closed convex subset and Open image in new window a given point, then Open image in new window satisfies the inequality
if and only if

where Open image in new window is the projection of Open image in new window onto Open image in new window .

Lemma 2.2 (see [10]).

The mapping Open image in new window defined by (2.24) is nonexpansive, that is,

Lemma 2.3 (see [10]).

If Open image in new window and Open image in new window is a closed convex subset, then for any Open image in new window , one has

Lemma 2.4 (see [21]).

Let Open image in new window , Open image in new window and Open image in new window be three nonnegative real sequences satisfying

Then

By Lemma 2.1, we know that the generalized strongly nonlinear implicit quasi-variational inequality (2.1) has a unique solution if and only if the mapping Open image in new window by

has a unique fixed point, where Open image in new window 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.

A mapping Open image in new window is said to be generalized pseudo-contractive if there exists a constant Open image in new window such that

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

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

Definition 3.2.

Let Open image in new window and Open image in new window be the mappings. The mapping Open image in new window is said to be as follows.

(i)Generalized pseudo-contractive with respect to Open image in new window in the first argument of Open image in new window , if there exists a constant Open image in new window such that
(ii)Lipschitz continuous with respect to the first argument of Open image in new window if there exists a constant Open image in new window such that

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

(iii) Open image in new window is also said to be Lipschitz continuous if there exists a constant Open image in new window such that

Definition 3.3.

Let Open image in new window be the mappings. A mapping Open image in new window is said to be the generalized Open image in new window -pseudo-contractive with respect to the second argument of Open image in new window , if there exists a constant Open image in new window such that

Definition 3.4.

Let Open image in new window be a set-valued mapping such that for each Open image in new window , Open image in new window is a nonempty closed convex subset of Open image in new window . The projection Open image in new window is said to be Lipschitz continuous if there exists a constant Open image in new window such that

Remark 3.5.

In many important applications, Open image in new window has the following form:

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

Now, we give the main result of this paper.

Theorem 3.6.

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

(1) Open image in new window

(2) Open image in new window

(3) Open image in new window

If the following conditions hold:

where Open image in new window and Open image in new window .

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

for Open image in new window .

Proof.

We first prove that the generalized strongly nonlinear implicit quasi-variational inequality (2.1) has a unique solution. By Lemma 2.1, it is sufficient to prove the mapping defined by

has a unique fixed point in Open image in new window .

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

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

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

It follows from (3.13)–(3.16) that

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

Now we prove that Open image in new window converges to Open image in new window . In fact, it follows from (3.11) and Open image in new window 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 Open image in new window , it follows from conditions (1) and (3) that

Therefore,

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

Remark 3.7.

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

Theorem 3.8.

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

(1) Open image in new window

(2) Open image in new window

(3) Open image in new window

If the following conditions holds:

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

for Open image in new window .

Remark 3.9.

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

Theorem 3.10.

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

(1) Open image in new window for Open image in new window ,

(2) Open image in new window ,

(3) Open image in new window and Open image in new window

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

for Open image in new window .

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

Theorem 3.11.

Let Open image in new window be a real Hilbert space and Open image in new window a set-valued mapping with nonempty closed convex values. Let Open image in new window be the Lipschitz continuous mapping with respect to positive constants Open image in new window and Open image in new window respectively. Let Open image in new window be the mapping such that Open image in new window and Open image in new window be Lipschitz continuous with respect to positive constants Open image in new window and Open image in new window respectively. A trimapping Open image in new window is generalized pseudo-contractive with respect to map Open image in new window in first argument of Open image in new window with constant Open image in new window and generalized Open image in new window -pseudo-contractive with respect to Open image in new window in the second argument of Open image in new window with constant Open image in new window , Lipschitz continuous with respect to first, second, and third arguments with positive constants Open image in new window , respectively. Suppose that Open image in new window is a Lipschitz continuous with positive constant Open image in new window . Let Open image in new window , Open image in new window and Open image in new window be three bounded sequences in Open image in new window satisfying the conditions (1)–(3) of Theorem 3.6. If the conditions of (3.10) hold for Open image in new window , then there exists a unique Open image in new window satisfying (2.2) and Open image in new window as Open image in new window , where Open image in new window is the three step iteration process with errors defined as follows:

for Open image in new window .

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

Theorem 3.12.

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

(1) Open image in new window for Open image in new window

(2) Open image in new window ,

(3) Open image in new window

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

for Open image in new window .

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

Theorem 3.13.

Let Open image in new window and Open image in new window be the same as in the Theorem 3.11. Let Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window and Open image in new window be six sequences in Open image in new window satisfying conditions (1)–(3) of Theorem 3.6. If the conditions of (3.10) hold for Open image in new window , then there exists Open image in new window satisfying the generalized strongly nonlinear implicit quasi-variational inequality (2.2) and Open image in new window as Open image in new window , where the three-step iteration process Open image in new window is defined by

for Open image in new window .

Remark 3.14.

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

Notes

Acknowledgment

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

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© Salahuddin and M. K. Ahmad. 2009

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Authors and Affiliations

  1. 1.Department of MathematicsAligarh Muslim UniversityAligarhIndia

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