Generalized Strongly Nonlinear Implicit Quasivariational Inequalities

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.

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.

Find , such that and

(21)

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

Special Cases

• (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

(22)

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

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

(23)

which is known as general implicit nonlinear quasi-variational inequality problem.

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

(24)

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

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

(25)

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

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

(26)

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

• (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

(27)

which is a nonlinear variational inequality, considered by Verma [17].

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

(28)

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) Let be a nonempty convex subset of and a mapping. The sequence , defined by

  

  (29)

(210)

, is called the three-step iterative process, where , , and are three real sequences in [ 0,1] satisfying some conditions.

• (2) In particular, if for all , then , defined by

  

  (211)

, is called the Ishikawa iterative process, where and are two real sequences in [ 0,1] satisfying some conditions.

• (3) In particular, if for all , then defined by

  

  (212)

for , 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.

• (4) For a nonempty subset of a Banach spaces and a mapping , the sequence , defined by

  

  (213)

, is called the three-step iterative process with errors. Here , and are three summable sequences in (i.e., , and ), and , and are three sequences in [ 0,1] satisfying certain restrictions.

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

(214)

, is called the Ishikawa iterative process with errors. Here and are two summable sequences in (i.e., and ; and are two sequences in [ 0,1] satisfying certain restrictions.

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

(215)

(216)

for , is called the Mann iterative process with errors, where is a summable sequence in and 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.

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

(217)

(218)

for , is called the three-step iterative process with errors, where , and are three bounded sequences in and , , , , , , , and are nine sequences in [ 0,1] satisfying the conditions

(219)

 (8) If for the sequence defined by

(220)

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

(221)

 (9) If for , the sequence defined by

(222)

for , is called the Mann iterative process with errors.

For our main results, we need the following lemmas.

Lemma 2.1 (see [3]).

If is a closed convex subset and a given point, then satisfies the inequality

(223)

if and only if

(224)

where is the projection of onto .

Lemma 2.2 (see [10]).

The mapping defined by (2.24) is nonexpansive, that is,

(225)

Lemma 2.3 (see [10]).

If and is a closed convex subset, then for any , one has

(226)

Lemma 2.4 (see [21]).

Let , and be three nonnegative real sequences satisfying

(227)

Then

(228)

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 by

(229)

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.

A mapping is said to be generalized pseudo-contractive if there exists a constant such that

(31)

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

(32)

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

(33)

Definition 3.2.

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

(i)Generalized pseudo-contractive with respect to in the first argument of , if there exists a constant such that

(34)

(ii)Lipschitz continuous with respect to the first argument of if there exists a constant such that

(35)

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

(iii) is also said to be Lipschitz continuous if there exists a constant such that

(36)

Definition 3.3.

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

(37)

Definition 3.4.

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

(38)

Remark 3.5.

In many important applications, has the following form:

(39)

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:

(310)

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:

(311)

for .

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

(312)

has a unique fixed point in .

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

(313)

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

(314)

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

(315)

(316)

It follows from (3.13)–(3.16) that

(317)

where

(318)

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

(319)

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

(320)

Similarly, we have

(321)

Again,

(322)

Let

(323)

Then and

(324)

Similarly, we deduce from (3.21) the following:

(325)

(326)

From the above inequalities, we get

(327)

where

(328)

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

(329)

(330)

Therefore,

(331)

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:

(332)

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:

(333)

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:

(334)

for .

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

Theorem 3.11.

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

(335)

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

(336)

for .

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

Theorem 3.13.

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

(337)

for .

Remark 3.14.

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