An New Iterative Scheme for Variational Inequalities and Nonexpansive Mappings in Hilbert Spaces

Qiqiong Chen^{1}^{, *}, Congjun Zhang^{2}

^{1}Department of Applied Mathematics, Nanjing University of Science and Technology, Nanjing, Jiangsu, China

^{2}College of Applied Mathematics, Nanjing University of Finance and Economics, Nanjing, Jiangsu, China

Abstract

In this paper, a new three-step iterative scheme is introduced for finding the common element of the set of fixed points of a nonexpansive mapping and the set of solutions of a variational inequality for -inverse-strongly monotone mappings. The result reveals that the proposed iterative sequence converges strongly to the common element of this two. And our studies can be regarded as an extension of the existing results, which we illustrate one by one in our remarks.

Keywords

Variational Inequality, Nonexpansive Mapping, Fixed Point

Received:May 18, 2015

Accepted: May 27, 2015

Published online: July 9, 2015

@ 2015 The Authors. Published by American Institute of Science. This Open Access article is under the CC BY-NC license. http://creativecommons.org/licenses/by-nc/4.0/

1. Introduction

Variational inequalities first studied by Stampacchia [1] in 1960s have played an important role in the development of pure and applied mathematics. They have also witnessed an explosive growth in theoretical progression, algorithmic development, etc.; see e.g. [2-14]. Let be a real Hilbert space, whose inner product and norm are denoted by and ‖.‖, respectively. Let be a nonempty closed convex subset of and a mapping from to . The classical variational inequality problem is to find a vector such that

,

For all or The set of solutions of the variational inequality is denoted by . A mapping of to is called α-inverse-strongly monotone [6] if there exists a positive real number α such that

for any . A mapping of into itself is called nonexpansive [6] if

for all . We denote the set of fixed points of by .

In order to seek for an element of Takahashi and Toyoda [4] introduced the following iterative scheme

(1)

for every , where is a sequence in and is a sequence in (0,2 is the metric projection of onto They proved that the iterative consequences generated by (1) converge weakly to an element For convenience, we will use through the whole paper.

On the other hand, Iiduka and Takahashi [5] put forward another iterative scheme:

(2)

for every , where is a sequence in is a sequence in (0, 2 is the metric projection of onto It was proved that the iterative consequences generated by (2) converge strongly to an element .

Furthermore, Yao and Yao [6] proposed the following mixed gradient method:

(3)

for every , where , are sequences in satisfied and is a sequence in (0, 2 is the metric projection of onto They proved that the iterative consequences defined by (3) converge strongly to , where was the metric projection of onto

In recent years, many authors have studied some different iterative schemes both in Hilbert spaces and Banach spaces, see e.g. [2-14]. Inspired and motivated by those previous researches, we suggest and analyze a new iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping and the solution set of a variational inequality for -inverse-strongly monotone mapping in real Hilbert spaces. Strong convergence theorems are established and the iterative methods considered by [4,5,7,10] are included in our results.

2. Preliminary

For convenience, we would like to list some definitions and fundamental lemmas which are useful in the following consequent analysis. They can be found in any standard functional analysis books such as [15,16].

*Definition 2.1* A mapping is a contraction on if there exists a constant such that

*Definition 2.2* A set-valued mapping** ** is called monotone if for all and imply

*Definition 2.** 3* A monotone mapping is maximal if its graph is not properly contained in the graph of any other monotone mapping.

It is known that a monotone mapping is maximal if and only if, for for every implies Let be a monotone mapping of into and let be the normal cone operator to defined by . Define

(4)

Then is maximal monotone and if and only if (see [11]).

*Definition 2.4* For every point , there exists a unique nearest point in denoted by , such that for all is called the metric projection of onto

It is well known that is a nonexpansive mapping of onto and satisifies for every

Moreover, is characterized by the following properties:

for all (5)

It is easy to see that the following is true:

(6)

Note that satisfies Opial's condition [17], i.e., for any sequence with the inequality

holds for every with

Next we present some useful lemmas.

The following lemma is an immediate consequence of equality:

*Lemma 2.** 1* Let be a real Hilbert space. Then the following inequality holds:

*Lemma 2.** 2 (Osilike [14])* Let be an inner space. Then for all and with we have

=

*Lemma 2.** 3 (Xu [10])* Assume is a sequence of nonnegative real numbers such that )+, where is a sequence in and is a sequence such that

(i). =

(ii). or

Then

For convenience, we use for strong convergence and for weak convergence in the following analysis.

3. Main Results

In this section, we suggest and analyze a new iterative scheme for finding the common element of the fixed points of a nonexpansive mapping and the solution set of variational inequalities for an -inverse-strongly monotone mapping in a real Hilbert space. Strong convergence theorems are established and several special cases are also discussed.

*Theorem 3.** 1* Let be a nonempty closed convex subset of a real Hilbert space Let be an -inverse-strongly monotone mapping of into and let be a nonexpansive mapping of into itself such that be a contraction mapping with coefficient Suppose and are given by

(7)

where ,{} are three sequences in and is a sequence in . Assume that ,{} are chosen so that for some with and

(C1) =,

(C2) =,

(C3)

Then the sequence converges strongly to where or equivalently satisfies the following variational inequality:

*Proof*: We first show that is a nonexpansive mapping. For all and , we have

which implies that** **** **is nonexpansive .

For convenience, we set

Then the iterative scheme (7) can be written as:

(8)

Let Then we have by (6) and Since the proof of the theorem is rather long, it will be more convenient to divide the process into several steps.

*Step 1.* We claim that is bounded.

Since both and are nonexpansive mappings, we have

(9)

Similarly, we obtain that

(10)

Combining (8) and (9), together with that is nonexpansive mapping, we see that

(11)

By (10) and (11), we get

Hence

(12)

From (12), we arrive at

(13)

By the method of induction, we have

(14)

Therefore is bounded. Consequently, all those sequences are bounded.

*Step 2.* We now in the position to prove that

Since both and are nonexpansive mappings, we first have

(15)

By similar method, we have

(16)

In view of (15), after simple calculation, we see that

(). (17)

By (16) and (17), we get

()

+()

+ (18)

In view of (18), we have

(19)

Where

=[]+()

+()

+. (20)

By the conditions (C1), (C2) and (C3), we see that and which combining with Lemma 2.3, yields

. (21)

Since together with (21) and the condition (C1) imply that

. (22)

Since =(1- and is bounded, we have

. (23)

Furthermore, combining Lemma 2.2 with that is nonexpansive, is -inverse-strongly monotone mapping, , and we obtain that

(24)

From (24), together with Lemma 2.2, we see that

(25)

which implies that

(26)

It follows from conditions (C1), (C2) and (21) that

(27)

*Step 3.* We show that

Since , is bounded, and we have

(28)

We now show that

Since

= {

}

[

], (29)

we get that

+ (30)

Hence

(31)

It follows from (30) that

(32)

Hence

(33)

Since

and are bounded,

we have

(34)

It follows from (28), (22),(23) and (34) , together with

that

(35)

*Step 4*. We prove that

As is bounded, there exists a subsequence of converges weakly to

Since

combining (23) and (34) we know that

Then

Next we show that

Let

where Then is maximal monotone. Let where

Since and we have

On the other hand, from (5) and

we see that Then

Thus

(36)

Putting , we have

Since is maximal, we have Hence

Now let us show that Assume that From Opial’s condition, we have

. (37)

This is a contradiction. Thus we obtain that

Since is a contraction mapping, by Banach's contraction theorem, there exists a unique fixed point of , that's

*Step 5*. We prove that

From (5), we know

(38)

*Step 6.* We claim that From Lemma 2.1 and Lemma 2.2, we obtain that

=

. (39)

Then we have

. (40)

That is

(41)

where , and {}.

From (38) and conditions (C1), (C2) and (C3), letting yields

(42)

Let

Then +

It is easy to check that

By Lemma 2.3, we see that

(43)

The proof is finished.

As an implication of Theorem 3.1, we have the following corollary:

*Corollary 3.1* Let be a nonempty closed convex subset of a real Hilbert space Let be an -inverse-strongly monotone mapping of into and let be a be a nonexpansive mapping of into itself such that be a contraction mapping with coefficient k Suppose that and are given by

where ,{} are three sequences in and is a sequence in . Assume that ,{} are chosen so that for some with and

(C1) =,

(C2) =,

(C3)

Then the sequence converges strongly to where or equivalently satisfies the following variational inequality:

Proof: The conclusion follows from Theorem 3.1 by setting

Theorem 3.1 extends the corresponding results of [4,5,7,10].

*Remark 3.1* Putting in Theorem 3.1, we can get the iterative scheme provided by [4].

*Remark 3.2* Putting in Theorem 3.1, we can get the iterative scheme provided by [5].

*Remark 3.3* The proposition 3.1 of [7] is a special case of our result.

In fact, letting in Theorem 3.1, we get

Then

by Theorem 3.1.

*Remark 3.4* Putting in Theorem 3.1, we can get the iterative scheme provided by [10].

*Remark 3.5* The conditions in Theorem 3.1 can be easily satisfied, for example

4. Conclusion

By introducing a new iterative scheme for variational inequalities and nonexpansive mappings in Hilbert spaces, we proved that the sequences generated by the iterative scheme strongly converge to a common element of the fixed points of a nonexpansive mapping and the solution set of variational inequality for -inverse-strongly monotone mapping.

Acknowledgements

This work is supported by the National Natural Science Foundation of China (No. 11071109) and the China Scholarship Council (No. 201406840039). The authors are so grateful for Professor Yuanguo Zhu’s valuable suggestions to improve this paper. It is accomplished during the first author’s visit to Professor Jinlu Li at Shawnee State University, USA. The authors also would like to express their deep gratitude for the warm hospitality from Shawnee State University.

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