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# A viscosity hybrid steepest-descent method for a system of equilibrium and fixed point problems for an infinite family of strictly pseudo-contractive mappings

Uamporn Witthayarat1, Jong Kyu Kim2 and Poom Kumam1*

Author Affiliations

1 Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi, KMUTT, Bangkok, 10140, Thailand

2 Department of Mathematics Education, Kyungnam University, Masan, 631-701, Korea

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Journal of Inequalities and Applications 2012, 2012:224  doi:10.1186/1029-242X-2012-224

 Received: 26 July 2012 Accepted: 21 September 2012 Published: 9 October 2012

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

Based on a viscosity hybrid steepest-descent method, in this paper, we introduce an iterative scheme for finding a common element of a system of equilibrium and fixed point problems of an infinite family of strictly pseudo-contractive mappings which solves the variational inequality for . Furthermore, we also prove the strong convergence theorems for the proposed iterative scheme and give a numerical example to support and illustrate our main theorem.

MSC: 46C05, 47D03, 47H05,47H09, 47H10, 47H20.

##### Keywords:
common fixed point; equilibrium problem; hybrid steepest-descent method; iterative algorithm; nonexpansive mapping; variational inequality

### 1 Introduction

Throughout this paper, we assume that H is a real Hilbert space with inner product and norm . Let C be a nonempty closed convex subset of H. A self-mapping is said to be a contraction on C if there exists a constant such that , . We denote by the collection of mappings f verifying the above inequality and note that each has a unique fixed point in C.

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

(1.1)

and we denote by the set of fixed points of the mapping T; that is, .

Note that T is the class of λ-strictly pseudo-contractive mappings including the class of nonexpansive mappings T on C (that is, , ) as a subclass. That is, T is nonexpansive if and only if T is 0-strictly pseudo-contractive.

A mapping is called k-Lipschitzian if there exists a positive constant k such that

(1.2)

F is said to be η-strongly monotone if there exists a positive constant η such that

(1.3)

Definition 1.1 A bounded linear operator A is said to be , if there exists a constant such that

In 2006, Marino and Xu [1] introduced the following iterative scheme: for ,

(1.4)

They proved that under appropriate conditions of the sequence , the sequence generated by (1.4) converges strongly to the unique solution of the variational inequality , , which is the optimality condition for the minimization problem

where h is a potential function for γf (i.e., for ).

In 2010, Jung [2] extended the result of Marino and Xu [1] to the class of k-strictly pseudo-contractive mappings with and introduced the following iterative scheme: for ,

(1.5)

where is a mapping defined by . He proved that the sequence generated by (1.5) converges strongly to a fixed point q of T, which is the unique solution of the variational inequality

Later, Tian [3] considered the following iterative method for a nonexpansive mapping with ,

(1.6)

where F is a k-Lipschitzian and η-strongly monotone operator. He proved that the sequence generated by (1.6) converges to a fixed point q in , which is the unique solution of the variational inequality

In 2010, Saeidi [4] introduced the following modified hybrid steepest-descent iterative algorithm for finding a common element of the set of solutions of a system of equilibrium problems for a family and the set of common fixed points for a family of infinitely nonexpansive mappings with respect to W-mappings (see [5]):

(1.7)

where B is a relaxed -cocoercive, k-Lipschitzian mapping such that . Then, under weaker hypotheses on coefficients, he proved the strongly convergence of the proposed iterative algorithm to the unique solution of the variational inequality.

Recently, Wang [6] extended and improved all the above results. He introduced a new iterative scheme: for ,

(1.8)

where is a mapping defined by (2.3), and F is a k-Lipschitzian and η-strongly monotone operator with . He proved that the sequence generated by (1.7) converges strongly to a common fixed point of an infinite family of -strictly pseudo-contractive mappings, which is a unique solution of the variational inequality

Very recently, He, Liu and Cho [7] introduced an explicit scheme which was defined by the following suitable sequence:

They generated -mapping by and where is a family of nonexpansive mappings from H into itself. They found that if , and satisfy appropriate conditions and , then converges strongly to , which satisfies the variational inequality for all .

In this paper, we introduce a new iterative scheme in a Hilbert space H which is a mixed iterative scheme of (1.7) and (1.8). We prove that the sequence converges strongly to a common element of the set of solutions of the system of equilibrium problems and the set of common fixed points of an infinite family of strictly pseudo-contractive mappings by using a viscosity hybrid steepest-descent method. The results obtained in this paper improved and extended the above mentioned results and many others. Finally, we give a simple numerical example to support and illustrate our main theorem in the last part.

### 2 Preliminaries

Let H be a real Hilbert space and C be a nonempty closed convex subset of H. We have

(2.1)

Recall that the nearest projection from H to C assigns to each the unique point satisfying the property

We recall some lemmas which will be needed in the rest of this paper.

Lemma 2.1In a Hilbert spaceH, the following inequality holds:

Lemma 2.2LetBbe ak-Lipschitzian andη-strongly monotone operator on a Hilbert spaceHwith, , and. Thenis a contraction with a contractive coefficientand.

Proof From (1.2), (1.3) and (2.1), we have

where , and so, .

Hence, S is a contraction with a contractive coefficient . □

Lemma 2.3LetHbe a Hilbert space. For a givenand,

Lemma 2.4LetHbe a real Hilbert space. Forqwhich solves the variational inequality, , , the following statement is true:

(2.2)

where.

Proof From Lemma (2.3), it follows that

□

Lemma 2.5[8]

LetCbe a closed convex subset of a Hilbert spaceHandbe a nonexpansive mapping with; if the sequenceweakly converges toxandconverges strongly to y, then.

Lemma 2.6[9]

Letandbe bounded sequences in a Banach spaceEandbe a sequence inwhich satisfies the following condition:

Suppose that, and. Then.

Lemma 2.7[10,11]

Letbe a sequence of non-negative real numbers satisfying

where, andsatisfy the following conditions:

(i) and;

(ii) or;

(iii) (), .

Then.

Lemma 2.8[12]

LetCbe a nonempty closed convex subset of a real Hilbert spaceHandbe aλ-strictly pseudo-contractive mapping. Define a mappingbyfor alland. ThenSis a nonexpansive mapping such that.

In this work, we defined the mapping by

(2.3)

where are real numbers such that , where is a -strictly pseudo-contractive mapping of C into itself and . By Lemma 2.8, we know that is a nonexpansive mapping and . As a result, it can be easily seen that is also a nonexpansive mapping.

Lemma 2.9[5]

LetCbe a nonempty closed convex subset of a strictly convex Banach space E. Letbe nonexpansive mappings ofCinto itself such thatandbe real numbers such thatfor each. Then for anyand, the limitexists.

By using Lemma 2.8, one can define the mapping W of C into itself as follows:

(2.4)

Such a mapping W is called the modified W-mapping generated by  , and  .

Lemma 2.10[5]

LetCbe a nonempty closed convex subset of a strictly convex Banach spaceE. Letbe nonexpansive mappings ofCinto itself such thatandbe real numbers such thatfor each. Then.

Combining Lemmas 2.7-2.9, one can get that .

Lemma 2.11[13]LetCbe a nonempty closed convex subset of a Hilbert spaceH, be a family of infinite nonexpansive mappings with, be a real sequence such that, for each. IfKis any bounded subset ofC, then

(2.5)

For solving the equilibrium problem, let us give the following assumptions on a bifunction , which were imposed in [14]:

(A1) for all ;

(A2) F is monotone, i.e., for all ;

(A3) for each , ;

(A4) for each , is convex and lower semicontinuous.

Lemma 2.12[14]

LetCbe a nonempty closed convex subset ofH, and letFbe a bifunction ofintosatisfying (A1)-(A4). Then forand, there existssuch that

(2.6)

Lemma 2.13[15]

LetCbe a nonempty closed convex subset ofH, and letFbe a bifunction ofintosatisfying (A1)-(A4). For, define a mappingas follows:

(2.7)

for all. Then the following conclusions hold:

(1) is single-valued;

(2) is firmly nonexpansive, i.e., for any,

(3) ;

(4) is closed and convex.

Lemma 2.14[5]

LetCbe a nonempty closed convex subset of a real Hilbert spaceH. Letbe an infinite family of nonexpanxive mappings withandbe a real sequence such thatfor each. Then:

(1) is nonexpansive andfor each;

(2) for eachand for each positive integerk, the limitexists;

(3) the mappingdefined byis a nonexpansive mapping satisfyingand it is called theW-mapping generated byand;

(4) ifKis any bounded subset ofC, then.

### 3 Main results

In this section, we will introduce an iterative scheme by using a viscosity hybrid steepest-descent method for finding a common element of the set of variational inequalities, fixed points for an infinite family of strictly pseudo-contractive mappings and the set of solutions of a system of equilibrium problems in a real Hilbert space.

Theorem 3.1LetCbe a nonempty closed convex subset of a real Hilbert spaceH, letbe a-strictly pseudo-contractive mapping with, be a finite family of bifunctionsintosatisfying (A1)-(A4) andbe a real sequence such thatfor each. LetBbe a k-Lipschitzian andη-strongly monotone operator on C withandwithand. Assume that. Let the mappingbe defined by (2.3). Letbe the sequence generated byand

(3.1)

whereandare sequences inwhich satisfy the following conditions:

(C1) and;

(C2) for some constant;

(C3) , for each.

Then the sequenceconverges strongly to, where, which is the unique solution of the variational inequality

(3.2)

or equivalently, qis the unique solution of the minimization problem

wherehis a potential function forγf.

Proof We will divide the proof of Theorem 3.1 into several steps.

Step 1. We show that is bounded. Let . Since for each , is nonexpansive. Given for and , for each , we have

Consider,

(3.3)

From Lemma 2.2, (3.1) and (3.3), it follows that

(3.4)

By mathematical induction, we have

(3.5)

and we obtain is bounded. So are , and .

Step 2. We claim that if is a bounded sequence in C, then

(3.6)

for every . From Step 2 of the proof in [[16], Theorem 3.1], we have for ,

(3.7)

Note that for every , we have

So, we note that

(3.8)

Now, applying (3.7) to (3.8), we conclude (3.6).

Step 3. We show that .

We define a sequence by , so that . We now observe that

(3.9)

It follows from (3.9) that

(3.10)

We observe that

(3.11)

and compute

(3.12)

Consider,

(3.13)

where is a constant such that for all .

Substituting (3.11) and (3.13) into (3.10), we can obtain

(3.14)

where .

It follows from (3.14) that

Hence, we have

From and the condition and for some , it follows that

(3.15)

By Lemma 2.5, we obtain

(3.16)

From and by (3.16), we get

(3.17)

Hence,

Step 4. We claim that .

It follows that

By the conditions (C1) and (C2), we obtain

(3.18)

Step 5. We show that

(3.19)

for any and . We note that is firmly nonexpansive by Lemma 2.12, then we observe that

and hence

(3.20)

It follows that

where

(3.21)

It follows from the condition (C1) that

(3.22)

So, we obtain

Using the condition (C1), (3.17) and (3.22), we obtain

(3.23)

Step 6. We show that , where .

The Banach contraction principle guarantees that has a unique fixed point q which is the unique solution of (3.1). Let be a subsequence of such that

Since is bounded, then there exists a subsequence which converges weakly to . Without loss of generality, we can assume that . We claim that .

Next, we need to show that . First, by (A2) and given and , we have

Thus,

(3.24)

From (A4), is a lower semicontinuous and convex, and thus weakly semicontinuous. The condition (C3) and (3.23) imply that

(3.25)

in norm. Therefore, letting in (3.24) yields

for all and . Replacing y with with and using (A1) and (A4), we obtain

Hence, , for all and . Letting and using (A3), we conclude that for all and . Therefore,

(3.26)

that is,

(3.27)

Next, we show that . By Lemma 2.6, we have

(3.28)

and . Assume that , then . Therefore, from the Opial property of a Hilbert space, (3.27), (3.28) and Step 4, we have

It is a contradiction. Thus z belongs to . Hence, .

Hence, by Lemma 2.4, we obtain

Step 7. We claim that converges strongly to . We observe that

where . Put and . It follows that

From (C1), (C2) and Step 5, it follows that and . Hence, by Lemma 2.7, the sequence converges strongly to q. □

Using Theorem 3.1, we obtain the following corollaries.

Corollary 3.2LetCbe a nonempty closed convex subset of a real Hilbert spaceH. Letbe an infinite family of nonexpansive mappings with, be a finite family of bifunctionsintosatisfying (A1)-(A4) andbe a real sequence such thatfor each. LetBbe a k-Lipschitzian andη-strongly monotone operator on C withandwithand. Assume that. Let the mappingbe defined by (2.3). Letbe the sequence generated byand

whereandare the sequences inwhich satisfy the following conditions:

(C1) and;

(C2) for some constant;

(C3) , for each.

Then the sequenceconverges strongly towhere, which is the unique solution of the variational inequality

Remark 3.3 Corollary 3.2 extends and improves Theorem 3.1 from f an infinite family of nonexpansive mappings to a family of strictly pseudo contractive mappings.

If in Theorem 3.1, we obtain the following corollary.

Corollary 3.4LetCbe a nonempty closed convex subset of a real Hilbert spaceH. Letbe an infinite family of nonexpansive mappings with, be a finite family of bifunctionsintosatisfying (A1)-(A4) andbe a real sequence such thatfor each. LetBbe a k-Lipschitzian andη-strongly monotone operator on C withandwithand. Assume that. Let the mappingbe defined by (2.3). Letbe the sequence generated byand

whereandare the sequences inwhich satisfy the following conditions:

(C1) and;

(C2) for some constant;

(C3) , for each.

Then the sequenceconverges strongly to, where, which is the unique solution of the variational inequality

If , , , and in Theorem 3.1, we obtain the following corollary:

Corollary 3.5LetCbe a nonempty closed convex subset of a real Hilbert spaceH. Letbe an infinite family of nonexpansive mappings with, be a finite family of bifunctionsintosatisfying (A1)-(A4) andbe a real sequence such thatfor each. LetBbe a k-Lipschitzian andη-strongly monotone operator on C withandwithand. Assume that. Let the mappingbe defined by (2.3). Letbe the sequence generated byand

whereandare the sequences inwhich satisfy the following conditions:

(C1) and;

(C2) for some constant;

(C3) , for each.

Then the sequenceconverges strongly towhere, which is the unique solution of the variational inequality

### 4 Numerical example

In this section, we give a real numerical example of Theorem 3.1 as follows.

Example 4.1 Let , , . , , , , , , , for every and . Then is the sequence generated by

(4.1)

and as , where 0 is the unique solution of the minimization problem

(4.2)

where is a constant.

Proof We divide the proof into four steps.

Step 1. Using the idea in [7], we can show that

(4.3)

where

(4.4)

Since , , , with the definition of , in Lemma 2.13, we have

(4.5)

By the equivalent property of the nearest projection from H to C, we can conclude that if we take , . By (3) in Lemma 2.13, we have

(4.6)

Step 2. We show that

(4.7)

Since , where is a -strictly pseudo-contractive mapping and , it can be easily seen that is a nonexpansive mapping. By (2.3), we have

and we compute (2.3) in the same way as above, so we obtain

Since , , , hence,

Step 3. We prove

where 0 is the unique solution of the minimization problem

Since we let , γ is a real number, so we choose . From (4.3), (4.4) and (4.7), we can obtain a special sequence of Theorem 3.1 as follows:

Since , , we have

Combining it with (4.6), we obtain

It is obvious that , 0 is the unique solution of the minimization problem , where is a constant number.

Step 4. In this step, we give the numerical results that support our main theorem as shown by plotting graphs using Matlab 7.11.0. We choose two different initial values as and in Table 1, Figure 1, and Figure 2, respectively. From the example, we can see that converges to 0.  □

Figure 1. The initial valueand iteration steps.

Figure 2. The initial valueand iteration steps.

Table 1. The sequence values on each different iteration step

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

All authors read and approved the final manuscript.

### Acknowledgements

This research was partially finished during the last visit at Kyungnam University, South Korea. The first author would like to thank the Office of the Higher Education Commission, Thailand, for the financial support of the PhD Program at KMUTT. The third author would like to thank the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission (under the project NRU-CSEC No.55000613) for financial support during the preparation of this manuscript. Finally, the authors would like to thank the referees for reading this paper carefully, providing valuable suggestions and comments, and improving the original version of this paper.

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