Open Access

A System of Generalized Mixed Equilibrium Problems and Fixed Point Problems for Pseudocontractive Mappings in Hilbert Spaces

Fixed Point Theory and Applications20102010:361512

https://doi.org/10.1155/2010/361512

Received: 2 April 2010

Accepted: 11 June 2010

Published: 4 July 2010

Abstract

We introduce and analyze a new iterative algorithm for finding a common element of the set of fixed points of strict pseudocontractions, the set of common solutions of a system of generalized mixed equilibrium problems, and the set of common solutions of the variational inequalities with inverse-strongly monotone mappings in Hilbert spaces. Furthermore, we prove new strong convergence theorems for a new iterative algorithm under some mild conditions. Finally, we also apply our results for solving convex feasibility problems in Hilbert spaces. The results obtained in this paper improve and extend the corresponding results announced by Qin and Kang (2010) and the previously known results in this area.

1. Introduction

Let be a real Hilbert space with inner product and norm and let be a nonempty closed convex subset of . We denote weak convergence and strong convergence by notations and , respectively. Let be a mapping. In the sequel, we will use to denote the set of fixed points of , that is, .

Definition 1.1.

Let be a mapping. Then is called

contraction if there exists a constant such that

(1.1)

nonexpansive if

(1.2)

Remark 1.2.

It is well known that if is nonempty, bounded, closed, and convex and is a nonexpansive mapping on then is nonempty; see, for example, [1].

strongly pseudocontractive with the coefficient if

(1.3)
strictly pseudocontractive with the coefficient if
(1.4)

for such a case, is also said to be a -strict pseudocontraction, and if , then is a nonexpansive mapping,

pseudocontractive if

(1.5)

The class of strict pseudocontractions falls into the one between classes of nonexpansive mappings and pseudocontractions. Within the past several decades, many authors have been devoting to the studies on the existence and convergence of fixed points for strict pseudocontractions.

In 1967, Browder and Petryshyn [2] introduced a convex combination method to study strict pseudocontractions in Hilbert spaces. On the other hand, Marino and Xu [3] and Zhou [4] introduced and researched some iterative scheme for finding a fixed point of a strict pseudocontraction mapping. More precisely, take and define a mapping by
(1.6)

where is a strict pseudocontraction. Under appropriate restrictions on , it is proved the mapping is nonexpansive. Therefore, the techniques of studying nonexpansive mappings can be applied to study more general strict pseudocontractions.

The domain of the function is the set
(1.7)

Let be a proper extended real-valued function and let be a bifunction of into such that , where is the set of real numbers.

There exists the generalized mixed equilibrium problem for finding such that
(1.8)
The set of solutions of (1.8) is denoted by that is,
(1.9)

We see that is a solution of problem (1.8) implies that

Special Examples

(1)If , problem (1.8) is reduced into the mixed equilibrium problem for finding such that
(1.10)
Problem (1.10) was studied by Ceng and Yao [5]. The set of solutions of (1.10) is denoted by .
  1. (2)
    If , problem (1.8) is reduced into the generalized equilibrium problem for finding such that
    (1.11)
     
Problem (1.11) was studied by Takahashi and Toyoda [6]. The set of solutions of (1.11) is denoted by .
  1. (3)
    If and , problem (1.8) is reduced into the equilibrium problem for finding such that
    (1.12)
     
Problem (1.12) was studied by Blum and Oettli [7]. The set of solutions of (1.12) is denoted by .
  1. (4)
    If , problem (1.8) is reduced into the mixed variational inequality of Browder type for finding such that
    (1.13)
     
Problem (1.13) was studied by Browder [8]. The set of solutions of (1.13) is denoted by .
  1. (5)
    If and , problem (1.8) is reduced into the variational inequality problem for finding such that
    (1.14)
     
Problem (1.14) was studied by Hartman and Stampacchia [9]. The set of solutions of (1.14) is denoted by . The variational inequality has been extensively studied in the literature. See, for example, [7, 10, 11] and the references therein.
  1. (6)
    If and , problem (1.8) is reduced into the minimize problem for finding such that
    (1.15)
     

The set of solutions of (1.15) is denoted by .

The generalized mixed equilibrium problems include fixed point problems, variational inequality problems, optimization problems, Nash equilibrium problems, and the equilibrium problem as special cases. Numerous problems in physics, optimization, and economics reduce to find a solution of (1.8). In 1997, Combettes and Hirstoaga [12] introduced an iterative scheme of finding the best approximation to initial data when is nonempty and proved a strong convergence theorem. Many authors have proposed some useful methods for solving the , and ; see, for instance, [5, 1223].

Definition 1.3.

Let be a nonlinear mapping. Then is called

(1)monotone if
(1.16)
(2) -strongly monotone if there exists a constant such that
(1.17)
  1. (3)
    -Lipschitz continuous if there exists a positive real number such that
    (1.18)
     

(4) -inverse-strongly monotone if there exists a constant such that

(1.19)

Remark 1.4.

It is obvious that any -inverse-strongly monotone mappings are monotone and -Lipschitz continuous.

For finding a common element of the set of fixed points of a nonexpansive mapping and the set of solution of variational inequalities for a -inverse-strongly monotone mapping, Takahashi and Toyoda [6] introduced the following iterative scheme:
(1.20)

where is the metric projection of onto , is a -inverse-strongly monotone mapping, is a sequence in , and is a sequence in . They showed that if is nonempty, then the sequence generated by (1.20) converges weakly to some .

On the other hand, Y. Yao and J.-C Yao [24] introduced the following iterative process defined recursively by
(1.21)

where is a -inverse-strongly monotone mapping, and are sequences in the interval , and is a sequence in . They showed that if is nonempty, then the sequence generated by (1.21) converges strongly to some .

Let be a strongly positive linear bounded operator on if there is a constant with property
(1.22)
A typical problem is to minimize a quadratic function over the set of the fixed points a nonexpansive mapping on a real Hilbert space :
(1.23)
where is a linear bounded operator, is the fixed point set of a nonexpansive mapping on and is a given point in Moreover, it is shown in [25] that the sequence defined by the scheme
(1.24)
converges strongly to Recently, Plubtieng and Punpaeng [26] proposed the following iterative algorithm:
(1.25)
They proved that if the sequences and of parameters satisfy appropriate condition, then the sequences and both converge to the unique solution of the variational inequality:
(1.26)
which is the optimality condition for the minimization problem:
(1.27)

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

Very recently, Ceng et al. [27] introduced iterative scheme for finding a common element of the set of solutions of equilibrium problems and the of fixed points of a -strict pseudocontraction mapping defined in the setting of real Hilbert space : and let
(1.28)

where for some and satisfies . Further, they proved that and converge weakly to , where .

On the other hand, for finding a common element of the set of fixed points of a -strict pseudocontraction mapping and the set of solutions of an equilibrium problems in a real Hilbert space, Liu [28] introduced the following iterative scheme:
(1.29)

where is a -strict pseudocontraction mapping and and are sequences in They proved that under certain appropriate conditions over , , and , the sequences and both converge strongly to some , which solves some variational inequality problems (1.26).

In 2008, Ceng and Yao [5] introduced an iterative scheme for finding a common fixed point of a finite family of nonexpansive mappings and the set of solutions of a problem (1.8) in Hilbert spaces and obtained the strong convergence theorem which used the following condition.

 :  is -strongly convex with constant and its derivative is sequentially continuous from the weak topology to the strong topology. We note that the condition for the function : is a very strong condition. We also note that the condition does not cover the case and for each . Very recently, Wangkeeree and Wangkeeree [29] introduced a general iterative method for finding a common element of the set of solutions of the mixed equilibrium problems, the set of fixed point of a -strict pseudocontraction mapping, and the set of solutions of the variational inequality for an inverse-strongly monotone mapping in Hilbert spaces. They obtained a strong convergence theorem except the condition for the sequences generated by these processes.

In 2009, Qin and Kang [30] introduced an explicit viscosity approximation method for finding a common element of the set of fixed points of strict pseudocontraction and the set of solutions of variational inequalities with inverse-strongly monotone mappings in Hilbert spaces. Let be a sequence generated by the following iterative algorithm:
(1.30)

Then, they proved that under certain appropriate conditions imposed on , , , , , and , the sequence generated by (1.30) converges strongly to , where .

In the present paper, motivated and inspired by Qin and Kang [30], Peng and Yao [21], Plubtieng and Punpaeng [26], and Liu [28], we introduce a new general iterative scheme for finding a common element of the set of fixed points of strict pseudocontractions, the set of common solutions of the system of generalized mixed equilibrium problems, and the set of common solutions of the variational inequalities for inverse-strongly monotone mappings in Hilbert spaces. We obtain a strong convergence theorem for the sequences generated by these processes under some parameter controlling conditions. The results in this paper extend and improve the corresponding recent results of Qin and Kang [30], Peng and Yao [21], Plubtieng and Punpaeng [26], and Liu [28] and many others.

2. Preliminaries

Let be a real Hilbert space and let be a nonempty closed convex subset of . In a real Hilbert space , it is well known that
(2.1)
For any , there exists a unique nearest point in , denoted by , such that
(2.2)

The mapping is called the metric projection of onto

It is well known that is a firmly nonexpansive mapping of onto , that is,
(2.3)
Moreover, is characterized by the following properties: and
(2.4)

for all .

Lemma 2.1.

Let be a nonempty closed convex subset of a real Hilbert space . Given and then,
(2.5)

Lemma 2.2.

Let be a Hilbert space, let be a nonempty closed convex subset of and let be a mapping of into Let . Then for ,
(2.6)

where is the metric projection of onto .

A set-valued mapping is called amonotone if for all , and imply . A monotone mapping is called maximal if the graph of 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 map of into , -Lipschitz continuous mappings and let be the normal cone to when , that is,
(2.7)
and define a mapping on by
(2.8)

Then is the maximal monotone and if and only if see [31].

Lemma 2.3.

Let be a Hilbert space, let be a nonempty closed convex subset of and let be -inverse-strongly monotone. It , then is a nonexpansive mapping in

Proof.

For all and , we have
(2.9)

So, is a nonexpansive mapping of into .

Lemma 2.4 (see [32]).

Let be an inner product space. Then, for all and with one has
(2.10)

Lemma 2.5 (see [25]).

Let be a nonempty closed convex subset of let be a contraction of into itself with , and let be a strongly positive linear bounded operator on with coefficient . Then, for
(2.11)

That is, is strongly monotone with coefficient .

Lemma 2.6 (see [25]).

Assume that is a strongly positive linear bounded operator on with coefficient and . Then .

Lemma 2.7 (see [4]).

Let be a nonempty closed convex subset of a real Hilbert space and let be a -strict pseudocontraction mapping with a fixed point. Then is closed and convex. Define by for each . Then is nonexpansive such that .

Lemma 2.8 (see [33]).

Let be a closed convex subset of a Hilbert space and let  :  be a nonexpansive mapping. Then is demiclosed at zero, that is,
(2.12)

Lemma 2.9 (see [34]).

Let be a nonempty closed convex subset of a strictly convex Banach space . Let be a sequence of nonexpansive mappings on . Suppose that is nonempty. Let be a sequence of positive number with . Then a mapping on defined by
(2.13)

for is well defined and nonexpansive and holds.

For solving the mixed equilibrium problem, let us give the following assumptions for the bifunction , the function and the set :

for all

is monotone, that is, for all

for each   

for each is convex and lower semicontinuous;

for each is weakly upper semicontinuous;

for each and , there exists a bounded subset and such that for any ,
(2.14)

is a bounded set.

By similar argument as in the proof of Lemma in [35], we have the following lemma appearing.

Lemma 2.10.

Let be a nonempty closed convex subset of . Let be a bifunction satisfies (A1)–(A5) and let be a proper lower semicontinuous and convex function. Assume that either (B1) or (B2) holds. For and , define a mapping as follows:
(2.15)

for all . Then, the following holds:

(i)for each ;

(ii) is single-valued;

(iii) is firmly nonexpansive, that is, for any
(2.16)

(iv) 

(v) is closed and convex.

Remark 2.11.

We remark that Lemma 2.10 is not a consequence of Lemma in [5], because the condition of the sequential continuity from the weak topology to the strong topology for the derivative of the function does not cover the case .

Lemma 2.12 (see [36]).

Let and be bounded sequences in a Banach space and let be a sequence in with Suppose for all integers and Then,

Lemma 2.13 (see [37]).

Assume that is a sequence of nonnegative real numbers such that
(2.17)

where is a sequence in and is a sequence in such that

(1)

(2) or

Then

Lemma 2.14.

Let be a real Hilbert space. Then for all ,
(2.18)

3. Main Results

In this section, we will use the new approximation iterative method to prove a strong convergence theorem for finding a common element of the set of fixed points of strict pseudocontractions, the set of common solutions of the system of generalized mixed equilibrium problems, and the set of a common solutions of the variational inequalities for inverse-strongly monotone mappings in a real Hilbert space.

Theorem 3.1.

Let be a nonempty closed convex subset of a real Hilbert space . Let and be two bifunctions from to satisfying and let be a proper lower semicontinuous and convex function with either (B1) or (B2). Let  :  be a -inverse-strongly monotone mapping, let  :  be a -inverse-strongly monotone mapping, let  :  be an -inverse-strongly monotone mapping, and let  :  be a -inverse-strongly monotone mapping. Let  :  be an -contraction with coefficient and let be a strongly positive linear bounded operator on with coefficient and . Let  :  be a -strict pseudocontraction with a fixed point. Define a mapping  :  by for all . Suppose that
(3.1)
Let be a sequence generated by the following iterative algorithm:
(3.2)

where , , , and are sequences in , where , , , and and are positive sequences. Assume that the control sequences satisfy the following restrictions:

,

and

and , where are two positive constants,

, where .

Then, converges strongly to a point which is the unique solution of the variational inequality:
(3.3)

Equivalently, one has

Proof.

Since , as , we may assume, without loss of generality, that for all . By Lemma 2.6, we know that if , then . We will assume that . Since is a strongly positive bounded linear operator on we have
(3.4)
Observe that
(3.5)
and so this shows that is positive. It follows that
(3.6)
Since is a contraction of into itself with , then, we have
(3.7)

Since , it follows that is a contraction of into itself. Therefore the Banach Contraction Mapping Principle implies that there exists a unique element such that

Next, we will divide the proof into five steps.

Step 1.

We claim that is bounded.

Indeed, let and by Lemma 2.10, we obtain
(3.8)
Note that dom and dom ; we have
(3.9)
Put and . For each and by Lemma 2.3, we get that and are nonexpansive. Thus, we have
(3.10)
From Lemma 2.7, we have that is nonexpansive with . It follows that
(3.11)
which yields that
(3.12)

Hence, is bounded, and so are , , , , , , and .

Step 2.

We claim that and

Observing that and dom , by the nonexpansiveness of , we get
(3.13)
Similarly, let dom and dom ; we have
(3.14)
From and ; thus, we compute
(3.15)
Similarly, we have
(3.16)
Also noticing that
(3.17)
we compute
(3.18)
Substitution of (3.13), (3.14), (3.15), and (3.16) into (3.18) yields that
(3.19)
where is an appropriate constant such that
(3.20)
Putting for all , we have
(3.21)
Then, we compute
(3.22)
It follows from (3.19) and (3.22) that
(3.23)
This together with (C2), (C3), (C4), and (C6) implies that
(3.24)
Hence, by Lemma 2.12, we obtain as . It follows that
(3.25)
Moreover, we also get
(3.26)
Observe that
(3.27)
By conditions (C2), (C3), and (3.25), we have
(3.28)

Step 3.

We claim that the following statements hold:

;

;

;

.

For , we compute
(3.29)
By the same way, we can get
(3.30)
We note that
(3.31)
Similarly, we have
(3.32)
Observe that
(3.33)
Substituting (3.29), (3.30), (3.31), and (3.32) into (3.33), we obtain
(3.34)
It follows from (3.2) and (3.34) that
(3.35)
It follows from (C5) that
(3.36)
From (C2), (C6), and (3.25), we have
(3.37)
Since , we also have
(3.38)
From (C2), (C6), and (3.25), we obtain
(3.39)
Similarly, from (3.37) and (3.39), we can prove that
(3.40)
On the other hand, let for each we get . By Lemma 2.10(iii), that is, is firmly nonexpansive, we obtain
(3.41)
So, we obtain
(3.42)
Observe that
(3.43)
and hence
(3.44)
By using the same argument in (3.42) and (3.44), we can prove that
(3.45)
Substituting (3.42), (3.44), and (3.45) into (3.33), we obtain
(3.46)
From Lemma 2.4, (3.2), and (3.46), we obtain
(3.47)
It follows that
(3.48)
From (C2), (C6), (3.37), (3.39), (3.40), and as , we also have
(3.49)
From (3.47) and by using the same argument above, we can prove that
(3.50)
Applying (3.28), (3.49), and (3.50), we obtain
(3.51)

Step 4.

We claim that where is the unique solution of the variational inequality for all

To show the above inequality, we choose a subsequence of such that
(3.52)

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

That is, we will prove that
(3.53)

Assume also that and .

Define a mapping by
(3.54)
where , where . Since and by Lemma 2.9, we have that is nonexpansive and
(3.55)
Notice that
(3.56)
where is an appropriate constant such that
(3.57)
From (C4), (C6), and (3.28), we obtain
(3.58)
Since is a contraction with the coefficient , we have that there exists a unique fixed point. We use to denote the unique fixed point to the mapping . That is, . Since is bounded, there exists a subsequence of which converges weakly to . Without loss of generality, we may assume that . It follows from (3.58), that
(3.59)

It follows from Lemma 2.8 that . By (3.55), we have .

Hence from (3.52) and (2.4), we arrive at
(3.60)
On the other hand, we have
(3.61)
From (3.25) and (3.60), we obtain that
(3.62)

Step 5.

We claim that .

Indeed, by (3.2) and using Lemmas 2.6 and 2.14, we observe that
(3.63)
which implies that
(3.64)
Taking
(3.65)
then, we can rewrite (3.64) as
(3.66)

and we can see that and . Applying Lemma 2.13 to (3.66), we conclude that converges strongly to in norm. This completes the proof.

If the mapping is nonexpansive, then . We can obtain the following result from Theorem 3.1 immediately.

Corollary 3.2.

Let be a nonempty closed convex subset of a real Hilbert space . Let and be two bifunctions from to satisfying (A1)–(A5) and let  :  be a proper lower semicontinuous and convex function with either (B1) or (B2). Let  :  be a -inverse-strongly monotone mapping, let  :  be a -inverse-strongly monotone mapping, let  :  be an -inverse-strongly monotone mapping and let  :  be a -inverse-strongly monotone mapping. Let be an -contraction with coefficient and let be a strongly positive linear bounded operator on with coefficient and . Let be a nonexpansive mapping with a fixed point. Suppose that
(3.67)
Let be a sequence generated by the following iterative algorithm (3.2), where , , , and are sequences in , where , , , and and are positive sequences. Assume that the control sequences satisfy (C1)–(C6) in Theorem 3.1. Then, converges strongly to a point which is the unique solution of the variational inequality:
(3.68)

Equivalently, one has

If and in Theorem 3.1, then we can obtain the following result immediately.

Corollary 3.3.

Let be a nonempty closed convex subset of a real Hilbert space . Let and be two bifunctions from to satisfying (A1)–(A4). Let be a -inverse-strongly monotone mapping and let be a -inverse-strongly monotone mapping. Let be an -contraction with coefficient and let be a strongly positive linear bounded operator on with coefficient and . Let be a -strict pseudocontraction with a fixed point. Define a mapping by for all . Suppose that
(3.69)
Let be a sequence generated by the following iterative algorithm:
(3.70)

where , , , and are sequences in , where , , , and and are positive sequences. Assume that the control sequences satisfy the condition (C1)–(C6) in Theorem 3.1 and . Then, converges strongly to a point , where

If and in Corollary 3.3, then and we get and ; hence we can obtain the following result immediately.

Corollary 3.4.

Let be a nonempty closed convex subset of a real Hilbert space . Let be a -strict pseudocontraction with a fixed point. Define a mapping by for all . Suppose that Let be a sequence generated by the following iterative algorithm:
(3.71)

where , , , and are sequences in . Assume that the control sequences satisfy the conditions (C2) and (C3), in Theorem 3.1, and . Then, converges strongly to a point , where .

Finally, we consider the following Convex Feasibility Problem :
(3.72)

where is an integer and each is assumed to be the of solutions of equilibrium problem with the bifunction and the solution set of the variational inequality problem. There is a considerable investigation on in the setting of Hilbert spaces which captures applications in various disciplines such as image restoration [38, 39], computer tomography [40], and radiation therapy treatment planning [41].

The following result can be obtained from Theorem 3.1. We, therefore, omit the proof.

Theorem 3.5.

Let be a nonempty closed convex subset of a real Hilbert space . Let be a bifunction from to satisfying (A1)–(A5) and let  :  be a proper lower semicontinuous and convex function with either (B1) or (B2). Let  :  be an -inverse-strongly monotone mapping for each . Let  :  be a contraction mapping with coefficient and let be a strongly positive linear bounded operator on with coefficient and . Let  :  be a -strict pseudocontraction with a fixed point. Define a mapping  :  by for all . Suppose that
(3.73)
Let be a sequence generated by the following iterative algorithm:
(3.74)

where such that , are positive sequences, and and are sequences in . Assume that the control sequences satisfy the following restrictions:

and

for each ,

, where is some positive constant for each ,

, for each .

Then, converges strongly to a point which is the unique solution of the variational inequality:
(3.75)

Equivalently, one has

Declarations

Acknowledgments

The authors are grateful to the anonymous referees for their helpful comments which improved the presentation of the original version of this paper. The first author was supported by the Thailand Research Fund and the Commission on Higher Education under Grant No. MRG5380044. The second author was supported by Rajamangala University of Technology Rattanakosin Research and Development Institute.

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi, KMUTT
(2)
Department of Mathematics, Faculty of Liberal Arts, Rajamangala University of Technology Rattanakosin, RMUTR

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© Poom Kumam and Chaichana Jaiboon. 2010

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