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A Shrinking Projection Method for Generalized Mixed Equilibrium Problems, Variational Inclusion Problems and a Finite Family of QuasiNonexpansive Mappings
Journal of Inequalities and Applicationsvolume 2010, Article number: 458247 (2010)
Abstract
The purpose of this paper is to consider a shrinking projection method for finding a common element of the set of solutions of generalized mixed equilibrium problems, the set of fixed points of a finite family of quasinonexpansive mappings, and the set of solutions of variational inclusion problems. Then, we prove a strong convergence theorem of the iterative sequence generated by the shrinking projection method under some suitable conditions in a real Hilbert space. Our results improve and extend recent results announced by Peng et al. (2008), Takahashi et al. (2008), S.Takahashi and W. Takahashi (2008), and many others.
1. Introduction
Throughout this paper, we assume that is 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.
Recall that the following definitions.
(1)A mapping is said to be nonexpansive if
(2)A mapping is said to be quasinonexpansive if
We denote be the set of fixed points of .
Let be a singlevalued nonlinear mapping and to a setvalued mapping. The variational inclusion problem is to find such that
where is the zero vector in . The set of solutions of problem (1.3) is denoted by .
Definition 1.1.
A mapping is said to be a inversestrongly monotone if there exists a constant with the property
Remark 1.2.
It is obvious that any inversestrongly monotone mapping is monotone and Lipschitz continuous. It is easy to see that if any constant is in , then the mapping is nonexpansive, where is the identity mapping on
A setvalued mapping is called monotone if for all , and implying . A monotone mapping; is maximal if its 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 all imply .
Definition 1.3.
Let be a setvalued maximal monotone mapping, then the singlevalued mapping defined by
is called the resolvent operator associated with , where is any positive number and is the identity mapping.
Remark 1.4.
(R1) The resolvent operator is singlevalued and nonexpansive for all , that is,
(R2)The resolvent operator is 1inverse strongly monotone; see [1], that is,
(R3)The solution of problem (1.3) is a fixed point of the operator for all ; see also [2], that is,
(R4)If , then the mapping is nonexpansive.
(R5) is closed and convex.
Let be a nonlinear mapping, let be a realvalued function and a bifunction from to . We consider the following generalized mixed equilibrium problem.
Finding such that
The set of such is denoted by that is,
It is easy to see that is solution of problem (1.9) implies that .
(i)In the case of (:the zero mapping), then the generalized mixed equilibrium problem (1.9) is reduced to the mixed equilibrium problem. Finding such that
The set of solution of (1.11) isdenoted by
(ii)In the case of , then the generalized mixed equilibrium problem (1.9) is reduced to the generalized equilibrium problem. Finding such that
The set of solution of (1.12) is denoted by
(iii)In the case of (:the zero mapping) and , then the generalized mixed equilibrium problem (1.9) is reduced to the equilibrium problem. Finding such that
The set of solution of (1.13) is denoted by
(iv)In the case of , and then the generalized mixed equilibrium problem (1.9) is reduced to the variational inequality problem. Finding such that
The set of solution of (1.14) is denoted by
The generalized mixed equilibrium problem include fixed point problems, optimization problems, variational inequalities problems, Nash equilibrium problems, noncooperative games, economics and the equilibrium problems as special cases (see, e.g., [3–8]). Some methods have been proposed to solve the generalized mixed equilibrium problems, generalized equilibrium problems and equilibrium problems; see, for instance, [9–22].
In 2007, Takahashi et al. [23] proved the following strong convergence theorem for a nonexpansive mapping by using the shrinking projection method in mathematical programming. For a and , they defined a sequence as follows
where . They proved that the sequence generated by (1.15) converges weakly to , where
In 2008, S. Takahashi and W. Takahashi [24] introduced the following iterative scheme for finding a common element of the set of solutions of mixed equilibrium problems and the set of fixed points of a nonexpansive mapping in a Hilbert space. Starting with arbitrary , define sequences , and by
They proved that under certain appropriate conditions imposed on , and , the sequence generated by (1.16) converges strongly to
In 2008, Zhang et al. [25] introduced the following new iterative scheme for finding a common element of the set of solutions to the problem (1.3) and the set of fixed points of a nonexpansive mapping in a real Hilbert space. Starting with an arbitrary , define sequences and by
where is the resolvent operator associated with and a positive number is a sequence in the interval .
In 2008, Peng et al. [26] introduced the following iterative scheme by the viscosity approximation method for finding a common element of the set of solutions to the problem (1.3), the set of solutions of an equilibrium problems and the set of fixed points of nonexpansive mappings in a Hilbert space. Starting with an arbitrary , define sequences and by
They proved that under certain appropriate conditions imposed on and , the sequence generated by (1.18) converges strongly to
In 2010, Katchang and Kumam [27] introduced an iterative scheme for finding a common element of the set of solutions for mixed equilibrium problems, the set of solutions of the variational inclusions with setvalued maximal monotone mappings, and inverse strongly monotone mappings and the set of fixed points of a finite family of nonexpansive mappings in a real Hilbert space.
In this paper, motivated and inspired by the previously mentioned results, we introduce an iterative scheme by the shrinking projection method for finding a common element of the set of solutions of generalized mixed equilibrium problems, the set of fixed points of a finite family of quasinonexpansive mappings and the set of solutions of variational inclusion problems in a real Hilbert space. Then, we prove a strong convergence theorem of the iterative sequence generated by the proposed shrinking projection method under some suitable conditions. The results obtained in this paper extend and improve several recent results in this area.
2. Preliminaries
Let be a real Hilbert space and let be a nonempty closed convex subset of Recall that the (nearest point) projection from onto assigns to each the unique point in satisfying the property
We recall some lemmas which will be needed in the rest of this paper.
Lemma 2.1.
For a given and ,
It is well known that is a firmly nonexpansive mapping of onto and satisfies
Lemma 2.2 (see [1]).
Let be a maximal monotone mapping and let be a Lipshitz continuous mapping. Then the mapping is a maximal monotone mapping.
Lemma 2.3 (see [28]).
Let be a closed convex subset of and let be a bounded sequence in . Assume that
(1)the weak limit set ,
(2)for each , exists.
Then is weakly convergent to a point in .
Lemma 2.4 (see [29]).
Each Hilbert space satisfies Opial's condition, that is, for any sequence with , the inequality
holds for each with .
Lemma 2.5 (see [30]).
Each Hilbert space satisfies the KadecKlee property, that is, for any sequence with and together imply .
For solving the generalized equilibrium problems, let us give the following assumptions for and the set :
(A1) for all
(A2) is monotone, that is, for all
(A3)for each is weakly upper semicontinuous;
(A4)for each is convex and lower semicontinuous;
(B1)for each and there exists a bounded subset and such that for any
(B2)C is bounded set.
Lemma 2.6 (see [31]).
Let be a nonempty closed convex subset of and let be a bifunction of into satisfying (A1)–(A4). Let be a proper lower semicontinuous and convex function such that . For and , define a mapping as follows:
Assume that either or holds. Then, the following conclusions hold:
(1)for each
(2) is singlevalued;
(3) is firmly nonexpansive, that is, for any
(4)
(5) is closed and convex.
Remark 2.7.
Replacing with in (2.5), then there exists , such that
3. Main Results
In this section, we will introduce an iterative scheme by using shrinking projection method for finding a common element of the set of solutions of generalized mixed equilibrium problems, the set of fixed points of a finite family of quasinonexpansive mappings and the set of solutions of variational inclusion problems in a real Hilbert space.
Let be a finite family of nonexpansive mappings of into itself, and let be real numbers such that for every . We define a mapping as follows:
Such a mapping is called the Kmapping generated by and ; see [32].
We have the following crucial Lemma 3.1 and Lemma 3.2 concerning mapping which can be found in [14]. Now we only need the following similar version in Hilbert spaces.
Lemma 3.1.
Let be a nonempty closed convex subset of a real Hilbert space . Let be a finite family of quasinonexpansive mappings and Lipschitz mappings of into itself with and let be real numbers such that for every , and . Let be the Kmapping generated by and . Then, the followings hold:
(1) is quasinonexpansive and Lipschitz,
(2).
Lemma 3.2.
Let be a nonempty closed convex subset of a real Hilbert space . Let be a finite family of quasinonexpansive mappings and Lipschitz mappings of into itself and sequences in such that Moreover, for every , let and be the Kmappings generated by and , and and , respectively. Then, for every , we have
Now we study the strong convergence theorem concerning the shrinking projection method.
Theorem 3.3.
Let be a nonempty closed convex subset of a real Hilbert space , let be a bifunction from to satisfying (A1)–(A4), and let be a proper lower semicontinuous and convex function with assumption (B1) or (B2). Let be a finite family of quasinonexpansive and Lipschitz mappings of into itself, and let be a inversestrongly monotone mapping of into , let a inversestrongly monotone mapping of into and be a maximal monotone mapping. Assume that
Let be the mapping generated by and . Let , , , and be sequences generated by , , and let
where satisfy the following conditions:
(i) for some with ;
(ii), for some with ;
(iii) for some with .
Then, and converge strongly to
Proof.
In the light of the definition of the resolvent, can be rewritten as . Let and using the fact be a sequence of mappings defined as in Lemma 2.6, is an inversestrongly monotone and that , where for some with , we can write
Next, we will divide the proof into six steps.
Step 1.
We first show that is well defined and is closed and convex for any .
From the assumption, we see that is closed and convex. Suppose that is closed and convex for some . Next, we show that is closed and convex for some . For any , we obtain that
is equivalent to
Thus is closed and convex. Then, is closed and convex for any . This implies that is well defined.
Step 2.
Next, we show by induction that for each .
Taking and by condition (ii), we get that is nonexpansive for all . From the assumption, we see that . Suppose for some . For any , we have
Thus, we have
It follows that This implies that for each .
Step 3.
Next, we show that and .
From , we have
for each . Using we also have
So, for , we have
This implies that
From and , we obtain
From (3.13), we have, for
It follows that
Thus the sequence is a bounded and nonincreasing sequence, so exists, that is,
Indeed, from (3.13), we get
From (3.16), we obtain
Since we have
By (3.18), we obtain
Step 4.
Next, we show that
For any given , . It is easy to see that . As is nonexpansive, we have
Similarly, we can prove that
Observe that
Substituting (3.21) into (3.23), and using conditions (i) and (ii), we have
It follows that
Since , we obtain
Since the resolvent operator is 1inverse strongly monotone, we obtain
which yields that
Similarly, we obtain
Substituting (3.28) into (3.23), and using condition (i), we have
It follows that
Applying and as to the last inequality, we get
Note that
Substituting (3.22) into (3.33), and using conditions (i) and (ii), we have
It follows that
Since , we obtain
Substituting (3.29) into (3.33), and using conditions (i) and (ii), we have
It follows that
Applying and as to the last inequality, we get
From (3.32) and (3.39), we have
From (3.33), (3.4), and condition (iii), we have
It follows that
Since , we obtain
On the other hand, in the light of Lemma 2.6(3), is firmly nonexpansvie, so we have
which implies that
Using (3.41) again and (3.45), we have
It follows from the condition (i) that
Since and , it is implied that
From (3.39) and (3.48), we have
By (3.3), we get
Since for some with , and as , we also have
From (3.40) and (3.48), we have
Furthermore, by the triangular inequality, we also have
Applying (3.51) and (3.52), we obtain
Let be the mapping defined by (3.1). Since is bounded, applying Lemma 3.2 and (3.54), we have
Step 5.
Next, we show that
Since is bounded, there exists a subsequence of which converges weakly to . Without loss of generality, we can assume that . Since and is closed and convex, is weakly closed and hence . From we obtain

(a)
First, we prove that .
We observe that is a Lipschitz monotone mapping and . From Lemma 2.2, we know that is maximal monotone. Let , that is, . Since , we have
that is,
By virtue of the maximal monotonicity of , we have
and so
It follows from , and that
It follows from the maximal monotonicity of that , that is, .

(b)
Next, we show that . Since dom , we have
(3.61)
From (A2), we also have
And hence
For with and let Since and we have So, from (3.63), we have
Since we have . Further, from the inverse strongly monotonicity of we have So, from (A5), the weakly lower semicontinuity of and , we have
as From (A1), (A4) and (3.65), we also get
Letting we have, for each
This implies that

(c)
Now, we prove that .
Assume Since and we know that and it follows by the Opial's condition (Lemma 2.4) that
which is a contradiction. Thus, we get .
The conclusion is .
Step 6.
Finally, we show that and , where
Since is nonempty closed convex subset of , there exists a unique such that Since and , we have
for all . From (3.69), is bounded, so . By the weak lower semicontinuity of the norm, we have
However, Since , we have
Using (3.69) and (3.70), we obtain . Thus and So, we have
Thus, we obtain that
From , we obtain . Using the KadecKlee property (Lemma 2.5) of , we obtain that
and hence in norm. Finally, noticing we also conclude that in norm. This completes the proof.
Corollary 3.4.
Let be a nonempty closed convex subset of a real Hilbert space , let be a bifunction from to satisfying (A1)–(A4), and let be a proper lower semicontinuous and convex function with assumption (B1) or (B2). Let be a finite family of nonexpansive mappings of into itself, let be a inversestrongly monotone mapping of into , let be a inversestrongly monotone mapping of into and a maximal monotone mapping. Assume that
Let be the mapping generated by and . Let , , , and be sequences generated by (3.3) satisfying the following conditions in Theorem 3.3. Then, and converge strongly to
From Theorem 3.3, we can obtain the following results.
Theorem 3.5.
Let be a nonempty closed convex subset of a real Hilbert space , let be a bifunction from to satisfying (A1)–(A4), and let be a proper lower semicontinuous and convex function with assumption (B1) or (B2). Let be a finite family of quasinonexpansive and Lipschitz mappings of into itself, let be a inversestrongly monotone mapping of into and let be a inversestrongly monotone mapping of into . Assume that
Let be the mapping generated by and . Let , , , and be sequences generated by , , and let
where satisfy the following conditions:
(i) for some with ;
(ii), for some with ;
(iii) for some with .
Then, and converge strongly to
Proof.
In Theorem 3.3 take , where is the indicator function of , that is,
Then the variational inclusion problem (1.3) is equivalent to variational inequality problem (1.14), that is, to find such that
Again, since , then
and so we have
We can obtain the desired conclusion from Theorem 3.3 immediately.
Next, we consider another class of important nonlinear mappings: strict pseudocontractions.
Definition 3.6.
A mapping is called strictly pseudocontraction if there exists a constant such that
If , then is nonexpansive.
In this case, let a strictly pseudocontraction. Putting , then is a inversestrongly monotone mapping. In fact, from (3.82) we have
Observe that
Hence, we obtain
This shows that is inversestrongly monotone mapping.
Now, we get the following result.
Theorem 3.7.
Let be a nonempty closed convex subset of a real Hilbert space , let be a bifunction from to satisfying (A1)–(A4) and let be a proper lower semicontinuous and convex function with assumption (B1) or (B2). Let be a finite family of quasinonexpansive and Lipschitz mappings of into itself, let be a strictly pseudocontraction mapping of into and let be a strictly pseudocontraction mapping of into . Assume that
Let be the mapping generated by and . Let , , , and be sequences generated by , , and let
where satisfy the following conditions:
(i) for some with ;
(ii), for some with ;
(iii) for some with .
Then, and converge strongly to
Proof.
Taking and , respectively. Then we see that is inversestrongly monotone and is inversestrongly monotone, respectively. We have and
By using Theorem 3.5, it is easy to obtain the desired conclusion.
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Acknowledgments
The authors would like to express their thank to the referees for helpful suggestions. The first author was supported by the National Research Council of Thailand and the Faculty of Science and Technology RMUTT Research Fund. The second author was supported by Rajamangala University of Technology Rattanakosin Research and Development Institute. The third author was supported by the Thailand Research Fund and the Commission on Higher Education under Grant No. MRG5380044.
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Keywords
 Equilibrium Problem
 Nonexpansive Mapping
 Real Hilbert Space
 Variational Inequality Problem
 Nonempty Closed Convex Subset