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A New Hybrid Algorithm for a System of Mixed Equilibrium Problems, Fixed Point Problems for Nonexpansive Semigroup, and Variational Inclusion Problem
Fixed Point Theory and Applications volume 2011, Article number: 217407 (2011)
Abstract
The purpose of this paper is to consider a shrinking projection method for finding the common element of the set of common fixed points for nonexpansive semigroups, the set of common fixed points for an infinite family of a strict pseudocontraction, the set of solutions of a system of mixed equilibrium problems, and the set of solutions of the variational inclusion problem. Strong convergence of the sequences generated by the proposed iterative scheme is obtained. The results presented in this paper extend and improve some wellknown results in the literature.
1. Introduction
Throughout this paper, we assume that 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 countable family of bifunctions from to , where is the set of real numbers and is an arbitrary index set. Let be a proper extended realvalued function. The system of mixed equilibrium problems is to find such that
The set of solutions of (1.1) is denoted by , that is,
If is a singleton, the problem (1.1) reduces to find the following mixed equilibrium problem (see also the work of FloresBazán in [1]). For finding such that,
the set of solutions of (1.3) is denoted by . Combettes and Hirstoaga [2] introduced the following system of equilibrium problems. For finding such that,
the set of solutions of (1.4) is denoted by , that is,
If is a singleton, the problem (1.4) becomes the following equilibrium problem. For finding such that
The set of solution of (1.6) is denoted by EP.
The equilibrium problem include fixed point problems, optimization problems, variational inequalities problems, Nash equilibrium problems, noncooperative games, economics and the (mixed) equilibrium problems as special cases (see, e.g., [3–8]). Some methods have been proposed to solve the equilibrium problem, see, for instance, [9–17].
Recall that, a mapping is said to be nonexpansive if
We denote the set of fixed points of by , that is .
Definition 1.1.
A family of mappings of into itself is called a nonexpansive semigroup on if it satisfies the following conditions:
(1), for all ;
(2), for all ;
(3), for all and ;
(4)for all , is continuous.
We denoted by the set of all common fixed points of , that is, . It is know that is closed and convex.
Let be a singlevalued nonlinear mapping and be a setvalued mapping. The variational inclusion problem is to find such that
where is the zero vecter in . The set of solutions of problem (1.8) is denoted by . A setvalued mapping is called monotone if for all and imply . 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 1.2.
A mapping is said to be a Lipschitz continous if there exists a constant such that
Definition 1.3.
A mapping is said to be a inversestrongly monotone if there exists a constant with the property
Remark 1.4.
It is obvious that any inversestrongly monotone mappings is monotone and Lipschitz continuous. It is easy to see that for any constant is in , then the mapping is nonexpansive, where is the identity mapping on .
Definition 1.5.
Let is called Lipschitz continuous, if there exists a constant such that
Let be a differentiable functional on a convex set , which is called:
(1)convex [18] if
where is the Fréchet derivative of at ;
(2)strongly convex [19] if there exists a constant such that
In particular, if for all , then is said to be strongly convex.
Definition 1.6.
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. The following characterizes the resolvent operator.
(R1)The resolvent operator is singlevalued and nonexpansive for all , that is,
(R2)The resolvent operator is 1inversestrongly monotone; see [20], that is,
(R3) The solution of problem (1.8) is a fixed point of the operator for all ; see also [21], that is,
(R4) If , then the mapping is nonexpansive.
(R5) is closed and convex.
In 2007, Takahashi et al. [22] proved the following strong convergence theorem for a nonexpansive mapping by using the shrinking projection method in mathematical programming. For and , they define a sequence as follows:
where . They proved that the sequence generated by (1.18) converges weakly to , where .
In 2008, S. Takahashi and W. Takahashi [23] introduced the following iterative scheme for finding a common element of the set of solution of generalized equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space. They proved the strong convergence theorems under certain appropriate conditions imposed on parameters. Next, Zhang et al. [24] introduced the following new iterative scheme for finding a common element of the set of solution to the problem (1.8) and the set of fixed points of a nonexpansive mapping in a real Hilbert space. Starting with an arbitrary , define a sequence by
where is the resolvent operator associated with and a positive number and is a sequence in the interval . Peng et al. [25] introduced the iterative scheme by the viscosity approximation method for finding a common element of the set of solutions to the problem (1.8), the set of solutions of an equilibrium problem, and the set of fixed points of a nonexpansive mapping in a Hilbert space.
In 2009, Saeidi [26] introduced a more general iterative algorithm for finding a common element of the set of solution for a system of equilibrium problems and the set of common fixed points for a finite family of nonexpansive mappings and a nonexpansive semigroup. In 2010, Katchang and Kumam [27] obtained a strong convergence theorem for finding a common element of the set of fixed points of a family of finitely nonexpansive mappings, the set of solutions of a mixed equilibrium problem and the set of solutions of a variational inclusion problem for an inversestrongly monotone mapping. Let be mapping (defined by (2.8)), be a contraction mapping and be inversestrongly monotone mappings. Let be the resolvent operator associated with and a positive number . Starting with arbitrary initial , defined a sequence by
They proved that under certain appropriate conditions imposed on , and , the sequence generated by (1.20) converges strongly to , where . Later, Kumam et al. [28] proved a strongly convergence theorem of the iterative sequence generated 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.
Liu et al. [29] introduced a hybrid iterative scheme for finding a common element of the set of solutions of mixed equilibrium problems, the set of common fixed points for nonexpansive semigroup and the set of solution of quasivariational inclusions with multivalued maximal monotone mappings and inversestrongly monotone mappings. Recently, Jitpeera and Kumam [30] considered a shrinking projection method of finding the common element of the set of common fixed points for a finite family of a strict pseudocontraction, the set of solutions of a systems of equilibrium problems and the set of solutions of variational inclusions. Then, they proved strong convergence theorems of the iterative sequence generated by the shrinking projection method under some suitable conditions in a real Hilbert space. Very recently, Hao [18] introduced a general iterative method for finding a common element of solution set of quasi variational inclusion problems and of the common fixed point set of an infinite family of nonexpansive mappings.
In this paper, motivated and inspired by the previously mentioned results, we introduce an iterative scheme by the shrinking projection method for finding the common element of the set of common fixed points for nonexpansive semigroups, the set of common fixed points for an infinite family of a strict pseudocontraction, the set of solutions of a systems of mixed equilibrium problems and the set of solutions of the variational inclusions problem. Then, we prove a strong convergence theorem of the iterative sequence generated by the 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 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 .
The following characterizes the projection . We recall some lemmas which will be needed in the rest of this paper.
Lemma 2.1.
For a given , , , for all .
It is well known that is a firmly nonexpansive mapping of onto and satisfies
Moreover, is characterized by the following properties: and for all ,
Lemma 2.2 (see [20]).
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 [31]).
Let be a closed convex subset of . 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 [32]).
Each Hilbert space satisfies Opial's condition, that is, for any sequence with , the inequality , hold for each with .
Lemma 2.5 (see [33]).
Each Hilbert space , satisfies the KadecKlee property, that is, for any sequence with and together imply .
For solving the system of mixed equilibrium problem, let us assume that function , satisfies the following conditions:
is monotone, that is, , for all ;
for each fixed , is convex and upper semicontinuous;
for each fixed , is convex.
Lemma 2.6 (see [34]).
Let be a nonempty closed convex subset of a real Hilbert space and let be a lower semicontinuous and convex functional from to . Let be a bifunction from to satisfying (H1)–(H3). Assume that
(i) is Lipschitz continuous with constant such that;
(a), for all ,
(b) is affine in the first variable,
(c)for each fixed , is sequentially continuous from the weak topology to the weak topology,
(ii) is strongly convex with constant and its derivative is sequentially continuous from the weak topology to the strong topology;
(iii)for each , there exist a bounded subset and such that for any ,
For given , Let be the mapping defined by:
for all . Then the following hold
(1) is singlevalued;
(2) is nonexpansive if is Lipschitz continuous with constant such that ;
(3);
(4) is closed and convex.
Lemma 2.7 (see [35]).
Let be a strict pseudocontraction, then
(1)the fixed point set of is closed convex so that the projection is well defined;
(2)define a mapping by
If , then is a nonexpansive mapping such that .
A family of mappings is called a family of uniformlystrict pseudocontractions, if there exists a constant such that
Let be a countable family of uniformly strict pseudocontractions. Let be the sequence of nonexpansive mappings defined by (2.5), that is,
Let be a sequence of nonexpansive mappings of into itself defined by (2.7) and let be a sequence of nonnegative numbers in . For each , define a mapping of into itself as follows:
Such a mapping is nonexpansive from to and it is called the mapping generated by and . For each , let the mapping be defined by (2.8). Then we can have the following crucial conclusions concerning .
Lemma 2.8 (see [36]).
Let be a nonempty closed convex subset of a real Hilbert space . Let be nonexpansive mappings of into itself such that is nonempty, let be real numbers such that for every . Then, for every and , exists.
Using this lemma, one can define a mapping and as follows and
Such a mapping is called the mapping. Since is nonexpansive and , is also nonexpansive. Indeed, observe that for each such that
Lemma 2.9 (see [36]).
Let be a nonempty closed convex subset of a Hilbert space , be a countable family of nonexpansive mappings with , be a real sequence such that , for all . Then .
Lemma 2.10 (see [37]).
Let be a nonempty closed convex subset of a Hilbert space , be a countable family of nonexpansive mappings with , be a real sequence such that , for all . If is any bounded subset of , then
Lemma 2.11 (see [38]).
Let be a nonempty bounded closed convex subset of a Hilbert space and let be a nonexpansive semigroup on , then for any ,
Lemma 2.12 (see [39]).
Let C be a nonempty bounded closed convex subset of H, be a sequence in C and be a nonexpansive semigroup on C. If the following conditions are satisfied:
(1);
(2), then .
3. Main Results
In this section, we will introduce an iterative scheme by using a shrinking projection method for finding the common element of the set of common fixed points for nonexpansive semigroups, the set of common fixed points for an infinite family of strict pseudocontraction, the set of solutions of a systems of mixed equilibrium problems and the set of solutions of the variational inclusions problem in a real Hilbert space.
Theorem 3.1.
Let be a nonempty closed convex subset of a real Hilbert space , let , be a finite family of mixed equilibrium functions satisfying conditions (H1)–(H3). Let be a nonexpansive semigroup on and let be a positive real divergent sequence. Let be a countable family of uniformly strict pseudocontractions, be the countable family of nonexpansive mappings defined by , for all , for all , , be the mapping defined by (2.8) and be a mapping defined by (2.9) with . Let be inversestrongly monotone mappings and be maximal monotone mappings such that
Let , , which are constants. Let , , , , and be sequences generated by , , , and
where , is the mapping defined by (2.4) and be a sequence in for all . Assume the following conditions are satisfied:
is Lipschitz continuous with constant such that
(a), for all ,
(b) is affine,
(c)for each fixed , is sequentially continuous from the weak topology to the weak topology;
is strongly convex with constant and its derivative is not only sequentially continuous from the weak topology to the strong topology but also Lipschitz continuous with a Lipschitz constant such that ;
for each and for all , there exist a bounded subset and such that for any ,
, for some ;
, for some ;
, for some ;
, for each .
Then, and converge strongly to .
Proof.
Pick any . Taking for and for all . From the definition of is nonexpansive for each , then also and , we note that . If follows that
Next, we will divide the proof into eight steps.
Step 1.
We first show by induction that for each .
Taking , we get that . Since are nonexpansive. From the assumption, we see that . Suppose for some . For any , we have
which yields
Applying (3.5) and (3.6), we get
Hence . This implies that for each .
Step 2.
Next, we show that is well defined and is closed and convex for any .
It is obvious that is closed and convex. Suppose that is closed and convex for some . Now, we show that is closed and convex for some . For any , we obtain
is equivalent to
Thus is closed and convex. Then, is closed and convex for any . This implies that is welldefined.
Step 3.
Next, we show that is bounded and exists. From , we have
for each . Using , we also have
So, for , we observe that
This implies that
Hence, we get is bounded. It follows by (3.5)–(3.7), that , and are also bounded. From , and , we obtain
It follows that, we have for each
It follows that
Thus, since the sequence is a bounded and nondecreasing sequence, so exists, that is
Step 4.
Next, we show that and .
Applying (3.15), we get
Thus, by (3.18), we obtain
On the other hand, from , which implies that
It follows by (3.21), we also have
By (3.20), we obtain
Step 5.
Next, we show that
for every . Indeed, for , note that is the firmly nonexpansie, so we have
Thus, we get
It follows that
By (3.5), (3.6), (3.7), and (3.27), we have for each
Consequently, we have
Equation (3.23) implies that for every
Step 6.
Next, we show that and , where .
For any given , , and . Since and are nonexpansive, we have
Similarly, we can show that
Observe that
Substituting (3.31) into (3.33) and using conditions (C4) and (C5), we have
It follows that
By (3.23), we obtain
Since the resolvent operator is 1inversestrongly monotone, we obtain
which yields
Similarly, we can obtain
Substituting (3.38) into (3.33), and using condition (C4) and (C5), we have
It follows that
By (3.23) and (3.36), we get
From (3.8) and (C4), we also have
Since , we obtain (3.23), we have
Since is a bounded sequence in , from Lemma 2.11 for all , we have
From (3.44) and (3.45), we get
So, we have
Step 7.
Next, we show that .
Since is bounded, there exists a subsequence of which converges weakly to . Without loss of generality, we can assume that .

(1)
First, we prove that . Indeed, from Lemma 2.12 and (3.47), we get , that is, , for all .

(2)
We show that , where , for all and . Assume that , then there exists a positive integer such that and so . Hence for any , , that is, . This together with , for all shows , for all , therefore we have , for all . It follows from the Opial's condition and (3.44) that
(3.48)
which is a contradiction. Thus, we get .

(3)
We prove that . Since , and , we have
(3.49)
It follows that
for all . From (3.30) and by conditions (C1)(c) and (C2), we get
By the assumption and by condition (H1), we know that the function and the mapping both are convex and lower semicontinuous, hence they are weakly lower semicontinuous.
These together with and , we have
Then, we obtain
Therefore .

(4)
Lastly, we prove that .
We observe that is an 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
By (3.42), and is inversestrongly monotone, we obtain that and it follows that
It follows from the maximal monotonicity of that , that is, . Since is bounded, there exists a subsequence of which converges weakly to . Without loss of generality, we can assume that . In similar way, we can obtain , hence .
Step 8.
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 (3.59) and is bounded, so .
By the weakly lower semicontinuous of the norm, we have
However, since , we have
Using (3.59) and (3.60), we obtain . Thus and . So, we have
Thus, we obtain that
From , we obtain . Using the KadecKlee property, we obtain that
and hence in norm. Finally, noticing . We also conclude that in norm. This completes the proof.
Theorem 3.2.
Let be a nonempty closed convex subset of a real Hilbert space , let be a finite family of mixed equilibrium functions satisfying conditions (H1)–(H3). Let be a nonexpansive semigroup on and let be a positive real divergent sequence. Let be a countable family of uniformly strict pseudocontractions, be the countable family of nonexpansive mappings defined by , for all , for all , be the mapping defined by (2.8) and be a mapping defined by (2.9) with . Let be inversestrongly monotone mapping. Such that
Let , , which are constants. Let , , , , and be sequences generated by , , , and
where is the mapping defined by (2.4) and be a sequence in for all . Assume the following conditions are satisfied:
is Lipschitz continuous with constant such that
(a), for all ,
(b) is affine,
(c)for each fixed , is sequentially continuous from the weak topology to the weak topology;
is strongly convex with constant and its derivative is not only sequentially continuous from the weak topology to the strong topology but also Lipschitz continuous with a Lipschitz constant such that ;
for each and for all , there exist a bounded subset and such that for any ,
, for some ;
, for some ;
, for some ;
, for each .
Then, and converge strongly to .
Proof.
In Theorem 3.1, take , where is the indicator function of , that is,
for . Then (1.8) is equivalent to variational inequality problem, that is, to find such that
Again, since , for , then
So, we have
Hence, we can obtain the desired conclusion from Theorem 3.1 immediately.
Next, we consider another class of important mappings.
Definition 3.3.
A mapping is called strictly pseudocontraction if there exists a constant such that
If , then is nonexpansive. In this case, we say that is a strictly pseudocontraction. Putting . Then, we have
Observe that
Hence, we obtain
Then, is inversestrongly monotone mapping.
Now, we obtain the following result.
Theorem 3.4.
Let be a nonempty closed convex subset of a real Hilbert space , let , be a finite family of mixed equilibrium functions satisfying conditions (H1)–(H3). Let be a nonexpansive semigroup on and let be a positive real divergent sequence. Let be a countable family of uniformly strict pseudocontractions, be the countable family of nonexpansive mappings defined by , for all , for all , , be the mapping defined by (2.8) and be a mapping defined by (2.9) with . Let be inversestrongly monotone mapping and be strictly pseudocontraction mapping of into for some , such that
Let , , which are constants. Let , , , and be sequences generated by , , , and
where , is the mapping defined by (2.4) and be a sequence in for all . Assume the following conditions are satisfied:
is Lipschitz continuous with constant such that
(a), for all ,
(b) is affine,
(c)for each fixed , is sequentially continuous from the weak topology to the weak topology;
is strongly convex with constant and its derivative is not only sequentially continuous from the weak topology to the strong topology but also Lipschitz continuous with a Lipschitz constant such that ;
for each and for all , there exist a bounded subset and such that for any ,
, for some ;
, for some ;
, for some ;
, for each
Then, and converge strongly to .
Proof.
Taking and , then we see that is inversestrongly monotone mapping, respectively. We have and . So, we have
By using Theorem 3.2, it is easy to obtain the desired conclusion.
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Acknowledgments
The authors would like to thank the Faculty of Science, King Monkut's University of Technology Thonburi for its financial support. Moreover, P. Kumam was supported by the Commission on Higher Education and the Thailand Research Fund under Grant MRG5380044.
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Jitpeera, T., Kumam, P. A New Hybrid Algorithm for a System of Mixed Equilibrium Problems, Fixed Point Problems for Nonexpansive Semigroup, and Variational Inclusion Problem. Fixed Point Theory Appl 2011, 217407 (2011). https://doi.org/10.1155/2011/217407
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Keywords
 Equilibrium Problem
 Nonexpansive Mapping
 Real Hilbert Space
 Nonempty Closed Convex Subset
 Strong Convergence Theorem