A System of Generalized Mixed Equilibrium Problems and Fixed Point Problems for Pseudocontractive Mappings in Hilbert Spaces
© Poom Kumam and Chaichana Jaiboon. 2010
Received: 2 April 2010
Accepted: 11 June 2010
Published: 4 July 2010
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.
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, .
Let be a mapping. Then is called
contraction if there exists a constant such that
It is well known that if is nonempty, bounded, closed, and convex and is a nonexpansive mapping on then is nonempty; see, for example, .
strongly pseudocontractive with the coefficient if
for such a case, is also said to be a -strict pseudocontraction, and if , then is a nonexpansive mapping,
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.
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.
Let be a proper extended real-valued function and let be a bifunction of into such that , where is the set of real numbers.
We see that is a solution of problem (1.8) implies that
- (2)If , problem (1.8) is reduced into the generalized equilibrium problem for finding such that(1.11)
- (3)If and , problem (1.8) is reduced into the equilibrium problem for finding such that(1.12)
- (4)If , problem (1.8) is reduced into the mixed variational inequality of Browder type for finding such that(1.13)
- (5)If and , problem (1.8) is reduced into the variational inequality problem for finding such that(1.14)
- (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  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, 12–23].
Let be a nonlinear mapping. Then is called
- (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
It is obvious that any -inverse-strongly monotone mappings are monotone and -Lipschitz continuous.
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 .
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 .
where is a potential function for (i.e., for ).
where for some and satisfies . Further, they proved that and converge weakly to , where .
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  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  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.
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 , Peng and Yao , Plubtieng and Punpaeng , and Liu , 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 , Peng and Yao , Plubtieng and Punpaeng , and Liu  and many others.
The mapping is called the metric projection of onto
for all .
where is the metric projection of onto .
Then is the maximal monotone and if and only if see .
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
So, is a nonexpansive mapping of into .
Lemma 2.4 (see ).
Lemma 2.5 (see ).
That is, is strongly monotone with coefficient .
Lemma 2.6 (see ).
Assume that is a strongly positive linear bounded operator on with coefficient and . Then .
Lemma 2.7 (see ).
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 ).
Lemma 2.9 (see ).
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 :
is monotone, that is, for all
for each is convex and lower semicontinuous;
for each is weakly upper semicontinuous;
is a bounded set.
By similar argument as in the proof of Lemma in , we have the following lemma appearing.
for all . Then, the following holds:
(i)for each ;
(ii) is single-valued;
(v) is closed and convex.
We remark that Lemma 2.10 is not a consequence of Lemma in , 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 ).
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 ).
where is a sequence in and is a sequence in such that
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.
where , , , and are sequences in , where , , , and and are positive sequences. Assume that the control sequences satisfy the following restrictions:
and , where are two positive constants,
, where .
Equivalently, one has
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.
We claim that is bounded.
Hence, is bounded, and so are , , , , , , and .
We claim that and
We claim that the following statements hold:
We claim that where is the unique solution of the variational inequality for all
Since is bounded, there exists a subsequence of which converges weakly to . Without loss of generality, we can assume that We claim that .
Assume also that and .
It follows from Lemma 2.8 that . By (3.55), we have .
We claim that .
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.
Equivalently, one has
If and in Theorem 3.1, then we can obtain the following result immediately.
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.
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 .
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 , and radiation therapy treatment planning .
The following result can be obtained from Theorem 3.1. We, therefore, omit the proof.
where such that , are positive sequences, and and are sequences in . Assume that the control sequences satisfy the following restrictions:
for each ,
, where is some positive constant for each ,
, for each .
Equivalently, one has
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.
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