- Research Article
- Open Access
Convergence Analysis for a System of Generalized Equilibrium Problems and a Countable Family of Strict Pseudocontractions
© P. Cholamjiak and S. Suantai. 2011
- Received: 18 October 2010
- Accepted: 27 December 2010
- Published: 4 January 2011
We introduce a new iterative algorithm for a system of generalized equilibrium problems and a countable family of strict pseudocontractions in Hilbert spaces. We then prove that the sequence generated by the proposed algorithm converges strongly to a common element in the solutions set of a system of generalized equilibrium problems and the common fixed points set of an infinitely countable family of strict pseudocontractions.
- Convex Subset
- Nonexpansive Mapping
- Real Hilbert Space
- Common Fixed Point
- Convex Banach Space
The solutions set of (1.2) is denoted by . If , then the solutions set of (1.2) is denoted by , and if , then the solutions set of (1.2) is denoted by . The problem (1.2) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, and the Nash equilibrium problem in noncooperative games; see also [1, 2]. Some methods have been constructed to solve the system of equilibrium problems (see, e.g., [3–7]). Recall that a mapping is said to be
It is easy to see that if is α-inverse-strongly monotone, then is monotone and -Lipschitz.
For solving the equilibrium problem, let us assume that satisfies the following conditions:
(A1) for all ,
(A2) is monotone, that is, for all ,
(A3)for each , ,
(A4)for each , is convex and lower semicontinuous.
It is worth mentioning that the class of strict pseudocontractions includes properly the class of nonexpansive mappings. It is also known that every -strict pseudocontraction is -Lipschitz; see .
where . If is a nonexpansive mapping with a fixed point and the control sequence is chosen so that , then the sequence defined by (1.7) converges weakly to a fixed point of (this is also valid in a uniformly convex Banach space with the Fréchet differentiable norm ).
In 1967, Browder and Petryshyn  introduced the class of strict pseudocontractions and proved existence and weak convergence theorems in a real Hilbert setting by using Mann iterative algorithm (1.7) with a constant sequence for all . Recently, Marino and Xu  and Zhou  extended the results of Browder and Petryshyn  to Mann's iteration process (1.7). Since 1967, the construction of fixed points for pseudocontractions via the iterative process has been extensively investigated by many authors (see, e.g., [13–22]).
Let be a nonempty, closed, and convex subset of a real Hilbert space . Let be a nonexpansive mapping, a bifunction, and let be an inverse-strongly monotone mapping.
where and . He proved that the sequence generated by (1.8) converges weakly to an element in under suitable conditions.
Due to the weak convergence, recently, S. Takahashi and W. Takahashi  introduced another modification iterative method of (1.8) for finding a common element of the fixed points set of a nonexpansive mapping and the solutions set of a generalized equilibrium problem in the framework of a real Hilbert space. To be more precise, they proved the following theorem.
Theorem 1.1 (see ).
where , and satisfy
(i) and ,
Then, converges strongly to .
Recently, Yao et al.  introduced a new modified Mann iterative algorithm which is different from those in the literature for a nonexpansive mapping in a real Hilbert space. To be more precise, they proved the following theorem.
Theorem 1.2 (see ).
Suppose that the following conditions are satisfied:
(i) and ,
then, the sequence generated by (1.10) strongly converges to a fixed point of .
We know the following crucial lemmas concerning the equilibrium problem in Hilbert spaces.
Lemma 1.3 (see ).
Lemma 1.4 (see ).
Then, the following statements hold:
(1) is single-valued,
(4) is closed and convex.
Let be a nonempty, closed, and convex subset of a real Hilbert space . Let for each . Let be a family of bifunctions, let be a family of -inverse-strongly monotone mappings, and let be a countable family of -strict pseudocontractions. For each , denote the mapping by , where is the mapping defined as in Lemma 1.4.
where , and .
In this paper, we first prove a path convergence result for a nonexpansive mapping and a system of generalized equilibrium problems. Then, we prove a strong convergence theorem of the iteration process (1.14) for a system of generalized equilibrium problems and a countable family of strict pseudocontractions in a real Hilbert space. Our results extend the main results obtained by Yao et al.  in several aspects.
for all and .
In the sequel, we need the following lemmas.
Let be a real uniformly convex Banach space, and let be a nonempty, closed, and convex subset of , and let be a nonexpansive mapping such that , then is demiclosed at zero.
Lemma 2.2 (see ).
where satisfies conditions: . If , then as .
Lemma 2.3 (see ).
where is a sequence in and is a sequence in such that
(a) ; (b) or .Then, .
Lemma 2.4 (see ).
for all . In particular, if , then is nonexpansive.
Lemma 2.5 (see ).
Let be a closed, and convex subset of a Hilbert space . Suppose that is a family of -strictly pseudocontractive mappings from into with and is a real sequence in such that . Then, the following conclusions hold:
(1) is a -strictly pseudocontractive mapping,
Lemma 2.7 (see ).
where is a family of nonnegative numbers satisfying
(i) for all ,
(ii) for all ,
(1)Each is a -strictly pseudocontractive mapping.
(2) satisfies -condition.
then and .
In the sequel, we will write satisfies the -condition if satisfies the -condition and is defined by Lemma 2.5 with .
Let be a nonempty, closed, and convex subset of a real Hilbert space . Let be a nonexpansive mapping. Let be a family of bifunctions, let be a family of -inverse-strongly monotone mappings, and let . For each , let the mapping be defined by (3.1). Assume that . For each , let the net be generated by (3.3). Then, as , the net converges strongly to an element in .
as since is bounded.
Since is bounded, without loss of generality, we may assume that converges weakly to . Applying Lemma 2.1 to (3.21), we can conclude that .
Since , we have as . By using the same argument as in the proof of Theorem 3.1 of , we can show that as . This completes the proof.
Let be a nonempty, closed and convex subset of a real Hilbert space . Let be a family of bifunctions, let be a family of -inverse-strongly monotone mappings and let be a countable family of -strict pseudocontractions for some such that . Assume that , , and for each satisfy the following conditions:
(i) and ,
Suppose that satisfies the -condition. Then, generated by (1.14) converges strongly to an element in .
Hence, is nonexpansive for each and so is .
Hence, by induction, is bounded and so are and .
for each .
By (i) and (4.24), it follows from Lemma 2.3 that . This completes the proof.
As a direct consequence of Lemmas 2.6 and 2.7 and Theorem 4.1, we obtain the following result.
where and are real sequences in which satisfy (i)-(ii) of Theorem 4.1 and is a real sequence which satisfies (i)–(iii) of Lemma 2.7. Then, converges strongly to an element in .
Theorems 4.1 and 4.2 extend the main results in  from a nonexpansive mapping to an infinite family of strict pseudocontractions and a system of generalized equilibrium problems.
If we take and for each , then Theorems 3.1, 4.1, and 4.2 can be applied to a system of equilibrium problems and to a system of variational inequality problems, respectively.
Let be an infinite family of nonexpansive mappings of into itself, and let be real numbers such that for all . Moreover, let and be the -mappings  generated by and and and . Then, we know from [7, 35] that satisfies the -condition. Therefore, in Theorem 4.1, the mapping can be also replaced by .
The authors would like to thank the referees for valuable suggestions. This research is supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, and the Thailand Research Fund. The first author is supported by the Royal Golden Jubilee Grant no. PHD/0261/2551 and by the Graduate School, Chiang Mai University, Thailand.
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