- Research Article
- Open Access
Strong Convergence Theorems for an Infinite Family of Equilibrium Problems and Fixed Point Problems for an Infinite Family of Asymptotically Strict Pseudocontractions
© Shenghua Wang et al. 2011
- Received: 12 October 2010
- Accepted: 29 January 2011
- Published: 23 February 2011
We prove a strong convergence theorem for an infinite family of asymptotically strict pseudo-contractions and an infinite family of equilibrium problems in a Hilbert space. Our proof is simple and different from those of others, and the main results extend and improve those of many others.
- Hilbert Space
- Convex Subset
- Equilibrium Problem
- Nonexpansive Mapping
- Iterative Scheme
Let be a closed convex subset of a Hilbert space . Let be a mapping and if there exists an element such that , then is called a fixed point of . The set of fixed points of is denoted by . Recall that
(1) is called nonexpansive if
(2) is called asymptotically nonexpansive  if there exists a sequence with such that
(3) is called to be a -strict pseudo-contraction  if there exists a constant with such that
It is clear that every asymptotically nonexpansive mapping is an asymptotically 0-strict pseudo-contraction and every -strict pseudo-contraction is an asymptotically -strict pseudo-contraction with for all . Moreover, every asymptotically -strict pseudo-contraction with sequence is uniformly -Lispchitzian, where and the fixed point set of asymptotically -strict pseudo-contraction is closed and convex; see [3, Proposition 2.6].
Let be a bifunction from to , where is the set of real numbers. The equilibrium problem for is to find such that for all . The set of such solutions is denoted by .
In 2007, S. Takahashi and W. Takahashi  first introduced an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of the equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space and proved a strong convergence theorem which is connected with Combettes and Hirstoaga's result  and Wittmann's result . More precisely, they gave the following theorem.
Theorem 1.1 (see ).
Let be a nonempty closed convex subset of . Let be a bifunction from to satisfying the following assumptions:
(A1) for all ;
(A2) is monotone, that is, for all ;
(A3)for all ,
(A4)for all , is convex and lower semicontinuous.
Then, the sequences and converge strongly to , where .
In , Tada and Takahashi proposed a hybrid algorithm to find a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of an equilibrium problem and proved the following strong convergence theorem.
Theorem 1.2 (see ).
where for some and satisfies . Then converges strongly to .
Recently, Kim and Xu  proposed a hybrid algorithm for finding a fixed point of an asymptotically -strict pseudo-contraction and proved a strong convergence theorem in a Hilbert space.
Theorem 1.3 (see ).
Assume that the control sequence is chosen such that . Then converges strongly to .
In this paper, motivated by [3, 8], we propose a new algorithm for finding a common element of the set of fixed points of an infinite family of asymptotically strict pseudo-contractions and the set of solutions of an infinite family of equilibrium problems and prove a strong convergence theorem. Our proof is simple and different from those of others, and the main results extend and improve those Kim and Xu , Tada and Takahashi , and many others.
for all and with .
Let denote the identity operator of , and let be a sequence in a Hilbert space and . Throughout the rest of the paper, denotes the strong convergence of to .
We need the following lemmas for our main results in this paper.
Lemma 2.1 (see ).
Lemma 2.2 (see ).
Then the following hold:
(1) is single-valued,
(2) is firmly nonexpansive, that is, for any ,
(3) , and
(4) is closed and convex.
Now, we are ready to give our main results.
This completes the proof.
Let be a nonempty closed subset of a Hilbert space . Let be an asymptotically -strict pseudo-contraction with sequence satisfying as . Let be a sequence in such that and as . Then as .
It follows from and as that . This completes the proof.
Lemma 2.2 shows that every ( ) is a firmly nonexpansive mapping and hence nonexpansive and .
Then converges strongly to .
is convex for all . So is also closed and convex for all .
So we have and hence for all . This shows that for all . This implies that the sequence is well defined.
Therefore, is bounded. From (3.13) and (3.14), and are also bounded.
This implies that there exists an element such that as .
Next we show that and .
Now, (3.42), (3.44), and Lemma 3.2 imply (3.43).
Since each is uniformly continuous and as , one get for each and hence .
Now we show .
Since every is nonexpansive, from (3.33) and , we have and hence . Lemma 2.2 shows that .
Hence . This completes the proof.
As direct consequences of Theorem 3.3, we can obtain the following corollaries.
where is defined by (3.8) and is a strictly decreasing sequence in . Then converges strongly to .
Putting for all and for all in Theorem 3.3, we obtain Corollary 3.4.
where , with , and with . Then converges strongly to .
Put for all and set for all in Theorem 3.3. By Lemma 2.2, we have for each . Hence, by Theorem 3.3, we obtain Corollary 3.5.
Our algorithms are of interest because the sequence in Theorem 3.3 is very different from the known manner. The proof is simple and different from those of others. The main results extend and improve those of Kim and Xu , Tada and Takahashi , and many others.
Put , , , , , , , , for all and all , , and . Then these control sequences satisfy all the conditions of Theorem 3.3.
The authors thank the referees for useful comments and suggestions. This study was supported by research funds from Dong-A University.
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