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.
(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].
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 ).
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 ).
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 ).
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.
We need the following lemmas for our main results in this paper.
Lemma 2.1 (see ).
Lemma 2.2 (see ).
Then the following hold:
3. Main Results
Now, we are ready to give our main results.
This completes the proof.
Now, (3.42), (3.44), and Lemma 3.2 imply (3.43).
As direct consequences of Theorem 3.3, we can obtain the following corollaries.
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.
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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