We introduce a new hybrid iterative scheme for finding a common element in the solutions set of a system of equilibrium problems and the common fixed points set of an infinitely countable family of relatively quasi-nonexpansive mappings in the framework of Banach spaces. We prove the strong convergence theorem by the shrinking projection method. In addition, the results obtained in this paper can be applied to a system of variational inequality problems and to a system of convex minimization problems in a Banach space.
Let
Combettes and Hirstoaga [
In 2004, Matsushita and Takahashi [
In 2008, Takahashi and Zembayashi [
Let
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The aim of this paper is to introduce a new hybrid projection algorithm for finding a common element in the solutions set of a system of equilibrium problems and the common fixed points set of an infinitely countable family of closed and relatively quasi-nonexpansive mappings in the frameworks of Banach spaces.
We will need the following lemmas.
Let
Let
Let
Let
Let
For solving the equilibrium problem, let us assume that a bifunction for all for all
Let
Let
Let
Let
Then the sequence
We divide our proof into six steps as follows.
From Lemma It is obvious that From Since Denote From ( From
Theorem from the case of an equilibrium problem to a finite family of equilibrium problems; from a single relatively nonexpansive mapping to an infinitely countable family of relatively quasi-nonexpansive mappings; if
The iteration ( We use the composition of mappings We construct the set
If we take
Let
In this section, we give several applications of Theorem
Let
Let
Then the sequence
Define
Next, we show that Consider For each For each Let
If we take
Let
Let
Let
Then the sequence
Define
Next, we show that
Let
If we take
Let
As a direct consequence of Theorem
Let
Then the sequence
Taking
Theorem from the case of an equilibrium problem to a finite family of equilibrium problems; from the class of nonexpansive mappings to the class of an infinitely countable family of quasi-nonexpansive mappings.
The authors would like to thank Professor Simeon Reich and the referee for valuable suggestions on improving the manuscript and the Thailand Research Fund for financial support. The first author was supported by the Royal Golden Jubilee Grant PHD/0261/2551 and by the Graduate School, Chiang Mai University, Thailand.