We propose an explicit iterative scheme for finding a common element of the set of fixed points of infinitely many strict pseudocontractive mappings and the set of solutions of an equilibrium problem by the general iterative method, which solves the variational inequality. In the setting of real Hilbert spaces, strong convergence theorems are proved. The results presented in this paper improve and extend the corresponding results reported by some authors recently. Furthermore, two numerical examples are given to demonstrate the effectiveness of our iterative scheme.
Let
Let
Generally,
A mapping
A mapping
Obviously, the class of strict pseudocontractions strictly includes the class of nonexpansive mappings. We denote the set of fixed points of
Let
The equilibrium problem for
Many problems in applied sciences such as physics, optimization, and economics reduce into finding some element of
In 2006, Marino and Xu [
The mapping
In this paper, we combine the operator
In the sequel, we will make use of the following lemmas in a real Hilbert space
Let
Let
Let
Let
Since
Assume that
Let
Let
Let
For solving the equilibrium problem, let us assume that the bifunction
for each
We recall some lemmas which will be needed in the rest of this paper.
Let
For
Let
Let
Let
We adopt the following notations:
Recall that, given a nonempty closed convex subset
Throughout the rest of this paper, we always assume that
Define a mapping
For simplicity, we will write
Let
The proof is divided into several steps.
Taking any
Hence, we have
It follows from (
Hence we get
From conditions (i) and (iii) and Lemma
By Lemma
Observe that
On the other hand, we have
Since
By the same argument as in the proof of Theorem
Since
If we extend the equilibrium problem to be system of equilibrium problems, we still obtain the desired result by the similar proof of Theorem
In this section, we consider the following two simple examples to demonstrate the effectiveness, realization, and convergence of the algorithm in Theorem
First, we give an example as follows.
In Theorem
Let


Errors ( 

94  0.9969 

150  0.9981 

450  0.9994 

Let
Next, we consider another simple example.
In Theorem
For analysis of the rate of convergence, we use the concept introduced by Rhoades [
Let
Now we turn to numerical simulation using the algorithm (
(a)


Errors ( 

10  (0.8289, 0.5521) 

50  (0.8308, 0.5559) 

100  (0.8310, 0.5561) 



Errors ( 

10  (0.8271, 0.5521) 

50  (0.8308, 0.5558) 

100  (0.8309, 0.5561) 

(a)


Errors ( 

10  (0.8341, 0.5534) 

50  (0.8308, 0.5559) 

100  (0.8310, 0.5561) 



Errors ( 

10  (0.8359, 0.5531) 

50  (0.8308, 0.5558) 

100  (0.8309, 0.5561) 

It is easy to see that the approximation values obtained by the algorithm (
The author would like to thank the referee for valuable suggestions to improve the paper and the Fundamental Research Funds for the Central Universities (Grant ZXH2012K001).