Iterative algorithms based on hybrid method and Cesàro mean of asymptotically nonexpansive mappings for equilibrium problems
© Zhang and Cui; licensee Springer. 2014
Received: 29 September 2013
Accepted: 5 January 2014
Published: 21 January 2014
Using Cesàro means of a mapping, we modify the progress of Mann’s iteration in hybrid method for asymptotically nonexpansive mappings in Hilbert spaces. Under suitable conditions, we prove that the iterative sequence converges strongly to a fixed point of an asymptotically nonexpansive mapping. We also introduce a new hybrid iterative scheme for finding a common element of the set of common fixed points of asymptotically nonexpansive mappings and the set of solutions of an equilibrium problem in Hilbert spaces.
MSC:47H10, 47J25, 90C33.
when , T is called nonexpansive.
The concept of asymptotically nonexpansive mapping was introduced by Goebel and Kirk  in 1972. We denote by the set of fixed points of T. It is well known that if is asymptotically nonexpansive, then is nonempty convex.
where denotes the metric projection from H onto a closed convex subset K of H. The above method is also called CQ method or hybrid method.
Shimizu and Takahashi  proved a strong convergence theorem of the above iteration for an asymptotically nonexpansive mapping in Hilbert spaces.
The set of solutions of (1.4) is denoted by . Given a mapping , let , , then if and only if , , i.e., z is the solution of the variational inequality.
There are several other problems, for example, the complementarity problem, fixed point problem and optimization problem, which can also be written in the form of an equilibrium problem. So, equilibrium problems provide us with a systematic framework to study a wide class of problems arising in financial economics, optimization and operation research etc., which motivates the extensive concern. See, for example, [10–14]. In recent years, equilibrium problems have been deeply and thoroughly researched. See, for example, [15–20]. Some methods have been proposed to solve the equilibrium problem in a Hilbert space; see, for instance, [21–23]. In 2011, Jitpeera, Katchang, and Kumam  found a common element of the set of solutions for mixed equilibrium problem, the set of solutions of the variational inequality for a β-inverse strongly monotone mapping, and the set of fixed points of a family of finitely nonexpansive mappings in a real Hilbert space by using the viscosity and Cesàro mean approximation method.
We shall prove that the above iterative sequence converges strongly to a fixed point of T under some proper conditions. In addition, we also introduce a new hybrid iterative scheme for finding a common element of the set of common fixed points of asymptotically nonexpansive mappings and the set of solutions of an equilibrium problem in Hilbert spaces.
We will use the notation ⇀ for weak convergence and → for strong convergence.
- (1)Opial’s condition, that is, for any sequence with , the inequality
holds for every with .
The Kadec-Klee property, that is, for any sequence with and together implies .
We also need the following lemmas.
Lemma 2.1 (See )
Let T be an asymptotically nonexpansive mapping defined on a bounded closed convex subset C of a Hilbert space H. Assume that is a sequence in C with the properties: (i) ; (ii) . Then .
Lemma 2.2 (See )
The set of solutions of the above inequality is denoted by . For solving the equilibrium problem, let us assume that a bifunction f satisfies the following conditions:
(A1) for all ;
(A2) f is monotone, i.e., , for all ;
(A4) for each , is convex and lower semi-continuous.
The following lemma appears implicitly in .
Lemma 2.3 ()
The following lemma was also given in .
Lemma 2.4 ()
- (2)is firmly nonexpansive, i.e.,
is closed and convex.
Lemma 2.5 ()
3 Strong convergence theorem of modified Mann iteration based on hybrid method
Inspired by Kim and Xu’s results (see ), Mann-type iteration (1.3) is modified to obtain the strong convergence theorem as follows.
Then converges strongly to .
Proof First note that T has a fixed point in C (see ); that is, is nonempty. We divide the proof of this theorem into four steps as below.
Step 1. We show that and are closed and convex for each .
it follows that is convex.
Step 2. We show that , .
It follows that and hence for each n. Thus we obtain , . This means that is well defined.
Step 3. We show that is bounded and .
Step 4. We show that converges strongly to .
Thus, we obtain that . Using the Kadec-Klee property of H, we get . Since is an arbitrary subsequence of , we can conclude that converges strongly to . □
Remark 3.2 It is not difficult to see from the proof above that the boundedness of C can be discarded if T is a nonexpansive mapping.
4 Strong convergence theorem for equilibrium problems
In this section, we prove a strong convergence theorem for finding a common element of the set of zero points of an asymptotically nonexpansive mapping T and the set of solutions of an equilibrium problem in a Hilbert space.
Suppose that and there exists such that , , and such that . Then converges strongly to .
Proof We divide the proof of this theorem into four steps as below.
Step 1. Similar to the proof of Step 1 in Theorem 3.1, it is easy to see that and are closed convex sets for each .
Step 2. We show that , .
From the proof of Step 2 in Theorem 3.1, we have . Thus , hence . Similar to the proof of Step 2 in Theorem 3.1, it is easy to see that . Therefore we have , . This means that is well defined.
Step 3. We show that is bounded and .
Step 4. We show that converges strongly to .
Since is bounded, there exists a subsequence of such that . By Lemma 2.1, we have . Next we show .
Let . From Condition (A3), we obtain that , . Therefore, .
Hence . Using the Kadec-Klee property of H, we get . Since is an arbitrary subsequence of , we can conclude that converges strongly to . □
The authors would like to thank the anonymous referee for some suggestions to improve the manuscript. This work was supported by Research Fund for the Doctoral Program of Harbin University of Commerce (13DL002), Scientific Research Project Fund of the Education Department of Heilongjiang Province and Foundation of Heilongjiang Educational Committee (12521070).
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