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Iterative algorithms based on hybrid method and Cesàro mean of asymptotically nonexpansive mappings for equilibrium problems
Fixed Point Theory and Applications volume 2014, Article number: 16 (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.
Let H be a real Hilbert space with the inner product and the norm . Let C be a nonempty closed convex subset of H. A mapping is said to be asymptotically nonexpansive if for each , there exists a nonnegative real number satisfying such that
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.
In 1953, Mann  introduced the iteration as follows: a sequence defined by
In an infinite-dimensional Hilbert space, Mann iteration could conclude only weak convergence . Attempts to modify the Mann iteration method (1.1) so that strong convergence is guaranteed have recently been made. Nakajo and Takahashi  proposed the following modification of Mann iteration method for a nonexpansive mapping T in a Hilbert space:
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.
In 2006, Kim and Xu  adapted the iteration (1.2) in a Hilbert space. More precisely, they introduced the following iteration process for asymptotically nonexpansive mappings:
Baillon  first proved that the following Cesàro mean iterative sequence weakly converges to a fixed point of a nonexpansive mapping in Hilbert spaces:
Shimizu and Takahashi  proved a strong convergence theorem of the above iteration for an asymptotically nonexpansive mapping in Hilbert spaces.
Let C be a nonempty closed convex subset of a real Hilbert space H, let be a functional, where ℝ is the set of real numbers. The equilibrium problem is to find such that
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.
Motivated by the above-mentioned results, in this paper we introduce the following iteration process for asymptotically nonexpansive mappings T with C a closed convex bounded subset of a real Hilbert space:
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.
Let H be a real Hilbert space. Then
for all and . It is also known that H satisfies
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 .
Let C be a nonempty closed convex subset of H. Then, for any , there exists a unique nearest point in C, denoted by , such that
Such a mapping is called the metric projection of H onto C. We know that is nonexpansive. Furthermore, for and ,
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 )
Let C be a nonempty bounded subset of a Hilbert space. Let T be an asymptotically nonexpansive mapping from C into itself such that is nonempty. Then, for any , there exists a positive integer such that for any integer , there is a positive integer satisfying
The equilibrium problem is to find such that
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 ;
(A3) for each ,
(A4) for each , is convex and lower semi-continuous.
The following lemma appears implicitly in .
Lemma 2.3 ()
Let C be a nonempty closed convex subset of H, let f be a bifunction of into ℝ satisfying (A1)-(A4), and let and . Then there exists such that
The following lemma was also given in .
Lemma 2.4 ()
Assume that satisfies (A1)-(A4). For and , define a mapping as follows:
for all . Then the following hold:
is firmly nonexpansive, i.e.,
is closed and convex.
Lemma 2.5 ()
Let C be a nonempty closed convex subset of H, let be a functional, satisfying (A1)-(A4), and let . Then, for and ,
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.
Theorem 3.1 Let C be a nonempty bounded closed convex subset of a real Hilbert space H and let be an asymptotically nonexpansive mapping with , denote . Assume that is a sequence in such that for all n and for some . Define a sequence in C by the following algorithm:
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 .
From the definition of and , it is obvious that is closed and is closed and convex for each . We prove that is convex. Since is equivalent to
it follows that is convex.
Step 2. We show that , .
Let and . Then from
we have . Next, we show that , . We prove this by induction. For , we have . Suppose that , then and there exists a unique element such that . Then
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 .
It follows from the definition of that . Therefore
Let for all . Then
On the other hand, from , we have
Therefore, the sequence is nondecreasing. Since C is bounded, we obtain that exists. This implies that is bounded. Noticing again that and , we have . It follows from (2.1) that
for all . This implies that
Since , we have . Noticing that , then as . Hence
Let . Also since for all n, then
From Lemma 2.2, we have
Put , then
as . Thus, we have
Step 4. We show that converges strongly to .
Put . Since is bounded, let be a subsequence of such that . By Lemma 2.1, we get . Since and , we have . It follows from and the weak lower semicontinuity of the norm that
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.
Theorem 4.1 Let H be a real Hilbert space, and let C be a nonempty bounded closed convex subset of H, let be a functional, satisfying (A1)-(A4). Let be an asymptotically nonexpansive mapping with , denote such that . Let be a sequence generated by
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 , .
Let . Putting , , by (2) of Lemma 2.4, we have is relatively nonexpansive. Noticing that relatively nonexpansive mappings are nonexpansive in Hilbert spaces, then for any ,
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 .
Similar to the proof of Step 3 in Theorem 3.1, we may obtain that is bounded and
Since , then . Noticing that , we have
For , we have
Since , by (4.4) and Lemma 2.5, we get that
Let . Since , , then by (4.3) and (4.5), we have
From Lemma 2.2, it follows that
Put , then
Let , we get that . Thus, by (4.3) and (4.5), we have
Therefore, by (4.8) and (4.2), we obtain that
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 .
From (4.3) and (4.5), we get that , . Since , then
Replacing n by , we have from Condition (A2) that
Let , since , by (4.5) and Condition (A4), we get that
For , , let , then , thus . By Condition (A1), we get that
Dividing by t, we have
Let . From Condition (A3), we obtain that , . Therefore, .
Denote . Since , , then . Since the norm is weakly lower semicontinuous, we have
Hence . Using the Kadec-Klee property of H, we get . Since is an arbitrary subsequence of , we can conclude that converges strongly to . □
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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).
The authors declare that they have no competing interests.
The main idea of this paper is proposed by JZ and YC. JZ prepared the manuscript initially and performed all the steps of the proofs in this research. All authors read and approved the final manuscript.
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Cite this article
Zhang, J., Cui, Y. Iterative algorithms based on hybrid method and Cesàro mean of asymptotically nonexpansive mappings for equilibrium problems. Fixed Point Theory Appl 2014, 16 (2014) doi:10.1186/1687-1812-2014-16
- strong convergence theorem
- asymptotically nonexpansive mapping
- hybrid method
- equilibrium problem
- Cesàro means