- Research
- Open Access
Shrinking projection methods for solving split equilibrium problems and fixed point problems for asymptotically nonexpansive mappings in Hilbert spaces
- Uamporn Witthayarat^{1},
- Afrah A N Abdou^{2}Email author and
- Yeol Je Cho^{2, 3}Email author
https://doi.org/10.1186/s13663-015-0448-5
© Witthayarat et al. 2015
- Received: 10 August 2015
- Accepted: 27 October 2015
- Published: 4 November 2015
Abstract
In this paper, we propose a new iterative sequence for solving common problems which consist of split equilibrium problems and fixed point problems for asymptotically nonexpansive mappings in the framework of Hilbert spaces and prove some strong convergence theorems of the generated sequence \(\{x_{n}\}\) by the shrinking projection method. Our results improve and extend the previous results given in the literature.
Keywords
- split equilibrium problem
- asymptotically nonexpansive mapping
- fixed point problem
- Hilbert space
MSC
- 54E70
- 47H25
1 Introduction
Throughout this paper, let \(\mathbb{R}\) and \(\mathbb{N}\) denote the set of all real numbers and the set of all positive integers, respectively. Let H be a real Hilbert space and C be a nonempty closed convex subset of H.
In 1997, Combettes and Hirstoaga [13] proposed an iterative method for solving problem (1.1) by the assumption that \(EP(F)\neq \emptyset\). Moreover, there are many new iteratively generated sequences for solving this problem together with fixed point problems (see [14–17]).
The solution sets of problems (1.2) and (1.3) are symbolized by \(EP(F_{1})\) and \(EP(F_{2})\), respectively. Therefore, we denote \(\Omega=\{v\in C: v\in EP(F_{1}) \mbox{ such that } Av\in EP(F_{2})\}\) as the solution set of \(SEP\).
In this paper, motivated and inspired by the results [18, 20, 21] and the recent works in this field, we introduce the shrinking projection method for solving split equilibrium problems and fixed point problems for asymptotically nonexpansive mappings in the framework of Hilbert spaces and prove some strong convergence theorems for the proposed new iterative method. In fact, our results improve and extend the results given by some authors.
2 Preliminaries
In this section, we recall some concepts including the assumption which will be needed for the proof of our main result.
Lemma 2.1
Lemma 2.2
[23]
- (1)
\(x_{n}\rightharpoonup z\);
- (2)
\(Tx_{n}-x_{n}\rightarrow0\).
Assumption 2.3
[24]
- (A1)
\(F(x,x)=0\) for all \(x\in C\);
- (A2)
F is monotone, i.e., \(F(x,y)+F(y,x)\leq0\) for all \(x,y\in C\);
- (A3)
for each \(x,y,z\in C\), \(\lim_{t\downarrow 0}F(tz+(1-t)x,y)\leq F(x,y)\);
- (A4)
for each \(x\in C\), \(y\mapsto F(x,y)\) is convex and lower semi-continuous.
Lemma 2.4
[24]
- (1)
\(T_{r}^{F}\) is single-valued;
- (2)\(T_{r}^{F}\) is firmly nonexpansive, i.e., for any \(x,y\in H\),$$\begin{aligned} \bigl\Vert T_{r}^{F}x-T_{r}^{F}y \bigr\Vert ^{2}\leq\bigl\langle T_{r}^{F}x-T_{r}^{F}y,x-y \bigr\rangle ; \end{aligned}$$
- (3)
\(F(T_{r}^{F})=EP(F)\);
- (4)
\(EP(F)\) is closed and convex.
3 Main results
In this section, we prove some strong convergence theorems of an iterative algorithm for solving a split equilibrium together with a fixed point problem revolving an asymptotically nonexpansive mapping in the framework of Hilbert spaces.
Theorem 3.1
Proof
In Theorem 3.1, if the mapping T is a nonexpansive mapping, then we immediately have the following.
Corollary 3.2
If \(H_{1}=H_{2}\), \(C=Q\) and \(A=I\) in Theorem 3.1, then we have the following.
Corollary 3.3
4 Applications
4.1 Applications to split variational inequality problems
Setting \(F_{1}(x,y)=\langle f(x),y-x\rangle\) and \(F_{2}(x,y)=\langle g(x),y-x\rangle\), it is clear that \(F_{1}\), \(F_{2}\) satisfy conditions (A1)-(A4), where f and g are \(\eta_{1}\)- and \(\eta_{2}\)-inverse strongly monotone mappings, respectively. Then, by Theorem 3.1, we get the following.
Theorem 4.1
Proof
The desired result can be proved directly through Theorem 3.1. □
4.2 Applications to split optimization problems
- (1)
for each \(x,y\in C\), \(f(tx+(1-t)y)\leq f(y)\), and for each \(u,v\in Q\), \(g(tu+(1-t)v)\leq g(v)\);
- (2)
\(f(x)\) is concave and upper semi-continuous for all \(x\in C\) and \(g(u)\) is concave and upper semi-continuous for all \(u\in Q\).
Let \(F_{1}(x,y)=f(x)-f(y)\) for all \(x,y\in C\) and \(F_{2}(u,v)=g(u)-g(v)\) for all \(u,v\in Q\). If f and g satisfy conditions (1) and (2), then it is clear that \(F_{1}:C\times C\to\mathbb{R}\) and \(F_{2}:Q\times Q\to \mathbb{R}\) are two bifunctions satisfying conditions (A1)-(A4). Therefore, by Theorem 3.1, we have the following.
Theorem 4.2
Proof
The desired result can be proved directly through Theorem 3.1. □
Declarations
Acknowledgements
This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, under grant No. (18-130-36-HiCi). The authors, therefore, acknowledge with thanks DSR technical and financial support. Also, Yeol Je Cho was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (2014R1A2A2A01002100).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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