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
MSC
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
References
- Inchan, I: Strong convergence theorems of modified Mann iteration methods for asymptotically nonexpansive mappings in Hilbert spaces. Int. J. Math. Anal. 2, 1135-1145 (2008) MATHMathSciNetGoogle Scholar
- Kim, JK, Nam, YM, Sim, JY: Convergence theorem of implicit iterative sequences for a finite family of asymptotically quasi-nonexpansive type mappings. Nonlinear Anal. 71, 2839-2848 (2009) MathSciNetView ArticleGoogle Scholar
- Kim, TH, Xu, HK: Strong convergence of modified Mann iterations for asymptotically nonexpansive mapping and semigroups. Nonlinear Anal. 64, 1140-1152 (2006) MATHMathSciNetView ArticleGoogle Scholar
- Opial, Z: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 73, 591-597 (1967) MATHMathSciNetView ArticleGoogle Scholar
- Blum, E, Oettli, W: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123-145 (1994) MATHMathSciNetGoogle Scholar
- Choudhury, BS, Kundu, S: A viscosity type iteration by weak contraction for approximating solutions of generalized equilibrium problem. J. Nonlinear Sci. Appl. 5, 243-251 (2012) MATHMathSciNetGoogle Scholar
- Kang, SM, Cho, SY, Qin, X: Hybrid projection algorithms for approximating fixed points of asymptotically quasi-pseudocontractive mappings. J. Nonlinear Sci. Appl. 5, 466-474 (2012) MATHMathSciNetGoogle Scholar
- Witthayarat, U, Cho, YJ, Kumam, P: Approximation algorithm for fixed points of nonlinear operators and solutions of mixed equilibrium problems and variational inclusion problems with applications. J. Nonlinear Sci. Appl. 5, 475-494 (2012) MATHMathSciNetGoogle Scholar
- Chang, SS, Lee, HWJ, Chan, CK: A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization. Nonlinear Anal. 70, 3307-3319 (2009) MATHMathSciNetView ArticleGoogle Scholar
- Katchang, P, Kumam, P: A new iterative algorithm of solution for equilibrium problems, variational inequalities and fixed point problems in a Hilbert space. J. Appl. Math. Comput. 32, 19-38 (2010) MATHMathSciNetView ArticleGoogle Scholar
- Plubtieng, S, Punpaeng, R: A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl. 336, 455-469 (2007) MATHMathSciNetView ArticleGoogle Scholar
- Qin, X, Shang, M, Su, Y: A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces. Nonlinear Anal. 69, 3897-3909 (2008) MATHMathSciNetView ArticleGoogle Scholar
- Combettes, PL, Hirstoaga, SA: Equilibrium programming using proximal like algorithms. Math. Program. 78, 29-41 (1997) View ArticleGoogle Scholar
- Agarwal, RP, Chen, JW, Cho, YJ: Strong convergence theorems for equilibrium problems and weak Bregman relatively nonexpansive mappings in Banach spaces. J. Inequal. Appl. 2013, Article ID 119 (2013) MathSciNetView ArticleGoogle Scholar
- Tada, A, Takahashi, W: Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem. J. Optim. Theory Appl. 133, 359-370 (2007) MATHMathSciNetView ArticleGoogle Scholar
- Takahashi, S, Takahashi, W: Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space. Nonlinear Anal. 69, 1025-1033 (2008) MATHMathSciNetView ArticleGoogle Scholar
- Takahashi, S, Takahashi, W: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl. 331, 506-515 (2007) MATHMathSciNetView ArticleGoogle Scholar
- He, Z: The split equilibrium problem and its convergence algorithms. J. Inequal. Appl. 2012, Article ID 162 (2012) View ArticleGoogle Scholar
- Censor, Y, Gibali, A, Reich, S: Algorithm for split variational inequality problems. Numer. Algorithms 59, 301-323 (2012) MATHMathSciNetView ArticleGoogle Scholar
- Kazmi, KR, Rizvi, SH: Iterative approximation of a common solution of a split equilibrium problem, a variational inequality problem and a fixed point problem. J. Egypt. Math. Soc. 21, 44-51 (2013) MATHMathSciNetView ArticleGoogle Scholar
- Bnouhachem, A: Strong convergence algorithm for split equilibrium problems and hierarchical fixed point problems. Sci. World J. 2014, Article ID 390956 (2014) View ArticleGoogle Scholar
- Iiduka, H, Takahashi, W: Strong convergence theorems for nonexpansive mappings and inverse strongly monotone mappings. Nonlinear Anal. 61, 341-350 (2005) MATHMathSciNetView ArticleGoogle Scholar
- Lin, PK, Tan, KK, Xu, HK: Demiclosedness principle and asymptotic behavior for asymptotically nonexpansive mappings. Nonlinear Anal. 24, 929-946 (1995) MATHMathSciNetView ArticleGoogle Scholar
- Combette, PL, Hirstoaga, SA: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 6, 117-136 (2005) MathSciNetGoogle Scholar