A strong convergence theorem for equilibrium problems and split feasibility problems in Hilbert spaces
© Tang et al.; licensee Springer. 2014
Received: 16 June 2013
Accepted: 31 January 2014
Published: 13 February 2014
The main purpose of this paper is to introduce an iterative algorithm for equilibrium problems and split feasibility problems in Hilbert spaces. Under suitable conditions we prove that the sequence converges strongly to a common element of the set of solutions of equilibrium problems and the set of solutions of split feasibility problems. Our result extends and improves the corresponding results of some others.
MSC:90C25, 90C30, 47J25, 47H09.
Keywordsequilibrium problems split feasibility problems strong convergence bounded linear operator fixed point
The problem (1.1) is very general in the sense that it includes, as special cases, optimization problems, variational inequality problems, the Nash equilibrium problems and others, see, for instance, [1–3]. Some methods have been proposed to solve the EP, see, e.g., [4–6] and [7, 8].
where A is a given real matrix, and C and Q are nonempty, closed and convex subsets in and , respectively.
Due to its extraordinary utility and broad applicability in many areas of applied mathematics (most notably, fully discretized models of problems in image reconstruction from projections, in image processing, and in intensity-modulated radiation therapy), algorithms for solving convex feasibility problems have been received great attention (see, for instance [10–13] and also [14–18]).
where and are the orthogonal projection onto C and Q, respectively, is any positive constant and denotes the adjoint of A.
Motivated and inspired by the research going on in the sections of equilibrium problems and split feasibility problems, the purpose of this article is to introduce an iterative algorithm for equilibrium problems and split feasibility problems in Hilbert spaces. Under suitable conditions we prove the sequence converges strongly to a common element of the set of solutions of equilibrium problems and the set of solutions of split feasibility problems. Our result extends and improves the corresponding results of He et al.  and some others.
2 Preliminaries and lemmas
the weak ω-limit set of .
A mapping is said to be demi-closed at origin, if for any sequence with and , then .
It is easy to prove that if is a firmly nonexpansive mapping, then T is demi-closed at the origin.
∇f is -Lipschitz, i.e., , .
Lemma 2.2 (See, for example, )
T is firmly nonexpansive.
is firmly nonexpansive.
This implies that is firmly nonexpansive.
(iii) ⇒ (i): From (iii) we immediately know that T is firmly nonexpansive.
Throughout this paper, let us assume that a bifunction satisfies the following conditions:
(A1) , ;
(A2) F is monotone, i.e., , ;
(A3) , ;
(A4) for each , is convex and lower semicontinuous.
is firmly nonexpansive;
Ω is closed and convex.
The following results play an important role in this paper.
Lemma 2.4 ()
Lemma 2.5 ()
Lemma 2.6 ()
, or .
3 Main results
We are now in a position to prove the following theorem.
then the sequence converges strongly to .
Proof Since the solution set Ω of EP and the solution set of SPF (1.2) are both closed and convex, Γ (≠∅) is closed and convex. Thus, the metric projection is well defined.
This implies that the sequence is bounded. From (3.2) and (3.6) we know that and both are bounded.
Now, we prove by employing the technique studied by Maingé . For the purpose we consider two cases.
Since T is demi-closed at origin, from (3.14) and (3.15) we have , i.e., .
It follows from (3.19), (3.20), and (3.22) that . In view of and that S is demi-closed at origin, we get .
Applying Lemma 2.6 to (3.25), from the condition (i) we obtain , that is, .
Noting the inequality (3.26), this shows that , that is, . This completes the proof of Theorem 3.1. □
This study was supported by the Scientific Research Fund of Sichuan Provincial Education Department (13ZA0199) and the Scientific Research Fund of Sichuan Provincial Department of Science and Technology (2012JYZ011) and by the National Natural Science Foundation of China (Grant No. 11361070).
- Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. Math. Stud. 1994, 63: 123–145.MathSciNetGoogle Scholar
- Chadli O, Wong NC, Yao JC: Equilibrium problems with applications to eigenvalue problems. J. Optim. Theory Appl. 2003, 117(2):245–266. 10.1023/A:1023627606067View ArticleMathSciNetGoogle Scholar
- Chadli O, Schaible S, Yao JC: Regularized equilibrium problems with an application to noncoercive hemivariational inequalities. J. Optim. Theory Appl. 2004, 121: 571–596.View ArticleMathSciNetGoogle Scholar
- Combettes PL, Hirstoaga SA: Equilibrium programming in Hilbert space. J. Nonlinear Convex Anal. 2005, 6: 117–136.MathSciNetGoogle Scholar
- Ceng LC, Yao JC: A hybrid iterative scheme for mixed equilibrium problems and fixed point problems. J. Comput. Appl. Math. 2008, 214: 186–201. 10.1016/j.cam.2007.02.022View ArticleMathSciNetGoogle Scholar
- Takahashi S, Takahashi W: Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space. Nonlinear Anal. 2008, 69: 1025–1033. 10.1016/j.na.2008.02.042View ArticleMathSciNetGoogle Scholar
- Reich S, Sabach S: Three strong convergence theorems regarding iterative methods for solving equilibrium problems in reflexive Banach spaces, optimization theory and related topics. Contemp. Math. 2012, 568: 225–240.View ArticleMathSciNetGoogle Scholar
- Kassay G, Reich S, Sabach S: Iterative methods for solving systems of variational inequalities in reflexive Banach spaces. SIAM J. Optim. 2011, 21: 1319–1344. 10.1137/110820002View ArticleMathSciNetGoogle Scholar
- Censor Y, Elfving T: A multiprojection algorithm using Bregman projection in product space. Numer. Algorithms 1994, 8: 221–239. 10.1007/BF02142692View ArticleMathSciNetGoogle Scholar
- Aleyner A, Reich S: Block-iterative algorithms for solving convex feasibility problems in Hilbert and in Banach. J. Math. Anal. Appl. 2008, 343(1):427–435. 10.1016/j.jmaa.2008.01.087View ArticleMathSciNetGoogle Scholar
- Bauschke HH, Borwein JM: On projection algorithms for solving convex feasibility problems. SIAM Rev. 1996, 38(3):367–426. 10.1137/S0036144593251710View ArticleMathSciNetGoogle Scholar
- Moudafi A: A relaxed alternating CQ-algorithm for convex feasibility problems. Nonlinear Anal. 2013, 79: 117–121.View ArticleMathSciNetGoogle Scholar
- Masad E, Reich S: A note on the multiple-set split convex feasibility problem in Hilbert space. J. Nonlinear Convex Anal. 2007, 8: 367–371.MathSciNetGoogle Scholar
- Yao Y, Chen R, Marino G, Liou YC: Applications of fixed point and optimization methods to the multiple-sets split feasibility problem. J. Appl. Math. 2012., 2012: Article ID 927530Google Scholar
- Xu HK: A variable Krasnosel’skii-Mann algorithm and the multiple-set split feasibility problem. Inverse Probl. 2006, 22: 2021–2034. 10.1088/0266-5611/22/6/007View ArticleGoogle Scholar
- Xu HK: Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces. Inverse Probl. 2010., 26(10): Article ID 105018Google Scholar
- Yang Q: The relaxed CQ algorithm for solving the split feasibility problem. Inverse Probl. 2004, 20: 1261–1266. 10.1088/0266-5611/20/4/014View ArticleGoogle Scholar
- Zhao J, Yang Q: Several solution methods for the split feasibility problem. Inverse Probl. 2005, 21: 1791–1799. 10.1088/0266-5611/21/5/017View ArticleGoogle Scholar
- López G, Martín-Márquez V, Wang FH, Xu HK: Solving the split feasibility problem without prior knowledge of matrix norms. Inverse Probl. 2012., 28: Article ID 085004 10.1088/0266-5611/28/8/085004Google Scholar
- He S, Zhao Z: Strong convergence of a relaxed CQ algorithm for the split feasibility problem. J. Inequal. Appl. 2013., 2013: Article ID 197 10.1186/1029-242X-2013-197Google Scholar
- Goebel K, Reich S: Uniform Convexity, Hyperbolic Geometry and Nonexpansive Mappings. Marcel Dekker, New York; 1984.MATHGoogle Scholar
- Chang SS: On Chidume’s open questions and approximate solutions for multi-valued strongly accretive mapping equations in Banach spaces. J. Math. Anal. Appl. 1997, 216: 94–111. 10.1006/jmaa.1997.5661View ArticleMathSciNetGoogle Scholar
- Suzuki T: Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochner integrals. J. Math. Anal. Appl. 2005, 305: 227–239. 10.1016/j.jmaa.2004.11.017View ArticleMathSciNetGoogle Scholar
- Xu HK: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 2002, 66: 240–256. 10.1112/S0024610702003332View ArticleGoogle Scholar
- Maingé PE: New approach to solving a system of variational inequalities and hierarchical problems. J. Optim. Theory Appl. 2008, 138: 459–477. 10.1007/s10957-008-9433-zView ArticleMathSciNetGoogle Scholar
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