Iterative Methods for Finding Common Solution of Generalized Equilibrium Problems and Variational Inequality Problems and Fixed Point Problems of a Finite Family of Nonexpansive Mappings
© Atid Kangtunyakarn. 2010
Received: 7 October 2010
Accepted: 2 November 2010
Published: 21 November 2010
We introduce a new method for a system of generalized equilibrium problems, system of variational inequality problems, and fixed point problems by using -mapping generated by a finite family of nonexpansive mappings and real numbers. Then, we prove a strong convergence theorem of the proposed iteration under some control condition. By using our main result, we obtain strong convergence theorem for finding a common element of the set of solution of a system of generalized equilibrium problems, system of variational inequality problems, and the set of common fixed points of a finite family of strictly pseudocontractive mappings.
Let be a real Hilbert space, and let be a nonempty closed convex subset of . Let be a nonlinear mapping, and let be a bifunction. A mapping of into itself is called nonexpansive if for all . We denote by the set of fixed points of (i.e., ). Goebel and Kirk  showed that is always closed convex, and also nonempty provided has a bounded trajectory.
The set of solutions of (1.2) is denoted by . Many problems in physics, optimization, and economics are seeking some elements of , see [2, 3]. Several iterative methods have been proposed to solve the equilibrium problem, see, for instance, [2–4]. In 2005, Combettes and Hirstoaga  introduced an iterative scheme of finding the best approximation to the initial data when is nonempty and proved a strong convergence theorem.
The problem of finding a common fixed point of a family of nonexpansive mappings has been studied by many authors. The well-known convex feasibility problem reduces to finding a point in the intersection of the fixed point sets of a family of nonexpansive mapping (see [6, 7]).
The ploblem of finding a common element of and the set of all common fixed points of a family of nonexpansive mappings is of wide interdisciplinary interest and importance. Many iterative methods are purposed for finding a common element of the solutions of the equilibrium problem and fixed point problem of nonexpansive mappings, see [8–10].
where is an -inverse strongly monotone mapping of into with positive real number , and , , , and proved strong convergence of the scheme (1.7) to , where in the framework of a Hilbert space, under some suitable conditions on , , and bifunction .
where is a contraction mapping and is -mapping generated by infinite family of nonexpansive mappings and infinite real number. Under suitable conditions of these parameters they proved strong convergence of the scheme (1.8) to , where .
In this section, we collect and give some useful lemmas that will be used for our main result in the next section.
Lemma 2.1 (see ).
Lemma 2.2 (see ).
Lemma 2.3 (see ).
Lemma 2.4 (see ).
Lemma 2.5 (see ).
The following lemma appears implicitly in .
Lemma 2.6 (see ).
Lemma 2.7 (see ).
In 2009, Kangtunyakarn and Suantai  defined a new mapping and proved their lemma as follows.
Let be a nonempty closed convex subset of strictly convex. Let be a finite family of nonexpanxive mappings of into itself with , and let , , where , , for all , for all . Let be the mapping generated by and . Then .
Let be a nonempty closed convex subset of Banach space. Let be a finite family of nonexpansive mappings of into itself and , , where , and such that as for and Moreover, for every , let and be the -mappings generated by and and and , respectively. Then for every .
Lemma 2.11 (see ).
3. Main Result
We will divide our proof into 6 steps.
To prove strong convergence theorem in this section, we need definition and lemma as follows.
Lemma 4.2 (see ).
The authors would like to thank Professor Dr. Suthep Suantai for his suggestion in doing and improving this paper.
- Goebel K, Kirk WA: Topics in Metric Fixed Point Theory, Cambridge Studies in Advanced Mathematics. Volume 28. Cambridge University Press, Cambridge, UK; 1990:viii+244.View ArticleMATHGoogle Scholar
- Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. The Mathematics Student 1994,63(1–4):123–145.MathSciNetMATHGoogle Scholar
- Combettes PL, Hirstoaga SA: Equilibrium programming in Hilbert spaces. Journal of Nonlinear and Convex Analysis 2005,6(1):117–136.MathSciNetMATHGoogle Scholar
- Moudafi A, Théra M: Proximal and dynamical approaches to equilibrium problems. In Ill-Posed Variational Problems and Regularization Techniques, Lecture Notes in Econom. and Math. Systems. Volume 477. Springer, Berlin, Germany; 1999:187–201.View ArticleGoogle Scholar
- Iiduka H, Takahashi W: Weak convergence theorems by Cesáro means for nonexpansive mappings and inverse-strongly-monotone mappings. Journal of Nonlinear and Convex Analysis 2006,7(1):105–113.MathSciNetMATHGoogle Scholar
- Bauschke HH, Borwein JM: On projection algorithms for solving convex feasibility problems. SIAM Review 1996,38(3):367–426. 10.1137/S0036144593251710MathSciNetView ArticleMATHGoogle Scholar
- Combettes PL: The foundations of set theoretic estimation. Proceedings of the IEEE 1993, 81: 182–208.View ArticleGoogle Scholar
- Kangtunyakarn A, Suantai S: A new mapping for finding common solutions of equilibrium problems and fixed point problems of finite family of nonexpansive mappings. Nonlinear Analysis: Theory, Methods & Applications 2009,71(10):4448–4460. 10.1016/j.na.2009.03.003MathSciNetView ArticleMATHGoogle Scholar
- Colao V, Marino G, Xu H-K: An iterative method for finding common solutions of equilibrium and fixed point problems. Journal of Mathematical Analysis and Applications 2008,344(1):340–352. 10.1016/j.jmaa.2008.02.041MathSciNetView ArticleMATHGoogle Scholar
- Takahashi S, Takahashi W: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. Journal of Mathematical Analysis and Applications 2007,331(1):506–515. 10.1016/j.jmaa.2006.08.036MathSciNetView ArticleMATHGoogle Scholar
- Takahashi S, Takahashi W: Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space. Nonlinear Analysis: Theory, Methods & Applications 2008,69(3):1025–1033. 10.1016/j.na.2008.02.042MathSciNetView ArticleMATHGoogle Scholar
- Qin X, Cho YJ, Kang SM: Viscosity approximation methods for generalized equilibrium problems and fixed point problems with applications. Nonlinear Analysis: Theory, Methods & Applications 2010,72(1):99–112. 10.1016/j.na.2009.06.042MathSciNetView ArticleMATHGoogle Scholar
- Takahashi W: Nonlinear Functional Analysis: Fixed Point Theory and Its Applications. Yokohama Publishers, Yokohama, Japan; 2000:iv+276.MATHGoogle Scholar
- Xu H-K: Iterative algorithms for nonlinear operators. Journal of the London Mathematical Society. Second Series 2002,66(1):240–256. 10.1112/S0024610702003332MathSciNetView ArticleMATHGoogle Scholar
- Bruck RE Jr.: Properties of fixed-point sets of nonexpansive mappings in Banach spaces. Transactions of the American Mathematical Society 1973, 179: 251–262.MathSciNetView ArticleMATHGoogle Scholar
- Browder FE: Nonlinear operators and nonlinear equations of evolution in Banach spaces. In Proceedings of Symposia in Pure Mathematics. Amer. Math. Soc., Providence, RI, USA; 1976:1–308.Google Scholar
- Suzuki T: Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals. Journal of Mathematical Analysis and Applications 2005,305(1):227–239. 10.1016/j.jmaa.2004.11.017MathSciNetView ArticleMATHGoogle Scholar
- Kangtunyakarn A, Suantai S: Hybrid iterative scheme for generalized equilibrium problems and fixed point problems of finite family of nonexpansive mappings. Nonlinear Analysis: Hybrid Systems 2009,3(3):296–309. 10.1016/j.nahs.2009.01.012MathSciNetMATHGoogle Scholar
- Takahashi W: Introduction to Nonlinear and Convex Analysis. Yokohama Publishers, Yokohama, Japan; 2009:iv+234.MATHGoogle Scholar
- Zhou H: Convergence theorems of fixed points for κ -strict pseudo-contractions in Hilbert spaces. Nonlinear Analysis: Theory, Methods & Applications 2008,69(2):456–462. 10.1016/j.na.2007.05.032MathSciNetView ArticleMATHGoogle Scholar
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