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A new algorithm for mixed equilibrium problem and Bregman strongly nonexpansive mappings in Banach spaces  [PDF]
Vahid Darvish
Mathematics , 2015,
Abstract: In this paper, we study a new iterative method for a common fixed point of a finite family of Bregman strongly nonexpansive mappings in the frame work of reflexive real Banach spaces. Moreover, we prove the strong convergence theorem for finding common fixed points with the solutions of a mixed equilibrium problem.
Existence and Strong Convergence Theorems for Generalized Mixed Equilibrium Problems of a Finite Family of Asymptotically Nonexpansive Mappings in Banach Spaces
Rabian Wangkeeree,Hossein Dehghan,Pakkapon Preechasilp
Journal of Applied Mathematics , 2012, DOI: 10.1155/2012/859492
Abstract: We first prove the existence of solutions for a generalized mixed equilibrium problem under the new conditions imposed on the given bifunction and introduce the algorithm for solving a common element in the solution set of a generalized mixed equilibrium problem and the common fixed point set of finite family of asymptotically nonexpansive mappings. Next, the strong convergence theorems are obtained, under some appropriate conditions, in uniformly convex and smooth Banach spaces. The main results extend various results existing in the current literature.
Strong Convergence Theorem for Solving Generalized Mixed Equilibrium Problems and Fixed Point Problems for Total Quasi--Asymptotically Nonexpansive Mappings in Banach Spaces
Zhaoli Ma,Lin Wang,Yunhe Zhao
Journal of Applied Mathematics , 2012, DOI: 10.1155/2012/506210
Abstract: We introduce an iterative scheme for finding a common element of the set of solutions of generalized mixed equilibrium problems and the set of fixed points for countable families of total quasi--asymptotically nonexpansive mappings in Banach spaces. We prove a strong convergence theorem of the iterative sequence generated by the proposed iterative algorithm in an uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property. The results presented in this paper improve and extend some recent corresponding results.
Approximation of fixed points for a representation of nonexpansive mappings in Banach spaces  [PDF]
Ebrahim Soori
Mathematics , 2015,
Abstract: The purpose of this paper is to study an implicit scheme for a representation of nonexpansive mappings on a closed convex subset of a smooth and uniformly convex Banach space with respect to a left regular sequence of means defined on an appropriate space of bounded real valued functions of the semigroup. This algorithm extends the algorithm that introduced in [N. Hussain, M. L. Bami and E. Soori, An implicit method for finding a common fixed point of a representation of nonexpansive mappings in Banach spaces, Fixed Point Theory and Applications 2014, 2014:238].
Approximation of Fixed Points of Weak Bregman Relatively Nonexpansive Mappings in Banach Spaces  [PDF]
Jiawei Chen,Zhongping Wan,Liuyang Yuan,Yue Zheng
International Journal of Mathematics and Mathematical Sciences , 2011, DOI: 10.1155/2011/420192
Abstract: We introduce a concept of weak Bregman relatively nonexpansive mapping which is distinct from Bregman relatively nonexpansive mapping. By using projection techniques, we construct several modification of Mann type iterative algorithms with errors and Halpern-type iterative algorithms with errors to find fixed points of weak Bregman relatively nonexpansive mappings and Bregman relatively nonexpansive mappings in Banach spaces. The strong convergence theorems for weak Bregman relatively nonexpansive mappings and Bregman relatively nonexpansive mappings are derived under some suitable assumptions. The main results in this paper develop, extend, and improve the corresponding results of Matsushita and Takahashi (2005) and Qin and Su (2007). 1. Introduction Throughout this paper, without other specifications, we denote by the set of real numbers. Let be a real reflexive Banach space with the dual space . The norm and the dual pair between and are denoted by and , respectively. Let be proper convex and lower semicontinuous. The Fenchel conjugate of is the function defined by We denote by the domain of , that is, . Let be a nonempty closed and convex subset of a nonlinear mapping. Denote by , the set of fixed points of . is said to be nonexpansive if for all . In 1967, Brègman [1] discovered an elegant and effective technique for the using of the so-called Bregman distance function (see, Section 2, Definition 2.1) in the process of designing and analyzing feasibility and optimization algorithms. This opened a growing area of research in which Bregman's technique is applied in various ways in order to design and analyze iterative algorithms for solving not only feasibility and optimization problems, but also algorithms for solving variational inequalities, for approximating equilibria, for computing fixed points of nonlinear mappings, and so on (see, e.g., [1–25], and the references therein). Nakajo and Takahashi [26] introduced the following modification of the Mann iteration method for a nonexpansive mapping in a Hilbert space as follows: where and is the metric projection from onto a closed and convex subset of . They proved that generated by (1.2) converges strongly to a fixed point of under some suitable assumptions. Motivated by Nakajo and Takahashi [26], Matsushita and Takahashi [27] introduced the following modification of the Mann iteration method for a relatively nonexpansive mapping in a Banach space as follows: where , for all , is the duality mapping of and is the generalized projection (see, e.g., [2, 3, 28]) from onto a closed and convex subset of
A New Hybrid Algorithm for Solving a System of Generalized Mixed Equilibrium Problems, Solving a Family of Quasi- -Asymptotically Nonexpansive Mappings, and Obtaining Common Fixed Points in Banach Space  [PDF]
J. F. Tan,S. S. Chang
International Journal of Mathematics and Mathematical Sciences , 2011, DOI: 10.1155/2011/106323
Abstract: The main purpose of this paper is to introduce a new hybrid iterative scheme for finding a common element of set of solutions for a system of generalized mixed equilibrium problems, set of common fixed points of a family of quasi- -asymptotically nonexpansive mappings, and null spaces of finite family of -inverse strongly monotone mappings in a 2-uniformly convex and uniformly smooth real Banach space. The results presented in the paper improve and extend the corresponding results announced by some authors. 1. Introduction Throughout this paper, we assume that is a real Banach space with a dual , is a nonempty closed convex subset of , and is the duality pairing between members of and . The mapping defined by is called the normalized duality mapping. Let be a bifunction, let be a nonlinear mapping, and let be a proper extended real-valued function. The “so-called” generalized mixed equilibrium problem for , , is to find such that The set of solutions of (1.2) is denoted by , that is, Special Examples (1)If , then the problem (1.2) is reduced to the generalized equilibrium problem (GEP), and the set of its solutions is denoted by (2)If , then the problem (1.2) is reduced to the mixed equilibrium problem (MEP), and the set of its solutions is denoted by These show that the problem (1.2) is very general in the sense that numerous problems in physics, optimization, and economics reduce to finding a solution of (1.2). Recently, some methods have been proposed for the generalized mixed equilibrium problem in Banach space (see, e.g., [1–3]). Let be a smooth, strictly convex, and reflexive Banach space, and let be a nonempty closed convex subset of . Throughout this paper, the Lyapunov function is defined by Following Alber [4], the generalized projection is defined by Let be a nonempty closed convex subset of , let be a mapping, and let be the set of fixed points of . A point is said to be an asymptotic fixed point of if there exists a sequence such that and . We denoted the set of all asymptotic fixed points of by . A point is said to be a strong asymptotic fixed point of if there exists a sequence such that and . We denoted the set of all strongly asymptotic fixed points of by . A mapping is said to be nonexpansive if A mapping is said to be relatively nonexpansive [5] if , and A mapping is said to be weak relatively nonexpansive [6] if , and A mapping is said to be closed if for any sequence with and , then . A mapping is said to be quasi-?-nonexpansive if and A mapping is said to be quasi-?-asymptotically nonexpansive, if and there exists a real sequence
Existence and iterative approximation for generalized equilibrium problems for a countable family of nonexpansive mappings in banach spaces  [cached]
Kamraksa Uthai,Wangkeeree Rabian
Fixed Point Theory and Applications , 2011,
Abstract: We first prove the existence of a solution of the generalized equilibrium problem (GEP) using the KKM mapping in a Banach space setting. Then, by virtue of this result, we construct a hybrid algorithm for finding a common element in the solution set of a GEP and the fixed point set of countable family of nonexpansive mappings in the frameworks of Banach spaces. By means of a projection technique, we also prove that the sequences generated by the hybrid algorithm converge strongly to a common element in the solution set of GEP and common fixed point set of nonexpansive mappings. AMS Subject Classification: 47H09, 47H10
New Iterative Approximation Methods for a Countable Family of Nonexpansive Mappings in Banach Spaces  [cached]
Nammanee Kamonrat,Wangkeeree Rabian
Fixed Point Theory and Applications , 2011,
Abstract: We introduce new general iterative approximation methods for finding a common fixed point of a countable family of nonexpansive mappings. Strong convergence theorems are established in the framework of reflexive Banach spaces which admit a weakly continuous duality mapping. Finally, we apply our results to solve the the equilibrium problems and the problem of finding a zero of an accretive operator. The results presented in this paper mainly improve on the corresponding results reported by many others.
Viscosity Approximation Methods for Generalized Mixed Equilibrium Problems and Fixed Points of a Sequence of Nonexpansive Mappings  [cached]
Zeng Wei-You,Huang Nan-Jing,Zhao Chang-Wen
Fixed Point Theory and Applications , 2008,
Abstract: We introduce an iterative scheme by the viscosity approximation method for finding a common element of the set of common solutions for generalized mixed equilibrium problems and the set of common fixed points of a sequence of nonexpansive mappings in Hilbert spaces. We show a strong convergence theorem under some suitable conditions.
Viscosity Approximation Methods for Generalized Mixed Equilibrium Problems and Fixed Points of a Sequence of Nonexpansive Mappings  [cached]
Wei-You Zeng,Nan-Jing Huang,Chang-Wen Zhao
Fixed Point Theory and Applications , 2008, DOI: 10.1155/2008/714939
Abstract: We introduce an iterative scheme by the viscosity approximation method for finding a common element of the set of common solutions for generalized mixed equilibrium problems and the set of common fixed points of a sequence of nonexpansive mappings in Hilbert spaces. We show a strong convergence theorem under some suitable conditions.
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