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Common Solutions of Generalized Mixed Equilibrium Problems, Variational Inclusions, and Common Fixed Points for Nonexpansive Semigroups and Strictly Pseudocontractive Mappings  [PDF]
Poom Kumam,Usa Hamphries,Phayap Katchang
Journal of Applied Mathematics , 2011, DOI: 10.1155/2011/953903
Abstract: We introduce a new iterative scheme by shrinking projection method for finding a common element of the set of solutions of generalized mixed equilibrium problems, the set of common solutions of variational inclusion problems with set-valued maximal monotone mappings and inverse-strongly monotone mappings, the set of solutions of fixed points for nonexpansive semigroups, and the set of common fixed points for an infinite family of strictly pseudocontractive mappings in a real Hilbert space. We prove that the sequence converges strongly to a common element of the above four sets under some mind conditions. Furthermore, by using the above result, an iterative algorithm for solution of an optimization problem was obtained. Our results improve and extend the corresponding results of Martinez-Yanes and Xu (2006), Shehu (2011), Zhang et al. (2008), and many authors. 1. Introduction Throughout this paper, we assume that is a real Hilbert space with inner product and norm are denoted by and , respectively. Let denote the family of all subsets of , and let be a closed-convex subset of . Recall that a mapping is said to be a -strict pseudocontraction [1] if there exists such that where denotes the identity operator on . When , is said to be nonexpansive [2] if And when , is said to be pseudocontraction if Clearly, the class of -strict pseudocontraction falls into the one between classes of nonexpansive mappings and pseudocontraction mapping. We denote the set of fixed points of by . A family of mappings of into itself is called a nonexpansive semigroup on if it satisfies the following conditions: (i) for all ,(ii) for all ,(iii) for all and ,(iv)for all is continuous. We denote by the set of all common fixed points of , that is, . It is known that is closed and convex. Let be a single-valued nonlinear mapping, and let be a set-valued mapping. We consider the following variational inclusion problem, which is to find a point such that where is the zero vector in H. The set of solutions of problem (1.4) is denoted by . Let the set-valued mapping be a maximal monotone. We define the resolvent operator associate with and as follows: where is a positive number. It is worth mentioning that the resolvent operator is single-valued, nonexpansive, and 1-inverse-strongly monotone ([3, 4]). Let be a bifunction of into , where is the set of real numbers, let be a mapping, and let be a real-valued function. The generalized mixed equilibrium problem is for finding such that The set of solutions of (1.6) is denoted by , that is, If , then the problem (1.6) is reduced into the
Common fixed points of commutative semigroups of nonexpansive mappings  [PDF]
T. Suzuki
Mathematics , 2004,
Abstract: In this paper, we discuss characterizations of common fixed points of commutative semigroups of nonexpansive mappings. We next prove convergence theorems to a common fixed point. We finally discuss nonexpansive retractions onto the set of common fixed points. In our discussion, we may not assume the strict convexity of the Banach space.
Strong convergence theorems for asymptotically nonexpansive mappings and asymptotically nonexpansive semigroups  [cached]
Yongfu Su,Xiaolong Qin
Fixed Point Theory and Applications , 2006, DOI: 10.1155/fpta/2006/96215
Abstract: Strong convergence theorems are obtained from modified Halpern iterative scheme for asymptotically nonexpansive mappings and asymptotically nonexpansive semigroups, respectively. Our results extend and improve the recent ones announced by Nakajo, Takahashi, Kim, Xu, and some others.
Strong convergence theorems for asymptotically nonexpansive mappings and asymptotically nonexpansive semigroups  [cached]
Su Yongfu,Qin Xiaolong
Fixed Point Theory and Applications , 2006,
Abstract: Strong convergence theorems are obtained from modified Halpern iterative scheme for asymptotically nonexpansive mappings and asymptotically nonexpansive semigroups, respectively. Our results extend and improve the recent ones announced by Nakajo, Takahashi, Kim, Xu, and some others.
The Meir-Keeler Type for Solving Variational Inequalities and Fixed Points of Nonexpansive Semigroups in Banach Spaces
Phayap Katchang,Poom Kumam
Journal of Applied Mathematics , 2012, DOI: 10.1155/2012/376421
Abstract: The aim of this paper is to introduce a new iterative scheme for finding common solutions of the variational inequalities for an inverse strongly accretive mapping and the solutions of fixed point problems for nonexpansive semigroups by using the modified viscosity approximation method associate with Meir-Keeler type mappings and obtain some strong convergence theorem in a Banach spaces under some parameters controlling conditions. Our results extend and improve the recent results of Li and Gu (2010), Wangkeeree and Preechasilp (2012), Yao and Maruster (2011), and many others.
Hybrid Algorithms of Nonexpansive Semigroups for Variational Inequalities
Peixia Yang,Yonghong Yao,Yeong-Cheng Liou,Rudong Chen
Journal of Applied Mathematics , 2012, DOI: 10.1155/2012/634927
Abstract: Two hybrid algorithms for the variational inequalities over the common fixed points set of nonexpansive semigroups are presented. Strong convergence results of these two hybrid algorithms have been obtained in Hilbert spaces. The results improve and extend some corresponding results in the literature.
Fixed point properties for semigroups of nonlinear mappings and amenability  [PDF]
A. T. -M. Lau,Yong Zhang
Mathematics , 2012,
Abstract: In this paper we study fixed point properties for semitopological semigroup of nonexpansive mappings on a bounded closed convex subset of a Banach space. We also study a Schauder fixed point property for a semitopological semigroup of continuous mappings on a compact convex subset of a separated locally convex space. Such semigroups properly include the class of extremely left amenable semitopological semigroups, the free commutative semigroup on one generator and the bicyclic semigroup $S_1 = < a, b: ab = 1 >$.
Strong Convergence to Common Fixed Points for Countable Families of Asymptotically Nonexpansive Mappings and Semigroups  [cached]
Wattanawitoon Kriengsak,Kumam Poom
Fixed Point Theory and Applications , 2010,
Abstract: We prove strong convergence theorems for countable families of asymptotically nonexpansive mappings and semigroups in Hilbert spaces. Our results extend and improve the recent results of Nakajo and Takahashi (2003) and of Zegeye and Shahzad (2008) from the class of nonexpansive mappings to asymptotically nonexpansive mappings.
Generalized Variational Inequalities Involving Relaxed Monotone Mappings and Nonexpansive Mappings
Xiaolong Qin,Meijuan Shang
Journal of Inequalities and Applications , 2008, DOI: 10.1155/2007/20457
Abstract: We consider the solvability of generalized variational inequalities involving multivalued relaxed monotone operators and single-valued nonexpansive mappings in the framework of Hilbert spaces. We also study the convergence criteria of iterative methods under some mild conditions. Our results improve and extend the recent ones announced by many others.
Generalized Variational Inequalities Involving Relaxed Monotone Mappings and Nonexpansive Mappings  [cached]
Qin Xiaolong,Shang Meijuan
Journal of Inequalities and Applications , 2007,
Abstract: We consider the solvability of generalized variational inequalities involving multivalued relaxed monotone operators and single-valued nonexpansive mappings in the framework of Hilbert spaces. We also study the convergence criteria of iterative methods under some mild conditions. Our results improve and extend the recent ones announced by many others.
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