oalib

Publish in OALib Journal

ISSN: 2333-9721

APC: Only $99

Submit

Search Results: 1 - 10 of 39 matches for " Rattanaporn Wangkeeree "
All listed articles are free for downloading (OA Articles)
Page 1 /39
Display every page Item
Strong Convergence Theorems of the General Iterative Methods for Nonexpansive Semigroups in Banach Spaces
Rattanaporn Wangkeeree
International Journal of Mathematics and Mathematical Sciences , 2011, DOI: 10.1155/2011/643740
Abstract: Let be a real reflexive Banach space which admits a weakly sequentially continuous duality mapping from to . Let be a nonexpansive semigroup on such that , and is a contraction on with coefficient . Let be -strongly accretive and -strictly pseudocontractive with and a positive real number such that . When the sequences of real numbers and satisfy some appropriate conditions, the three iterative processes given as follows: , , , , and , converge strongly to , where is the unique solution in of the variational inequality , . Our results extend and improve corresponding ones of Li et al. (2009) Chen and He (2007), and many others. 1. Introduction Let be a real Banach space. A mapping of into itself is said to be nonexpansive if for each . We denote by the set of fixed points of . A mapping is called -contraction if there exists a constant such that for all . 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 , the mapping is continuous. We denote by the set of all common fixed points of , that is, In [1], Shioji and Takahashi introduced the following implicit iteration in a Hilbert space where is a sequence in and is a sequence of positive real numbers which diverges to . Under certain restrictions on the sequence , Shioji and Takahashi [1] proved strong convergence of the sequence to a member of . In [2], Shimizu and Takahashi studied the strong convergence of the sequence defined by in a real Hilbert space where is a strongly continuous semigroup of nonexpansive mappings on a closed convex subset of a Banach space and . Using viscosity method, Chen and Song [3] studied the strong convergence of the following iterative method for a nonexpansive semigroup with in a Banach space: where is a contraction. Note however that their iterate at step is constructed through the average of the semigroup over the interval . Suzuki [4] was the first to introduce again in a Hilbert space the following implicit iteration process: for the nonexpansive semigroup case. In 2002, Benavides et al. [5], in a uniformly smooth Banach space, showed that if satisfies an asymptotic regularity condition and fulfills the control conditions , , and , then both the implicit iteration process (1.5) and the explicit iteration process (1.6), converge to a same point of . In 2005, Xu [6] studied the strong convergence of the implicit iteration process (1.2) and (1.5) in a uniformly convex Banach space which admits a weakly sequentially continuous duality mapping.
The Shrinking Projection Method for Solving Variational Inequality Problems and Fixed Point Problems in Banach Spaces
Rabian Wangkeeree,Rattanaporn Wangkeeree
Abstract and Applied Analysis , 2009, DOI: 10.1155/2009/624798
Abstract: We consider a hybrid projection algorithm based on the shrinking projection method for two families of quasi--nonexpansive mappings. We establish strong convergence theorems for approximating the common element of the set of the common fixed points of such two families and the set of solutions of the variational inequality for an inverse-strongly monotone operator in the framework of Banach spaces. As applications, at the end of the paper we first apply our results to consider the problem of finding a zero point of an inverse-strongly monotone operator and we finally utilize our results to study the problem of finding a solution of the complementarity problem. Our results improve and extend the corresponding results announced by recent results.
A General Iterative Method for Variational Inequality Problems, Mixed Equilibrium Problems, and Fixed Point Problems of Strictly Pseudocontractive Mappings in Hilbert Spaces
Wangkeeree Rattanaporn,Wangkeeree Rabian
Fixed Point Theory and Applications , 2009,
Abstract: We introduce an iterative scheme for finding a common element of the set of fixed points of a -strictly pseudocontractive mapping, the set of solutions of the variational inequality for an inverse-strongly monotone mapping, and the set of solutions of the mixed equilibrium problem in a real Hilbert space. Under suitable conditions, some strong convergence theorems for approximating a common element of the above three sets are obtained. As applications, at the end of the paper we first apply our results to study the optimization problem and we next utilize our results to study the problem of finding a common element of the set of fixed points of two families of finitely -strictly pseudocontractive mapping, the set of solutions of the variational inequality, and the set of solutions of the mixed equilibrium problem. The results presented in the paper improve some recent results of Kim and Xu (2005), Yao et al. (2008), Marino et al. (2009), Liu (2009), Plubtieng and Punpaeng (2007), and many others.
A General Composite Algorithms for Solving General Equilibrium Problems and Fixed Point Problems in Hilbert Spaces
Rattanaporn Wangkeeree,Uthai Kamraksa,Rabian Wangkeeree
Abstract and Applied Analysis , 2011, DOI: 10.1155/2011/976412
Abstract: We introduce a general composite algorithm for finding a common element of the set of solutions of a general equilibrium problem and the common fixed point set of a finite family of asymptotically nonexpansive mappings in the framework of Hilbert spaces. Strong convergence of such iterative scheme is obtained which solving some variational inequalities for a strongly monotone and strictly pseudocontractive mapping. Our results extend the corresponding recent results of Yao and Liou (2010).
Viscosity Approximation Methods for Nonexpansive Nonself-Mappings in Hilbert Spaces
Rabian Wangkeeree
International Journal of Mathematics and Mathematical Sciences , 2007, DOI: 10.1155/2007/48648
Abstract: Viscosity approximation methods for nonexpansive nonself-mappings are studied. Let C be a nonempty closed convex subset of Hilbert space H, P a metric projection of H onto C and let T be a nonexpansive nonself-mapping from C into H. For a contraction f on C and {tn}⊆(0,1), let xn be the unique fixed point of the contraction x↦tnf(x)
The General Hybrid Approximation Methods for Nonexpansive Mappings in Banach Spaces
Rabian Wangkeeree
Abstract and Applied Analysis , 2011, DOI: 10.1155/2011/854360
Abstract: We introduce two general hybrid iterative approximation methods (one implicit and one explicit) for finding a fixed point of a nonexpansive mapping which solving the variational inequality generated by two strongly positive bounded linear operators. Strong convergence theorems of the proposed iterative methods are obtained in a reflexive Banach space which admits a weakly continuous duality mapping. The results presented in this paper improve and extend the corresponding results announced by Marino and Xu (2006), Wangkeeree et al. (in press), and Ceng et al. (2009).
An Extragradient Approximation Method for Equilibrium Problems and Fixed Point Problems of a Countable Family of Nonexpansive Mappings
Wangkeeree Rabian
Fixed Point Theory and Applications , 2008,
Abstract: We introduce a new iterative scheme for finding the common element of the set of common fixed points of nonexpansive mappings, the set of solutions of an equilibrium problem, and the set of solutions of the variational inequality. We show that the sequence converges strongly to a common element of the above three sets under some parameters controlling conditions. Moreover, we apply our result to the problem of finding a common fixed point of a countable family of nonexpansive mappings, and the problem of finding a zero of a monotone operator. This main theorem extends a recent result of Yao et al. (2007) and many others.
Strong Convergence of Ces ro Mean Iterations for Nonexpansive Nonself-Mappings in Banach Spaces
Rabian Wangkeeree
Fixed Point Theory and Applications , 2007, DOI: 10.1155/2007/59262
Abstract: Let E be a real uniformly convex Banach space which admits a weakly sequentially continuous duality mapping from E to E*, C a nonempty closed convex subset of E which is also a sunny nonexpansive retract of E, and T:C ¢ ’E a non-expansive nonself-mapping with F(T) ¢ ‰ ¢ …. In this paper, we study the strong convergence of two sequences generated by xn+1= ±nx+(1 ¢ ’ ±n)(1/n+1) ¢ ‘j=0n(PT)jxn and yn+1=(1/n+1) ¢ ‘j=0nP( ±ny+(1 ¢ ’ ±n)(TP)jyn) for all n ¢ ‰ ¥0, where x,x0,y,y0 ¢ C, { ±n} is a real sequence in an interval [0,1], and P is a sunny non-expansive retraction of E onto C. We prove that {xn} and {yn} converge strongly to Qx and Qy, respectively, as n ¢ ’ ¢ , where Q is a sunny non-expansive retraction of C onto F(T). The results presented in this paper generalize, extend, and improve the corresponding results of Matsushita and Kuroiwa and many others.
Strong Convergence of Cesàro Mean Iterations for Nonexpansive Nonself-Mappings in Banach Spaces
Wangkeeree Rabian
Fixed Point Theory and Applications , 2007,
Abstract: Let be a real uniformly convex Banach space which admits a weakly sequentially continuous duality mapping from to , a nonempty closed convex subset of which is also a sunny nonexpansive retract of , and a non-expansive nonself-mapping with . In this paper, we study the strong convergence of two sequences generated by and for all , where , is a real sequence in an interval , and is a sunny non-expansive retraction of onto . We prove that and converge strongly to and , respectively, as , where is a sunny non-expansive retraction of onto . The results presented in this paper generalize, extend, and improve the corresponding results of Matsushita and Kuroiwa and many others.
An Extragradient Approximation Method for Equilibrium Problems and Fixed Point Problems of a Countable Family of Nonexpansive Mappings
Rabian Wangkeeree
Fixed Point Theory and Applications , 2008, DOI: 10.1155/2008/134148
Abstract: We introduce a new iterative scheme for finding the common element of the set of common fixed points of nonexpansive mappings, the set of solutions of an equilibrium problem, and the set of solutions of the variational inequality. We show that the sequence converges strongly to a common element of the above three sets under some parameters controlling conditions. Moreover, we apply our result to the problem of finding a common fixed point of a countable family of nonexpansive mappings, and the problem of finding a zero of a monotone operator. This main theorem extends a recent result of Yao et al. (2007) and many others.
Page 1 /39
Display every page Item


Home
Copyright © 2008-2017 Open Access Library. All rights reserved.