Rattanaporn Wangkeeree

The general iterative methods for nonexpansive mappings in Banach spaces

In this paper, we introduce a general iterative approximation method for finding a common fixed point of a countable family of nonexpansive mappings which is a unique solution of some variational inequality. We prove the strong convergence theorems of such iterative scheme in a reflexive Banach space which admits a weakly continuous duality mapping. As applications, at the end of the paper, we apply our results to the problem of finding a zero of an accretive operator. The main result extends various results existing in the current literature.

A General Iterative Method for Variational Inequality Problems, Mixed Equilibrium Problems, and Fixed Point Problems of Strictly Pseudocontractive Mappings in Hilbert Spaces

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.

Continuity of the solution mappings to parametric generalized vector equilibrium problems

Abstract The aim of this paper is to establish the continuity of the efficient solution mappings to a parametric generalized strong vector equilibrium problem, by using the Holder relation. Our result extends and improves some recent results in the references therein.

The Shrinking Projection Method for Solving Variational Inequality Problems and Fixed Point Problems in Banach Spaces

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 VISCOSITY APPROXIMATION METHOD OF FIXED POINT SOLUTIONS OF VARIATIONAL INEQUALITIES FOR NONEXPANSIVE SEMIGROUPS IN HILBERT SPACES

Abstract. Let H be a real Hilbert space and S = fT ( s ) : 0 • s 0. Let 0 < ° < ° „fi . It is proved that the sequences fx t g and fx n g generated by the iterative method x t = t°f ( x t ) + ( I i tA )1 ‚ t Z ‚ t 0 T ( s ) x t ds; and x n +1 = fi n °f ( x n ) + ( I i fi n A )1 t n Z t n 0 T ( s ) x n ds; where ftg;ffi n g ‰ (0 ; 1) and f‚ t g;ft n g are positive real divergent se-quences, converges strongly to a common flxed point ~ x 2 F ( S ) whichsolves the variational inequality h ( °f i A )~ x;x i ~x i • 0 for x 2 F ( S ). 1. IntroductionLet H be a real Hilbert space, and let C be a nonempty closed convex subsetof H . A mapping T of C into itself is said to be nonexpansive if kTxiTyk •kxiyk for each x;y 2 C: We denote F ( T ) the set of flxed points of T . A family S = fT ( s ) : 0 • s < 1g of mapping of

Strong Convergence Theorems of the General Iterative Methods for Nonexpansive Semigroups in Banach Spaces

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.

A General Composite Algorithms for Solving General Equilibrium Problems and Fixed Point Problems in Hilbert Spaces

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).