Viscosity Approximation Methods For The Common Fixed Points Of An Infinite Family Of Asymptotically Nonexpansive Mappings

The aim of this paper is to study common fixed points for an infinite family of asymptotically nonexpansive mappings and a system of the infinite variational inequal-ities for inverse-strongly accretive mappings.The first result is introduced to a triple composite modified iteration{xn} for nonexpansive mappings in a Banach space with a uniformly Gateaux differentiable norm.The iteration {xn}is defined by and we prove the strong convergence of it. The second result is introduced a new iteration{xn} Let{Ti}∞n=1be an infinite family of asymptotically nonexpansive mapping and uniform Lipschitzian from K into itself,F=∩∞i=1F(Ti)≠Φ.{xn}convergences strongly to x∈F,if and only if for any i,limnâ†'∞||Tixn-xn||=0holdsï¼›And x is also a solution of the following variational inequality:<x-f(x),J(x-p)>≤0,(?)p∈F.In the uniformly convex and q-uniformly smooth Banach space E with the best smooth constant K, the third result is that we define a sequence{xn} by We introducethe sequence convergences strongly to x=QFu, and (x, y) is a solution of the of the following variational inequality: where y=Qc(x-μnBx). The results in the present paper extend some resent results of other authors[24-32].In the first chapter, we introduce some related research backgrounds, some rele-vant knowledge, some basic concepts and notations.In the second chapter, we prove strong convergence of the triple composite modi-fied iteration for nonexpansive mappings.In the third chapter, we discuss on viscosity approximation methods of common fixed points for an infinite family of asymptotically nonexpansive mappings.In the fourth chapter, we study iterative approximation to the common solutions of an infinite variational inequality system for inverse-strongly accretive mappings.