Approximation of Fixed Points of Nonexpansive Mappings and Solutions of Variational Inequalities

Abstract

Let E be a real q -uniformly smooth Banach space with constant [subscript]dq[/subscript] , q≥2 . Let T:E[arrow right]E and G:E[arrow right]E be a nonexpansive map and an η -strongly accretive map which is also κ -Lipschitzian, respectively. Let {[subscript]λn[/subscript] } be a real sequence in [0,1] that satisfies the following condition: C1:lim[subscript]λn[/subscript] =0 and ∑[subscript]λn[/subscript] =∞ . For δ∈(0,[superscript](qη/[subscript]dq[/subscript] [superscript]kq[/superscript] )1/(q-1)[/superscript] ) and σ∈(0,1) , define a sequence {[subscript]xn[/subscript] } iteratively in E by [subscript]x0[/subscript] ∈E , [subscript]xn+1[/subscript] =[superscript]T[subscript]λn+1[/subscript] [/superscript] [subscript]xn[/subscript] =(1-σ)[subscript]xn[/subscript] +σ[T[subscript]xn[/subscript] -δ[subscript]λn+1[/subscript] G(T[subscript]xn[/subscript] )] , n≥0 . Then, {[subscript]xn[/subscript] } converges strongly to the unique solution [superscript]x*[/superscript] of the variational inequality problem VI(G,K) (search for [superscript]x*[/superscript] ∈K such that ...G[superscript]x*[/superscript] ,[subscript]jq[/subscript] (y-[superscript]x*[/superscript] )...≥0 for all y∈K) , where K:=Fix(T)={x∈E:Tx=x}≠∅ . A convergence theorem related to finite family of nonexpansive maps is also proved.