Convergence to Common Fixed Point for Generalized Asymptotically Nonexpansive Semigroup in Banach Spaces

摘要

LetKbe a nonempty closed convex subset of a reflexive and strictly convex BanachspaceEwith a uniformlyGâteauxdifferentiable norm,ℱ={T(h):h≥0}a generalized asymptoticallynonexpansive self-mapping semigroup ofK, andf:K→Ka fixed contractive mappingwith contractive coefficientβ∈(0,1). We prove that the following implicit and modified implicitviscosity iterative schemes{xn}defined byxn=αnf(xn)+(1−αn)T(tn)xnandxn=αnyn+(1−αn)T(tn)xn,yn=βnf(xn−1)+(1−βn)xn−1strongly converge top∈Fasn→∞andpis the unique solution to the following variationalinequality:〈f(p)−p,j(y−p)〉≤0for ally∈F.