Convergence Theorem for a Family of Generalized Asymptotically Nonexpansive Semigroup in Banach Spaces

Abstract EN

Let E be a real reflexive and strictly convex Banach space with a uniformly Gâteaux differentiable norm.

Let ?={T(t):t≥0} be a family of uniformly asymptotically regular generalized asymptotically nonexpansive semigroup of E, with functions u,v:[0,∞)→[0,∞).

Let F:=F(?)=∩t≥0F(T(t))≠∅ and f:K→K be a weakly contractive map.

For some positive real numbers λ and δ satisfying δ+λ>1, let G:E→E be a δ-strongly accretive and λ-strictly pseudocontractive map.

Let {tn} be an increasing sequence in [0,∞) with limn→∞tn=∞, and let {αn} and {βn} be sequences in (0,1] satisfying some conditions.

Strong convergence of a viscosity iterative sequence to common fixed points of the family ? of uniformly asymptotically regular asymptotically nonexpansive semigroup, which also solves the variational inequality 〈(G-γf)p,j(p-x)〉≤0, for all x∈F, is proved in a framework of a real Banach space.