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

Let be a real reflexive and strictly convex Banach space with a uniformly Gâteaux differentiable norm. Let ={()∶≥0} be a family of uniformly asymptotically regular generalized asymptotically nonexpansive semigroup of , with functions ,∶[0,∞)→[0,∞). Let ∶=()=∩≥0(())≠∅ and ∶→ be a weakly contractive map. For some positive real numbers and satisfying +>1, let ∶→ be a -strongly accretive and -strictly pseudocontractive map. Let {} be an increasing sequence in [0,∞) with lim→∞=∞, and let {} and {} 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 ⟨(−),(−)⟩≤0, for all ∈, is proved in a framework of a real Banach space