Convergence Theorem for a Family of Generalized Asymptotically Nonexpansive Semigroup in Banach Spaces. (22nd July 2012) - Ali, Bashir
Abstract : Let E be a real reflexive and strictly convex Banach space with a uniformly Gâteaux differentiable norm. Let 𝔍 = { 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 ( 𝔍 ) = ∩ t ≥ 0 F ( 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 { t n } be an increasing sequence in [ 0, ∞ ) with l i m n → ∞ t n = ∞, and let { α n } and { β n } be sequences in ( 0, 1 ] satisfying some conditions. Strong convergence of a viscosity iterative sequence to common fixed points of the family 𝔍 of uniformly asymptotically regular asymptotically nonexpansive semigroup, which also solves the variational inequality 〈 ( G - γ f ) p, j ( p - x ) 〉 ≤ 0, for all x ∈ F, is proved in a framework of a real Banach space.