Convergence Theorem for a Family of Generalized Asymptotically Nonexpansive Semigroup in Banach Spaces - Bashir Ali; G. C. Ugwunnadi (International Journal of Mathematics and Mathematical Sciences)
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≥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 𝔍 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.