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

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.