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

Let ð ¸ be a real reflexive and strictly convex Banach space with a uniformly Gâteaux differentiable norm. Let ð ” = { 𠑇 ( ð ‘¡ ) ∶ ð ‘¡ ≥ 0 } be a family of uniformly asymptotically regular generalized asymptotically nonexpansive semigroup of ð ¸ , with functions ð ‘¢ , ð ‘£ ∶ [ 0 , ∞ ) → [ 0 , ∞ ) . Let ð ¹ âˆ¶ = ð ¹ ( ð ” ) = ∩ ð ‘¡ ≥ 0 ð ¹ ( 𠑇 ( ð ‘¡ ) ) ≠∅ and ð ‘“ ∶ ð ¾ â†’ ð ¾ be a weakly contractive map. For some positive real numbers 𠜆 and ð ›¿ satisfying ð ›¿ + 𠜆 > 1 , let ð º âˆ¶ ð ¸ â†’ ð ¸ be a ð ›¿ -strongly accretive and 𠜆 -strictly pseudocontractive map. Let { ð ‘¡ ð ‘› } be an increasing sequence in [ 0 , ∞ ) with l i m ð ‘› → ∞ ð ‘¡ ð ‘› = ∞ , and let { ð ›¼ ð ‘› } and { ð ›½ ð ‘› } be sequences in ( 0 , 1 ] satisfying some conditions. Strong convergence of a viscosity iterative sequence to common fixed points of the family ð ” of uniformly asymptotically regular asymptotically nonexpansive semigroup, which also solves the variational inequality ⟨ ( ð º âˆ’ ð ›¾ ð ‘“ ) ð ‘ , ð ‘— ( ð ‘ âˆ’ ð ‘¥ ) ⟩ ≤ 0 , for all ð ‘¥ ∈ ð ¹ , is proved in a framework of a real Banach space.