International Journal of Mathematics and Mathematical Sciences
Volume 2012 (2012), Article ID 986426, 16 pages
http://dx.doi.org/10.1155/2012/986426
Research Article

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

Bashir Ali1 and G. C. Ugwunnadi2

1Department of Mathematical Sciences, Bayero University, P.M.B. 3011 Kano, Nigeria
2Department of Mathematics, University of Nigeria, Nsukka, Nigeria

Received 15 March 2012; Revised 8 June 2012; Accepted 8 June 2012

Academic Editor: Ram U. Verma

Copyright © 2012 Bashir Ali and G. C. Ugwunnadi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.