Table of Contents
ISRN Mathematical Analysis
Volume 2011 (2011), Article ID 684158, 14 pages
http://dx.doi.org/10.5402/2011/684158
Research Article

Common Fixed Points Approximation for Asymptotically Nonexpansive Semigroup in Banach Spaces

Department of Mathematical Sciences, Bayero University, P.M.B. 3011, Kano, Nigeria

Received 3 May 2011; Accepted 9 June 2011

Academic Editors: J. S. Jung and C. Zhu

Copyright © 2011 Bashir Ali. 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 Banach space satisfying local uniform Opial's condition, whose duality map is weakly sequentially continuous. Let π’₯ ∢ = { 𝑇 ( 𝑑 ) 𝑑 β‰₯ 0 } be a uniformly asymptotically regular family of asymptotically nonexpansive semigroup of 𝐸 with function π‘˜ ∢ [ 0 , ∞ ) β†’ [ 0 , ∞ ) . Let β‹‚ β„± ∢ = 𝑑 β‰₯ 0 𝐹 ( 𝑇 ( 𝑑 ) ) β‰  βˆ… and 𝑓 ∢ 𝐸 β†’ 𝐸 be weakly contractive map. Let 𝐺 ∢ 𝐸 β†’ 𝐸 be 𝛿 -strongly accretive and πœ† -strictly pseudocontractive map with 𝛿 + πœ† > 1 . Let { 𝑑 𝑛 } be an increasing sequence in [ 0 , ∞ ) and let { 𝛼 𝑛 } and { 𝛽 𝑛 } be sequences in [ 0 , 1 ] satisfying some conditions. For some positive real number 𝛾 appropriately chosen, let { π‘₯ 𝑛 } be a sequence defined by π‘₯ 0 ∈ 𝐸 , π‘₯ 𝑛 + 1 = 𝛽 𝑛 π‘₯ 𝑛 + ( 1 βˆ’ 𝛽 𝑛 ) 𝑦 𝑛 ,   𝑦 𝑛 = ( 𝐼 βˆ’ 𝛼 𝑛 𝐺 ) 𝑇 ( 𝑑 𝑛 ) π‘₯ 𝑛 + 𝛼 𝑛 𝛾 𝑓 ( π‘₯ 𝑛 ) ,   𝑛 β‰₯ 0 . It is proved that { π‘₯ 𝑛 } converges strongly to a common fixed point π‘ž of the family π’₯ which is also the unique solution of the variational inequality ⟨ ( 𝐺 βˆ’ 𝛾 𝑓 ) π‘ž , 𝑗 ( π‘ž βˆ’ π‘₯ ) ⟩ β‰₯ 0 , f o r a l l π‘₯ ∈ β„± .