Fixed Point Theory and Applications
Volume 2010 (2010), Article ID 618767, 9 pages
doi:10.1155/2010/618767
Research Article

Ishikawa Iterative Process for a Pair of Single-valued and Multivalued Nonexpansive Mappings in Banach Spaces

K. Sokhuma1 and A. Kaewkhao2

1Department of Mathematics, Faculty of Science, Burapha University, Chonburi 20131, Thailand
2Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand

Received 8 August 2010; Accepted 24 September 2010

Academic Editor: T. D. Benavides

Copyright © 2010 K. Sokhuma and A. Kaewkhao. 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 nonempty compact convex subset of a uniformly convex Banach space 𝑋 , and let 𝑡 ∶ 𝐸 → 𝐸 and 𝑇 ∶ 𝐸 → 𝐾 𝐶 ( 𝐸 ) be a single-valued nonexpansive mapping and a multivalued nonexpansive mapping, respectively. Assume in addition that F i x ( 𝑡 ) ∩ F i x ( 𝑇 ) ≠ ∅ and 𝑇 𝑤 = { 𝑤 } for all 𝑤 ∈ F i x ( 𝑡 ) ∩ F i x ( 𝑇 ) . We prove that the sequence of the modified Ishikawa iteration method generated from an arbitrary 𝑥 0 ∈ 𝐸 by 𝑦 𝑛 = ( 1 − 𝛽 𝑛 ) 𝑥 𝑛 + 𝛽 𝑛 𝑧 𝑛 , 𝑥 𝑛 + 1 = ( 1 − 𝛼 𝑛 ) 𝑥 𝑛 + 𝛼 𝑛 𝑡 𝑦 𝑛 , where 𝑧 𝑛 ∈ 𝑇 𝑥 𝑛 and { 𝛼 𝑛 } , { 𝛽 𝑛 } are sequences of positive numbers satisfying 0 < 𝑎 ≤ 𝛼 𝑛 , 𝛽 𝑛 ≤ 𝑏 < 1 , converges strongly to a common fixed point of 𝑡 and 𝑇 ; that is, there exists 𝑥 ∈ 𝐸 such that 𝑥 = 𝑡 𝑥 ∈ 𝑇 𝑥 .