International Journal of Mathematics and Mathematical Sciences
Volume 2011 (2011), Article ID 643740, 21 pages
http://dx.doi.org/10.1155/2011/643740
Research Article

Strong Convergence Theorems of the General Iterative Methods for Nonexpansive Semigroups in Banach Spaces

Rattanaporn Wangkeeree1,2

1Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand
2Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand

Received 4 February 2011; Accepted 22 March 2011

Academic Editor: Yonghong Yao

Copyright © 2011 Rattanaporn Wangkeeree. 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 Banach space which admits a weakly sequentially continuous duality mapping from 𝐸 to 𝐸 ∗ . Let 𝒮 = { 𝑇 ( 𝑠 ) ∶ 0 ≤ 𝑠 < ∞ } be a nonexpansive semigroup on 𝐸 such that ⋂ F i x ( 𝒮 ) ∶ = 𝑡 ≥ 0 F i x ( 𝑇 ( 𝑡 ) ) ≠ ∅ , and 𝑓 is a contraction on 𝐸 with coefficient 0 < 𝛼 < 1 . Let 𝐹 be 𝛿 -strongly accretive and 𝜆 -strictly pseudocontractive with 𝛿 + 𝜆 > 1 and 𝛾 a positive real number such that √ 𝛾 < 1 / 𝛼 ( 1 − 1 − 𝛿 / 𝜆 ) . When the sequences of real numbers { 𝛼 𝑛 } and { 𝑡 𝑛 } satisfy some appropriate conditions, the three iterative processes given as follows: 𝑥 𝑛 + 1 = 𝛼 𝑛 𝛾 𝑓 ( 𝑥 𝑛 ) + ( 𝐼 − 𝛼 𝑛 𝐹 ) 𝑇 ( 𝑡 𝑛 ) 𝑥 𝑛 , 𝑛 ≥ 0 , 𝑦 𝑛 + 1 = 𝛼 𝑛 𝛾 𝑓 ( 𝑇 ( 𝑡 𝑛 ) 𝑦 𝑛 ) + ( 𝐼 − 𝛼 𝑛 𝐹 ) 𝑇 ( 𝑡 𝑛 ) 𝑦 𝑛 , 𝑛 ≥ 0 , and 𝑧 𝑛 + 1 = 𝑇 ( 𝑡 𝑛 ) ( 𝛼 𝑛 𝛾 𝑓 ( 𝑧 𝑛 ) + ( 𝐼 − 𝛼 𝑛 𝐹 ) 𝑧 𝑛 ) , 𝑛 ≥ 0 converge strongly to ̃ 𝑥 , where ̃ 𝑥 is the unique solution in F i x ( 𝒮 ) of the variational inequality ⟨ ( 𝐹 − 𝛾 𝑓 ) ̃ 𝑥 , 𝑗 ( 𝑥 − ̃ 𝑥 ) ⟩ ≥ 0 , 𝑥 ∈ F i x ( 𝒮 ) . Our results extend and improve corresponding ones of Li et al. (2009) Chen and He (2007), and many others.