Journal of Inequalities and Applications 
Volume 2008 (2008), Article ID 284345, 12 pages
doi:10.1155/2008/284345
Research Article

Approximation of Fixed Points of Nonexpansive Mappings and Solutions of Variational Inequalities

C. E. Chidume,1 C. O. Chidume,2 and Bashir Ali3

 1The Abdus Salam International Centre for Theoretical Physics, 34014 Trieste, Italy
2Department of Mathematics and Statistics, College of Sciences and Mathematics, Auburn University, Auburn, AL 36849, USA
3Department of Mathematical Sciences, Bayero University, 3011 Kano, Nigeria
Received 3 July 2007; Accepted 17 October 2007

Recommended by Siegfried Carl

Abstract

Let E be a real q-uniformly smooth Banach space with constant dq, q≥2. Let T:E→E and G:E→E be a nonexpansive map and an η-strongly accretive map which is also κ-Lipschitzian, respectively. Let {λn} be a real sequence in [0,1] that satisfies the following condition: C1:limλn=0 and ∑λn=∞. For δ∈(0,(qη/dqkq)1/(q−1)) and σ∈(0,1), define a sequence {xn} iteratively in E by x0∈E, xn+1=Tλn+1xn=(1−σ)xn+σ[Txn−δλn+1G(Txn)], n≥0. Then, {xn} converges strongly to the unique solution x* of the variational inequality problem VI(G,K) (search for x*∈K such that 〈Gx*,jq(y−x*)〉≥0 for all y∈K), where K:=Fix(T)={x∈E:Tx=x}≠∅. A convergence theorem related to finite family of nonexpansive maps is also proved.