Journal of Applied Mathematics
Volume 2012 (2012), Article ID 341953, 11 pages
http://dx.doi.org/10.1155/2012/341953
Research Article

An Iterative Algorithm on Approximating Fixed Points of Pseudocontractive Mappings

Youli Yu

School of Mathematics and Information Engineering, Taizhou University, Linhai 317000, China

Received 4 September 2011; Accepted 16 September 2011

Academic Editor: Yonghong Yao

Copyright © 2012 Youli Yu. 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 with a uniformly Gâteaux differentiable norm. Let K be a nonempty bounded closed convex subset of E, and every nonempty closed convex bounded subset of K has the fixed point property for non-expansive self-mappings. Let 𝑓 ∶ 𝐾 → 𝐾 a contractive mapping and 𝑇 ∶ 𝐾 → 𝐾 be a uniformly continuous pseudocontractive mapping with 𝐹 ( 𝑇 ) ≠ ∅ . Let { 𝜆 𝑛 } ⊂ ( 0 , 1 / 2 ) be a sequence satisfying the following conditions: (i) l i m 𝑛 → ∞ 𝜆 𝑛 = 0 ; (ii) ∑ ∞ 𝑛 = 0 𝜆 𝑛 = ∞ . Define the sequence { 𝑥 𝑛 } in K by 𝑥 0 ∈ 𝐾 , 𝑥 𝑛 + 1 = 𝜆 𝑛 𝑓 ( 𝑥 𝑛 ) + ( 1 − 2 𝜆 𝑛 ) 𝑥 𝑛 + 𝜆 𝑛 𝑇 𝑥 𝑛 , for all 𝑛 ≥ 0 . Under some appropriate assumptions, we prove that the sequence { 𝑥 𝑛 } converges strongly to a fixed point 𝑝 ∈ 𝐹 ( 𝑇 ) which is the unique solution of the following variational inequality: ⟨ 𝑓 ( 𝑝 ) − 𝑝 , 𝑗 ( 𝑧 − 𝑝 ) ⟩ ≤ 0 , for all 𝑧 ∈ 𝐹 ( 𝑇 ) .