Fixed Point Theory and Applications
Volume 2005 (2005), Issue 1, Pages 67-77
doi:10.1155/FPTA.2005.67
Convergence theorems for fixed points of demicontinuous pseudocontractive mappings
The Abdus Salam International Centre for Theoretical Physics, Trieste 34014, Italy
Received 26 August 2004
Copyright © 2005 C. E. Chidume and H. Zegeye. 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 D be an open subset of a real uniformly smooth Banach space E. Suppose T:D¯→E is a demicontinuous pseudocontractive mapping satisfying an appropriate condition, where D¯ denotes the closure of D. Then, it is proved that (i) D¯⊆ℛ(I+r(I−T))
for every r>0; (ii) for a given y0∈D, there exists a unique path t→yt∈D¯, t∈[0,1], satisfying yt:=tTyt+(1−t)y0. Moreover, if F(T)≠∅ or there exists y0∈D such that the set K:={y∈D:Ty=λy+(1−λ)y0 for λ>1} is bounded, then it is proved that, as t→1−, the path {yt} converges strongly to a fixed point of T. Furthermore, explicit iteration procedures with bounded error terms are proved to converge strongly to a fixed point of T.