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

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.