Convergence Theorems for Fixed Points of Demicontinuous Pseudocontractive Mappings

Let D be an open subset of a real uniformly smooth Banach space E. Suppose T : D̄ → E is a demicontinuous pseudocontractive mapping satisfying an appropriate condition, where D̄ denotes the closure ofD. Then, it is proved that (i) D̄ ⊆ (I + r(I −T)) for every r > 0; (ii) for a given y0 ∈ D, there exists a unique path t → yt ∈ D̄, t ∈ [0,1], satisfying yt := tT yt + (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 .