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 of D. Then, it is proved that (i) D̄ ⊆ ℛ (I + r (I-T)) for every r > 0; (ii) for a given y₀ ∈ D, there exists a unique path t → y_t ∈ D̄, t ∈ [0, 1], satisfying y_t := tTy_t + (1-t) y₀. Moreover, if F (T) ≠ ∅ or there exists y₀ ∈ D such that the set K := {y ∈ D: T y = λ y + (1-λ) y₀ for λ >1} is bounded, then it is proved that, as t → 1-, the path {y_t} 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.