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 0$"> r > 0 ; (ii) for a given y 0 ∈ D , there exists a unique path t → y t ∈ D ¯ , t ∈ [ 0 , 1 ] , satisfying y t : = t T y t + ( 1 − t ) y 0 . Moreover, if F ( T ) ≠ ∅ or there exists y 0 ∈ D such that the set 1\}$"> K : = { y ∈ D : T y = λ y + ( 1 − λ ) y 0  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 .