Viscosity approximation methods for pseudocontractive mappings in Banach spaces

Let K be a closed convex subset of a Banach space E and let T : K → E be a continuous weakly inward pseudocontractive mapping. Then for t ∈ (0, 1), there exists a sequence {y_t} ⊂ K satisfying y_t = (1 - t)f(y_t) + tT(y_t), where f ∈ Π_K {colon equals} {f : K → K, a contraction with a suitable contractive constant}. Suppose further that F(T) ≠ ∅ and E is reflexive and strictly convex which has uniformly Gâteaux differentiable norm. Then it is proved that {y_t} converges strongly to a fixed point of T which is also a solution of certain variational inequality. Moreover, an explicit iteration process which converges strongly to a fixed point of T and hence to a solution of certain variational inequality is constructed provided that T is Lipschitzian.