Viscosity approximation methods for pseudocontractive mappings in Banach spaces (Q870202)

Let \(K\) be a closed convex subset of a Banach space \(E\) and let \(T : K \to E\) be a continuous weakly inward pseudocontractive mapping. Then for \(t\in(0,1)\), there exists a sequence \(\{y_t\} \subset K\) satisfying \(y_t = (1 - t)f(y_t) + tT(y_t)\), where \(f \in \Pi_K := \{f : K\to K,\text{ a contraction with a suitable contractive constant}\}\). Suppose further that \(F(T) \neq \emptyset\) 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.