VISCOSITY APPROXIMATION METHODS FOR MULTIVALUED MAPPINGS IN BANACH SPACES

Let K be a nonempty closed and convex subset of a real reflexive Banach space X that has weakly sequentially continuous duality mapping J. Let T : K -> K be a multivalued non-expansive non-self-mapping satisfying the weakly inwardness condition as well as the condition T(y) = {y} for any y is an element of F(T) and such that for a contraction f : K -> K and any t is an element of (0, 1), there exists x(t) is an element of K satisfying x(t) is an element of tf (x(t)) + (1 - t)Tx(t). Then it is proved that {xt} subset of K converges strongly to a fixed point of T, which is also a solution of certain variational inequality. Moreover, the convergence of two explicit methods are also investigated.