The paper gives convergence theorems for approximating fixed points of multivalued nonexpansive nonself mappings by means of viscosity type methods. The main result (Theorem 1) goes as follows. Let be a uniformly convex Banach space with uniformly Gâteaux differentiable norm, be a nonempty closed convex subset of and be a nonself nonexpansive multivalued mapping (here, denotes the set of all nonempty compact subsets of ). Suppose that is a nonexpansive retract of and has only strict fixed points, that is, for all fixed points of . For each and , consider the multivalued contraction defined by
and assume that has a fixed point . Then has a fixed point if and only if remains bounded as and, in this case, converges strongly as to a fixed point of . Several other related results are obtained in the same way or as corollaries.