The purpose of the paper is to study the existence and approximation of fixed points of firmly nonexpansive mappings in Banach spaces. The following definition is used. Let be a smooth Banach space, be a nonempty closed convex subset of , and be the normalized duality mapping from into . The mapping is of firmly nonexpansive type if, for all ,
A fixed point theorem for firmly nonexpansive-type mappings in Banach spaces is obtained. It is shown that every nonexpansive-type mapping which has a fixed point is strongly relatively nonexpansive. A weak convergence theorem is presented. The obtained results are applied to the proximal point algorithm for monotone operators satisfying the range condition in Banach spaces.