Generalized Asymptotic Pointwise Contractions and Nonexpansive Mappings Involving Orbits

Four recent papers 1–4 present simple and elegant proofs of fixed point results for pointwise contractions, asymptotic pointwise contractions, and asymptotic nonexpansive mappings. Kirk and Xu 1 study these mappings in the context of weakly compact convex subsets of Banach spaces, respectively, in uniformly convex Banach spaces. Hussain and Khamsi 2 consider these problems in the framework of metric spaces and CAT 0 spaces. In 3 , the authors prove coincidence results for asymptotic pointwise nonexpansive mappings. Espı́nola et al. 4 examine the existence of fixed points and convergence of iterates for asymptotic pointwise contractions in uniformly convex metric spaces. In this paper we do not consider more general spaces, but instead we formulate less restrictive conditions for the mappings and show that the conclusions of the theorems still stand even in such weaker settings.