Generalized Acyclic Contractions, Acyclic Contractions, and Reverse Acyclic Contractions

This article introduces a notion of a reverse acyclic contraction which is defined on the union of two subsets A and B of a metric space, and explores the existence of a fixed point x of the mapping in A and a fixed point y of the mapping in B such that in the setting of complete metric spaces. Further, we introduce a new class of generalized acyclic contractions that includes the class of acyclic contractions as well as the class of reverse acyclic contractions. Indeed, for the class of generalized acyclic contractions which are defined on the union of two subsets A and B of a metric space, we investigate the existence of a fixed point x of the mapping in A and a fixed point y of the mapping in B such that , and deduce some fixed point theorems for acyclic contractions and reverse acyclic contractions.