Quasi-Cauchy sequences on asymmetric metric spaces

Abstract

In recent years, quasi-Cauchy sequences have been studied by many authors in the real line and in metric spaces. In this paper, we investigate the concepts of quasi Cauchyness of sequences, ward compactness and ward continuity in asymmetric metric spaces. We prove that forward totally boundedness
coincides with upward compactness, backward totally boundedness coincides with downward compactness, an upward continuous function on a subset E of an asymmetric metric space X to an asymmetric metric space Y is forward continuous under the condition that forward convergence implies backward
convergence on X. We also prove some other interesting theorems.