Existence of Periodic Fixed Point Theorems in the Setting of Generalized Quasi-Metric Spaces

Very recently, Lin et al. [1] introduced the notion of generalized quasi-metric inspired from the notion of generalized metric, defined by Branciari [2]. It is a very well-known fact that the concept of generalizedmetric can be derived from the definition ofmetric by replacing the triangle inequality with a weaker condition, namely, quadrilateral inequality. In spite of the analogy between the definitions ofmetric and generalized metric, the topological structure of these spaces is completely different. It was proved that the topologies of these two spaces are incomparable [3]. In what follows that we recall the basic definitions and results on the topics for the sake of completeness.Throughout the paper, the symbolsR,N, andN 0 denote the real numbers, the natural numbers, and the positive integers, respectively. A quite natural generalization of the notion of a metric was introduced by Branciari [2] in 2000 by replacing the triangle inequality assumption of a metric with a weaker condition, quadrilateral inequality.