On convergent sequences and fixed point theorems inD-metric spaces

Abstract

Examples of completeD-metric spaces are given in which every convergent sequence has at most two limits and in which there are convergent sequences with exactly two limits. Also an example of a completeD-metric space having a convergent sequence with infinitely many limits is given and, using the example, several fixed point theorems inD-metric spaces are shown to be false. Modifications of some of these theorems and their generalizations are obtained either by imposing restrictions on the number of limits of certain convergent sequences in the space or by assuming the sequential continuity of theD-metric in any two variables and the theorems so obtained are illustrated by means of examples.