Fixed point theorems by combining Jleli and Samet s, and Branciari s inequalities

The aim of this paper is to introduce a new class of generalized metric spaces (called RS-spaces) that unify and extend, at the same time, Branciari’s generalized metric spaces and Jleli and Samet’s generalized metric spaces. Both families of spaces seen to be different in nature: on the one hand, Branciari’s spaces are endowed with a rectangular inequality and their metrics are finite valued, but they can contain convergent sequences with two different limits, or convergent sequences that are not Cauchy; on the other hand, in Jleli and Samet’s spaces, although the limit of a convergent sequence is unique, they are not endowed with a triangular inequality and we can found two points at infinite distance. However, we overcome such drawbacks and we illustrate that many abstract metric spaces (like dislocated metric spaces, b-metric spaces, rectangular metric spaces, modular metric spaces, among others) can be seen as particular cases of RS-spaces. In order to show its great applicability, we present some fixed point theorems in the setting of RS-spaces that extend well-known results in this line of research.