# Fixed point theorems by combining Jleli and Samets, and Branciaris inequalities

Volume 9, Issue 6, pp 3822--3849 Publication Date: June 10, 2016
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### Authors

Antonio Francisco Roldán López de Hierroa - Department of Quantitative Methods for Economics and Business, University of Granada, Granada, Spain. - PAIDI Research Group FQM-268, University of Jaén, Jaén, Spain. Naseer Shahzad - Operator Theory and Applications Research Group, Department of Mathematics, King Abdulaziz University, P. O. Box 80203, Jeddah, 21589, Saudi Arabia.

### Abstract

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.

### Keywords

• Generalized metric space
• Branciari metric space
• fixed point
• contractive mapping.

•  47H10
•  54H25

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