The Meir-Keeler type contractions in extended modular b-metric spaces with an application

Abstract

In this paper, we introduce the notion of a modular $ p $-metric space (an extended modular $ b $-metric space) and establish some fixed point results for $ \alpha $-$ \widehat{\nu} $-Meir-Keeler contractions in this new space. Using these results, we deduce some new fixed point theorems in extended modular metric spaces endowed with a graph and in partially ordered extended modular metric spaces. Also, we develop an important relation between fuzzy-Meir-Keeler and extended fuzzy $ p $-metric with modular $ p $-metric and get certain new fixed point results in triangular fuzzy $ p $-metric spaces. We provide an example and an application to support our results which generalize several well known results in the literature.

Keywords

• fixed point
• extended modular metric space
• $ \alpha $-$ \widehat{\nu} $-meir-keeler contraction
• integral equation
• triangular fuzzy p-metric space