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

Abdolsattar Gholidahneh; Shaban Sedghi; Ozgur Ege; Zoran D. Mitrovic; Manuel de la Sen; Abdolsattar Gholidahneh; Shaban Sedghi; Ozgur Ege; Zoran D. Mitrovic; Manuel de la Sen

doi:10.3934/math.2021107

AIMS Mathematics

2021, Volume 6, Issue 2: 1781-1799. doi: 10.3934/math.2021107

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Research article

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

1.
Department of Mathematics, Qaemshahr Branch, Islamic Azad University, Qaemshahr, Iran
2.
Department of Mathematics, Faculty of Science, Ege University, Bornova, Izmir 35100, Turkey
3.
University of Banja Luka, Faculty of Electrical Engineering, Patre 5, Banja Luka 78000, Bosnia and Herzegovina
4.
Institute of Research and Development of Processes University of the Basque Country, Leioa 48940, Spain

Received: 13 September 2020 Accepted: 22 November 2020 Published: 30 November 2020
MSC : 47H10, 54H25

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.
- fixed point,
- extended modular metric space,
- $ \alpha $-$ \widehat{\nu} $-Meir-Keeler contraction,
- integral equation,
- triangular fuzzy $ p $-metric space
Citation: Abdolsattar Gholidahneh, Shaban Sedghi, Ozgur Ege, Zoran D. Mitrovic, Manuel de la Sen. The Meir-Keeler type contractions in extended modular $ b $-metric spaces with an application[J]. AIMS Mathematics, 2021, 6(2): 1781-1799. doi: 10.3934/math.2021107

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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.

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