Recent advancements in $ M $-cone metric space over Banach algebra endowed with binary relation

Muhammad Tariq; Saber Mansour; Abdullah Assiry; Jalil Ur Rehman; Muhammad Tariq; Saber Mansour; Abdullah Assiry; Jalil Ur Rehman

doi:10.3934/math.2025582

AIMS Mathematics

2025, Volume 10, Issue 6: 12935-12955. doi: 10.3934/math.2025582

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Recent advancements in $ M $-cone metric space over Banach algebra endowed with binary relation

1.
Department of Mathematics, MY University, Islamabad 44000, Pakistan
2.
Department of Mathematics, College of Sciences, Umm Al-Qura University, Mecca 21955, Saudi Arabia

Received: 25 January 2025 Revised: 01 May 2025 Accepted: 20 May 2025 Published: 05 June 2025
MSC : 36A07, 48H11, 58H25

The objectives of this paper are three steps: first, to generalize the idea of complete $ M $-cone metric spaces over Banach algebra; second, to present a new topological structure utilizing the concept introduced by Fernandez et al. (Fixed point results in M-cone metric space over Banach algebra with an application, Filomat, 36 (2022), 5547–5562.); and third, to explore the idea of Banach algebra type relational theoretic contractions and cyclic Banach algebra type contractions in $ M $-cone metric spaces, establishing several fixed point results for these contractions. To illustrate the discussed concepts and results, several examples are provided. As an application, we discuss a solution of the nonlinear integral equation based on the main results.
- fixed point,
- $ M $-cone metric spaces over Banach algebra,
- relation-theoretic contractions,
- nonlinear integral equation
Citation: Muhammad Tariq, Saber Mansour, Abdullah Assiry, Jalil Ur Rehman. Recent advancements in $ M $-cone metric space over Banach algebra endowed with binary relation[J]. AIMS Mathematics, 2025, 10(6): 12935-12955. doi: 10.3934/math.2025582

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Abstract

The objectives of this paper are three steps: first, to generalize the idea of complete $ M $-cone metric spaces over Banach algebra; second, to present a new topological structure utilizing the concept introduced by Fernandez et al. (Fixed point results in M-cone metric space over Banach algebra with an application, Filomat, 36 (2022), 5547–5562.); and third, to explore the idea of Banach algebra type relational theoretic contractions and cyclic Banach algebra type contractions in $ M $-cone metric spaces, establishing several fixed point results for these contractions. To illustrate the discussed concepts and results, several examples are provided. As an application, we discuss a solution of the nonlinear integral equation based on the main results.

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