A fresh look at Volterra integral equations: a fixed point approach in $ \mathfrak{F} $-bipolar metric spaces

Mohammed H. Alharbi; Jamshaid Ahmad; Mohammed H. Alharbi; Jamshaid Ahmad

doi:10.3934/math.2025409

AIMS Mathematics

2025, Volume 10, Issue 4: 8926-8945. doi: 10.3934/math.2025409

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Research article

A fresh look at Volterra integral equations: a fixed point approach in $ \mathfrak{F} $-bipolar metric spaces

Mohammed H. Alharbi ,
Jamshaid Ahmad ^,

Department of Mathematics and Statistics, College of Science, University of Jeddah, Jeddah 23218, Saudi Arabia

Received: 05 December 2024 Revised: 10 April 2025 Accepted: 15 April 2025 Published: 18 April 2025
MSC : 46S40, 47H10, 54H25

Our aim of the research article was to obtain fixed point results for generalized contractions equipped with precisely defined control functions in the framework of $ \mathfrak{F} $-bipolar metric spaces. As a consequence, some well-known results from the literature were derived as special cases of our leading theorems. To validate the theoretical findings and demonstrate their practical relevance, a non-trivial illustrative example is included. Furthermore, we explored the applicability of our major results by addressing the existence and uniqueness of solutions to Volterra integral equations, a class of problems frequently arising in mathematical modeling of real-world systems. In addition, we extended the utility of our fixed point theorems to homotopy theory, where they served as a foundational tool in analyzing the existence of continuous deformations between mappings, thereby contributing to the study of topological invariants and the structural properties of function spaces.
- fixed point,
- $ \mathfrak{F} $-bipolar metric space,
- rational contraction,
- Volterra integral equation,
- homotopy theory
Citation: Mohammed H. Alharbi, Jamshaid Ahmad. A fresh look at Volterra integral equations: a fixed point approach in $ \mathfrak{F} $-bipolar metric spaces[J]. AIMS Mathematics, 2025, 10(4): 8926-8945. doi: 10.3934/math.2025409

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Abstract

Our aim of the research article was to obtain fixed point results for generalized contractions equipped with precisely defined control functions in the framework of $ \mathfrak{F} $-bipolar metric spaces. As a consequence, some well-known results from the literature were derived as special cases of our leading theorems. To validate the theoretical findings and demonstrate their practical relevance, a non-trivial illustrative example is included. Furthermore, we explored the applicability of our major results by addressing the existence and uniqueness of solutions to Volterra integral equations, a class of problems frequently arising in mathematical modeling of real-world systems. In addition, we extended the utility of our fixed point theorems to homotopy theory, where they served as a foundational tool in analyzing the existence of continuous deformations between mappings, thereby contributing to the study of topological invariants and the structural properties of function spaces.

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