Existence and uniqueness of solutions for fractional Volterra-Fredholm equations in Banach spaces of order $ \eta\in(1, 2) $

Mdi Begum Jeelani; Farva Hafeez; Nouf AbdulRahman Alqahtani; Mdi Begum Jeelani; Farva Hafeez; Nouf AbdulRahman Alqahtani

doi:10.3934/math.2025976

AIMS Mathematics

2025, Volume 10, Issue 9: 21916-21928. doi: 10.3934/math.2025976

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Existence and uniqueness of solutions for fractional Volterra-Fredholm equations in Banach spaces of order $ \eta\in(1, 2) $

1.
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh
2.
Department of Mathematics and Statistics, University of Lahore, Sargodha, Pakistan

Received: 18 May 2025 Revised: 17 August 2025 Accepted: 09 September 2025 Published: 22 September 2025
MSC : 45J05, 45M10, 46E15, 47G20

The primary objective of this paper is to investigate and establish existence and uniqueness results for solutions of nonlinear Volterra-Fredholm integro-differential equations (VFIDEs) of fractional order, specifically for $ 1 < \eta < 2 $. By leveraging fixed-point theorems and contraction mapping principles within Banach spaces, we derive comprehensive results for both one-dimensional and two-dimensional nonlinear fractional-order equations. By presenting sufficient conditions, we ensure the existence and uniqueness of a fixed point associated with the operator form of the VFIDEs. Our analysis provides a rigorous framework for understanding the behavior of such equations, and the results obtained in this study enhance our knowledge of fractional integro-differential equations (FIDEs). To illustrate the practical application of these theoretical results, two examples are provided that demonstrate the uniqueness of solutions.
- Volterra-Fredholm integro-differential equations,
- fixed point theorem,
- Riemann-Liouville derivative,
- Caputo derivative
Citation: Mdi Begum Jeelani, Farva Hafeez, Nouf AbdulRahman Alqahtani. Existence and uniqueness of solutions for fractional Volterra-Fredholm equations in Banach spaces of order $ \eta\in(1, 2) $[J]. AIMS Mathematics, 2025, 10(9): 21916-21928. doi: 10.3934/math.2025976

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Abstract

The primary objective of this paper is to investigate and establish existence and uniqueness results for solutions of nonlinear Volterra-Fredholm integro-differential equations (VFIDEs) of fractional order, specifically for $ 1 < \eta < 2 $. By leveraging fixed-point theorems and contraction mapping principles within Banach spaces, we derive comprehensive results for both one-dimensional and two-dimensional nonlinear fractional-order equations. By presenting sufficient conditions, we ensure the existence and uniqueness of a fixed point associated with the operator form of the VFIDEs. Our analysis provides a rigorous framework for understanding the behavior of such equations, and the results obtained in this study enhance our knowledge of fractional integro-differential equations (FIDEs). To illustrate the practical application of these theoretical results, two examples are provided that demonstrate the uniqueness of solutions.

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