On integrable and approximate solutions for Hadamard fractional quadratic integral equations

Saud Fahad Aldosary; Mohamed M. A. Metwali; Manochehr Kazemi; Ateq Alsaadi; Saud Fahad Aldosary; Mohamed M. A. Metwali; Manochehr Kazemi; Ateq Alsaadi

doi:10.3934/math.2024279

AIMS Mathematics

2024, Volume 9, Issue 3: 5746-5762. doi: 10.3934/math.2024279

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On integrable and approximate solutions for Hadamard fractional quadratic integral equations

1.
Department of Mathematics, College of Science and Humanities in Alkharj, Prince Sattam bin Abdulaziz University, Alkharj 11942, Saudi Arabia
2.
Department of Mathematics and Computer Science, Faculty of Science, Damanhour University, Damanhour, Egypt
3.
Department of Mathematics, Ashtian Branch, Islamic Azad University, Ashtian, Iran
4.
Department of Mathematics and Statistics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia

Received: 18 December 2023 Revised: 19 January 2024 Accepted: 22 January 2024 Published: 30 January 2024
MSC : 45G10, 47H10, 47H30, 47N20

This article addressed the integrable and approximate solutions of Hadamard-type fractional Gripenberg's equation in Lebesgue spaces $ L_1[1, e] $. It is well known that the Gripenberg's equation has significant applications in mathematical biology. By utilizing the fixed point (FPT) approach and the measure of noncompactness (MNC), we demonstrated the presence of monotonic integrable solutions as well as the uniqueness of the solution for the studied equation in spaces that are not Banach algebras. Moreover, the method of successive approximations was successfully applied and, as a result, we obtained the approximate solutions for these integral equations. To validate the obtained results, we provided several numerical examples.
- measure of noncompactness (MNC),
- Hadamard's fractional operator,
- Gripenberg's equation,
- approximate solution,
- fixed point theorem (FPT)
Citation: Saud Fahad Aldosary, Mohamed M. A. Metwali, Manochehr Kazemi, Ateq Alsaadi. On integrable and approximate solutions for Hadamard fractional quadratic integral equations[J]. AIMS Mathematics, 2024, 9(3): 5746-5762. doi: 10.3934/math.2024279

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Abstract

This article addressed the integrable and approximate solutions of Hadamard-type fractional Gripenberg's equation in Lebesgue spaces $ L_1[1, e] $. It is well known that the Gripenberg's equation has significant applications in mathematical biology. By utilizing the fixed point (FPT) approach and the measure of noncompactness (MNC), we demonstrated the presence of monotonic integrable solutions as well as the uniqueness of the solution for the studied equation in spaces that are not Banach algebras. Moreover, the method of successive approximations was successfully applied and, as a result, we obtained the approximate solutions for these integral equations. To validate the obtained results, we provided several numerical examples.

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