Riemann-Liouville fractional-order pantograph differential equation constrained by nonlocal and weighted pantograph integral equations

Ahmed M. A. El-Sayed; Wagdy G. El-Sayed; Kheria M. O. Msaik; Hanaa R. Ebead; Ahmed M. A. El-Sayed; Wagdy G. El-Sayed; Kheria M. O. Msaik; Hanaa R. Ebead

doi:10.3934/math.2025228

AIMS Mathematics

2025, Volume 10, Issue 3: 4970-4991. doi: 10.3934/math.2025228

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Research article

Riemann-Liouville fractional-order pantograph differential equation constrained by nonlocal and weighted pantograph integral equations

1.
Department of Mathematics, Faculty of Science, Alexandria University, Alexandria, Egypt
2.
Department of Mathematics, Faculty of Science, Zintan University, Zintan, Libya

Received: 15 December 2025 Revised: 30 January 2025 Accepted: 18 February 2025 Published: 06 March 2025
MSC : 26A33, 34B10, 45M10, 47H09, 47H10

In this research, we investigated the Riemann-Liouville fractional-order pantograph differential equation constrained by nonlocal and weighted pantograph integral constraints. We presented novel sufficient conditions for the uniqueness of the solution. Moreover, we analyzed the continuous dependence of the solution on some functions and parameters. Additionally, we proved the Hyers-Ulam stability of the problem. To demonstrate the applicability of our results, we included several examples. The present study was located in the space $ L_1[0, T] $. The techniques of Schauder's fixed point theorem and Kolmogorov's compactness criterion were the primary tools utilized in this work. These contributions offer a comprehensive framework for understanding the qualitative behavior of the fractional-order pantograph equation.
Citation: Ahmed M. A. El-Sayed, Wagdy G. El-Sayed, Kheria M. O. Msaik, Hanaa R. Ebead. Riemann-Liouville fractional-order pantograph differential equation constrained by nonlocal and weighted pantograph integral equations[J]. AIMS Mathematics, 2025, 10(3): 4970-4991. doi: 10.3934/math.2025228

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Abstract

In this research, we investigated the Riemann-Liouville fractional-order pantograph differential equation constrained by nonlocal and weighted pantograph integral constraints. We presented novel sufficient conditions for the uniqueness of the solution. Moreover, we analyzed the continuous dependence of the solution on some functions and parameters. Additionally, we proved the Hyers-Ulam stability of the problem. To demonstrate the applicability of our results, we included several examples. The present study was located in the space $ L_1[0, T] $. The techniques of Schauder's fixed point theorem and Kolmogorov's compactness criterion were the primary tools utilized in this work. These contributions offer a comprehensive framework for understanding the qualitative behavior of the fractional-order pantograph equation.

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