Existence and uniqueness theorems for fractional differential equations with proportional delay

Document Type : Research Paper

Authors

1 Department of Mathematics, Shivaji University, Kolhapur - 416004, India.

2 Department of Mathematics, Jaysingpur College, Jaysingpur (Affiliated to Shivaji University, Kolhapur) - 416101, India.

Abstract

In this paper, we apply the successive approximation method (SAM) to solve nonlinear differential equations (DEs) with proportional delay. Utilizing SAM, we establish results on existence and uniqueness. Differential equations (DEs) with proportional delay represent a particular case of time-dependent delay differential equations (DDEs). We demonstrate that the equilibrium solution of time-dependent DDEs is asymptotically stable over finite time intervals. We obtained a series solution for the pantograph and Ambartsumian equations and proved their convergence. Furthermore, we prove that the zero solution of the pantograph and Ambartsumian equations is asymptotically stable. The outcomes of integer order obtained for DEs with proportional delay and time-dependent DDEs have been extended to the initial value problem (IVP) for fractional DDEs and a system of fractional DDEs involving the Caputo fractional derivative. Finally, we illustrate SAMs efficacy using particular non-linear DEs with proportional delay. The results obtained for non-linear DEs with proportional delay by SAM are compared with exact solutions and other iterative methods. It is noted that SAM is easier to use than other techniques, and the solutions obtained using SAM are consistent with the exact solution.

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Main Subjects

Computational Methods for Differential Equations
Volume 13, Issue 3
July 2025
Pages 904-918

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History

  • Receive Date: 22 July 2023
  • Revise Date: 10 January 2024
  • Accept Date: 06 May 2024

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