Gronwall’s inequality and stability analysis of nonlinear fractional difference equations

Anshul Sharma; Suyash Narayan Mishra; Anurag Shukla

doi:10.1515/jncds-2024-0049

Article

Gronwall’s inequality and stability analysis of nonlinear fractional difference equations

Anshul Sharma , Suyash Narayan Mishra and Anurag Shukla

Published/Copyright: July 22, 2025

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From the journal Journal of Nonlinear, Complex and Data Science Volume 26 Issue 2

Abstract

This paper investigates the existence and uniqueness of solutions for nonlinear fractional difference equations of the Hilfer type using Brouwer’s and Banach’s fixed-point theorems. The study builds on the fundamental properties of linear fractional difference equations, the discrete comparison principle, and key concepts in fractional calculus. Hilfer-type nabla fractional differences, which generalize the Riemann–Liouville and Caputo nabla differences, are analyzed. Solutions for linear Hilfer-like fractional difference equations are derived using the successive approximation method. Gronwall’s inequality and its generalized form are presented and applied to examine asymptotic stability. The theoretical results are validated through numerical examples, simulations, and the Newton’s iteration method, demonstrating the practical relevance of the findings.

Keywords: discrete equations; Hilfer discrete fractional operator; Gronwall’s inequality; asymptotic stability; Newton’s iteration method

Corresponding author: Anurag Shukla, Department of Applied Sciences, Rajkiya Engineering College, Kannauj, 209732, Uttar Pradesh, India, E-mail: anuragshukla259@gmail.com

Research ethics: Followed.
Informed consent: Followed.
Author contributions: Equally.
Use of Large Language Models, AI and Machine Learning Tools: None declared.
Conflict of interest: Authors have no conflict of interest.
Research funding: None declared.
Data availability: My manuscript has no associated data.

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Received: 2024-04-28

Accepted: 2025-07-04

Published Online: 2025-07-22

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Articles in the same Issue

https://doi.org/10.1515/jncds-2024-0049

Keywords for this article

discrete equations; Hilfer discrete fractional operator; Gronwall’s inequality; asymptotic stability; Newton’s iteration method