Ulam type stabilities for (k,ψ)-fractional order quadratic integral equations

Abstract

The primary objective of this paper is to comprehensively estab-lish the Hyers-Ulam, generalized Hyers-Ulam, Hyers-Ulam-Rassias,and generalized Hyers–Ulam stability properties for (k, ψ)-fractionalorder quadratic integral equations. These stability concepts play acrucial role in understanding the persistence, resilience, and responseof solutions to small perturbations, providing insight into the behav-ior and reliability of solutions within complex systems. Our analysisis grounded in the application of Gronwall’s lemma, an essential toolthat we adapt specifically for the unique structure of (k, ψ)-fractionalorder systems. This approach not only enriches the theoretical un-derstanding of stability within these fractional order integral equa-tions but also broadens the applicability of Gronwall’s lemma to newcontexts.To substantiate our findings, we provide two illustrative ex-amples, carefully chosen to demonstrate the stability characteristicsacross a range of conditions and parameter settings. These examplesare further supplemented by detailed 2D and 3D graphical represen-tations generated in MATLAB, allowing for a visual examination ofstability and solution dynamics. These visualizations not only com-plement the analytical proofs but also offer an intuitive validation ofthe stability results. Through this integrated approach the paper aims to present a well-rounded and thorough assessment of stability in (k,ψ)-fractional order quadratic integral equations.