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

The primary objective of this paper is to comprehensively establish the Hyers-Ulam, generalized Hyers-Ulam, Hyers-Ulam-Rassias, and generalized Hyers-Ulam stability properties for (k, ?)-fractional order quadratic integral equations. These stability concepts play a crucial role in understanding the persistence, resilience, and response of solutions to small perturbations, providing insight into the behavior and reliability of solutions within complex systems. Our analysis is grounded in the application of Gronwall?s lemma, an essential tool that we adapt specifically for the unique structure of (k, ?)-fractional order systems. This approach not only enriches the theoretical understanding of stability within these fractional order integral equations but also broadens the applicability of Gronwall?s lemma to new contexts. To substantiate our findings, we provide two illustrative examples, carefully chosen to demonstrate the stability characteristics across a range of conditions and parameter settings. These examples are further supplemented by detailed 2D and 3D graphical representations generated in MATLAB, allowing for a visual examination of stability and solution dynamics. These visualizations not only complement the analytical proofs but also offer an intuitive validation of the stability results. Through this integrated approach the paper aims to present a well-rounded and thorough assessment of stability in (k, ?)-fractional order quadratic integral equations.