Ulam–Hyers–Rassias Stability Results for Nonlinear Mixed Partial Integro-Differential Equations with Discontinuous Kernels

Abstract

This paper presents a novel stability analysis of nonlinear mixed partial integro-differential equations with discontinuous kernels. The paper fills a significant gap in the literature by offering, for the first time, a rigorous proof of stability in both the Ulam–Hyers and Ulam– Hyers–Rassias frameworks for such equations under general conditions. Discontinuous kernels present a great deal of difficulty due to the complex behavior they introduce into the system’s dynamics and their intrinsic singularities. This work not only specifies the stability criteria but also provides insights into the underlying mechanisms governing the behavior of the solution by creating a new analytical structure. This will be done by using fixed-point arguments within the framework of continuous function spaces, equipped with a generalized Bielecki metric. Additionally, in order to provide insight into the stability behavior under slight perturbations, we investigate the σ–semi–Ulam–Hyers stability of the nonlinear mixed partial integrodifferential equations with discontinuous kernels. The results deepen our knowledge of partial integro-differential equations and may find use in a variety of areas where discontinuous kernels are significant. In contrast to many previous studies, our method allows the system’s solution to exist in metric space instead of normed space. Furthermore, this work is innovative and significant because no previous study has been done on the stability of this kind of nonlinear mixed partial integro-differential equations with discontinuous kernels. Finally, for verification, we provide several examples and include 2D and 3D graphs of specific variables and functions, which are generated using MATLAB.