Qualitative Stability Analysis of Oscillations in Carleman Volterra Integral Equations

Abstract

This paper investigates the stability properties of Carleman Volterra integral equations with oscillatory kernels characterized by sine and cosine functions, structures frequently encountered in modeling physical phenomena such as wave propagation, resonance, and signal transmission. Drawing upon the stability concepts introduced by Ulam–Hyers and Ulam–Hyers–Rassias, we establish novel and explicit criteria ensuring Ulam–Hyers stability, Ulam–Hyers–Rassias stability, and σ–semi–Ulam–Hyers stability for the considered class of integral equations. The analysis is carried out using fixed–point theory within the framework of a generalized Bielecki metric, which
enables the treatment of both finite and infinite domains. The main findings demonstrate that, under suitable conditions on the oscillatory kernels, the proposed integral equations admit stable solutions in the sense of the aforementioned stability concepts. In addition, the outcomes validate the robustness of the solutions to perturbation as well as offer verifiable criteria for establishing stability behavior. For a demonstration of the practicality and efficiency of the theoretical outcomes, some examples have been illustrated using graphical simulations. This information will lead to a more profound insight into the stability behavior of integral equations with oscillatory
kernels.