Nonlinear Delay Integral Inequalities of the Volterra-Fredholm Type on Time Scales

Abstract

The theory of integral inequalities constitutes a fundamental component of modern mathematical analysis. It plays a key role in examining the qualitative properties of solutions to complex differential and integral equations. As such, it is crucial for sta- bility analysis, the investigation of existence and uniqueness, and the estimation of solutions in various dynamic systems. This study aims to investigate the theory of time scales, then to provide some gener- alisations of the Volterra-Fredholm type integral inequalities on time scales with delay. Including a time delay element into these inequalities makes them more realistic in modeling natural and physical systems, as it reflects the natural delay in response and cumulative effects over time. Furthermore, thiswork will include illustrative examples and numerical simulation to demonstrate the practical utility and effectiveness of the derived inequalities.