Delayed Gronwall inequality with weakly singular kernel

Abstract: Delay Gronwall inequality with a weakly singular kernel has been a subject of interest in various mathematical studies. In this article, we will delve into the consideration of this inequality and its application in the study continuity of the state trajectory for a Volterra integral equation with delay. Using delay Gronwall inequality with a singular kernel, we investigate the behavior and properties of the state trajectory if there are delays. This analysis aims to improve our understanding of the dynamics associated in del Volterra integral equations with delay. In addition, we present a comprehensive example demonstrating the practical significance of our results.