Delayed Gronwall inequality with weakly singular kernel

TL;DR

This paper investigates a delay Gronwall inequality with a weakly singular kernel and applies it to analyze the continuity and behavior of state trajectories in delayed Volterra integral equations, enhancing understanding of their dynamics.

Contribution

It introduces a new form of delay Gronwall inequality with a singular kernel and applies it to study the continuity of solutions in delayed Volterra integral equations.

Findings

Established a delay Gronwall inequality with a weakly singular kernel.

Analyzed the continuity of state trajectories in delayed Volterra integral equations.

Provided a practical example demonstrating the application of the theoretical results.

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.