Existence and Uniqueness Theorems for Impulsive Delay Differential Equations

Abstract

This study investigates existence and uniqueness theorems for a class of impulsive delay differential equations (IDDEs), which represent a significant extension of classical differential equations by incorporating impulsive effects and time delays. Impulsive delay differential equations are utilized to model various phenomena in science and engineering where sudden changes and time-dependent effects are present, such as in control systems, biological populations, and economic models.

The research is grounded in the analysis of IDDEs, which combine elements of both impulsive differential equations and delay differential equations. Impulsive differential equations account for instantaneous changes at specific moments, while delay differential equations incorporate time lags in the system's response. The interaction of these two components adds complexity to the analysis, making it essential to develop robust theoretical tools to ensure the existence and uniqueness of solutions.

The primary objective of this study is to establish and prove existence and uniqueness theorems for IDDEs. These theorems provide crucial conditions under which a solution exists and is unique for a given set of initial conditions and parameters. The approach involves the use of advanced mathematical techniques, including fixed-point theorems, Lyapunov functions, and integral inequalities, to derive sufficient conditions for the existence and uniqueness of solutions.

To address the existence of solutions, the study employs the Banach fixed-point theorem and the Schauder fixed-point theorem. These theorems are instrumental in demonstrating that under certain conditions, there exists at least one solution to the IDDE. Specifically, the study considers an appropriate functional space where the IDDE is transformed into a corresponding integral equation. By applying the fixed-point theorems to this integral equation, conditions are derived that guarantee the existence of a solution.