Existence and Stability of Solutions of Differential Equations Involving Generalized Piecewise Constant Arguments

Abstract

This paper focuses on proving the existence of solutions for differential equations with a generalized piecewise constant argument by employing absolutely continuous functions. Accordingly, the concept of absolutely continuous solutions for differential equations with a piecewise constant argument of generalized type (DEPCAG) is introduced. Furthermore, by utilizing nonsmooth Lyapunov functions, Lyapunov’s second method is extended to the DEPCAG framework. One of the theorems developed herein is then applied to analyze the stability of a specific class of DEPCAG. The applicability of the results established is illustrated through a suitable example of the logistic growth model.