Levinson Theorem for Differential Equations with Piecewise Constant Argument Generalized

TL;DR

This paper extends Levinson's asymptotic theorem to generalized differential equations with piecewise constant arguments, providing a new theoretical foundation for analyzing their asymptotic behavior.

Contribution

It develops an adaptation of Levinson's theorem for generalized differential equations with piecewise constant arguments using fixed point methods.

Findings

Established a version of Levinson's theorem for these equations.

Proved the theorem using Banach fixed point theorem.

Highlighted key hypotheses for the adapted theorem.

Abstract

In this work, it is presented an adaptation of an asymptotic theorem of N. Levinson of 1948, to differential equation with piecewise constant argument generalized, which were introduced by M. Akhmet in 2007. By simplicity and without loss of generality, the case where the argument is delayed is considered. The N. Levinson's theorem which is adapted is that dealt by M. S. P. Eastham in his work which is present in this bibliography. The more relevant hypotheses of this theorem are highlighted an it is established a version of this theorem with these hypotheses for ordinary differential equations. Such a version is that which is adapted to differential equation with piecewise constant argument generalized. The adaptation is proved by mean the Banach fixed point where contractive operator is built form a suitable version of the constant variation formula.