Existence and uniqueness of Remotely Almost Periodic solutions of differential equations with piecewise constant argument

TL;DR

This paper establishes conditions for the existence and uniqueness of remotely almost periodic solutions in differential equations with piecewise constant argument, including perturbed models and biological applications.

Contribution

It provides new sufficient conditions for RAP solutions in DEPCA and perturbed systems, extending the theory to models with biological relevance.

Findings

Unique RAP solutions exist under exponential dichotomy and Lipschitz conditions.

Perturbed DEPCA admit RAP solutions within certain parameter ranges.

Application to biological models demonstrates the practical relevance of the theoretical results.

Abstract

We study differential equations with piecewise constant argument (DEPCA) and establish the existence and uniqueness of remotely almost periodic (RAP) solutions for \[ x'(t)=A(t)x(t)+B(t)x([t])+f(t). \] Under an exponential dichotomy for the associated linear hybrid system \(x'(t)=A(t)x(t)+B(t)x([t])\) and suitable RAP/Lipschitz assumptions on the data, we derive sufficient conditions guaranteeing a unique RAP solution. We further consider perturbed DEPCA of the form \[ \begin{aligned} x'(t)&=A(t)x(t)+B(t)x([t])+f(t)+\nu\,g_{\nu}\bigl(t,x(t),x([t])\bigr),\\ y'(t)&=\tilde f\bigl(t,y(t),y([t])\bigr)+\nu\,g_{\nu}\bigl(t,y(t),y([t])\bigr), \end{aligned} \] and prove the existence (and, when appropriate, uniqueness) of RAP solutions for \(\nu\) in a suitable range, under mild uniform Lipschitz and smallness conditions on \(g_{\nu}\). As an application, we obtain RAP solutions for nonautonomous Lasota-Wazewska type models with piecewise constant argument, and show the existence of a unique positive RAP solution under biologically meaningful hypotheses.