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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2602.00926 |
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Table of Contents:
- 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 \[ \begin{aligned} x'(t)&=A(t)x(t)+B(t)x([t])+f(t)+ν\,g_ν\bigl(t,x(t),x([t])\bigr),\\ y'(t)&=\tilde f\bigl(t,y(t),y([t])\bigr)+ν\,g_ν\bigl(t,y(t),y([t])\bigr), \end{aligned} \] and prove the existence (and, when appropriate, uniqueness) of RAP solutions for \(ν\) in a suitable range, under mild uniform Lipschitz and smallness conditions on \(g_ν\). 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.