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