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

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)+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.