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)+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.

💡 Research Summary

The paper investigates differential equations with piecewise constant argument (DEPCA) and establishes a comprehensive theory for the existence and uniqueness of remotely almost periodic (RAP) solutions. A RAP function is a bounded uniformly continuous function whose “remote translation numbers” are relatively dense in ℝ; that is, for every ε>0 there exists a relatively dense set of τ such that lim sup_{|t|→∞}‖f(t+τ)−f(t)‖<ε. This notion is weaker than classical almost periodicity and is well suited to hybrid systems that are continuous on each interval (n,n+1) but may be discontinuous at integer points.

The authors first introduce the spaces RAP(ℝ,ℂ^q) and Z‑RAP(ℝ,ℂ^q), the latter being the class of functions that are continuous on ℝ\ℤ, have finite one‑sided limits at every integer, and satisfy the RAP condition only for integer shifts τ∈ℤ. They also define a bounded‑with‑regular‑jumps space B_R(ℝ,ℝ^n) to handle the inherent discontinuities of the term x(