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

Diego Jaure, Christopher Maulen

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.

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 in a suitable range, under mild uniform Lipschitz and smallness conditions on . 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.
Paper Structure (10 sections, 21 theorems, 148 equations)

This paper contains 10 sections, 21 theorems, 148 equations.

Table of Contents

  1. Introduction
  2. Organization and main ideas of this work
  3. Preliminars on Remotely Almost Periodic Sequences
  4. Differential Equations with Piecewise Constant Argument
  5. Remotely Almost-Periodic Solutions in Difference Equations
  6. Existence of Remotely Almost Periodic Solutions for DEPCA
  7. Proof Theorem \ref{['existencia_solucion_depca_rap']}
  8. Existence of Remotely Almost Periodic Solutions for Perturbed DEPCA
  9. Proof Theorem \ref{['teo_perturbada_depca']}
  10. Lasota-Wazewska Model

Key Result

Theorem 1.1

Suppose that $A$ and $B$ are remotely almost periodic matrix valued functions, and $f \in \mathbb{Z}RAP(\mathbb{R}, \mathbb{R}^q)$. Assume that the discrete equation associated with DEPCA_lineal admits an exponential dichotomy on $\mathbb{Z}$ and that its Green's kernel is bi-remotely almost periodi

Theorems & Definitions (49)

  • Definition 1.1
  • Definition 1.2: pinto_remotelyzhang_depca
  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Theorem 2.1
  • proof
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.2
  • ...and 39 more