Asymptotically almost periodic solutions for certain differential equations with piecewise constant arguments

Zonghong Feng,Yong Wang,Xin Ma

doi:10.1186/s13662-020-02699-6

Abstract

It is well known that differential equations with piecewise constant arguments is a class of functional differential equations, which has fascinated many scholars in recent years. These delay differential equations have been successfully applied to diverse models in real life, especially in biology, physics, economics, etc. In this work, we are interested in the existence and uniqueness of asymptotically almost periodic solution for certain differential equation with piecewise constant arguments. Due to the particularity of the equations, we cannot use the traditional method to convert it into the difference equation with exponential dichotomy. Through constructing Cauchy matrix of the investigated system to find the corresponding Green matrix of the difference equation, we need the concept of exponential dichotomy and the Banach contraction fixed point theorem of the corresponding system. Then we give some sufficient conditions to obtain the existence and uniqueness of asymptotically almost periodic solutions for these systems.

Highlights

1 Introduction In recent years, the delay differential equations have been successfully applied to various models in many fields, especially in biology, physics, and economy
In 1977, Myshkis [29] proposed a differential equation with noncontinuous variables x (t) = g t, x(t), x h(t), where h is a deviated function with piecewise constant arguments such as h(t) = [t] or h(t) denotes the largest integer function
Motivated by the paper of Castillo [8], we study the above linear nonhomogeneous system (1.1) and nonlinear nonhomogeneous system (1.2) and get some sufficient conditions of the existence and uniqueness for asymptotically almost periodic solutions

Summary

Introduction

We can further prove the existence and uniqueness of asymptotically almost periodic solution for linear inhomogeneous system (1.1) (see Theorem 3.3). By using exponential dichotomy and the Banach contraction fixed point theorem, some sufficient conditions for the existence and uniqueness of asymptotically almost periodic solution for nonlinear nonhomogeneous system (1.2) are obtained (see Theorem 3.5). Definition 2.7 ([32]) Let C(n) be a q × q matrix and invertible, we say that the linear difference equation y(n + 1) = C(n)y(n) with exponential dichotomy on Z for all n ∈ Z.

Results

Conclusion