Oscillatory Periodic Solutions for Two Differential-Difference Equations Arisingin Applications

Abstract

We study the existence of oscillatory periodic solutions for two nonautonomous differential-difference equations which arise in a variety of applications with the following forms: $\dot{x}\left(t\right)=-f(t,x(t-r\left)\right)$ and $\dot{x}\left(t\right)=-f(t,x(t-s\left)\right)-f(t,x(t-2s\left)\right)$, where $f\in C(ℝ\times ℝ,ℝ)$ is odd with respect to $\mathrm{x}$, and are two given constants. By using a symplectic transformation constructed by Cheng (2010) and a result in Hamiltonian systems, the existence of oscillatory periodic solutions of the above-mentioned equations is established.