Oscillatory Periodic Solutions for Two Differential-Difference Equations Arising in Applications

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:ẋ(t)=-f(t,x(t-r))andẋ(t)=-f(t,x(t-s))-f(t,x(t-2s)), wheref∈C(R×R,R)is odd with respect tox,andr,s>0are 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.