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: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.
Highlights
Introduction and Statement of Main ResultsFurumochi 1 studied the following equation: xt a − sin x t − r, 1.1 with t ≥ 0, a ≥ 0, r > 0, which models phase-locked loop control of high-frequency generators and is widely applied in communication systems
Motivated by the lack of more results on periodic solutions for nonautonomous differential-difference equations, in the present paper, we study the following equations: xt −f t, x t − r, 1.7 xt −f t, x t − s − f t, x t − 2s, 1.8 where f t, x ∈ C Ê × Ê, Ê is odd with respect to x and r π/2, s π/3
We show that the reduction method in 7 can be used to study oscillatory periodic solutions of 1.7 and 1.8
Summary
College of Mathematics and Physics, Nanjing University of Information Science and Technology, Nanjing 210044, China. We study the existence of oscillatory periodic solutions for two nonautonomous differentialdifference equations which arise in a variety of applications with the following forms: xt. −f t, x t − r and xt −f t, x t − s − f t, x t − 2s , where f ∈ C Ê × Ê, Ê is odd with respect to x, and r, s > 0 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
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