tailieunhanh - Existence of periodic solutions for second order rayleigh equations with piecewise constant argument

Based on a continuation theorem of Mawhin, periodic solutions are found for the second-order Rayleigh equation with piecewise constant argument. In this paper we study a slightly more general second-order Rayleigh equation with piecewise constant argument of the form. | Turk J Math 30 (2006) , 57 – 74. ¨ ITAK ˙ c TUB Existence of Periodic Solutions for Second Order Rayleigh Equations With Piecewise Constant Argument Gen-qiang Wang, Sui Sun Cheng Abstract Based on a continuation theorem of Mawhin, periodic solutions are found for the second-order Rayleigh equation with piecewise constant argument. Key Words: Rayleigh equation, deviating argument, piecewise constant argument, periodic solution, Mawhin’s continuation theorem. 1. Introduction Qualitative behaviors of first order delay differential equations with piecewise constant arguments are the subject of many investigations (see, . [1–19]), while those of higher order equations are not. However, there are reasons for studying higher order equations with piecewise constant arguments. Indeed, as mentioned in [10], a potential application of these equations is in the stabilization of hybrid control systems with feedback delay, where a hybrid system is one with a continuous plant and with a discrete (sampled) controller. As an example, suppose a moving particle is subjected to damping and a restoring controller −φ(x[t − k]) which acts at sampled time [t − k], then the equation of motion is of the form x00 (t) + a (t) x0 (t) = −φ(x[t − k]). Mathematics Subject Classification: 34K13 57 WANG, CHENG In this paper we study a slightly more general second-order Rayleigh equation with piecewise constant argument of the form x00 (t) + f (t, x0 (t)) + g (t, x ([t − k])) = 0, (1) where [·] is the greatest-integer function, k is a positive integer, f (t, x) and g (t, x) are continuous on R2 such that for (t, x) ∈ R2 , f (t + ω, x) = f (t, x) and g (t + ω, x) = g (t, x) , for some positive integer ω. We also require f(t, 0) = 0 for all t in R. By a solution of (1) we mean a function x (t) which is defined on R and which satisfies the conditions (i) x0 (t) is continuous on R, (ii) x0 (t) is differentiable at each point t ∈ R, with the possible exception of the points [t] ∈ R where .