Oscillations of some delay differential equations with periodic coefficients
The author considers the following delay differential equations: \((E)\;x'(t)+a(t)[px(t-\tau)+qx(t-\sigma)]=0\), \((\overline E)\;x'(t)+f(t)x(t- \tau)-g(t)x(t-\sigma)=0\), \(t\geq 0\), where \(a,f,g\) are continuous periodic functions with a period \(\omega>0\), \(p,q,\tau(\geq 0)\), \(\sigma(\geq 0)\in R\) and \(\tau=\mu\omega\), \(\sigma=\nu\omega\) for some nonnegative integers \(\mu,\nu\). Denote \(A={1\over\omega}\int^ \omega_ 0a(t)dt\). Theorem 1. The following statements are equivalent: (I) All solutions of \((E)\) are oscillatory. (II) The equation \(\lambda+A(p\exp(- \lambda\tau)+q\exp(-\lambda\sigma))=0\) has no real roots. Moreover, there are given sufficient conditions for (i) every nonoscillatory solution of \((\overline E)\) to tend to zero as \(t\to\infty\); (ii) all solutions of \((\overline E)\) to be oscillatory.
scientific article; zbMATH DE number 55696
scientific article