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Zbl 0751.34038
Philos, Ch.G.
Oscillations of some delay differential equations with periodic coefficients.
(English)
[J] J. Math. Anal. Appl. 162, No.2, 452-475 (1991). ISSN 0022-247X

The author considers the following delay differential equations: $(E)\ x'(t)+a(t)[px(t-\tau)+qx(t-\sigma)]=0$, $(\overline E)\ x'(t)+f(t)x(t- \tau)-g(t)x(t-\sigma)=0$, $t\ge 0$, where $a,f,g$ are continuous periodic functions with a period $\omega>0$, $p,q,\tau(\ge 0)$, $\sigma(\ge 0)\in R$ and $\tau=\mu\omega$, $\sigma=\nu\omega$ for some nonnegative integers $\mu,\nu$. Denote $A={1\over\omega}\int\sp \omega\sb 0a(t)dt$.\par 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.
[P.Marušiak (Žilina)]
MSC 2000:
*34K99 Functional-differential equations
34C10 Qualitative theory of oscillations of ODE: Zeros, etc.
34C25 Periodic solutions of ODE

Keywords: delay differential equations; oscillatory; nonoscillatory solution

Cited in: Zbl 0885.39006

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