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Asymptotic behavior for linear delay-differential equations with periodically oscillatory coefficients. (English) Zbl 0870.34076

Consider the scalar equation \(x'(t)=p(t)x(t-\tau)\) where \(\tau\) is a positive constant. The function \(p:[0,\infty)\to\mathbb{R}\) is a continuous and periodic function with period \(\omega>0\) and has the property \(-p(\omega-t)\geq p(t)>0\) for \(t\in(0,\omega/2)\). The aim of the paper is to prove that every solution of this equation tends to zero as \(t\to\infty\) for sufficiently small \(\tau>0.\) More precisely the following result is established: Put \(\lambda=\max_{0\leq t\leq\omega}\int_t^{t+\tau}|p(s)|ds\), \(\mu=\min_{0\leq t\leq\omega/2-\tau}\int_t^{t+\tau}p(s) ds,\) and \(c\) is the root of \(c\exp(c\mu)=1\) (note that \(0<c<1\)). Suppose that \(\tau\leq\omega/12\) and \(q_0\exp\left\{c\int_0^{\omega/2}p(s) ds\right\}\}<1,\) where \(q_0=\lambda^2/2\) for \(\lambda\leq1\) and \(\lambda-1/2\) for \(\lambda>1.\) Then every solution tends to zero as \(t\to\infty\). A motivating example is the case of \(p(t)=\sin t\). If \(\tau=0\) no solution of the obtained ordinary differential equation \(x'=p(t)x\) tends to zero as \(t\to\infty\) but it is proved in the paper that for positive and sufficiently small \(\tau\) every solution converges to zero as \(t\to\infty\).
Reviewer: I.Ginchev (Varna)

MSC:

34K25 Asymptotic theory of functional-differential equations
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