## 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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