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Asymptotic behaviour of solutions of two-dimensional linear

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differential systems with deviating arguments

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*R. Koplatadze, N. Partsvania and I. P. Stavroulakis*

**Address.**

A. Razmadze Mathematical Institute of the Georgian Academy of Sciences,

1 M. Aleksidze St., Tbilisi 0193, Georgia.
A. Razmadze Mathematical Institute of the Georgian Academy of Sciences,

1 M. Aleksidze St., Tbilisi 0193, Georgia.

Department of Mathematics, University of Ioannina,

451 10 Ioannina, Greece

**E-mail:**

roman@rmi.acnet.ge

ninopa@rmi.acnet.ge

ipstav@cc.uoi.gr

**Abstract.**

Sufficient conditions are established for the oscillation of proper
solutions of the system

\begin{align*}

u_1'(t) & =p(t)u_2(\sigma(t))\,, \\

u_2'(t) & =-q(t)u_1(\tau(t))\,,

\end{align*}

where $p,\,q: R_{+}\to R_{+}$ are locally summable functions, while
$\tau$ and

$\sigma: R_{+}\to R_{+}$ are continuous and continuously differentiable
functions,

respectively, and $\lim\limits_{t\to +\infty} \tau(t)=+\infty$, $\lim\limits_{t\to
+\infty} \sigma(t)=+\infty$.

**AMSclassification.** 34K06, 34K11.

**Keywords.** Two-dimensional differential system, proper solution,
oscillatory system.