Asymptotic behaviour of solutions of two-dimensional linear

differential systems with deviating arguments

 

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.