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Oscillatory Behaviour of Solutions of Two-Dimensional Differential Systems with Deviated Arguments

Authors:

Koplatadze R Partsvania N

Institution:

(1) A. Razmadze Mathematical Institute, Georgian Academy of Sciences, 1, M. Aleksidze St., Tbilisi, 380093, Georgia

Abstract:

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

$\begin{gathered} u'_1 (t) = f_1 \left( {t,u_1 (\tau _1 (t)), \ldots u_1 (\tau _m (t)),u_2 (\sigma _1 (t)), \ldots ,u_2 (\sigma _m (t))} \right), \hfill \\ u'_2 (t) = f_2 \left( {t,u_1 (\tau _1 (t)), \ldots u_1 (\tau _m (t)),u_2 (\sigma _1 (t)), \ldots ,u_2 (\sigma _m (t))} \right), \hfill \\ \end{gathered}$

where f _i: Ropf

₊ ×

^2m

(i=1,2) satisfy the local Carathéodory conditions and tau

_i,

_i:

₊

₊(i=1,...,m) are continuous functions such that sgr

_i(t)

t for $\sigma _i (t) \leqslant t{\text{ for }}t \in \mathbb{R}_ + ,{\text{ }}\mathop {\lim }\limits_{t \to + \infty } {\text{ }}\tau _i (t) = + \infty ,{\text{ }}\mathop {\lim }\limits_{t \to + \infty } {\text{ }}\sigma _i (t) = + \infty (i = 1, \ldots ,m)$ .

Keywords:

Two-dimensional differential system with deviated arguments proper solution oscillatory solution

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