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

Authors:

Koplatadze R. Partsvania N.

Affiliation:

(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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