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On oscillatory solutions of the system of differential equations with deviating arguments. (English) Zbl 0625.34073
The paper deals with systems $y_ i'(t)=f_ i(t,y_ 1(\sigma_{i,1}(t)),...y_ n(\sigma_{i,n}(t))),\quad i=1,...,n,\quad n\geq 2$ in which $(-1)^{\nu_ i}f_ i(t,x_ 1,...,x_ n)sgn x_{i+1}\geq 0,\quad \nu_ i=0\quad or\quad 1,\quad x_{n+1}=x_ 1.$ Conditions are established under which each so-called proper solution is oscillatory or tends monotonically to infinity or to zero. The paper extends the method of demonstration from nth order equations to systems of first order equations.
Reviewer: T.Dlotko
##### MSC:
 34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument) 34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
##### Keywords:
oscillatory solution; monotonic solutions
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##### References:
 [1] Bartušek M.: On existence of oscillatory solution of the system of differential equations. Arch. Math., XVII, 7-10, 1981. · Zbl 0471.34022 [2] Kiguradze T. I.: Some singular boundary value problems for ordinary differential equations. (in Russian), Tbilisi Univ. Press, Tbilisi, 1975. [3] Chanturia T. A.: On a comparison theorem for linear differential equations. (in Russian). Izvestija AN SSSR, 40, No 5, 1129-1142, 1976. [4] Chanturia T. A.: Asymptotic properties of solutions of some sets of nonautonomous ordinary differential equations. Mat. Zametki, 32, No 4, 577-588, 1982 [5] Koplatadze R. T., Chanturia T. A.: On oscillatory properties of differential equations with deviating argument. Tbilisi Univ. Press, Tbilisi, 1977
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