On oscillation of solutions of second-order systems of deviated differential equations. (English) Zbl 0868.34054

Summary: Sufficient conditions are found for the oscillation of proper solutions of the system of differential equations \[ \begin{aligned} u_1'(t)=f_1(t,u_1(\tau_1(t)), \dots,u_1(\tau_m(t)),u_2(\sigma_1(t)),\dots,u_2(\sigma_m(t))),\\ u_2'(t)=f_2(t,u_1(\tau_1(t)),\dots,u_1(\tau_m(t)),u_2(\sigma_1(t)), \dots,u_2(\sigma_m(t))),\end{aligned} \] where \(f_i:\mathbb{R}_+\times \mathbb{R}^{2m}\to \mathbb{R}\) \((i=1,2)\) satisfy the local Carathéodory conditions and \(\sigma_i,\tau_i: \mathbb{R}_+\to \mathbb{R}\) \((i=1,\dots,m)\) are continuous functions such that \(\sigma_i(t)\leq t\) for \(t\in \mathbb{R}_+\), \(\lim_{t\to+\infty}\sigma_i(t)=+\infty\), \(\lim_{t\to+\infty}\tau_i(t)=+\infty\) (\(i=1,\dots,m\)).


34K11 Oscillation theory of functional-differential equations


[1] E. Hille, Non-oscillation theorems.Trans. Amer. math. Soc. 64(1948), 234–252. · Zbl 0031.35402
[2] R. Koplatadze, On oscillation of solutions of second-order retarded differential inequalities and equations. (Russian)Mathematica Balkania 5:29(1975), 163–172.
[3] R. Koplatadze, Criteria of oscillation of solutions of second-order retarded differential inequaliteis and equations. (Rusian)Proc. Vekua Inst. Appl. Math. (Tbiliss. Gos. Univ. Inst. Prikl. Mat. Trudy) 17 (1986), 104–120.
[4] R. Koplatadze, On differential equations with deviating arguments having propertiesA andB. (Russian)Differentsial’nye Uravneeniya 25 (1989), No. 11, 1897–1909. · Zbl 0693.34068
[5] R. Koplatadze, On oscillatory properties of solutions of functional differential equations.Memoirs on Differential Equations and Mathematical Physics 3(1994), 1–179,A. Razmadze Mathematical Institute of the Georgian Academy of Sciences, Tbilisi. · Zbl 0843.34070
[6] J. D. Mirzov, Asymptotic behavior of solutions of systems of nonlinear non-autonomous ordinary differential equations. (Russian)Maikop, 1993.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.