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Oscillatory solutions of systems of equations with linearly transformed argument. (Russian) Zbl 0612.34027

We deal with the problem: \[ (1)\quad y^{(N)}(t)=\sum_{m\in \phi}C_ m(t)y^{(m_ 1)}(\alpha_ mt+\beta_ m)+f(t),\quad t\in (- \infty,\infty) \]
\[ (2)\quad y^{(2j)}(K\pi)=0,\quad j=0,1,...,[\frac{N-1}{2}],\quad K=0,\pm 1,... \] where \(y(t)=(y_ 1(t),...,y_ n(t))\); \(\phi =\{m=(m_ 1,m_ 2)\); \(m_ 1,m_ 2\) are integers where \(m_ 1\in [0,N-1]\), \(m_ 2\in [0,T],T<\infty \}\); \(C_ m(t)\) are \(n\times n\) matrices.
The purpose of this article is to study the properties of the non- homogeneous problem (1)-(2) and to construct a method for solving this problem. For the homogeneous problem \((f(t)=0,C_ m(t)=C_ m(t,\lambda))\), it is necessary to clear up, for which \(\lambda\) the non- trivial solutions exist.

MSC:

34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
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