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On the structure of solutions of a system of three differential inequalities. (English) Zbl 0820.34007

The paper deals with the differential inequalities (1) \(\alpha_ i y_ i'(t) y_{i+ 1}(t)\geq 0\), \(y_{i+ 1}(t)= 0\Rightarrow y_ i'(t)= 0\), \(i= 1,2,3\), \(t\in J= (a, b)\), \(-\infty\leq a< b\leq \infty\), \(\alpha_ i\in \{-1, 1\}\), \(y_ 4= y_ 1\). \(y= (y_ 1, y_ 2, y_ 3)\) is a solution of (1) if \(y_ i: J\to R\), \(i= 1,2,3\), \(R= (-\infty, \infty)\) is locally absolutely continuous and (1) holds for all \(t\in J\) such that \(y_ i'(t)\) exists. The structure of solutions of (1) with respect to their zeros is investigated for \(\alpha_ 1 \alpha_ 2 \alpha_ 3= -1\) as well as for \(\alpha_ 1 \alpha_ 2 \alpha_ 3= 1\).

MSC:

34A40 Differential inequalities involving functions of a single real variable
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
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