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On the oscillation of a class of linear homogeneous third order differential equations. (English) Zbl 0973.34023

The authors consider the equation \[ y'''+a(t)y''+b(t)y'+c(t)y=0 \tag{\(*\)} \] with \(a(t)\leq 0\), \(b(t)\leq 0\), \(c(t)\leq 0\) and \(b(t)\not \equiv 0\) and \(c(t)\not \equiv 0\) in any neighbourhood of \(+\infty \). It is shown that if equation \((*)\) has at least one oscillatory solution than the set of oscillatory solutions forms a two-dimensional subspace of the solution space (see also J. M. Dolan [J. Differ. Equations 7, 367-388 (1970; Zbl 0191.10001)], W. J. Kim [Proc. Am. Math. Soc. 26, 286-293 (1970; Zbl 0206.09601)], F. Neuman [J. Differ. Equations 15, 589-596 (1974; Zbl 0287.34029)]). They also establish two sufficient conditions for the existence of oscillatory solutions to equation \((*)\).

MSC:

34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
34C11 Growth and boundedness of solutions to ordinary differential equations
34A30 Linear ordinary differential equations and systems
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