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Oscillatory and asymptotic properties of third and fourth order linear differential equations. (English) Zbl 0688.34018
The author considers the following equations \[ (1)\quad (r(t)(r(t)y'(t))')'+p(t)y(t)=0; \]
\[ (2)\quad (r(t)(r(t)(r(t)y'(t))')')'+p(t)y(t)=0, \] where \(r,p\in C([t_ 0,\infty))\), \(r(t)>0\), \(R(t)=\int^{t}_{t_ 0}(1/r(s))ds\). Using v- derivative of the function y at the point t [J. Ohriska, Čzech. Math. J. 39(114), 24-44 (1989; Zbl 0673.34044)] the author proves sufficient conditions under which the equation (1) has (has not) the property (A) and the equation (2) has (has not) the property (B).
The equation (1) has the property (A) if every solution y(t) of (1) is either oscillatory or \((3)\quad d^ iy(t)/dR^ i\to 0\) as \(t\to \infty\), \(i=0,1,...,n-1\). The equation (2) has the property (B) if every solution y(t) of (2) is either oscillatory or satisfies the condition (3) or \(| d^ iy(t)/dR^ i| \to \infty\) as \(t\to \infty\), \(i=0,1,...,n-1\).
Reviewer: P.Marusciak

34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34E05 Asymptotic expansions of solutions to ordinary differential equations
34A30 Linear ordinary differential equations and systems
Full Text: EuDML
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