×

zbMATH — the first resource for mathematics

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

MSC:
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
PDF BibTeX XML Cite
Full Text: EuDML
References:
[1] T. A. Čanturija: On a comparison theorem for linear differential equations. (Russian). Izv. Akad. Nauk SSSR, Ser. Mat. 40 (1976), 1128-1142.
[2] V. A. Kondratev: On the oscillation of solutions of the equation \(y^{(n)} + p(x) y = 0\). (Russian). Trudy Mosk. Mat. Obšč., 10 (1961), 419-436.
[3] J. Ohriska: Oscillation of differential equations and \(v\)-derivatives. Czech. Math. J., 39 (114), (1989), 24-44. · Zbl 0673.34044
[4] J. Ohriska: On the oscillation of a linear differential equation of second order. Czech. Math. J., 39 (114), (1989), 16-23. · Zbl 0673.34043
[5] Ch. G. Philos, Y. G. Sficas: Oscillatory and asymptotic behavior of second and third order retarded differential equations. Czech. Math. J., 32 (107), No. 2 (1982), 169-182. · Zbl 0507.34062
[6] W. T. Reid: Sturmian theory of ordinary differential equations. Springer-Verlag New York Inc., 1980. · Zbl 0459.34001
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.