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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

##### 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
##### Keywords:
nonoscillatory solutions; v-derivative
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##### 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
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