Oscillation of strongly noncanonical equations
Authors
Blanka Baculikova
 Department of Mathematics, Faculty of Electrical Engineering and Informatics, Technical University of Košice, Letná 9, 042 00 Košice, Slovakia
Abstract
New oscillation criteria for third order noncanonical differential equations of the form
\[
\left(r_2(t)\left(r_1(t)y'(t)\right)'\right)'+p(t)y(\tau(t))=0
\]
are established.
Our technique employs an equivalent canonical representation of the studied equation, which essentially simplifies the examination of noncanonical equations. The results obtained are supported by several illustrative examples.
Keywords
 Oscillation
 third order differential equations
 noncanonical operator
MSC
References

[1]
J. Džurina, Comparison theorems for nonlinear ODE’s, Math. Slovaca, 42 (1992), 299–315.

[2]
P. Hartman, Disconjugate nth order linear differential equations and principal solutions, Bull. Amer. Math. Soc., 74 (1968), 125–129.

[3]
P. Hartman, Principal solutions of disconjugate nth order linear differential equations, Amer. J. Math., 91 (1969), 306– 362.

[4]
I. T. Kiguradze, T. A. Chanturia , Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations, Kluwer Acad. Publ., Dordrecht (1993)

[5]
T. Kusano, M. Naito, Comparison theorems for functional differential equations with deviating arguments, J. Math. Soc. Japan, 33 (1981), 509–532.

[6]
A. Y. Levin , Nonoscillation of solutions of the equation \(x^{(n) }+ p_1(t)x^{(n1)} +... + p_n(t)x = 0\), Uspekhi Mat. Nauk, 24 (1969), 43–49.

[7]
G. S. Ladde, V. Lakshmikantham, B. G. Zhang, Oscillation Theory of Differential Equations with Deviating Arguments, Marcel Dekker, New York (1987)

[8]
G. Pólya, On the meanvalue theorem corresponding to a given linear homogeneous differential equations, Trans. Amer. Math. Soc., 24 (1922), 312–324.

[9]
W. F. Trench, Canonical forms and principal systems for general disconjugate equations, Trans. Amer. Math. Soc., 184 (1974), 319–327.

[10]
D. Willett, Asymptotic behaviour of disconjugate nth order differential equations, Canad. J. Math., 23 (1971), 293–314.