# Oscillation of strongly noncanonical equations

Volume 11, Issue 10, pp 1124--1128 Publication Date: June 27, 2018       Article History
• 236 Views

### 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

•  34C10
•  34K11

### References

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

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

• [3] P. Hartman, Principal solutions of disconjugate n-th 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 , Non-oscillation of solutions of the equation $x^{(n) }+ p_1(t)x^{(n-1)} +... + p_n(t)x = 0$, 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 mean-value 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 n-th order differential equations, Canad. J. Math., 23 (1971), 293–314.