Oscillating nonlinear third order differential equations. (English) Zbl 0871.34022

We consider a nonlinear third order differential equation of the form \[ y'''+ p(t)y''+ q(t)y'+r(t)y= f(t,y,y',y''),\tag{1} \] where \(p\) and \(r\) are continuous functions and \(q\) is a continuously differentiable function on the interval \(I=[a,+\infty)\), \(0<a<+\infty\), and \(f\in C(I\times \mathbb{R}^3,\mathbb{R})\) with \[ f(t,y_1,y_2,y_3)y_1<0 \quad\text{for all}\quad (t,y_1,y_2,y_3)\in I\times \mathbb{R}^3,\;y_1\neq 0. \] By a solution of (1) (proper solution) we mean a function \(y\) defined on an interval \([t_0,\infty)\subset I\), having a continuous third derivative with \(\sup\{|y(s)|: s>t\}>0\) for any \(t\in[t_0,+\infty)\), and which satisfies equation (1). By an oscillatory solution we mean a solution of (1) that has arbitrarily large zeros. Otherwise, the solution is said to be nonoscillatory. The aim of this paper is to study the properties of solutions of equation (1) in the case when the complete third order linear differential equation \(y'''+p(t)y''+ q(t)y'+r(t)y=0\) is oscillatory on \(I\), i.e. it has at least one nontrivial oscillatory solution.


34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
Full Text: DOI


[1] Kiguradze, I. T., An oscillation criterion for a class of ordinary differential equations, Differentsial’nye Uravneniya, 28, 207-219 (1992) · Zbl 0768.34018
[3] Gregus, M.; Gera, M., Some results in the theory of a third-order linear differential equation, Ann. Polon. Math., 42, 93-102 (1983) · Zbl 0534.34015
[4] Kondrat’ev, V. A., An elementary derivation of a necessary and sufficient condition for non-oscillation of the solutions of a second order linear differential equation, Uspehi Mat. Nauk, 12, 159-160 (1957)
[5] Rovder, Y., Oscillation criteria for third-order linear differential equations, Mat. Časopis, 25, 231-244 (1975) · Zbl 0309.34028
[6] Gera, M., Bedingungen für die Existens Oszillatorischer Lösungen der Gleichung \(x\)‴ + \(a\) (t) x″ + b (t) x′ + c (t) \(x =0 c (t) \)≥0, Mat. Čas., 25, 23-40 (1975) · Zbl 0316.34031
[7] Gregus, M., Third Order Linear Differential Equations (1987), Riedel · Zbl 0878.34025
[8] Lazer, A. C., The behavior of solutions of the differential equation \(y\)‴ + p (x) y′ + q (x) y = 0, Pacific J. Math., 7, 435-466 (1966) · Zbl 0143.31501
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.