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

Greguš, M.; Graef, J. R.; Gera, M.

Nonlinear Anal., Theory Methods Appl. 28, No. 10, 1611-1622 (1997).

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.

MSC:

34C10

Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations

Keywords:

asymptotic behavior; proper solution; oscillatory solution; nonoscillatory

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