Oscillatory properties of solutions of certain nonlinear third order differential equations (Q2763907)

The authors study the third-order nonlinear differential equation NEWLINE\[NEWLINE y'''+p(t,y,y',y'')y''+q(t,y,y',y'')y'+r(t,y,y',y'')y=0,\tag{1} NEWLINE\]NEWLINE where \(p,q,r: (a,\infty)\times \mathbb{R}^3\to\mathbb{R}\) are continuous functions and obtain new results on the behavior of proper solutions. Here, sufficient conditions are presented, which ensure that every solution to (1) is either oscillatory or it is nonoscillatory and \(y(t)y'(t)>0\) and \(y'(t)y''(t)>0\) for large \(t\). The main tool is a comparison of equation (1) with the third-order linear ordinary differential equation. The authors present also an application to the forced differential equation NEWLINE\[NEWLINE y'''+p(t,y,y',y'')y''+q(t,y,y',y'')y'+r(t,y,y',y'')y=f(t,y,y',y''),NEWLINE\]NEWLINE which is viewed as a perturbation of (1).