Oscillation of \(n\)th order equations with perturbations
Two oscillation theorems are proved for the equation \[ x^{(n)}+H(t,x)=Q(t,x), \tag{1} \] \(n\) even. In Theorem 1 it is shown that if the homogeneous equation (2) \(x^{(n)}+H(t,x)=0\) boundedly oscillates and \[ |Q(t,x)|\leq Q_0(t)|x|^{r}, \] where \(r \geq 1\) and \(Q_0(t)\) is small, then every bounded solution of (1) oscillates. Theorem 2 relates the nonoscillatory solutions of (1) to a class of solutions of the equation \(y^{(n)} =Q(t,y)\). Thus, estimates on the growth of the nonoscillatory solutions of (1) are proved. Several examples are given to illustrate the theorems.
scientific article; zbMATH DE number 3558250