Oscillation and nonoscillation of forced second order dynamic equations. (English) Zbl 1160.34029

Bohner, Martin; Tisdell, Christopher C.

Pac. J. Math. 230, No. 1, 59-71 (2007).

Summary: Oscillation and nonoscillation properties of second order Sturm-Liouville dynamic equations on time scales – for example, second order self-adjoint differential equations and second order Sturm-Liouville difference equations – have attracted much interest. Here we consider a given homogeneous equation and a corresponding equation with forcing term. We give new conditions implying that the latter equation inherits the oscillatory behavior of the homogeneous equation. We also give new conditions that introduce oscillation of the inhomogeneous equation while the homogeneous equation is nonoscillatory. Finally, we explain a gap in a result given in the literature for the continuous and the discrete case. A more useful result is presented, improving the theory even for the corresponding continuous and discrete cases. Examples illustrating the theoretical results are supplied.

MSC:

34C10	Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
39A10	Additive difference equations

Keywords:

dynamic equation; generalized zero; oscillation; nonoscillation; inhomogeneous equation; time scale