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.
Keywords
dynamic equation, generalized zero, oscillation,
nonoscillation, inhomogeneous equation, time scale
