Oscillation problems for Hill's equation with periodic damping

This paper deals with the second-order linear differential equation x″+a(t)x′+b(t)x=0, where a and b are periodic coefficients. The main purpose is to present new criteria which guarantee that all nontrivial solutions are nonoscillatory and that those are oscillatory. Our nonoscillation theorem and oscillation theorem are proved by using the Riccati technique. In our theorem, the composite function of an indefinite integral of b and a suitable multiple-valued continuously differentiable function are focused, and the composite function of them plays an important role. The results obtained here include a result by Kwong and Wong [15] and a result by Sugie and Matsumura [26]. An application to a equation of Whittaker-Hill type is given to show the usefulness of our results. Finally, simulations are also attached to illustrate that our oscillation criterion is sharp.