Oscillation and Nonoscillation of Linear Ordinary Differential Equations

Abstract

The oscillation and nonoscillation property of solutions of second order linear differential equations is of special interest, and therefore, it has been the subject of many investigations. The interest in second order linear oscillations is due, in a large part, to the fact that many physical systems are modelled by such equations. In this chapter we shall discuss some of the most basic results in the theory of oscillations of linear ordinary differential equations of second order. In Section 2.1, we shall present Sturm and Sturm-Picone comparison theorems which are useful in oscillation theory. In Section 2.2, we shall provide some necessary and sufficient conditions for the nonoscillation as well as some comparison theorems of Sturm’s type. Sufficiency criteria for the nonoscillation are given in Section 2.3. In Section 2.4, we shall establish sufficient conditions for the oscillation of second order differential equations with alternating coefficients. Integral averaging techniques as well as interval criteria for the oscillation are discussed in Section 2.5. In Section 2.6, several criteria for oscillation of linear second order differential equations with integrable coefficients are established. Finally, in Section 2.7 we shall discuss the problems of forced oscillations.