where p(x) is a continuous positive function for 0 0. Our main concern will be to obtain oscillation criteria (or nonoscillation criteria) for the equation (1), i.e. conditions on the function p(x) from which conclusions may be drawn as to the oscillatory, or nonoscillatory, character of (1). There exists an extensive literature on this subject [2-12, to name just a few], which goes back to a classical paper by Kneser [5]. Considerable progress was made in a relatively recent paper by Hille [4], who recognized the relevance of the expression

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Oscillation criteria for second-order linear differential equations

where, in the interval considered, r(x), q(x) and P(x) are functions of C\", C and C, respectively, and the prime denotes differentiation with respect to x. Unless there is an explicit statement to the contrary, it will be assumed that all these functions are defined, and belong to the classes indicated, in the interval (0, oo). Simple examples show that the general equation (1.1) includes equations of radically different oscillatory behavior. For a more penetrating study of the solutions of (1.1) it is therefore necessary to introduce distinctions between certain sub-classes of equations (1.1) which reflect these basic differences. Taking our cue from the analogous second-order equation

https://www.ams.org/tran/1958-089-02/S0002-9947-1958-0102639-X/S0002-9947-1958-0102639-X.pdf

On the oscillation of solutions of self-adjoint linear differential equations of the fourth order

Classification of second order linear differential equations with respect to oscillation

On etablit des conditions suffisantes pour que toutes les solutions soient oscillatoires. On utilise les techniques de Riccati et les principes variationnels

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Riccati techniques and variational principles in oscillation theory for linear systems

(1) x"+a(t)x = 0, t > 0, where a(t) is a locally integrable function of t. We call equation (1) oscillatory if all solutions of (1) have arbitrarily large zeros on [0, oo), otherwise, we say equation (1) is nonoscillatory. As a consequence of Sturm's Separation Theorem [21], if one of the solutions of (1) is oscillatory, then all of them are. The same is true for the nonoscillation of (1). The literature on second order linear oscillation is voluminous. The first such result is of course the classical theorem of Sturm which asserts that

https://www.ams.org/tran/1969-144-00/S0002-9947-1969-0251305-6/S0002-9947-1969-0251305-6.pdf

Oscillation and nonoscillation of solutions of second order linear differential equations with integrable coefficients

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Richard Allan Moore

Papers

The behavior of solutions of a linear differential equation of second order.