Helen A. Merrill
On Solutions of Differential Equations which Possess an Oscillation Theorem
Transactions of the American Mathematical Society, Vol. 4, No. 4 (Oct. 1903), 423-433
Introduction
The object of this paper is the study of differential equations possessing certain properties, later defined, which may be called Sturmian properties, since they characterize equations first discussed by Sturm in his celebrated Mémoire sur les équations linéaires du second ordre, in the first volume of Liouville's Journal. This memoir of Sturm's is chiefly devoted to the study of the equation
with a few slightly more general forms. At the close of the article he refers briefly to the fact that these results were first obtained by the use of equations in finite differences of the second order, but none of this work was ever published, owing probably to the fact that at that time it was impossible to treat the subject with sufficient rigor by this method. In the present state of analysis Sturm's original method may easily be placed on a satisfactory basis as regards rigor, and it is found to possess certain marked advantages, due to the ease with which recurrent relations satisfying the necessary conditions are set up, and to the possibility of generalizing much of the work. Sturm's famous theorem in regard to the location of the real roots of an equation is only a result of the application of this general method to polynomials.
In order to establish on a rigorous basis the method of finite differences, se will be made of a recent theorem of Painlévé and Picard in regard to the Cauchy-Lipschitz proof of the existence of a solution of a differential equation.
After recalling some familiar definitions and theorems of which use will be made, a method will be developed by which recurrent relations of the required character are constructed. Some properties of the functions which satisfy these relations will then be established, and it will be shown that these same properties characterize the differential equations, to which under certain conditions the recurrent relations lead.