Existence theorems for the general real self-adjoint linear system of the second order

Abstract

Sturmt in 1836 established many fundamental theorems concerning the properties of solutions of the linear differential equation (1) below, and the system formed by (1) and the boundary conditions (3), of which the oscillation theorems of ? 1 of the present paper are immediate consequences. In the special case of periodic conditions Masont and Bocher? with more or less generality established an oscillation theorem. Birkhoff II extended their work to the general self-adjoint linear boundary conditions (see (5) and (1)), where, however, he assumed K = 1, and that X does not enter into the boundary conditions, and established an oscillation theorem for up (x), the solution corresponding to the pth characteristic number. It will be the object of this paper to generalize these results to the most general real, self-adjoint linear system of the second order, where K and the coefficients of the boundary conditions are functions of X, by extending Bocher's and Birkhoff's methods,? which are based on the application of Sturm's theorems to this system.**