Oscillation theorems for the real, self-adjoint linear system of the second order

It is the object of this paper to determine the number of oscillations of a linear combination of the form (2) for the systems (3) and (4). From these results, an oscillation theorem for the solution up (x), corresponding to the pth characteristic number of (4), is obtained. Given the second order self-adjoint linear differential equation d du1 (1) dxL~~~K x,Xdx G (x,X) u 0 (1) ~~~dx [I z ) dZz,W and two linear combinations of a solution which does not vanish identically and its first derivative, (2) Li [ u (x,X) ] = ai (x,X) u (x,X) (i (x,X) K (x, ) ux (x ,X) (i = 1, 2), we shall impose the following conditions and shall assume that they -are satisfied throughout this paper: I. K(x,X), G(x,X), ai(x,X), i(x,X), aix(x,X), fix(x,X)t are continuous, real functions of x in the interval (X) (a _ x b) and for all real values of X in the interval (A) (-i 0 in (X, A) . III. For each value of x in (X), K and G decrease (or do not increase) * Presented to. the Society, Sept. 6, 1917. tfix (x x) a fi (x, x) 136