The boundary problem associated with a differential equation in which the coefficient of the parameter changes sign

Abstract

with coefficients real and continuous, and X a complex parameter, has been extensively studied. The results of these investigations (and of those dealing with the generalized system of order n) comprise an .extensive theory for the case in which the coefficient of the parameter maintains its sign in the interval determined by the boundary conditions. Not merely the existence, but the asymptotic forms of the characteristic values and functions are known, and the expansibility of a function, arbitrary within wide restrictions, in a series of characteristic functions has been established. Far less has been done with the cases in which k(x) either changes sign or otherwise vanishes in the given interval. The existence of infinitely many characteristic values has been variously established,t but their distribution and the form of the corresponding solutions has not been determined. If k(x) changes sign a finite number of times Hilbert's theory of the polar integral equation is applicable and yields the theorem that a function which is continuous together with its first four derivatives and which satisfies the boundary and certain auxiliary conditions is expansible in a uniformly convergent series of characteristic functions. The method of infinitely many variables has been applied with considerable success to the more general case in which k(x) is less restricted.: Still the expansion theorems obtained are hardly comparable with those which have been obtained under the hypothesis that the sign of k(x) does not change.