Compatibility of the coefficients of a~generalized second order linear differential equation

I. S. Kats

Sbornik. Mathematics, Tome 8 (1969) no. 3, pp. 345-356

Résumé

We consider a boundary value problem for the generalized second order differential equation \begin{equation} -\frac d{dM(x)}\biggl(y^+(x)-\int_{c+0}^{x+0}y(s)dQ(s)\biggr)-\lambda y(x)=0, \end{equation} where $M(x)$ is a nondecreasing function, and $Q(x)$ is the difference of two nondecreasing functions; $y^+(x)$ designates the right derivative of the function $y(x)$. Differential equation (1) is a generalization of the differential equation \begin{equation} -y''+q(x)y-\lambda\rho(x)y=0, \end{equation} where $\rho(x)\geqslant0$ and $q(x)$ are locally integrable real functions. Even when equation (1) is considered on a finite interval and the functions $M(x)$ and $Q(x)$ have bounded variation there (the regular case), it may turn out that not every function in $L_M^{(2)}$ can be expanded in solutions of equation (1) (for equation (2) this is exceptional). In this paper we find a condition which is necessary and sufficient for any function $f(x)\in L_M^{(2)}$ to expand in the solutions (“eigenfunctions”) of the boundary value problem with equation of the form (1); in the case when this condition is not fulfilled, we find the class of all functions in $L_M^{(2)}$ which can be expanded in these “eigenfunctions”. Bibliography: 5 titles.