OSCILLATION AND SPECTRAL THEORY OF STURM-LIOUVILLE DIFFERENTIAL EQUATIONS WITH NONLINEAR DEPENDENCE IN SPECTRAL PARAMETER

摘要

In this paper, we consider the eigenvalue problem for the second order Sturm-Liouville differential equation and the Dirichlet boundary conditions. Our setting is more general than in the current literature in two respects: (i) the coefficients depend on the spectral parameter λ in general nonlinearly, and (ii) the potential is merely monotone in λ and not necessarily strictly monotone in λ, so that the usual strict normality assumption is now removed. This general setting leads to new definitions of an eigenvalue and an eigenfunction - called a finite eigenvalue and a finite eigenfunction. With these new concepts we show that the finite eigenvalues are isolated, bounded from below, and establish an oscillation theorem, i.e., a result counting the zeros of the finite eigenfunctions. The traditional theory in which the potential is linear and strictly monotone in λ nicely follows from our results.