Oscillation and spectral theory of Sturm-Liouville differential equations with nonlinear dependence in spectral parameter
| Authors | |
|---|---|
| Year of publication | 2013 |
| Type | Article in Periodical |
| Magazine / Source | Dynamic Systems and Applications |
| MU Faculty or unit | |
| Citation | |
| Field | General mathematics |
| Keywords | Sturm-Liouville differential equation; nonlinear dependence on spectral parameter; finite eigenvalue; finite eigenfunction; oscillation theorem |
| Attached files | |
| Description | 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 lambda in general nonlinearly, and (ii) the potential is merely monotone in lambda and not necessarily strictly monotone in lambda, 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 lambda nicely follows from our results. |
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