Two-parameter right definite Sturm—Liouville problems with eigenparameter-dependent boundary conditions

Abstract

Linked equations are studied on [0,1] subject to boundary conditions of the form Results are given on existence, location, asymptotics and perturbation of the eigenvalues λj and oscillation of the eigenfunctions yi.

Right-Definite Multiparameter Sturm–liouville Problems With Eigenparameter-Dependent Boundary Conditions

“…This is a generalization of two-parameter results proved in [2]. The proofs here are similar and depend on results in [8].…”

“…The proofs of (4) and (5) follow by considering the corresponding asymptotic problems and are similar to the proof of [2,Lemma 3.4].…”

Two-Parameter Uniformly Elliptic Sturm–liouville Problems With Eigenparameter-Dependent Boundary Conditions

“…In § 2 we prove the oscillation theorem in the uniform-left-definite (ULD) case. As in [1], this result depends heavily on the asymptotic nature of the zeroth eigencurves of (1) and (2). In § 3, we remove the ULD assumption and retain only the uniform ellipticity (UE).…”

“…with the boundary conditions have been investigated and results about the existence and oscillation theory are known [6]; there are also parameter dependence results and asymptotic expansions [6]. Klein's oscillation theorem for equations (1) and 2 states that, for each non-negative integer pair (m, n), there is a unique eigenvalue (λ, µ) ∈ R 2 and (up to scalar multiples) a unique pair of eigenfunctions (y 1 , y 2 ) such that y 1 has m zeros and y 2 has n zeros in (0, 1). A special case was proved by Klein, and the general one (for continuous coefficients) was proved by Ince [9].…”

“…There it was shown that each eigenvalue has a unique oscillation count (m, n) where m and n are the number of zeros of the corresponding eigenfunctions y 1 and y 2 , respectively. In addition, there is an oscillation theorem [1,Theorem 4.4] that addresses the extent to which the converse is true.…”