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We show that the eigenvalues of a class of higher-order Sturm-Liouville problems depend not only continuously but also smoothly on boundary points and that the derivative of the nth eigenvalue as a function of an endpoint satisfies a first order differential equation. In addition, we prove that as the length of the interval shrinks to zero all 2kth-order Dirichlet eigenvalues march off to plus infinity; this is also true for the first (i.e., lowest) eigenvalue. [ABSTRACT FROM AUTHOR]

In this paper, we consider the dependence of eigenvalues of sixth-order boundary value problems on the boundary. We show that the eigenvalues depend not only continuously but also smoothly on boundary points, and that the derivative of the $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}n$th eigenvalue as a function of an endpoint satisfies a first-order differential equation. In addition, we prove that as the length of the interval shrinks to zero all higher eigenvalues of such boundary value problems march off to plus infinity. This is also true for the first (that is, lowest) eigenvalue. [ABSTRACT FROM PUBLISHER]

Abstract: We show that a class of regular self-adjoint fourth order boundary value problems (BVPs) is equivalent to a certain class of matrix problems. Conversely, for any given matrix problem in this class, there exist fourth order self-adjoint BVPs which are equivalent to the given matrix problem. Equivalent here means that they have exactly the same eigenvalues. Such an equivalence is considered with the special coupled boundary conditions, i.e. periodic boundary conditions. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the dependence of eigenvalues of a class of fourth-order Sturm–Liouville problems on the boundary. We show that the eigenvalues depend not only continuously but smoothly on boundary points, and that the derivative of the nth eigenvalue as a function of an endpoint satisfies a first order differential equation. In addition, we prove that as the length of the interval shrinks to zero all higher fourth-order Dirichlet eigenvalues march off to plus infinity, this is also true for the first (i.e., lowest) eigenvalue. [Copyright &y& Elsevier]

Abstract: The eigenvalues of a regular fourth-order Sturm–Liouville (SL) problems depend not only continuously but smoothly on the problem. An expression for the derivative of the eigenvalues with respect to a given parameter: an endpoint, a boundary condition, a coefficient, or the weight function, are found. [Copyright &y& Elsevier]