Singular eigenvalue problems for second order linear ordinary differential equations

Árpád Elbert; Takaŝi Kusano; Manabu Naito

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Displaying similar documents to “Singular eigenvalue problems for second order linear ordinary differential equations”

On the range of a closed operator in an $L_{1}$ -space of vector-valued functions

Ryotaro Sato (2005)

Commentationes Mathematicae Universitatis Carolinae

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Let $X$ be a reflexive Banach space and $A$ be a closed operator in an $L_{1}$ -space of $X$ -valued functions. Then we characterize the range $R (A)$ of $A$ as follows. Let $0 \neq λ_{n} \in ρ (A)$ for all $1 \leq n < \infty$ , where $ρ (A)$ denotes the resolvent set of $A$ , and assume that ${lim}_{n \to \infty} λ_{n} = 0$ and ${sup}_{n \geq 1} ∥ λ_{n} {(λ_{n} - A)}^{- 1} ∥ < \infty$ . Furthermore, assume that there exists $λ_{\infty} \in ρ (A)$ such that $∥ λ_{\infty} {(λ_{\infty} - A)}^{- 1} ∥ \leq 1$ . Then $f \in R (A)$ is equivalent to ${sup}_{n \geq 1} {∥ {(λ_{n} - A)}^{- 1} f ∥}_{1} < \infty$ . This generalizes Shaw’s result for scalar-valued functions.

The eigenvalues of symmetric Sturm-Liouville problem and inverse potential problem, based on special matrix and product formula

Chein-Shan Liu, Botong Li (2024)

Applications of Mathematics

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The Sturm-Liouville eigenvalue problem is symmetric if the coefficients are even functions and the boundary conditions are symmetric. The eigenfunction is expressed in terms of orthonormal bases, which are constructed in a linear space of trial functions by using the Gram-Schmidt orthonormalization technique. Then an $n$ -dimensional matrix eigenvalue problem is derived with a special matrix $𝐀 : = [a_{i j}]$ , that is, $a_{i j} = 0$ if $i + j$ is odd.Based on the product formula, an integration method with a fictitious...

On the spectrum and eigenfunctions of the operator $(V f) (x) = \int_{0}^{x^{α}} f (t) d t$

I. Yu. Domanov (2007)

Banach Center Publications

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Behaviour of the first eigenvalue of the p-Laplacian in a domain with a hole

M. Sango (2001)

Colloquium Mathematicae

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We investigate the behaviour of a sequence $λ_{s}$ , s = 1,2,..., of eigenvalues of the Dirichlet problem for the p-Laplacian in the domains $Ω_{s}$ , s = 1,2,..., obtained by removing from a given domain Ω a set $E_{s}$ whose diameter vanishes when s → ∞. We estimate the deviation of $λ_{s}$ from the eigenvalue of the limit problem. For the derivation of our results we construct an appropriate asymptotic expansion for the sequence of solutions of the original eigenvalue problem.

On graphs whose second least eigenvalue is at least $- 1$

M. Petrović (1990)

Matematički Vesnik

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Some properties complementary to Brualdi-Li matrices

Chuanlong Wang, Xuerong Yong (2015)

Czechoslovak Mathematical Journal

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In this paper we derive new properties complementary to an $2 n \times 2 n$ Brualdi-Li tournament matrix $B_{2 n}$ . We show that $B_{2 n}$ has exactly one positive real eigenvalue and one negative real eigenvalue and, as a by-product, reprove that every Brualdi-Li matrix has distinct eigenvalues. We then bound the partial sums of the real parts and the imaginary parts of its eigenvalues. The inverse of $B_{2 n}$ is also determined. Related results obtained in previous articles are proven to be corollaries.

Displaying similar documents to “Singular eigenvalue problems for second order linear ordinary differential equations”

On the range of a closed operator in an L 1 -space of vector-valued functions

The eigenvalues of symmetric Sturm-Liouville problem and inverse potential problem, based on special matrix and product formula

On the spectrum and eigenfunctions of the operator ( V f ) ( x ) = ∫ 0 x α f ( t ) d t

Behaviour of the first eigenvalue of the p-Laplacian in a domain with a hole

On graphs whose second least eigenvalue is at least - 1

Some properties complementary to Brualdi-Li matrices

On the range of a closed operator in an $L_{1}$ -space of vector-valued functions

On the spectrum and eigenfunctions of the operator $(V f) (x) = \int_{0}^{x^{α}} f (t) d t$

On graphs whose second least eigenvalue is at least $- 1$