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

Similarity:

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 ≠ λ n ∈ ρ ( A ) for all 1 ≤ n < ∞ , where ρ ( A ) denotes the resolvent set of A , and assume that lim n → ∞ λ n = 0 and sup n ≥ 1 ∥ λ n ( λ n - A ) - 1 ∥ < ∞ . Furthermore, assume that there exists λ ∞ ∈ ρ ( A ) such that ∥ λ ∞ ( λ ∞ - A ) - 1 ∥ ≤ 1 . Then f ∈ R ( A ) is equivalent to sup n ≥ 1 ∥ ( λ n - A ) - 1 f ∥ 1 < ∞ . 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

Similarity:

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...

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

M. Sango (2001)

Colloquium Mathematicae

Similarity:

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.

Some properties complementary to Brualdi-Li matrices

Chuanlong Wang, Xuerong Yong (2015)

Czechoslovak Mathematical Journal

Similarity:

In this paper we derive new properties complementary to an 2 n × 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.

Sublinear eigenvalue problems on compact Riemannian manifolds with applications in Emden-Fowler equations

Alexandru Kristály, Vicenţiu Rădulescu (2009)

Studia Mathematica

Similarity:

Let (M,g) be a compact Riemannian manifold without boundary, with dim M ≥ 3, and f: ℝ → ℝ a continuous function which is sublinear at infinity. By various variational approaches, existence of multiple solutions of the eigenvalue problem - Δ g ω + α ( σ ) ω = K ̃ ( λ , σ ) f ( ω ) , σ ∈ M, ω ∈ H₁²(M), is established for certain eigenvalues λ > 0, depending on further properties of f and on explicit forms of the function K̃. Here, Δ g stands for the Laplace-Beltrami operator on (M,g), and α, K̃ are smooth positive functions. These...