Characteristic determinant of the spectral problem for the ordinary differential operator with the boundary load

Abstract

In this paper, we consider a linear operator, generated by the ordinary differential expression with the boundary load and strongly regular boundary conditions of the general form. We prove the possibility of constructing the characteristic determinant of the spectral problem.

On Stability of Basis Property of Root Vectors System of the Sturm-Liouville Operator with an Integral Perturbation of Conditions in Nonstrongly Regular Samarskii-Ionkin Type Problems

“…Our aim is to show that the basis property in 2 (0, 1) of the E&AF system of problem (9)- (11) is not stable at small changes of kernel ( ) of integral perturbation. In [19] the construction method of the characteristic determinant of the spectral problem with integral perturbation of the boundary conditions has been suggested. The spectral properties of nonlocal problems have been considered in [20].…”

“…From analysis of (19) it is also easy to see that Δ 1 ( 0 ) = 0 for all > . That is, all eigenvalues 0 , > , of the Samarskii-Ionkin problem are eigenvalues of the perturbed spectral problem (9)- (11).…”

“…In this section we use the method of our paper [19] to construct the characteristic determinant of the problem with integral perturbation of the boundary condition.…”

Differential operator equations with interface conditions in modified direct sum spaces

“….. be any Cauchy sequence in the inner-product space H 2 . Since (8) we see that the first components (u n (.)) of the sequence (U n ) forms a Cauchy sequence of the Hilbert space W 2 2 (−π, 0)⊕W 2 2 (0, π), therefore is convergent.…”

Asymptotic distribution of eigenvalues for fourth-order boundary value problem with discontinuous coefficients and transmission conditions

“…There are many papers and books that the spectral properties of such problem are investigated; see [2], [3], [6]. Some spectral properties of such problems with discontinuous coe¢ cients and the eigenvalue parameter both in the di¤erential equation and boundary conditions have been studied by O. Sh.…”