Some properties of eigenvalues and generalized eigenvectors of one boundary value problem

Abstract

We investigate a discontinuous boundary value problem which consists of a Sturm-Liouville equation with piecewise continuous potential together with eigenparameter dependent boundary conditions and supplementary transmission conditions. We establish some spectral properties of the considered problem. In particular, it is shown that the problem under consideration has precisely denumerable many eigenvalues λ 1 , λ 2 , ..., which are real and tends to +∞. Moreover, it is proven that the generalized eigenvectors form a Riesz basis of the adequate Hilbert space.

Oscillation Properties for Non-Classical Sturm-Liouville Problems With Additional Transmission Conditions

“…In this work we will extend and generalize some well-known Sturm's comparison and oscillation theorems for problems of a new type, which consist of Sturm-Liouville equation In recent years, the authors of this study and some other mathematicians have studied many other spectral properties of similar BTVP's (see, for example, [1,2,4,5,6,7,9,13,17,22,23,27,28]). BVTP's also find many important application in the study of various phenomena in physics and technology, such as vibration of string involving different types of loads (see, for example, [12,14,18,21,25,26]), heat transfer through a solid-liquid interface(see, for example, [8,15,20]), water vapor diffusion through a porous membrane (see, for example, [24]).…”

Numerical Analysis for Sturm–Liouville Problems with Nonlocal Generalized Boundary Conditions

“…An eigenparameter-dependent boundary shape function is such a function that it satisfies the eigenparameter-dependent boundary conditions in Equations ( 2) and (3), automatically. Theorem 1 is crucial that the eigenparameter-dependent boundary shape function can be constructed from Equations (11) and (12) easily, with the help of two shape functions s 1 (x, λ) and s 2 (x, λ) and a new function v(x).…”

“…However, we emphasize the numerical aspect of the spectral problem of Equations ( 1)-( 3), which is not addressed in [1]. Currently, the Sturm-Liouville problems with eigenparameter-dependent boundary conditions and transmission conditions are also important issues in many studies [11,12]. Recently, many engineering issues and mathematical schemes have been presented resolving the GSLPs, e.g., an analytical method to acquire the sharp estimates for the lowest positive periodic eigenvalue and all Dirichlet eigenvalues of a GSLP [13]; a class of generalized discontinuous GSLPs with boundary conditions rationally dependent on the eigenparameter was solved by using operator theoretic formulation under the new inner product [14]; the GSLP had infinite eigenvalues and corresponding eigenfunctions existed such that the sequence of eigenvalues was increasing as shown in [15]; a thorough formulation of the weighted residual collocation method was based on the Bernstein polynomials for a class of Sturm-Liouville boundary value problems [16]; the infinitely dimensional minimization problem of the lowest positive Neumann eigenvalue for the SLP [17], and an optimal design of a structure was described by a GSLP with a spectral parameter in the boundary conditions [18].…”

Non-classical periodic boundary value problems with impulsive conditions

“…Recently, there has been an increasing interest in Sturm-Liouville boundary value problems defined on two or more disjoint segments with common ends, the so-called many-interval SLPs (see, for example, [9][10][11][12][13][14][15][16][17][18][19][20][21] and references cited therein). To deal with such multi-interval boundary value problems, naturally, additional conditions (the so-called transmission conditions, jump conditions, interface conditions, and impulsive conditions) are imposed at these common endpoints.…”