On one boundary value problem with a spectral parameter in the boundary conditions

Abstract

We consider the following boundary value problem with a spectral parameter in the equation and the boundary conditions:-u" + q(z)u = AZu, O<x<l,Here A is a spectral parameter, q(z) is a nonnegative continuous function on the interval [0,1], and ai and/3/are real constants. The present article is devoted to studying the properties of the eigenvalues and eigenfunctions of the boundary value problem (0.1)-(0.3). Boundary value problems for ordinary differential equations with a parameter in the boundary conditions in various statements were studied in many articles [1][2][3][4][5][6][7][8][9][10][11][12]. Problems with similar statements were studied in [10][11][12]. However, there is an essential difference between the problem (0.1)-(0.3) and those of [10-121 . Namely, in [10-121 the authors supposed that al, fll = 0, in which case the problem reduces to a linear one. In our case the boundary condition involves both A and As; therefore, it is natural to implement the operator interpretation of the problem in terms of quadratic operator pencils (see, for instance, A. G. Kostyuchenko and A. A. Shkalikov's article [13]). The main result of the present article is Theorem 2.1 on the number of zeros of eigenfunctions under their natural enumeration and Theorem 4.1 on asymptotic formulas for eigenvalues and eigenfunctions. It was A. A. Shkalikov who drew the authors' attention to nontriviality of this problem in the case when the boundary conditions involve A and A s.Henceforth we suppose that the following condition is satisfied:

On eigenvalues of a boundary value problem for a second order elliptic differential-operator equation

“…In these notation, from (2.5), it follows that the eigenvalue λ is a root of the equation As was noted, the problem (1.1), (1.2) is reduced to the study of spectral problem (2.1), (2.2), for every k. The similar lemma for boundary value problems of the form (2.1), (2.2) was proved in [11]. Therefore we omit the proof of Lemma 2.2.…”

“…In [11], a boundary value problem for a second order ordinary differential equation is considered when one, and the same, spectral parameter λ quadratically participates in the equation, while in the boundary conditions it appears as a quadratic trinomial (with respect to λ). In particular, the asymptotic behavior of eigenvalues of the considered boundary value problems is studied.…”

Inverse nodal problem for p–laplacian dirac system

“…In [21], eigenfunction expansion for one-dimensional Dirac operators describing the motion of a particle in quantum mechanics is investigated. Inverse nodal problem for Dirac system without spectral parameter in boundary condition has been studied by various authors [22][23][24][25][26][27][28][29].…”

Asymptotic Behaviour of Eigenvalues and Eigenfunctions of a Sturm-Liouville Problem with Retarded Argument

“…Also, differential equations and nonlinear differential equations have been studied by many mathematicians in several ways for a long time cf. [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20].…”

“…The asymptotic formulas for the eigenvalues and eigenfunctions of the Sturm-Liouville problem with the spectral parameter in the boundary condition were obtained in [17][18][19][20].…”