Sturm-Liouville Problem for Stationary Differential Operator With Nonlocal Integral Boundary Condition

Abstract

The Sturm‐Liouville problem with various types of nonlocal integral boundary conditions is considered in this paper. In the first part of paper we investigate Sturm‐Liouville problem with two cases of nonlocal integral boundary conditions. We prove general properties of the eigenfunctions and eigenvalues for such problem in the complex case. In the second part we investigate real eigenvalues case. The spectrum depends of these problems on boundary condition parameters is analyzed. Qualitative behaviour of all eigenvalues subject to nonlocal boundary condition parameters is described. Šiame straipsnyje nagrinejamas Šturmo‐Liuvilio uždavinys su viena nelokaliaja integralinio tipo kraštine salyga. Pirmoje straipsnio dalyje tiriamas Šturmo‐Liuvilio uždavinys su dvieju tipu integraline nelokaliaja salyga. Irodytos tikriniu funkciju ir tikriniu reikšmiu bendrosios savybes komplesineje plokštumoje. Antroje dalyje plačiau ištirtas realiuju tikriniu reikšmiu atvejis. Straipsnyje nagrinejama kaip Šturmo‐Liuvilio uždavinio spektras priklauso nuo kraštiniu salygu parametru. Priklausomai nuo nelokaliuju kraštiniu salygu parametru, aprašytas kokybinis tikriniu reikšmiu pasiskirstymas.

Semi-Implicit Difference Scheme for a Two-Dimensional Parabolic Equation With an Integral Boundary Condition

“…We have indicated the properties of the spectrum of a differential operator with the nonlocal condition. However, as noticed in [21], similar results are valid for discrete approximation of the diffusion operator (see also [18,26,35] and references therein).…”

Investigation of Negative Critical Points of the Characteristic Function for Problems with Nonlocal Boundary Conditions

“…Firstly, let us consider the case where ξ is fixed. We define a constant eigenvalue as the eigenvalue λ = q 2 that does not depend on the parameter γ ∈ C [11,17]. For any constant eigenvalue we define the constant eigenvalue point q ∈ C q := {z ∈ C : −π/2 < arg z π/2 or z = 0} and the constant eigenvalue γ-value point (q, γ) ∈ C q × C, respectively.…”

“…Gulin and V. A. Morozova [6], N. I. Ionkin and E. A. Valikova [7], M. Sapagovas and A.Štikonas [8],Štikonas [9], S. Pečiulytė [10][11][12][13]. Such problems with nonlocal integral boundary conditions are analyzed B. Bandyrskii, I. Lazurchak, V. Makarov and M. Sapagovas [14], R.Čiupaila, Z. Jesevičiūtė and M. Sapagovas [15], G. Infante [16], A.Štikonas and S. Pečiulytė [10,17], etc. In recent decades the number of differential problems with nonlocal boundary conditions and numerical methods for such problems have increased significantly.…”

“…For the fixed parameter ξ, dependence of spectra of these problems on the parameter γ in nonlocal boundary conditions has been investigated in the previous research (see, [8,11,13,15,17]) and S. Pečiulytė Doctoral Thesis [10]. Furthermore, conditions, where there exist constant, negative and only real eigenvalues have been obtained in these articles.…”

Numerical Analysis of the Eigenvalue Problem for One-Dimensional Differential Operator with Nonlocal Integral Conditions

“…In papers [7,8] the eigenvalue problems for one-dimensional differential operator with p(x) ≡ 1 and various nonlocal integral conditions are investigated analytically. However, such problems with a non-constant coefficient p(x) are met in the literature quite rarely and are considerably less investigated.…”