Characteristic Functions for Sturm—liouville Problems With Nonlocal Boundary Conditions

Abstract

Abstract. This paper presents some new results on a spectrum in a complex plane for the second order stationary differential equation with one Bitsadze-Samarskii type nonlocal boundary condition. In this paper, we survey the characteristic function method for investigation of the spectrum of this problem. Some new results on characteristic functions are proved. Many results of this investigation are presented as graphs of characteristic functions. A definition of constant eigenvalues and the characteristic function is introduced for the Sturm-Liouville problem with general nonlocal boundary conditions.

Nodal Solutions of Nonlocal Boundary Value Problems

“…The existence of positive solutions of BVPs with nonlocal BCs, including three-point, multi-point, and integral BCs, have been studied extensively, see, for example, [1,4,5,6,7,8,9,15,16,24,26,28,29,31] and the references therein. In recent years, progress has also been made to the study of nodal solutions, i.e., solutions with a specific zero-counting property in (a, b), for nonlinear BVPs consisting of Eq.…”

On the stability of a weighted finite difference scheme for wave equation with nonlocal boundary conditions

“…We plot a graph of function (31) in each interval: γ " γpαq, α P r0, π{hs; γ :" γ´pβq " γpıβq, β ě 0; γ :" γ`pβq " γpπ{h`ıβq, β ě 0. We combine them on one graph of real characteristic function [13]. Finally, we add vertical lines q " α " 2πk, k " 0, N 1 , which are the roots of the first type, and get generalized characteristic function.…”

“…A function γ " γpqq is called complex-real characteristic function [13]. The roots of the second type q 2k`1 , k " 0, N 1 , N 1 :" tN {2u can be found as γ-values of the characteristic function (31).…”

“…There were found eigenvalues of two types: the first type eigenvalues do not depending on γ, and the second type eigenvalues which do not depend on γ and exist only for some rational ξ. The authors introduced a method of generalized characteristic functions [13]. The complex eigenvalues exist for these problems.…”

Inverse Nodal Problem for a Class of Nonlocal Sturm‐liouville Operator

“…Eigenvalue problems with nonlocal conditions are closely linked to boundary problems for diffrential equations with nonlocal conditions [6,11,13,14]. In the papers [9,24,30,25,26] the similar problems are investigated for the operators with nonlocal condition of Bitsadze-Samarskii or integral type. Albeverio, Hryniv and Nizhnik [1] solve the inverse spectral problem for a class of Sturm-Liouville operators with singular non-local potentials and non-local boundary conditions.…”