Abstract
In this paper, relations between discrete Sturm--Liouville problem with nonlocal integral boundary condition characteristics (poles, critical points, spectrum curves) and graphs characteristics (vertices, edges and faces) were found. The previous article was devoted to the Sturm--Liouville problem in the case two-points nonlocal boundary conditions.
Highlights
Problem [3, 4, 5] with Integral Boundary Condition were found using Euler’s charakteristic formula [2]
In previous article [5] we have found relations between spectrum curve properties and graphs theory in the case of two-points nonlocal boundary conditions
All nonconstant eigenvalues are γ-points of (Complex-Real) Characteristic Function (CF)[6]
Summary
Particular properties of the spectrum of a discrete Sturm–Liouville. Problem (dSLP) [3, 4, 5] with Integral Boundary Condition were found using Euler’s charakteristic formula [2]. Z h (q ) γc(q) = γc(q; ξ) := Pξh(q) , q ∈ Chq. All nonconstant eigenvalues (which depend on the parameter γ) are γ-points of (Complex-Real) Characteristic Function (CF)[6]. We denote a set Poles P := {pi, i = 1, np}, where np is the number of poles at Chq. For our problems P ⊂ Rhx ∪ {0} and all poles are of the first order (we write deg+(p) = 1, p ∈ P). There exist Pole Points (PP) of the second order. They are described as follows: p1k2 = 2nk/ gcd (m+, m−). For poles and CP deg+(q), q ∈ P ∪B∪{∞}, corresponds to the number of outgoing Spectrum Curves at that point.
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