ON ROOT FUNCTIONS OF NONLOCAL DIFFERENTIAL SECOND-ORDER OPERATOR WITH BOUNDARY CONDITIONS OF PERIODIC TYPE
Dildabek G. Ivanova M.B. Sadybekov M.A.
31 December 2021al-Farabi Kazakh State National University
KazNU Bulletin. Mathematics, Mechanics, Computer Science Series
2021#112Issue 429 - 44 pp.
In this paper we consider one class of spectral problems for a nonlocal ordinary differential operator (with involution in the main part) with nonlocal boundary conditions of periodic type. Such problems arise when solving by the method of separation of variables for a nonlocal heat equation. We investigate spectral properties of the problem for the nonlocal ordinary differential equation Ly (x) ≡ −y′′ (x) + εy′′ (−x) = λy (x), −1 < x < 1. Here λ is a spectral parameter, |ε| < 1. Such equations are called nonlocal because they have a term y′′ (−x) with involutional argument deviation. Boundary conditions are nonlocal y′ (−1) + ay′ (1) = 0, y (−1) − y (1) = 0. Earlier this problem has been investigated for the special case a = −1. We consider the case a ≠ −1. A criterion for simplicity of eigenvalues of the problem is proved: the eigenvalues will be simple if and only if the number r =√(1 − ε) / (1 + ε) is irrational. We show that if the number r is irrational, then all the eigenvalues of the problem are simple, and the system of eigenfunctions of the problem is complete and minimal but does not form an unconditional basis in L2 (−1, 1). For the case of rational numbers r, it is proved that a (chosen in a special way) system of eigen-and associated functions forms an unconditional basis in L2 (−1, 1).
associated function , eigenfunction , eigenvalue , multiplicity of eigenvalues , Nonlocal differential operator , spectrum , unconditional basis
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Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan
South Kazakhstan Medical Academy, Shymkent, Kazakhstan
Institute of Mathematics and Mathematical Modeling
South Kazakhstan Medical Academy
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