Nonlocal multipoint problem
for an ordinary differential equations of even order involution

Ya. O. Baranetskij; P. I. Kalenyuk; L. I. Kolyasa; M. I. Kopach

doi:10.15330/ms.49.1.80-94

We study a nonlocal multipoint problem for an ordinary differential equation of even order with coefficients containing an involution operator. The spectral properties of a self-adjoint operator with boundary conditions generalizing the conditions of antiperiodicity are investigated. For a differential equation of even order, we consider a problem with multipoint conditions that are perturbations of self-adjoint boundary conditions. We study cases when multipoint conditions include boundary conditions that are regular, but not strongly regular according to Birkhoff, or irregular. The eigenvalues and elements of the system of the root functions of the operator of the problem are determined. It is proved that the system is complete and contains an infinite number of associated functions. Sufficient conditions are obtained for which this system is a Riesz basis. Similar results are obtained for the operator generated by the multipoint problem for an ordinary differential equation of even order with coefficients containing the involution operator.

1. Ya.O. Baranetskij, Boundary value problems with irregular conditions for differential-operator equations, Bukov. Mathemat. Journ., 3 (2015), №3–4, 33–40. (in Ukrainian)

2. Ya O. Baranetskij, A.A. Basha, Nonlocal multipoint problem for differential-operator equations of order 2n, J. Math. Sci., 217 (2016), №2, 176–186.

3. Ya.O. Baranetskij, P.I. Kalenyuk, L.I. Kolyasa, Boundary-value for abstract second-order differential equations with operators involution, Bukov. Mathemat. Journ., 4 (2016), №3–4, 22–30.

4. Ya.O. Baranetskij, P.I. Kalenyuk, L.I. Kolyasa, M.I. Kopach, The nonlocal problem for the differentialoperator equation of the even order with the involution, Carpathian Math. Publ., 9 (2017), №2, 109–119.

5. Ya. Baranetskij, L. Kolyasa, Boundary-value problem for abstract second-order differential equation with involution, Journal Visnyk of Lviv Polytechnic National University. Physical and mathematical sciences, 871 (2017), 20–27. (in Ukrainian)

6. G.D. Birkhoff, On the asymptotic character of the solutions of certain linear differential equations containing a parameter, Trans. Amer. Math. Soc., 9 (1908), 219–231.

7. G.D. Birkhoff, Boundary value and expansion problems of ordinary linear differential equations, Trans. Amer. Math. Soc., 91 (1908), №4, 373–395.

8. A. Cabada, F.A.F. Tojo, Solutions and Green’s function of the first order linear equation with reflection and initial conditions, Boundary Value Problems, 99 (2014), 1–16.

9. I.Ts. Gokhberg, M.G. Krein, Introduction to the Theory of Linear Not Self-Adjoint Operators, Nauka, Moscow, 1965, 448 p. (in Russian)

10. N.I. Ionkin, Solution of a boundary-value problem in heat conduction with a nonclassical boundary condition, Differ. Equat., 13 (1977), №2, 204–211.

11. P.I. Kalenyuk, Ya.E. Baranetskii, Z.N. Nitrebich, Generalized Method of the Separation of Variables, Naukova Dumka, Kiev, 1993, 231 p. (in Russian)

12. P.I. Kalenyuk, Ya.O. Baranetskij, L.I. Kolyasa, Nonlocal boundary value problems for the operator of differentiation of even order, Nonclassical problems of the theory of differential equations: A collection of scientific papers is dedicated to the 80th birthday of Bogdan Iosifovich Ptashnik, 2017, 91–110. (in Ukrainian)

13. G.M. Kesel’man, On the unconditional convergence of eigenfunction expansions of certain differential operators, Izv. Vyssh. Uchebn. Zaved. Mat., 39 (1964), 82–93. (in Russian)

14. V.P. Mikhailov, On Riesz bases in L2(0;1), Doklady Akad. Nauk SSSR, 144 (1962), 981–984. (in Russian)

15. A.Yu. Mokin, On the of initial boundary value problems for the heat equation, Differ. Equat., 45 (2009), №1, 126–141.

16. M.A. Naimark, Linear differential operators, Nauka, Moscow, 1969, 528 p. (in Russian)

17. D.O’Regan, Existence results for differential equations with reflection of the argument, J. Aust-tral. Math. Soc., 57 (1994), №2, 237–260.

18. B.Yo. Ptashnyk, V.S. Il’kiv, I.Ya. Kmit’, V.M. Polishchuk, Nonlocal boundary problems for partial differential equations, Naukova Dumka, Kyiv, 2002, 416 p. (in Ukrainian)

19. A.A. Shkalikov, On the basis problem of the eigenfunctions of an ordinary differential operator, Uspekhi Mat. Nauk, 34 (1979), №5, 235–236.

20. M.H. Stone, A comparison of the series of Fourier and Birkhoff, Trans. Amer. Math. Soc. 28 (1926), №4, 695-761.

21. M.H. Stone, Irregular differential systems of ordinary two and the related expansion problems, Trans. Amer. Soc., 29 (1927), №1, 23-53.

22. J. Tamarkin, Sur quelques points de la theorie des equations differentielles lineaires ordinaires et sur la generalisation de la serie de Fourier, Rend. Circ. Matem. Palerm., (1912), №3, 345-382.

23. J. Tamarkin, Some general problems of the theory of ordinary linear differential equations and expansions of an arbitrary function in series of fundamental functions, Math. Z., (1927), 1-54.

	Nonlocal multipoint problem for an ordinary differential equations of even order involution
Author	Ya. O. Baranetskij, P. I. Kalenyuk, L. I. Kolyasa, M. I. Kopach kolyasa.lubov@gmail.com Lviv Polytechnic National University, Lviv, Ukraine
Abstract	We study a nonlocal multipoint problem for an ordinary differential equation of even order with coefficients containing an involution operator. The spectral properties of a self-adjoint operator with boundary conditions generalizing the conditions of antiperiodicity are investigated. For a differential equation of even order, we consider a problem with multipoint conditions that are perturbations of self-adjoint boundary conditions. We study cases when multipoint conditions include boundary conditions that are regular, but not strongly regular according to Birkhoff, or irregular. The eigenvalues and elements of the system of the root functions of the operator of the problem are determined. It is proved that the system is complete and contains an infinite number of associated functions. Sufficient conditions are obtained for which this system is a Riesz basis. Similar results are obtained for the operator generated by the multipoint problem for an ordinary differential equation of even order with coefficients containing the involution operator.
Keywords	differential-operator equation; root function; operator of involution; essentially a non-self-adjoint operator; Riesz basis; nonlocal problem
DOI	doi:10.15330/ms.49.1.80-94
Reference	1. Ya.O. Baranetskij, Boundary value problems with irregular conditions for differential-operator equations, Bukov. Mathemat. Journ., 3 (2015), №3–4, 33–40. (in Ukrainian) 2. Ya O. Baranetskij, A.A. Basha, Nonlocal multipoint problem for differential-operator equations of order 2n, J. Math. Sci., 217 (2016), №2, 176–186. 3. Ya.O. Baranetskij, P.I. Kalenyuk, L.I. Kolyasa, Boundary-value for abstract second-order differential equations with operators involution, Bukov. Mathemat. Journ., 4 (2016), №3–4, 22–30. 4. Ya.O. Baranetskij, P.I. Kalenyuk, L.I. Kolyasa, M.I. Kopach, The nonlocal problem for the differentialoperator equation of the even order with the involution, Carpathian Math. Publ., 9 (2017), №2, 109–119. 5. Ya. Baranetskij, L. Kolyasa, Boundary-value problem for abstract second-order differential equation with involution, Journal Visnyk of Lviv Polytechnic National University. Physical and mathematical sciences, 871 (2017), 20–27. (in Ukrainian) 6. G.D. Birkhoff, On the asymptotic character of the solutions of certain linear differential equations containing a parameter, Trans. Amer. Math. Soc., 9 (1908), 219–231. 7. G.D. Birkhoff, Boundary value and expansion problems of ordinary linear differential equations, Trans. Amer. Math. Soc., 91 (1908), №4, 373–395. 8. A. Cabada, F.A.F. Tojo, Solutions and Green’s function of the first order linear equation with reflection and initial conditions, Boundary Value Problems, 99 (2014), 1–16. 9. I.Ts. Gokhberg, M.G. Krein, Introduction to the Theory of Linear Not Self-Adjoint Operators, Nauka, Moscow, 1965, 448 p. (in Russian) 10. N.I. Ionkin, Solution of a boundary-value problem in heat conduction with a nonclassical boundary condition, Differ. Equat., 13 (1977), №2, 204–211. 11. P.I. Kalenyuk, Ya.E. Baranetskii, Z.N. Nitrebich, Generalized Method of the Separation of Variables, Naukova Dumka, Kiev, 1993, 231 p. (in Russian) 12. P.I. Kalenyuk, Ya.O. Baranetskij, L.I. Kolyasa, Nonlocal boundary value problems for the operator of differentiation of even order, Nonclassical problems of the theory of differential equations: A collection of scientific papers is dedicated to the 80th birthday of Bogdan Iosifovich Ptashnik, 2017, 91–110. (in Ukrainian) 13. G.M. Kesel’man, On the unconditional convergence of eigenfunction expansions of certain differential operators, Izv. Vyssh. Uchebn. Zaved. Mat., 39 (1964), 82–93. (in Russian) 14. V.P. Mikhailov, On Riesz bases in L2(0;1), Doklady Akad. Nauk SSSR, 144 (1962), 981–984. (in Russian) 15. A.Yu. Mokin, On the of initial boundary value problems for the heat equation, Differ. Equat., 45 (2009), №1, 126–141. 16. M.A. Naimark, Linear differential operators, Nauka, Moscow, 1969, 528 p. (in Russian) 17. D.O’Regan, Existence results for differential equations with reflection of the argument, J. Aust-tral. Math. Soc., 57 (1994), №2, 237–260. 18. B.Yo. Ptashnyk, V.S. Il’kiv, I.Ya. Kmit’, V.M. Polishchuk, Nonlocal boundary problems for partial differential equations, Naukova Dumka, Kyiv, 2002, 416 p. (in Ukrainian) 19. A.A. Shkalikov, On the basis problem of the eigenfunctions of an ordinary differential operator, Uspekhi Mat. Nauk, 34 (1979), №5, 235–236. 20. M.H. Stone, A comparison of the series of Fourier and Birkhoff, Trans. Amer. Math. Soc. 28 (1926), №4, 695-761. 21. M.H. Stone, Irregular differential systems of ordinary two and the related expansion problems, Trans. Amer. Soc., 29 (1927), №1, 23-53. 22. J. Tamarkin, Sur quelques points de la theorie des equations differentielles lineaires ordinaires et sur la generalisation de la serie de Fourier, Rend. Circ. Matem. Palerm., (1912), №3, 345-382. 23. J. Tamarkin, Some general problems of the theory of ordinary linear differential equations and expansions of an arbitrary function in series of fundamental functions, Math. Z., (1927), 1-54.
Pages	80-94
Volume	49
Issue	1
Year	2018
Journal	Matematychni Studii
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Table of content of issue	html

Nonlocal multipoint problem for an ordinary differential equations of even order involution