Надіслати цю статтю Нелокальна задача з багатоточковими збуреннями сильно регулярних за Біркгофом крайових умов для диференціального оператора парного порядку
Анотація
Досліджено спектральні властивості несамоспряженої задачі для оператора диференціювання порядку 2n з нелокальними умовами, що є багатоточкови ми збуреннями сильно регулярних за Біркгофом самоспряжених умов. Встановлено достатні умови, за яких система власних функцій є повною і при деяких додаткових припущеннях утворює базис Рісса. Побудовано множину операторів перетворення, кожен елемент якої відображає систему власних функцій незбуреної задачі у систему власних функцій деякої ізоспектральної задачі. Вивчено випадки задач з регулярними та нерегулярними за Біркгофом двоточковими збуреннями. Визначено умови існування і єдиності розв’язку задачі.
Зразок для цитування: Я. О. Баранецький, І. І. Демків , П. І. Каленюк, “Нелокальна задача з багатоточковими збуреннями сильно регулярних за Біркгофом крайових умов для диференціального оператора парного порядку,” Мат. методи та фіз.-мех. поля, 63, No. 1, 21–36 (2020), https://doi.org/10.15407/mmpmf2020.63.1.21-36
Translation: Y. О. Baranetskij , І. І. Demkiv, P. І. Kalenyuk, “Nonlocal problem with multipoint perturbations of the Birkhoff strongly regular boundary conditions for an even-order differential operator,” J. Math. Sci., 270, No. 1, 19–38 (2023), https://doi.org/10.1007/s10958-023-06330-3
Ключові слова
Посилання
Ya. О. Baranetskij, P. І. Kalenyuk, “Boundary-value problems with Birkhoff regular but not strongly regular conditions for a second-order differential operator,” Mat. Met. Fiz.-Mekh. Polya, 59, No. 4, 7–23 (2016); English translation: J. Math. Sci., 238, No. 1, 1–21 (2019), https://doi.org/10.1007/s10958-019-04214-z
Ya. О. Baranetskij, P. І. Kalenyuk, “Nonlocal multipoint problem with multiple spectrum for an ordinary (2n)th order differential equation,” Mat. Met. Fiz.-Mekh. Polya, 60, No. 3, 32–45 (2017); English translation: J. Math. Sci., 246, No. 2, 152–169 (2020), https://doi.org/10.1007/s10958-020-04727-y
Ya. О. Baranetskij, P. І. Kalenyuk, “A non-local problem with multipoint perturbations of the boundary conditions of the Sturm-type for an ordinary differential equation of even order,” Mat. Met. Fiz.-Mekh. Polya, 62, No. 1, 25–36 (2019).
Ya. О. Baranetskij, P. І. Kalenyuk, L. I. Kolyasa, “Spectral properties of nonself-adjoint nonlocal boundary-value problems for the operator of differentiation of even order,” Ukr. Mat. Zh., 70, No. 6, 739–751 (2018); English translation: Ukr. Math. J., 70, No. 6, 851–865 (2018), https://doi.org/10.1007/s11253-018-1538-4
Ya. О. Baranetskij, P. І. Kalenyuk, M. I. Kopach, “Nonlocal multipoint problem for partial differential equations of even order with constant coefficients,“ Mat. Met. Fiz.-Mekh. Polya, 61, No. 1, 11–30 (2018); English translation: J. Math. Sci., 249, No. 3, 307–332 (2020), https://doi.org/10.1007/s10958-020-04945-4
Ju. M. Berezanskii, Expansions in Eigenfunctions of Selfadjoint Operators, Amer. Math. Soc., Providence (1968).
I. C. Gohberg, M. G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, Amer. Math. Soc., Providence (1969).
V. A. Il’in, L. V. Kritskov, “ Properties of spectral expansions corresponding to non-self-adjoint differential operators,” Itogi Nauk. Tekhn. Ser., 96, 5–105 (2006); English translation: J. Math. Sci., 116, No. 5, 3489–3550 (2003), https://doi.org/10.1023/A:1024180807502
P. Kalenyuk, Ya. Baranetskij, L. Kolyasa, “Nonlocal boundary-value problem for a differentiation operator of even order,” in: Nonclassical Problems of the Theory of Differential Equations: Collection of Scientific Works Devoted to the 80th Birthday of B. I. Ptashnyk [in Ukrainian], Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, National Academy of Sciences of Ukraine, Lviv (2017), pp. 91–109.
V. V. Katrakhov, S. M. Sitnik, “The transmutation method and boundary-value problems for singular elliptic equations,” Sovrem. Matem. Fundam. Napravl., 64, No. 2, 211–426 (2018) (in Russian), https://doi.org/10.22363/2413-3639-2018-64-2-211-426
G. M. Keselʹman, “On the unconditional convergence of eigenfunction expansions of certain differential operators,” Izv. Vysš. Učebn. Zaved. Matematika, No. 2 (39), 82–93 (1964) (in Russian).
V. P. Mikhailov, “Riesz basis in L_2[0; 1], Doklady Akad. Nauk SSSR, 144, No. 5, 981–984 (1962) (in Russian).
M. A. Naimark, Linear Differential Operators. Part I: Elementary Theory of Linear Differential Operators, Frederick Ungar Publ. Co., New York (1967).
A. A. Shkalikov, “Perturbations of self-adjoint and normal operators with discrete spectrum,” Uspekhi Mat. Nauk, 71, No. 5(431), 113–174 (2016), https://doi.org/10.4213/rm9740; English translation: Russ. Math. Surv., 71, No. 5, 907–964 (2016), https://doi.org/10.1070/RM9740
Ya. O. Baranetskij, I. I. Demkiv, I. Ya. Ivasiuk, M. I. Kopach, “The nonlocal problem for the 2n differential equations with unbounded operator coefficients and involution,” Carpath. Math. Publ., 10, No. 1, 14–30 (2018), https://doi.org/10.15330/cmp.10.1.14-30
Ya. O. Baranetskij, P. I. Kalenyuk, L. I. Kolyasa, M. I. Kopach, “Nonlocal multipoint problem for an ordinary differential equations of even order involution,” Mat. Stud., 49, No. 1, 80–94 (2018), https://doi.org/10.15330/ms.49.1.80-94
Ya. O. Baranetskij, P. I. Kalenyuk, L. I. Kolyasa, M. I. Kopach, “The nonlocal problem for the differential-operator equation of the even order with involution,” Carpath. Math. Publ., 9, No. 2, 109–119 (2017), https://doi.org/10.15330/cmp.9.2.109-119
Ya. O. Baranetskij, P. I. Kalenyuk, M. I. Kopach, A. V. Solomko, “The nonlocal boundary value problem with perturbations of mixed boundary conditions for an elliptic equation with constant coefficients. I,” Carpath. Math. Publ., 11, No. 2, 228–239 (2019), https://doi.org/10.15330/cmp.11.2.228-239
Ya. O. Baranetskij, P. I. Kalenyuk, M. I. Kopach, A. V. Solomko, “The nonlocal boundary value problem with perturbations of mixed boundary conditions for an elliptic equation with constant coefficients. II,” Carpath. Math. Publ., 12, No. 1, 173–188 (2020), https://doi.org/10.15330/cmp.12.1.173-188
Ya. O. Baranetskij, P. I. Kalenyuk, M. I. Kopach, A. V. Solomko, “The nonlocal multipoint problem with Dirichlet-type conditions for an ordinary differential equation of even order with involution,” Mat. Stud., 54, No. 1, 64–78 (2020), https://doi.org/10.30970/ms.54.1.64-78
G. D. Birkhoff, “Boundary value and expansion problems of ordinary linear differential equations,” Trans. Amer. Math. Soc., 9, No. 4, 373–395 (1908), https://doi.org/10.1090/S0002-9947-1908-1500818-6
G. D. Birkhoff, “On the asymptotic character of the solutions of certain linear differential equations containing a parameter,” Trans. Amer. Math. Soc., 9, No. 2, 219–231 (1908), https://doi.org/10.1090/S0002-9947-1908-1500810-1
G. Freiling, “Irregular boundary value problems revisited,” Results Math., 62, No. 3-4, 265–294 (2012), https://doi.org/10.1007/s00025-012-0281-7
V. V. Kravchenko, S. M. Sitnik (eds), Transmutation Operators and Applications, Birkhäuser, Basel (2020), http://doi.org/10.1007/978-3-030-35914-0
J. Locker, Eigenvalues and Completeness for Regular and Simply irregular Two-Point Differential Operators, Mem. Am. Math. Soc., Vol. 195, No. 911, Amer. Math. Soc. (2008), http://doi.org/10.1090/memo/0911
M. H. Stone, ”A comparison of the series of Fourier and Birkhoff,” Trans. Amer. Math. Soc., 28, 695–761 (1926), http://doi.org/10.2307/1989072
M. H. Stone, Linear Transformations in Hilbert Space and Their Applications to Analysis, Ser. Colloquium Publications. Vol. XV, Amer. Math. Soc., New York (1932).
J. Tamarkin, “Some general problems of the theory of ordinary linear differential equations and expansion of an arbitrary function in series of fundamental functions,” Math. Zeit., 27, No. 1, 1–54 (1928), https://doi.org/10.1007/BF01171084
Посилання
- Поки немає зовнішніх посилань.
Ця робота ліцензована Creative Commons Attribution 3.0 License.