Linear Noetherian boundary-value problem for a matrix difference equation

  • S. M. Chuiko Slov'yansky sovereign pedagogical university
  • Ya. V. Kalinichenko Slov'yansky sovereign pedagogical university
  • A. A. Boichuk Institute of Mathematics, National Academy of Sciences of Ukraine, Kiev

Abstract

UDC 517.9 <br>

We establish constructive conditions for the solvability and propose a scheme for constructing solutions of a linear Noetherian boundary-value problem for a matrix difference equation.  We suggest an original scheme of regularization of a linear Noetherian boundary-value problem for a linear degenerate system of difference equations. 

References

Boichuk, Alexander A.; Samoilenko, Anatolii M. Generalized inverse operators and Fredholm boundary-value problems. Second edition. Translated from the Russian by Peter V. Malyshev. Inverse and Ill-posed Problems Series, 59. De Gruyter, Berlin, 2016. xvi+296 pp. ISBN: 978-3-11-037839-9; 978-3-11-037844-3; 978-3-11-038735-3 doi: 10.1515/9783110378443

Boĭchuk, A. A.; Krivosheya, S. A. A critical periodic boundary value problem for the matrix Riccati equation. (Russian) ; translated from Differ. Uravn. 37 (2001), no. 4, 439--445, 573 Differ. Equ. 37 (2001), no. 4, 464--471 doi: 10.1023/A:1019267220924

A. A. Bojchuk, Краевые задачи для систем разностных уравнений (Russian) Kraevy`e zadachi dlya sistem raznostny`kh uravnenij, Ukr. mat. zhurn.,49, No 6, 832 – 835 (1997).

Martynyuk, D. I.; Martynjuk, D. I. Лекции по качественной теории разностных уравнений. (Russian) [[Lectures on the qualitative theory of difference equations]] ``Naukova Dumka'', Kiev, 246 pp. 1972. MR0611163

Boĭchuk, O. A.; Krivosheya, S. A. A criterion for the solvability of matrix equations of Lyapunov type. (Ukrainian) ; translated from Ukraïn. Mat. Zh. 50 (1998), no. 8, 1021--1026 Ukrainian Math. J. 50 (1998), no. 8, 1162--1169 (1999) doi: 10.1007/BF02513089

S. M. Chujko, Обобщенный оператор Грина линейной нетеровой краевой задачи для матричного разностного уравнения (Russian) Obobshhenny`j operator Grina linejnoj neterovoj kraevoj zadachi dlya matrichnogo raznostnogo uravneniya, Tavr. vestn. informatiki i matematiki, No 1 (26), 104 – 116 (2015).

S. M. Chujko,Обобщенное матричное дифференциально-алгебраическое уравнение (Russian) Obobshhennoe matrichnoe differenczial`no-algebraicheskoe uravnenie, Ukr. mat. visn.,12, No 1,11 – 26 (2015).

Chuĭko, S. M. The Green operator of a generalized matrix differential-algebraic boundary value problem. (Russian) ; translated from Sibirsk. Mat. Zh. 56 (2015), no. 4, 942--951 Sib. Math. J. 56 (2015), no. 4, 752--760 doi: 10.1134/s0037446615040175

Voevodin, V. V.; Kuznetsov, Yu. A. Матрицы и вычисления. (Russian) [[Matrices and calculations]] ``Nauka'', Moscow, 1984. 319 pp. MR0758446

Kreĭn, S. G. Линей ные уравнения в банаховом пространстве. (Russian) [[Linear equations in a Banach space]] Izdat. ``Nauka'', Moscow, 1971. 104 pp. MR0374949

Tikhonov, A. N.; Arsenin, V. Ya. Методы решения некорректных задач. (Russian) [[Methods for the solution of ill-posed problems]] Third edition. ``Nauka'', Moscow, 1986. 288 pp. MR0857101

S. M. Chuiko, E. V. Chuiko, A. V. Belushenko, On a regularization method for solving linear matrix equation, Bull.Taras Shevchenko Nat. Univ. Ser. Math.,1, 12 – 14 (2014).

Boichuk, A. A.; Pokutnyi, A. A.; Chistyakov, V. F. Application of perturbation theory to the solvability analysis of differential algebraic equations. (Russian) ; translated from Zh. Vychisl. Mat. Mat. Fiz. 53 (2013), no. 6, 958-969 Comput. Math. Math. Phys. 53 (2013), no. 6, 777--788 doi: 10.1134/S0965542513060043

Chuĭko, Sergeĭ M. On the regularization of a matrix differential-algebraic boundary value problem. (Russian) ; translated from Ukr. Mat. Visn. 13 (2016), no. 1, 76--90, 143 J. Math. Sci. (N.Y.) 220 (2017), no. 5, 591--602 doi: 10.1007/s10958-016-3202-6

Chuiko, S. Weakly nonlinear boundary value problem for a matrix differential equation. Miskolc Math. Notes 17 (2016), no. 1, 139--150. doi: 10.18514/MMN.2016.1312

Published
28.03.2020
How to Cite
ChuikoS. M., Kalinichenko Y. V., and Boichuk A. A. “Linear Noetherian Boundary-Value Problem for a Matrix Difference Equation”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, no. 3, Mar. 2020, pp. 340-54, doi:10.37863/umzh.v72i3.637.
Section
Research articles