Bounded in the mean of order $p$ solutions of a difference equation with a jump of the operator coefficient
Authors:
M. F. Gorodnii and I. V. Gonchar
Translated by:
S. V. Kvasko
Journal:
Theor. Probability and Math. Statist. 101 (2020), 103-108
MSC (2020):
Primary 60H99; Secondary 39A10
DOI:
https://doi.org/10.1090/tpms/1114
Published electronically:
January 5, 2021
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Abstract: We consider a linear difference equation in a Banach space, with a jump of the operator coefficient. We study the problem of the existence of a unique solution which is bounded on $\mathbb {Z}$ in the mean of order $p$.
References
- Daniel Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981. MR 610244
- V. E. Slyusarchuk, Invertibility of linear nonautonomous difference operators in the space of bounded functions on ${\bf Z}$, Mat. Zametki 37 (1985), no. 5, 662–666, 780 (Russian). MR 797706
- A. G. Baskakov, On the invertibility of linear difference operators with constant coefficients, Izv. Vyssh. Uchebn. Zaved. Mat. 5 (2001), 3–11 (Russian); English transl., Russian Math. (Iz. VUZ) 45 (2001), no. 5, 1–9. MR 1860652
- A. Ya. Dorogovtsev, Periodicheskie i statsionarnye rezhimy beskonechnomernykh determinirovannykh i stokhasticheskikh dinamicheskikh sistem, “Vishcha Shkola”, Kiev, 1992 (Russian, with Russian and Ukrainian summaries). MR 1206004
- T. Morozan, Bounded, periodic and almost periodic solutions of affine stochastic discrete-time systems, Rev. Roumaine Math. Pures Appl. 32 (1987), no. 8, 711–718. MR 917687
- I. V. Gonchar, On the bounded and summable solutions of a difference equation with a jump of an operator coefficient, Bull. Taras Shevchenko Nat. Univ. Kyiv Ser. Phys. Math. 2 (2016), 25–28. (In Ukrainian)
References
- D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin–Heidelberg–New York, 1981. MR 610244
- V. E. Slyusarchuk, Invertibility of nonautonomous linear difference operators in the space of bounded functions on $\mathbb {Z}$, Maten. Zametki 37 (1985), no. 5, 662–666; English transl. in Math. Notes 37 (1985), no. 5, 360–363. MR 797706
- A. G. Baskakov, On the invertibility of linear difference operators with constant coefficients, Izv. Vuzov Matem. 45 (2001), no. 5, 3–11; English transl. in Russian Mathematics 45 (2001), no. 5, 1–9. MR 1860652
- A. Ya. Dorogovtsev, Periodic and Stationary Conditions of Infinite-Dimensional Deterministic and Stochastic Dynamical Systems, “Vyshcha Shkola”, Kyiv, 1992. (In Russian) MR 1206004
- T. Morozan, Bounded, periodic and almost periodic solutions of stochastic discrete-time systems, Rev. Roumaine Math. Pure Appl. 32 (1987), no. 8, 711–718. MR 917687
- I. V. Gonchar, On the bounded and summable solutions of a difference equation with a jump of an operator coefficient, Bull. Taras Shevchenko Nat. Univ. Kyiv Ser. Phys. Math. 2 (2016), 25–28. (In Ukrainian)
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Additional Information
M. F. Gorodnii
Affiliation:
Faculty for Mechanics and Mathematics, Kyiv Taras Shevchenko National University, Volodymyrs’ka Street, 64/13, Kyiv 01601, Ukraine
Email:
gorodnii@univ.kiev.ua
I. V. Gonchar
Affiliation:
Faculty for Mechanics and Mathematics, Kyiv Taras Shevchenko National University, Volodymyrs’ka Street, 64/13, Kyiv 01601, Ukraine
Email:
goncharinna@ukr.net
Keywords:
Banach space,
difference equation,
jump operator coefficient,
bounded in the mean of order $p$ solution
Received by editor(s):
July 4, 2019
Published electronically:
January 5, 2021
Article copyright:
© Copyright 2020
American Mathematical Society
