Bounded in the mean and stationary solutions of second-order difference equations with operator coefficients

Author:
M. F. Horodnii

Journal:
Theor. Probability and Math. Statist. **110** (2024), 159-165

MSC (2020):
Primary 60H99; Secondary 39A10

DOI:
https://doi.org/10.1090/tpms/1211

Published electronically:
May 10, 2024

Full-text PDF

Abstract |
References |
Similar Articles |
Additional Information

Abstract: We study the question of the existence of a unique bounded in the mean solution for the second-order difference equation with piecewise constant operator coefficients and of the stationary solution of the corresponding difference equation with constant operator coefficients. The case is considered when the corresponding “algebraic” operator equations have separated roots.

References
- A. G. Baskakov,
*On the invertibility of linear difference operators with constant coefficients*, Izv. Vysš. Ucebn. Zaved., Mat. **5** (2001), 3–11. (Russian) MR1860652
- A. Ya. Dorogovtsev,
*Periodic and stationary regimes of infinite-dimensional deterministic and stochastic dynamical systems*,“Vyshcha Shkola”, Kiev, 1992. (Russian) MR1206004
- A. Ya. Dorogovtsev,
*Stationary and periodic solutions of a stochastic difference equation in a Banach space*, Theory Probab. Math. Stat. **42** (1991), 39–46. MR1069311
- M. Gorodnii, I. Gonchar,
*Bounded in the mean of order p solutions of a difference equation with a jump of the operator coefficient*, Theory Probab. Math. Stat. **101** (2020), 103–108. MR4060334
- D. Henry,
*Geometric theory of semilinear parabolic equations*, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981. MR0610244
- M. Horodnii, V. Kravets,
*Bounded in the mean solutions of a second-order difference equation*, Modern Stoch. Theory Appl. **8** (2021),no. 4, 465–473. MR4342879
- L. Y. Kabantsova,
*Second-order linear difference equations in a Banach space and the splitting of operators* Izv. Sarat. Univ. (N.S.), Ser. Mat. Mekh. Inform. **17**, (2017), no. 3, 285–293. MR3697884
- Daoxing Xia,
*On the semihyponormal $n$-tuple of operators*, Integral Equations Operator Theory **6** (1983), no. 6, 879–898. MR **719110**, DOI 10.1007/BF01691929
- 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. MR0917687
- V. E. Slyusarchuk,
*Invertibility of linear nonautonomous difference operators in the space of bounded functions on Z*, Mat. Zametki **37** (1985), no. 5, 662–666. (Russian) MR0797706

References
- A. G. Baskakov,
*On the invertibility of linear difference operators with constant coefficients*, Izv. Vysš. Ucebn. Zaved., Mat. **5** (2001), 3–11. (Russian) MR1860652
- A. Ya. Dorogovtsev,
*Periodic and stationary regimes of infinite-dimensional deterministic and stochastic dynamical systems*,“Vyshcha Shkola”, Kiev, 1992. (Russian) MR1206004
- A. Ya. Dorogovtsev,
*Stationary and periodic solutions of a stochastic difference equation in a Banach space*, Theory Probab. Math. Stat. **42** (1991), 39–46. MR1069311
- M. Gorodnii, I. Gonchar,
*Bounded in the mean of order p solutions of a difference equation with a jump of the operator coefficient*, Theory Probab. Math. Stat. **101** (2020), 103–108. MR4060334
- D. Henry,
*Geometric theory of semilinear parabolic equations*, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981. MR0610244
- M. Horodnii, V. Kravets,
*Bounded in the mean solutions of a second-order difference equation*, Modern Stoch. Theory Appl. **8** (2021),no. 4, 465–473. MR4342879
- L. Y. Kabantsova,
*Second-order linear difference equations in a Banach space and the splitting of operators* Izv. Sarat. Univ. (N.S.), Ser. Mat. Mekh. Inform. **17**, (2017), no. 3, 285–293. MR3697884
- A. S. Markus, I. V. Mereutsa,
*On the complete n-tuple of roots of the operator equation corresponding to a polynomial operator bundle*, Math. USSR, Izv. **7**, (1973), no. 5, 1105–1128. MR **719110**
- 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. MR0917687
- V. E. Slyusarchuk,
*Invertibility of linear nonautonomous difference operators in the space of bounded functions on Z*, Mat. Zametki **37** (1985), no. 5, 662–666. (Russian) MR0797706

Similar Articles

Retrieve articles in *Theory of Probability and Mathematical Statistics*
with MSC (2020):
60H99,
39A10

Retrieve articles in all journals
with MSC (2020):
60H99,
39A10

Additional Information

**M. F. Horodnii**

Affiliation:
Taras Shevchenko National University of Kyiv, 64 Volodymyrs’ka St., 01601, Kyiv, Ukraine

Email:
horodnii@gmail.com

Keywords:
Banach space,
second-order difference equation,
bounded in the mean solution,
stationary solution,
“algebraic” operator equation,
separated roots

Received by editor(s):
January 6, 2023

Accepted for publication:
June 25, 2023

Published electronically:
May 10, 2024

Additional Notes:
This work has been supported by Ministry of Education and Science of Ukraine: Grant of the Ministry of Education and Science of Ukraine for perspective development of a scientific direction “Mathematical sciences and natural sciences” at Taras Shevchenko National University of Kyiv.

Article copyright:
© Copyright 2024
Taras Shevchenko National University of Kyiv