Mathematica Bohemica, Vol. 142, No. 2, pp. 185-196, 2017

Initial data stability and admissibility of spaces for Itô linear difference equations

Ramazan Kadiev, Pyotr Simonov

Received August 18, 2014.  First published December 5, 2016.

Abstract:  The admissibility of spaces for Itô functional difference equations is investigated by the method of modeling equations. The problem of space admissibility is closely connected with the initial data stability problem of solutions for Itô delay differential equations. For these equations the $p$-stability of initial data solutions is studied as a special case of admissibility of spaces for the corresponding Itô functional difference equation. In most cases, this approach seems to be more constructive and expedient than other traditional approaches. For certain equations sufficient conditions of solution stability are given in terms of parameters of those equations.
Keywords:  Itô functional difference equation; stability of solutions; admissibility of spaces
Classification MSC:  39A60, 39A30, 60H25, 37H10, 93E15
DOI:  10.21136/MB.2016.0059-14

[1] D. L. Andrianov: Boundary value problems and control problems for linear difference systems with aftereffect. Russ. Math. 37 (1993), 1-12; translation from Izv. Vyssh. Uchebn. Zaved. Mat. 5 (1993), 3-16. MR 1265616 | Zbl 0836.34087
  [2] N. V. Azbelev, P. M. Simonov: Stability of Differential Equations with Aftereffect. Stability and Control: Theory, Methods and Applications 20. Taylor and Francis, London (2003). MR 1965019 | Zbl 1049.34090
  [3] S. Elaydi: Periodicity and stability of linear Volterra difference systems. J. Math. Anal. Appl. 181 (1994), 483-492. DOI 10.1006/jmaa.1994.1037 | MR 1260872 | Zbl 0796.39004
  [4] S. Elaydi, S. Zhang: Stability and periodicity of difference equations with finite delay. Funkc. Ekvacioj, Ser. Int. 37 (1994), 401-413. MR 1311552 | Zbl 0819.39006
  [5] N. Ikeda, S. Watanabe: Stochastic Differential Equations and Diffusion Processes. North-Holland Mathematical Library 24. North-Holland Publishing, Amsterdam; Kodansha Ltd., Tokyo (1981). MR 0637061 | Zbl 0495.60005
  [6] R. Kadiev: Sufficient stability conditions for stochastic systems with aftereffect. Differ. Equations 30 (1994), 509-517; translation from Differ. Uravn. 30 (1994), 555-564. MR 1299841 | Zbl 0824.93069
  [7] R. Kadiev: Stability of solutions of stochastic functional differential equations. Doctoral dissertation, DSc Habilitation thesis, Makhachkala (2000) (in Russian).
  [8] R. Kadiev, A. V. Ponosov: Stability of linear stochastic functional-differential equations under constantly acting perturbations. Differ. Equations 28 (1992), 173-179; translation from Differ. Uravn. 28 (1992), 198-207. MR 1184920 | Zbl 0788.60071
  [9] R. Kadiev, A. V. Ponosov: Relations between stability and admissibility for stochastic linear functional differential equations. Func. Diff. Equ. 12 (2005), 209-244. MR 2137849 | Zbl 1093.34046
  [10] R. Kadiev, A. V. Ponosov: The $W$-transform in stability analysis for stochastic linear functional difference equations. J. Math. Anal. Appl. 389 (2012), 1239-1250. DOI 10.1016/j.jmaa.2012.01.003 | MR 2879292 | Zbl 1248.93168

Access to the full text of journal articles on this site is restricted to the subscribers of Myris Trade. To activate your access, please contact Myris Trade at
Subscribers of Springer need to access the articles on their site, which is

Affiliations:   Ramazan Kadiev, Dagestan Research Center of the Russian Academy of Sciences and Department of Mathematics, Dagestan State University, M. Gadgiev st. 43 a, 367025, Makhachkala, Russia, e-mail:; Pyotr Simonov, Perm State National Research University, P. O. Box 7345, 614083, Perm, Russia, e-mail:

PDF available at: