Advances in Difference Equations
Volume 2008 (2008), Article ID 718408, 21 pages
doi:10.1155/2008/718408
Research Article
Stability of Equilibrium Points of Fractional Difference Equations with Stochastic Perturbations
1Dipartimento di Matematica e Informatica, Universita di Salerno, via Ponte Don Melillo, 84084 Fisciano (Sa), Italy
2Department of Higher Mathematics, Donetsk State University of Management, 163 a Chelyuskintsev street, 83015 Donetsk, Ukraine
Received 6 December 2007; Accepted 9 May 2008
Academic Editor: Jianshe Yu
Copyright © 2008 Beatrice Paternoster and Leonid Shaikhet. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
It is supposed that the fractional difference equation xn+1=(μ+∑j=0kajxn−j)/(λ+∑j=0kbjxn−j), n=0,1,…, has an equilibrium point x^ and is exposed to additive stochastic perturbations type of σ(xn−x^)ξn+1 that are directly proportional to the deviation of the system state xn from the equilibrium point x^. It is shown that known results in the theory of stability of stochastic difference equations that were obtained via V. Kolmanovskii and L. Shaikhet general method of Lyapunov functionals construction can be successfully used for getting of sufficient conditions for stability in probability of equilibrium points of the considered stochastic fractional difference equation. Numerous graphical illustrations of stability regions and trajectories of solutions are plotted.