Advances in Difference Equations
Volume 2008 (2008), Article ID 718408, 21 pages
doi:10.1155/2008/718408

Stability of equilibrium points of fractional difference equations with stochastic perturbations

Beatrice Paternoster1 and Leonid Shaikhet2

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

Abstract

It is supposed that the fractional difference equation xn+1=(μ+j=0kajxnj)/(λ+j=0kbjxnj), n=0,1,, has an equilibrium point x^ and is exposed to additive stochastic perturbations type of σ(xnx^)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.