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Abstract

In this chapter the procedure of the construction of Lyapunov functionals considered above is used for different types of stationary and nonstationary nonlinear systems and for different types of stability. It is shown that, using the procedure of the construction of Lyapunov functionals, investigation of stability in probability of a nonlinear stochastic difference equation with order of nonlinearity higher than 1 can be reduced to the investigation of asymptotic mean square stability of the linear part of this equation. This investigation is applied in particular for equilibrium points of fractional difference equations. Besides, almost sure asymptotic stability of the trivial solution of a nonlinear scalar stochastic difference equation is studied. Sufficient criteria for stability are obtained by virtue of the procedure of the construction of Lyapunov functionals, martingale decomposition and semi-martingale convergence theorems. Some demonstrative examples with illustrative figures of stability regions and trajectories of solutions are presented.

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Copyright information

© Springer-Verlag London Limited 2011

Authors and Affiliations

  1. 1.Department of Higher MathematicsDonetsk State University of ManagementDonetskUkraine

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