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Mean square stability of difference equations with a stochastic delay. (English) Zbl 1029.39005

The subject of the paper is a nonlinear nonautonomous delay difference equation \[ x(n+1) = f(n,x(n),x(n-1),\dots, x(n-\eta(n+1)),\quad n\in \mathbb N. \] The function \(\eta: \mathbb N \to \{ 1,2,\dots,r \}\) counts the number of delays, it is subject to a discrete Markov process.
Within this framework the authors prove two theorems on mean square asymptotic stability of the zero solution by applying Lyapunov stability methods. An elementary example illustrates the theory.

MSC:

39A11 Stability of difference equations (MSC2000)
60H25 Random operators and equations (aspects of stochastic analysis)
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References:

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