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A note on asymptotic stability of delay difference systems. (English) Zbl 1082.39007

The paper deals with linear delay difference system \(x_{n+1} - x_n = A x_{n-k}, n=0,1,\dots,\) where \(A\) is a \(2 \times 2\) real matrix and \(k\) is a nonnegative integer. The author gives explicit necessary and sufficient conditions for the asymptotic stability of the system in the term of the determinant of \(A\), the trace of \(A\) and the delay \(k.\) As an application, the local asymptotic stability of the positive equilibrium of the Lotka-Volterra difference system are investigated.

MSC:

39A11 Stability of difference equations (MSC2000)
39A12 Discrete version of topics in analysis
34K20 Stability theory of functional-differential equations
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