Criteria for an exponential dichotomy of difference equations. (English) Zbl 0693.39001

Consider the system of linear difference equations (1) \(x(n+1)=A(n)x(n)\), where A(n) is a \(k\times k\) invertible matrix for \(n\in N\) such that (2) \(| A(n)| \leq M\) for \(n=1,2,3,...\); \(M\geq 1\). The entries \(a_{ij}(n)\) of A(n) are real functions. The authors prove the following propositions. Proposition 1: Suppose that (1) has an exponential dichotomy. Then there exist constants \(0<\theta <1\), \(T>0\), \(T\in N\) such that \(| x(n)| \leq \theta \sup \{| x(u)|:| u-n| \leq T\), \(u,n\in N\), \(n\geq T\}\). Proposition 2: Suppose that there exist constants \(T\geq 1\), \(T\in N\), and \(0<\theta <1\) such that \(| x(n)| \leq \theta \sup \{| x(u)|:| u-n| \leq T\), \(u,n\in N\), \(n\geq T\}\). Then (1) has an exponential dichotomy. Proposition 3: Suppose that A(n) is a \(k\times k\) bounded upper triangular and invertible matrix for all \(n\in N\). Then (1) has an exponential dichotomy if and only if the corresponding diagonal system \(x(n+1)=diag(\alpha_{11}(n),...,\alpha_{kk}(n))x(n)\) has an exponential dichotomy.


39A10 Additive difference equations
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