## 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.