Poincaré types solutions of systems of difference equations (Q1856898)

Denote shortly by \(P\) the Poincaré property of a solution \(y(n)\) of equation \[ y(n+1)= \bigl[A+B(n) \bigr]y(n), \quad n=0,1,\dots, \] where \(A\) is a \(k\times k\) nonsingular matrix and \(B(n)\) is a \(k\times k\) matrix defined on \(\mathbb{Z}^+\). The solution \(y(n)\) has weak \(P\) if \[ \lim_{n\to\infty} \biggl[\bigl\|y(n)\bigr\|\biggr]^{1/n}=|\lambda| \] for some eigenvalue \(\lambda\) of matrix \(A\); resp. has \(P\) if \[ \lim_{n\to\infty} \bigl \|y(n+1)\bigr\|/ \bigl\|y(n)\bigr \|= |\lambda|; \] resp. has strong \(P\) if \[ \lim y(n)\lambda^{-n}=c\neq 0; \] resp. has ergodic \(P\) if \[ \lim_{n\to\infty} y(n)/\bigl\|y(n)\bigr\|= \xi\text{ (eigenvector)}. \] In the paper the relations between Poincaré properties of different types as well as sufficient condition for them are studied. Moreover the authors extend and generalize the known Poincaré theorem. Some illustrative examples are included.