A characterization of exponential separation in terms of reducibility for linear difference equations
Consider the difference equations (1) \(x(n+1)=A(n)x(n)\) and (2) \(y(n+1)=B(n)y(n)\), where A(n) and B(n) are \(k\times k\) matrix functions such that for \(n\in N=\{0,1,2,...\}\), \(| A(n)|,| B(n)| \leq M\), \(| A^{-1}(n)|\), \(| B^{-1}(n)| \leq M\). Equations (1) and (2) are said to be kinematically similar if there exists a \(k\times k\) invertible matrix function S(n), with \(| S(n)|\), \(| S^{-1}(n)| \leq M_ 1\), such that the change of variables \(x=S(n)y\) transforms (1) to (2), Equation (1) is said to be reducible to (2) if it is kinematically similar to (2) with \(B(n)=diag(B_ 1(n),...,B_ r(n)).\) The ordered pair \(V_ 1,V_ 2\) of subspaces of \(R^ k\) is said to be exponentially separated with respect to (1) if \(\dim V_ i\geq 1,\) \(i=1,2\), \(V_ 1\cap V_ 2=\{0\}\) and for \(n\geq m\), we have \(| x_ 2(n)| | x_ 2(m)|^{-1}\cdot | x_ 1(m)| | x_ 1(n)|^{-1}\geq Kp^{n-m},\) \(0<K\leq 1\), \(p>1\), where \(x_ i(n)\) is a solution of (1), \(x_ i(0)\in V_ i\), \(i=1,2.\) The author shows that equation (1) is exponentially separated if and only if all neighboring equations are reducible. The study is made using a matrix equation of the form \(Z(n+1)=A_ 1(n)Z(n)B_ 1^{-1}(n)\), \(n\in N\), where \(A_ 1(n),B_ 1(n)\) are \(k\times k\), \(s\times s\) invertible matrix functions, bounded together with their inverses, for which some intermediary results are derived.
scientific article; zbMATH DE number 4059998
scientific article