Quadratic characterizations for reciprocal linear difference equations (Q1318210)

Consider the real \(n\)-dimensional linear difference equation \[ x(l+1)=A(l)x(l),\;l \geq 0, \text{ integer }. \tag{1} \] Let \(x(l,p,x_ 0)\), \(l \geq p \geq 0\) be solution of (1) such that \(x(p,p,x_ 0)=x_ 0\). The origin \(x=0\) for (1) is said to be uniformly strictly stable (u.s.s.) if there exist values \(0<m \leq 1 \leq M\) where \(m | x_ 0 | \leq | x(l,p,x_ 0) | \leq M | x_ 0 |\) for all \(l \geq p \geq 0\). Let also, for a given \(l \geq 0\), \(S(l)\) represent a real symmetric \(n\times n\) matrix with ordered eigenvalues \(s_ 1(l) \leq \ldots \leq s_ n(l)\). The author proves that the origin for (1) is u.s.s. if and only if there exists a sequence a positive definite symmetric matrices \(S(l)\) where \(\sup_{l \geq 0} s_ n(l)= \gamma <\infty\), \(\inf_{l \geq 0} s_ 1(l)=\mu>0\), and \(A^ t(l)S(l+1) A(l)+S(l)\), \(l \geq 0\). Some applications of the above result for the stability of \(N\)-periodic and reciprocal systems of linear difference equations are presented.