On the order of exponential growth of the solution of the linear difference equation with periodic coefficient in Banach space
Denote by \(E_{\alpha}\) the set of all elements f(t) of a Banach space E such that \(\sup \{\| f(t)\| \exp (-\alpha t)\}<\infty\) and let A(t)\(\in {\mathcal L}(E)\) be such that \(A(t+\omega)=A(t)\). The author proves the following theorem: If \(f\in E_{\alpha}\), then the solution y of equation \(\Delta y-A(t)y=f(t)\), \(t\geq \delta\), where \(\Delta y=(y(t+\delta)-y(t))/\delta\), \(\delta >0\), belongs to \(E_{\beta}\) such that \(\beta =\alpha\), when \(\alpha >\alpha_ 0\), \(\beta >\alpha\), when \(\alpha =\alpha_ 0\), \(\beta =\alpha_ 0\), when \(\alpha <\alpha_ 0\) \((\alpha_ 0=(\ln | \lambda_ 0| /\omega\), \(\lambda_ 0\in \sigma_ B\), \(B=\prod^{0}_{k=\omega -1}[I+A(k)])\).
scientific article; zbMATH DE number 3916885