Exponential characteristic of a linear differential equation of first order in a Banach space (Q910912)

Let X be a Banach space, \(\alpha\geq 0\), and \(E_{\alpha}\) the set of all continuous maps \(\phi\) : [0,\(\infty)\to X\) for which \(\overline{\lim}_{t\to \infty}(1/t)\log \| \phi (t)\|_ X\leq \alpha.\) The author studies the problem of finding, for fixed \(f\in E_{\alpha}\), the smallest \(\beta\) with the property that all solutions to the initial problem \(dy/dt=A(t)y+f(t),\) \(y(0)=0\), belong to \(E_{\beta}\). Here A(t) is a family of strongly continuous operators in X with uniformly bounded norms.