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On uniform N-equistability. (English) Zbl 0597.34064
In the paper the following theorem is proved. Let X be a Banach space and let \(\{\) U(t,s)\(\}\) be an evolution operator acting in X. We assume that the evolution operator \(\{U(t,s)\}\) is uniformly bounded, i.e. \(\sup_{0\leq t-s\leq 1}\| U(t,s)\| =K<+\infty.\) Let N(a,u) be a continuous nondecreasing function of two arguments, \(a>0\), \(u\geq 0\), such that \(N(0,u)=0\) and \(N(a,u)>0\) for \(a>0\). Suppose that for each x there is a(x) such that \(\sup_{s}\int^{\infty}_{s}N(a(x),\quad \| U(t,s)\|)dt<+\infty\)
then there are \(M>0\) and \(b>0\) such that \(\| U(t,s)\| \leq Me^{- b(t-s)}.\)

MSC:
34G10 Linear differential equations in abstract spaces
34D05 Asymptotic properties of solutions to ordinary differential equations
47D03 Groups and semigroups of linear operators
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References:
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