# zbMATH — the first resource for mathematics

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
##### Keywords:
Banach space; evolution operator
Full Text:
##### References:
 [1] Daleckii, I; Krein, M, Stability of differential equations in a Banach space, (1970), [Russian] [2] Przyłuski, K.M; Rolewicz, S, On stability of linear time-varying infinite dimensional systems, Systems control lett., 5, 307-315, (1984) · Zbl 0543.93057 [3] {\scK. M. Przyłuski and S. Rolewicz}, On stability of linear time-varying infinite dimensional systems, in “Contribution to Operation Research” (D. Pallaschke, Ed.), Lecture Notes in Economics and Mathematical Systems 240, Springer-Verlag, Berlin, 159-173. [4] Zabczyk, J, Remarks on the contol of discrete-time distributed parameter systems, SIAM J. control optim., 12, 721-735, (1974) · Zbl 0254.93027
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.