Constantin Buse and Constantin P. Niculescu: An ergodic characterization of uniformly exponentially stable evolution families, p.33-40

Abstract:

Suppose that $\varphi :[0,\infty )\rightarrow \lbrack 0,\infty )$ is a nondecreasing function such that $\lim_{t\rightarrow \infty }\varphi (t)=\infty$ and U={U(t,s)}_t≥s≥0 is an exponentially bounded evolution family on a Banach space $X$ having the property that for each $x\in X$ and each $s\geq 0$ the map $t\rightarrow \vert\vert U(s+t,s)x\vert\vert$ is continuous on $(0,\infty )$. Then $\mathcal{U}$ is uniformly exponentially stable if and only if there exist two positive constants $\alpha$ and $C$ such that for all $t>0$, $s\geq 0$ and $x\in X.$

Key Words: Evolution family, one-parameter semigroup of operators, exponential stability.

2000 Mathematics Subject Classification: Primary: 47D06,
Secondary: 35B35

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