Electronic Journal of Differential Equations,
Vol. 2012 (2012), No. 218, pp. 1-10.
Title: Weak Rolewicz's theorem in Hilbert spaces
Authors: Constantin Buse (West Univ. of Timisoara, Romania)
Gul Rahmat (Government College Univ., Lahore, Pakistan)
Abstract:
Let $\phi:\mathbb{R}_+:=[0, \infty)\to \mathbb{R}_+$ be a nondecreasing
function which is positive on $(0, \infty)$ and let
$\mathcal{U} =\{U(t, s)\}_{t\ge s\ge 0}$ be a positive strongly continuous
periodic evolution family of bounded linear operators acting on a
complex Hilbert space $H$. We prove that $\mathcal{U}$ is uniformly
exponentially stable if for each unit vector $x\in H$, one has
$$
\int_0^\infty \phi(|\langle U(t, 0)x, x\rangle|)dt<\infty.
$$
The result seems to be new and it generalizes others of the same topic.
Moreover, the proof is surprisingly simple.
Submitted October 3, 2012. Published November 29, 2012.
Math Subject Classifications: 47A30, 46A30.
Key Words: Uniform exponential stability; Rolewicz's type theorems;
weak integral stability boundedness.