Electron. J. Diff. Equ., Vol. 2012 (2012), No. 218, pp. 1-10.

Weak Rolewicz's theorem in Hilbert spaces

Constantin Buse, Gul Rahmat

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.

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Constantin Buse
West University of Timisoara
Department of Mathematics
Bd. V. Parvan No. 4, 300223-Timisoara, Romania
email: buse@math.uvt.ro
Gul Rahmat
Government College University
Abdus Salam School of Mathematical Sciences
Lahore, Pakistan
email: gulassms@gmail.com

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