Electron. J. Diff. Eqns., Vol. 2003(2003), No. 14, pp. 1-10.

A new theorem on exponential stability of periodic evolution families on Banach spaces

Constantin Buse & Oprea Jitianu

We consider a mild solution $v_f(\cdot, 0)$ of a well-posed inhomogeneous Cauchy problem
$\dot v(t)=A(t)v(t)+f(t)$, $v(0)=0$
on a complex Banach space $X$, where $A(\cdot)$ is a 1-periodic operator-valued function. We prove that if $v_f(\cdot, 0)$ belongs to $AP_0(\mathbb{R}_+, X)$ for each $f\in AP_0(\mathbb{R}_+, X)$ then for each $x\in X$ the solution of the well-posed Cauchy problem
$\dot u(t)=A(t)v(t)$, $u(0)=x$
is uniformly exponentially stable. The converse statement is also true. Details about the space $AP_0(\mathbb{R}_+, X)$ are given in the section 1, below. Our approach is based on the spectral theory of evolution semigroups.

Submitted November 13, 2002. Published February 11, 2003.
Math Subject Classifications: 26D10, 34A35, 34D05, 34B15, 45M10, 47A06.
Key Words: Almost periodic functions, exponential stability, periodic evolution families of operators, integral inequality, differential inequality on Banach spaces.

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Constantin Buse
Department of Mathematics
West University of Timisoara
Bd. V. Parvan 4
1900 Timisoara, Romania
e-mail: buse@hilbert.math.uvt.ro
Oprea Jitianu
Department of Applied Mathematics
University of Craiova
Bd. A. I. Cuza 13,
1100-Craiova, Romania
e-mail: jitianu@ucv.netmasters.ro

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