Electronic Journal of Differential Equations,
Vol. 2003(2003), No. 14, pp. 1-10.
Title: A new theorem on exponential stability of
periodic evolution families on Banach spaces
Authors: Constantin Buse (West Univ. of Timisoara, Romania)
Oprea Jitianu (Univ. of Craiova, Romania)
Abstract:
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.