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.