Electronic Journal of Differential Equations,
Vol. 2003(2003), No. 125, pp. 1-7.

Title: Exponential stability of linear and almost periodic 
       systems on Banach spaces

Authors: Constantin Buse (West Univ. of Timisoara, Romania)
         Vasile Lupulescu (University of Tg. Jiu, Romania)

Abstract: 
 Let $v_f(\cdot, 0)$ the mild solution of the well-posed
 inhomogeneous Cauchy problem
 $$ 
 \dot v(t)=A(t)v(t)+f(t), \quad v(0)=0\quad t\ge 0 
 $$
 on a complex Banach space $X$, where $A(\cdot)$ is an almost
 periodic (possible unbounded) operator-valued function.
 We prove that $v_f(\cdot, 0)$ belongs to a suitable subspace
 of bounded and uniformly continuous functions if and only if
 for each $x\in X$ the solution of the homogeneous Cauchy problem
 $$ 
 \dot u(t)=A(t)u(t), \quad u(0)=x\quad t\ge 0 
 $$
 is uniformly exponentially stable. Our approach is based on the
 spectral theory of evolution semigroups.
 
Submitted November 13, 2003. Published December 16, 2003.
Math Subject Classifications: 35B10, 35B15, 35B40, 47A10, 47D03.
Key Words: Almost periodic functions; uniform exponential stability; 
           evolution semigroups.