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.