Electron. J. Diff. Eqns., Vol. 2003(2003), No. 125, pp. 1-7.

Exponential stability of linear and almost periodic systems on Banach spaces

Constantin Buse & Vasile Lupulescu

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.

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Constantin Buse
Department of Mathematics
West University of Timisoara
Bd. V. Parvan 4, Timisoara, Romania
e-mail: buse@hilbert.math.uvt.ro
Vasile Lupulescu
Department of Mathematics
"Constantin Brancusi"- University of Tg. Jiu
Bd. Republicii, No. 1, Tg. Jiu, Romania
email: vasile@utgjiu.ro

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