Geodesic
A new theorem on exponential stability of periodic evolution families on Banach spaces
Electronic journal of differential equations, Tome 2003 (2003)
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.
Classification : 26D10, 34A35, 34D05, 34B15, 45M10, 47A06
Keywords: almost periodic functions, exponential stability, periodic evolution families of operators, integral inequality, differential inequality on Banach spaces 
@article{EJDE_2003__2003__a117,
     author = {Bu\c{s}e,  Constantin and Jitianu,  Oprea},
     title = {A new theorem on exponential stability of periodic evolution families on {Banach} spaces},
     journal = {Electronic journal of differential equations},
     year = {2003},
     volume = {2003},
     zbl = {1053.47034},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2003__2003__a117/}
}
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%A Jitianu,  Oprea
%T A new theorem on exponential stability of periodic evolution families on Banach spaces
%J Electronic journal of differential equations
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%F EJDE_2003__2003__a117
Buşe,  Constantin; Jitianu,  Oprea. A new theorem on exponential stability of periodic evolution families on Banach spaces. Electronic journal of differential equations, Tome 2003 (2003). http://geodesic.mathdoc.fr/item/EJDE_2003__2003__a117/