Geodesic
A new theorem on exponential stability of periodic evolution families on Banach spaces
Electronic Journal of Differential Equations, Tome 2003 (2003).

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: 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},
     publisher = {mathdoc},
     volume = {2003},
     year = {2003},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2003__2003__a117/}
}
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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/