A new theorem on exponential stability of periodic evolution families on Banach spaces
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 $fin AP_0(mat...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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Texas State University
2003-02-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2003/14/abstr.html |
| 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 $fin AP_0(mathbb{R}_+, X)$ then for each $xin X$ 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. |
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| ISSN: | 1072-6691 |