Recurrent equations with sign and Fredholm alternative (Q317097)
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scientific article; zbMATH DE number 6631543
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Recurrent equations with sign and Fredholm alternative |
scientific article; zbMATH DE number 6631543 |
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Recurrent equations with sign and Fredholm alternative (English)
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30 September 2016
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recurrent and almost periodic linear equations
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bounded and recurrent solutions
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spectral theory
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Favard theory
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Fredholm alternative
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The authors consider the linear differential system NEWLINE\[NEWLINE\dot x=A(t)x+f(t),NEWLINE\]NEWLINE where the vector \(f\) is bounded and continuous and the matrix \(A\) is recurrent, i.e., \(A\) is bounded, uniformly continuous, and its hull is compact minimal with respect to the compact-open topology and the flow induced by translations \((A\tau)(t)=A(t+\tau)\). It is proved that a Fredholm alternative holds, extending the result [\textit{P. Cieutat} and \textit{A. Haraux}, Port. Math. (N.S.) 59, No. 2, 141--159 (2002; Zbl 1018.34049)]. The existence of exponential dichotomies in the nonrecurrent case is also characterized.
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