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Register a new userCharacterization of Fr{e}chet Spaces and Application to Hausdorff MNC
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Authors
Henning Wunderlich
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In this short note, we give a characterization of Fr{e}chet spaces via properties of their metric. This allows us to prove that the Hausdorff measure of noncompactness (MNC), defined over Fr{e}chet spaces, is indeed an MNC. As first applications, we lift well-known fixed-point theorems for contractive and condensing operators to the setting of Fr{e}chet spaces.
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We calculate exactly the Laplace transform of the Fr{e}chet distribution in the form $gamma x^{-(1+gamma)} exp(-x^{-gamma})$, $gamma > 0$, $0 leq x < infty$, for arbitrary rational values of the shape parameter $gamma$, i.e. for $gamma = l/k$ with $l, k = 1,2, ldots$. The method employs the inverse Mellin transform. The closed form expressions are obtained in terms of Meijer G functions and their graphical illustrations are provided. A rescaled Fr{e}chet distribution serves as a kernel of Fr{e}chet integral transform. It turns out that the Fr{e}chet transform of one-sided L{e}vy law reproduces the Fr{e}chet distribution.
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