The fixed point property in \(c_{0}\) with the alpha norm (Q2925795)
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scientific article; zbMATH DE number 6361925
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | The fixed point property in \(c_{0}\) with the alpha norm |
scientific article; zbMATH DE number 6361925 |
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27 October 2014
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fixed point property
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space \(c_{0\alpha}\)
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The fixed point property in \(c_{0}\) with the alpha norm (English)
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The authors prove the followingNEWLINENEWLINE Theorem 2.2. Let \(K\) be a nonempty closed convex and bounded subset of \(c_{0\alpha}= (c_0,\|\cdot\|_\alpha)\), where NEWLINE\[NEWLINE\| (x_i)\|_\alpha= \sup_i|x_i|+\alpha \sum_i{|x_i|\over 2^i},\quad \alpha\geq 0.NEWLINE\]NEWLINE Then \(K\) is weakly compact if and only if every nonempty closed convex subset \(M\subset K\) has the FPP (i.e., for every non-expansive mapping \(T: M\to M\) we have \(\text{Fix}(T)\neq\emptyset\)).
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