The fixed point property in \(c_{0}\) with the alpha norm
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 \(\text{Fix}(T)\neq\emptyset\)).
scientific article; zbMATH DE number 6361925