Applications of fixed point theorems in the theory of generalized IFS.

Mihail, Alexandru; Miculescu, Radu

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Given Banach spaces $X$ , $Y$ and a compact Hausdorff space $K$ , we use polymeasures to give necessary conditions for a multilinear operator from $C (K, X)$ into $Y$ to be completely continuous (resp. unconditionally converging). We deduce necessary and sufficient conditions for $X$ to have the Schur property (resp. to contain no copy of $c_{0}$ ), and for $K$ to be scattered. This extends results concerning linear operators.

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Kechris and Louveau in [5] classified the bounded Baire-1 functions, which are defined on a compact metric space $K$ , to the subclasses $ℬ_{1}^{ξ} (K)$ , $ξ < ω_{1}$ . In [8], for every ordinal $ξ < ω_{1}$ we define a new type of convergence for sequences of real-valued functions ( $ξ$ -uniformly pointwise) which is between uniform and pointwise convergence. In this paper using this type of convergence we obtain a classification of pointwise convergent sequences of continuous real-valued functions defined on a compact metric...

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