A Denjoy-Wolf theorem for compact holomorphic mappings in complex Banach spaces (Q2860895)

In [\textit{M. Budzyńska} et al., J. Math. Anal. Appl. 396, No. 2, 504--512 (2012; Zbl 1329.32007)], the authors have proven the following version of the Denjoy-Wolff theorem quoted in the introduction of the paper under review. \newline { Theorem}. If \(D\) is a bounded and strictly convex domain in a complex and reflexive Banach space \((X,\| \cdot \|)\) and \(f: D\rightarrow D\) is compact, holomorphic, and fixed-point-free, then there exists a point \(\xi \in \partial D\) such that the sequence \((f^n)\) of the iterates of \(f\) converges in the compact-open topology to the constant map taking the value \(\xi\). \newline Compactness means that \(f(D)\) is relatively compact in \(X\). The aim of the present paper is to prove that the hypotheses of reflexivity can be omitted.\newline The main tool of the proof is the horosphere. Consider the set of natural numbers as sequence \((n)_{n \geq 1}\) and fix a subultranet \((n_\gamma )_{\gamma \in \Gamma }\).\newline { Definition}. Let \(D\) be a bounded and convex domain in an arbitrary complex Banach space \((X,\| \cdot \|)\). Let \(x \in D\), \(\xi \in \partial D\), \(R>0\), \(x_n \in D\), \(n=1,2,\ldots\), and \(\lim_{n \rightarrow \infty } = \xi\). The horosphere \(H(x,\xi,R,(x_n))\) in \(D\) is defined as follows: NEWLINE\[NEWLINE H(x,\xi,R,(x_n)) := \{y \in D: \lim_{\gamma \in \Gamma }[k_D(y,x_{n_\gamma })-k_D(x,x_{n_\gamma })]<\frac12 \log R\}.NEWLINE\]NEWLINE Here \(k_D(x,y)\) denotes the Kobayashi distance in \(D\). A long proof shows that the horosphere is nonempty for each \(R>0\) under some assumptions. The main theorem is the following:\newline { Theorem}. If \(D\) is a bounded and strictly convex domain in an arbitrary complex Banach space \((X,\| \cdot \|)\) and \(f: D\rightarrow D\) is compact, \(k_D\)-nonexpansive, and fixed-point-free, then there exists a point \(\xi \in \partial D\) such that the sequence \((f^n)\) of the iterates of \(f\) converges in the bounded-open topology to the constant map taking the value \(\xi\), that is, the sequence \((f^n)\) tends to \(\xi\), uniformly on each \(k_D\)-bounded subset \(C\) of \(D\). \newline Since holomorphic mappings \(f:D\rightarrow D\) are \(k_D\)-nonexpansive, this theorem can be reformulated substituting ``\(k_D\)-nonexpansive'' with ``holomorphic''. Three corollarys conclude this paper.