Theorems of Denjoy–Wolff type

A Denjoy–Wolff theorem for compact holomorphic mappings in reflexive Banach spaces

“…Throughout this paper A denotes the closure of a set A ⊂ X in (X, ∥ · ∥). Next, all considered domains D ⊂ X in our paper are bounded and convex, and k D denotes the Kobayashi distance in D ( [3] and also [5,1,[6][7][8][9][10][11][12][13]). …”

“…The following version of the Denjoy-Wolff theorem for a bounded and strictly convex domain in C k has recently been established in [1] (see [2,1] for an up-to-date list of references concerning this topic). Since the proof of Theorem 1.1 has a metric character, it can be extended to the case of k D -nonexpansive mappings, where k D denotes the Kobayashi distance in D [3].…”

The Denjoy-Wolff Theorem for condensing mappings in a bounded and strictly convex domain in a complex Banach space

“…The proof is based on the ideas given in [2] and [14]. For the convenience of the reader we present it.…”

A Denjoy-Wolff theorem for compact holomorphic mappings in complex Banach spaces

“…Recall that this is valid for open unit balls in complex Banach spaces ( [5], [20], [21], [26], [27], [29]; see also [1], [2] and [3] for the case of bounded and convex domains in C k ). Now we prove this fact in the special case where the sequence {x n } stems from a compact, fixed-point-free and k D -nonexpansive self-mapping f of D; in particular, from a compact, fixed-point-free and holomorphic self-mapping of D. In contrast with [4], we no longer assume that the complex Banach space X is reflexive.…”

“…The following version of the Denjoy-Wolff theorem ( [8], [33], [34] and [35]) for bounded and strictly convex domains in complex and reflexive Banach spaces has recently been established in [4] (see [3] and [5] for an up-to-date list of references regarding this topic). Since the assumption that the complex Banach space (X, ∥ · ∥) is reflexive is essential in [4], it is natural to ask if Theorem 1.1 holds in all Banach spaces.…”