Fixed Points of Holomorphic Mappings: A Metric Approach

Kuczumow, Tadeusz; Reich, Simeon; Shoikhet, David

doi:10.1007/978-94-017-1748-9_14

Cited by 34 publications

(40 citation statements)

References 85 publications

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“…Now, we recall the fixed point theorem for nonexpansive mappings in product spaces. W. A. Kirk [9] used a retraction approach based on a method due to Bruck [4] to prove the following theorems (the analogous results for the product of convex weakly compact sets was obtained by the first author [10], see also [5], [6], [7], [8], [11], [12], [13] and [14]). …”

Section: Let S and Smentioning

confidence: 99%

The common fixed point set of commuting nonexpansive mappings in Cartesian products of separable spaces

Kuczumow¹,

Michalska²

2007

Fixed Point Theory and Its Applications

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Section: Let S and Smentioning

confidence: 99%

The common fixed point set of commuting nonexpansive mappings in Cartesian products of separable spaces

Kuczumow¹,

Michalska²

2007

Fixed Point Theory and Its Applications

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“…By k B we denote the Kobayashi distance on B [10], [14]. We now recall several useful properties of the Kobayashi distance k B , which are common for all bounded and convex domains in reflexive Banach spaces.…”

Section: Preliminariesmentioning

confidence: 99%

“…In the case of the Hilbert ball B H , when we consider averaged k BHnonexpansive self-mappings, the asymptotically regular mappings appear in a natural way [14], [17].…”

Section: Corollary 41 Theorem 41 Is Valid For Holomorphic Self-mapmentioning

confidence: 99%

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An example in holomorphic fixed point theory

Budzyńska¹

2003

Proc. Amer. Math. Soc.

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“…Another important class of nonlinear semigroups consists of semigroups of holomorphic self-mappings. Such semigroups occur in several diverse fields, including, for example, the theory of Markov stochastic branching processes [36,74], Krein spaces [83][84][85], fixed point theory [51], the geometry of complex Banach spaces [9,82], control theory and optimization [38]. These semigroups can be considered natural nonlinear analogs of the semigroups generated by bounded linear operators.…”

Section: Introductionmentioning

confidence: 99%

Complex dynamical systems and the geometry of domains in Banach spaces

Elin

Reich

Shoikhet

2004

Dissertationes Math.

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consisting of all the holomorphic self-mappings of D, is a semigroup with respect to composition. Definition 1.1.2. A mapping f ∈ Hol(D, Ω) is said to be univalent on D if for each pair of distinct points x 1 and x 2 in D we have f (x 1 ) = f (x 2 ).In this case one can define the inverse mappingIt is well known (see, for example, [39]) that if X = C n and Y = C m are finite dimensional complex spaces, then f :However, this fact is no longer true in the infinite dimensional case (see counterexamples in [79,4]). Therefore, in the general case a univalent mappingIt is also known (see, for example, [29,49]is a linear isomorphism between X and Y . In this situation we will say that D and Ω are biholomorphically equivalent.As we have already mentioned, in this case X and Y must be linearly isomorphic Banach spaces. Generally speaking, the converse is not true. That is, even if X and Y are isomorphic Banach spaces and f ∈ Hol(D, Y ) has at each point x in D a continuously invertible Fréchet derivative, the mapping f need not be univalent on D.Nevertheless, in this case the mapping f is biholomorphic on a neighborhood of each x ∈ D by the inverse function theorem. In this situation we will say that f ∈ Hol(D, Ω) is locally biholomorphic.The set of all univalent mappings from a domain D ⊂ X into X will be denoted by Univ(D). For the special case when D is the open unit ball of X, the subset of Univ(D) normalized by the conditions f (0) = 0 and f (0) = I will be denoted by S(D). This notation conforms to the one used in the classical onedimensional case, when D = ∆ = {z ∈ C : |z| < 1}. In this case we simply writeThat is, S consists of all the mappings f ∈ Univ(∆) such that f has the following Taylor series at the origin: Convex, star-shaped and spiral-shaped domainsDefinition 1.2.1. A set M in X is called convex if for each pair of points w 1 and w 2 in M , the line segment joining w 1 and w 2 is contained in M , i.e., for each t ∈ [0, 1], the point w = (1 − t)w 1 + tw 2 is also in M .If D is a domain in X, then f ∈ Univ(D) is said to be a convex mapping on D if its image f (D) = Ω is a convex domain in X.Geometry of domains in Banach spaces 9 Definition 1.2.2. A set M in X is called star-shaped (with respect to the origin) if given any w ∈ M , the point w t = tw also belongs to M for every t with 0 ≤ t ≤ 1. That is, if M contains w, then it also contains the entire line segment joining w to the origin. In this definition the origin is in Ω. If, in particular, the origin belongs to Ω, then we will say that f is star-like, to make our definition consistent with the classical one. In this case the mapping f has a null point τ in D. The set of all biholomorphic mappings on D which are star-shaped on D will be denoted by Star(D).If, in addition, there is τ ∈ D such that f (τ ) = 0, (1.2.1) then we will write f ∈ S * τ (D). Of course, in this case such a point τ is unique because. Again, in the one-dimensional case, when X = C, we will simply write S * to denote the family of all biholomorphic (univalent) star-like functions...

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Fixed Points of Holomorphic Mappings: A Metric Approach

Cited by 34 publications

References 85 publications

The common fixed point set of commuting nonexpansive mappings in Cartesian products of separable spaces

The common fixed point set of commuting nonexpansive mappings in Cartesian products of separable spaces

An example in holomorphic fixed point theory

Complex dynamical systems and the geometry of domains in Banach spaces

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