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Two Results on Jachymski-Schröder-Stein Contractions

100%
Reich S., Zaslavski A.
Bulletin of the Polish Academy of Sciences. Mathematics
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2008
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tom 56
|
nr 1
53-58
EN
We establish two fixed point theorems for certain mappings of contractive type.
2
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Two generic results in fixed point theory

100%
Reich S., Zaslavski A.
Banach Center Publications
|
2007
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tom 77
|
nr 1
215-225
EN
We give two examples of the generic approach to fixed point theory. The first example is concerned with the asymptotic behavior of infinite products of nonexpansive mappings in Banach spaces and the second with the existence and stability of fixed points of continuous mappings in finite-dimensional Euclidean spaces.
3
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Nonexpansive retracts in Banach spaces

100%
Kopecká E., Reich S.
Banach Center Publications
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2007
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tom 77
|
nr 1
161-174
EN
We study various aspects of nonexpansive retracts and retractions in certain Banach and metric spaces, with special emphasis on the compact nonexpansive envelope property.
4
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Zone and double zone diagrams in abstract spaces

100%
Reem D., Reich S.
Colloquium Mathematicum
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2009
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tom 115
|
nr 1
129-145
EN
A zone diagram of order n is a relatively new concept which was first defined and studied by T. Asano, J. Matoušek and T. Tokuyama. It can be interpreted as a state of equilibrium between n mutually hostile kingdoms. Formally, it is a fixed point of a certain mapping. These authors considered the Euclidean plane with finitely many singleton-sites and proved the existence and uniqueness of zone diagrams there. In the present paper we generalize this concept in various ways. We consider general sites in m-spaces (a simple generalization of metric spaces) and prove several existence and (non)uniqueness results in this setting. In contrast with previous works, our (rather simple) proofs are based on purely order-theoretic arguments. Many explicit examples are given, and some of them illustrate new phenomena which occur in the general case. We also re-interpret zone diagrams as a stable configuration in a certain combinatorial game, and provide an algorithm for finding this configuration in a particular case.
5
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Averages of holomorphic mappings and holomorphic retractions on convex hyperbolic domains

100%
Reich S., Shoikhet D.
Studia Mathematica
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1998
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tom 130
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nr 3
231-244
EN
Let D be a hyperbolic convex domain in a complex Banach space. Let the mapping F ∈ Hol(D,D) be bounded on each subset strictly inside D, and have a nonempty fixed point set ℱ in D. We consider several methods for constructing retractions onto ℱ under local assumptions of ergodic type. Furthermore, we study the asymptotic behavior of the Cesàro averages of one-parameter semigroups generated by holomorphic mappings.
6
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Reflexivity and approximate fixed points

100%
Matoušková E., Reich S.
Studia Mathematica
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2003
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tom 159
|
nr 3
403-415
EN
A Banach space X is reflexive if and only if every bounded sequence {xₙ} in X contains a norm attaining subsequence. This means that it contains a subsequence ${x_{n_k}}$ for which $sup_{f∈S_{X*}} lim sup_{k→∞} f(x_{n_k})$ is attained at some f in the dual unit sphere $S_{X*}$. A Banach space X is not reflexive if and only if it contains a normalized sequence {xₙ} with the property that for every $f ∈ S_{X*}$, there exists $g ∈ S_{X*}$ such that $lim sup_{n→∞}f(xₙ) < lim inf_{n→∞}g(xₙ)$. Combining this with a result of Shafrir, we conclude that every infinite-dimensional Banach space contains an unbounded closed convex set which has the approximate fixed point property for nonexpansive mappings.
7
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Complex dynamical systems and the geometry of domains in Banach spaces

89%
Elin M., Reich S., Shoikhet D.
Instytut Matematyczny Polskiej Akademii Nauk
EN
We consider semigroups of holomorphic self-mappings on domains in Hilbert and Banach spaces, and then develop a new dynamical approach to the study of geometric properties of biholomorphic mappings. We establish, for example, several flow invariance conditions and find parametric representations of semicomplete vector fields. In order to examine the asymptotic behavior of these semigroups, we use diverse tools such as hyperbolic metric theory and estimates of solutions of generalized differential equations. In addition, we introduce a new method involving admissible upper and lower bounds. Finally, we apply our dynamical approach to obtain several growth and covering theorems for star-like mappings on the open unit balls of Banach and Hilbert spaces.
8
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Convergence of iterates of asymptotically nonexpansive mappings in Banach spaces with the uniform Opial property

81%
Bruck R., Kuczumow T., Reich S.
Colloquium Mathematicum
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1993
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tom 65
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nr 2
169-179
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