Wyniki wyszukiwania

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A nonlinear Banach-Steinhaus theorem and some meager sets in Banach spaces

100%

Jachymski J.

Studia Mathematica

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2005

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tom 170

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nr 3

303-320

EN

We establish a Banach-Steinhaus type theorem for nonlinear functionals of several variables. As an application, we obtain extensions of the recent results of Balcerzak and Wachowicz on some meager subsets of L¹(μ) × L¹(μ) and c₀ × c₀. As another consequence, we get a Banach-Mazurkiewicz type theorem on some residual subset of C[0,1] involving Kharazishvili's notion of Φ-derivative.

2

Convergence of iterates of linear operators and the Kelisky-Rivlin type theorems

100%

Jachymski J.

Studia Mathematica

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2009

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tom 195

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nr 2

99-112

EN

Let X be a Banach space and T ∈ L(X), the space of all bounded linear operators on X. We give a list of necessary and sufficient conditions for the uniform stability of T, that is, for the convergence of the sequence $(Tⁿ)_{n∈ ℕ}$ of iterates of T in the uniform topology of L(X). In particular, T is uniformly stable iff for some p ∈ ℕ, the restriction of the pth iterate of T to the range of I-T is a Banach contraction. Our proof is elementary: It uses simple facts from linear algebra, and the Banach Contraction Principle. As a consequence, we obtain a theorem on the uniform convergence of iterates of some positive linear operators on C(Ω), which generalizes and subsumes many earlier results including, the Kelisky-Rivlin theorem for univariate Bernstein operators, and its extensions for multivariate Bernstein polynomials over simplices. As another application, we also get a new theorem in this setting giving a formula for the limit of iterates of the tensor product Bernstein operators.

3

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A covering criterion and its applications to product theorems

100%

Jachymski J.

Commentationes Mathematicae

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2005

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tom 45

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nr 1

EN

We establish a covering criterion involving a neighbourhood system and ideals of open sets which yields, in particular, a compactness criterion for an arbitrary topological space. As an application, we give new proofs of Tychonoff’s compactness theorem: we consider separately the case of a countable product, in a proof of which the ordinary mathematical induction is used, and the case of an uncountable product proved by the transfinite induction. Subsequently, the same argument is applied to obtain some results on products of Lindelöf spaces.

4

Nonlinear contractive conditions: A comparison and related problems

63%

Jachymski J., Jóźwik I.

Banach Center Publications

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2007

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tom 77

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nr 1

123-146

EN

We establish five theorems giving lists of nonlinear contractive conditions which turn out to be mutually equivalent. We derive them from some general lemmas concerning subsets of the plane which may be applied both in the single- or set-valued case as well as for a family of mappings. A separation theorem for concave functions is proved as an auxiliary result. Also, we discuss briefly the following problems for several classes of contractions: stability of procedure of successive approximations, existence of approximate fixed points, continuous dependence of fixed points on parameters, existence of invariant sets for iterated function systems. Moreover, James Dugundji's contribution to the metric fixed point theory is presented. Using his notion of contractions, we also establish an extension of a domain invariance theorem for contractive fields.

Ograniczanie wyników

2 Studia Mathematica

1 Banach Center Publications

1 Commentationes Mathematicae

4 Jachymski J.

1 Jóźwik I.

1 2009

1 2007

2 2005

A nonlinear Banach-Steinhaus theorem and some meager sets in Banach spaces

Convergence of iterates of linear operators and the Kelisky-Rivlin type theorems

A covering criterion and its applications to product theorems

Nonlinear contractive conditions: A comparison and related problems