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1
Content available remote

On the Converse of Caristi's Fixed Point Theorem

100%
Głąb S.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2004
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tom 52
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nr 4
411-416
EN
Let X be a nonempty set of cardinality at most $2^{ℵ₀}$ and T be a selfmap of X. Our main theorem says that if each periodic point of T is a fixed point under T, and T has a fixed point, then there exist a metric d on X and a lower semicontinuous map ϕ :X→ ℝ ₊ such that d(x,Tx) ≤ ϕ(x) - ϕ(Tx) for all x∈ X, and (X,d) is separable. Assuming CH (the Continuum Hypothesis), we deduce that (X,d) is compact.
2
Content available remote

Continuous dependence on parameters for second order discrete BVP’s

64%
Galewski M., Głąb S.
Open Mathematics
|
2012
|
tom 10
|
nr 3
1076-1083
EN
Using Fan’s Min-Max Theorem we investigate existence of solutions and their dependence on parameters for some second order discrete boundary value problem. The approach is based on variational methods and solutions are obtained as saddle points to the relevant Euler action functional.
3
Content available remote

Descriptive properties of density preserving autohomeomorphisms of the unit interval

64%
Głąb S., Strobin F.
Open Mathematics
|
2010
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tom 8
|
nr 5
928-936
EN
We prove that density preserving homeomorphisms form a Π11-complete subset in the Polish space ℍ of all increasing autohomeomorphisms of unit interval.
4
Content available remote

Large free subgroups of automorphism groups of ultrahomogeneous spaces

64%
Głąb S., Strobin F.
Colloquium Mathematicum
|
2015
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tom 140
|
nr 2
279-295
EN
We consider the following notion of largeness for subgroups of $S_{∞}$. A group G is large if it contains a free subgroup on 𝔠 generators. We give a necessary condition for a countable structure A to have a large group Aut(A) of automorphisms. It turns out that any countable free subgroup of $S_{∞}$ can be extended to a large free subgroup of $S_{∞}$, and, under Martin's Axiom, any free subgroup of $S_{∞}$ of cardinality less than 𝔠 can also be extended to a large free subgroup of $S_{∞}$. Finally, if Gₙ are countable groups, then either $∏_{n∈ℕ} Gₙ$ is large, or it does not contain any free subgroup on uncountably many generators.
5
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Algebraic and topological properties of some sets in ℓ₁

51%
Banakh T., Bartoszewicz A., Głąb S., Szymonik E.
Colloquium Mathematicum
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2012
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tom 129
|
nr 1
75-85
EN
For a sequence x ∈ ℓ₁∖c₀₀, one can consider the set E(x) of all subsums of the series $∑_{n=1}^{∞} x(n)$. Guthrie and Nymann proved that E(x) is one of the following types of sets: (𝓘) a finite union of closed intervals; (𝓒) homeomorphic to the Cantor set; 𝓜 𝓒 homeomorphic to the set T of subsums of $∑_{n=1}^{∞} b(n)$ where b(2n-1) = 3/4ⁿ and b(2n) = 2/4ⁿ. Denote by ℐ, 𝓒 and 𝓜 𝓒 the sets of all sequences x ∈ ℓ₁∖c₀₀ such that E(x) has the property (ℐ), (𝓒) and (𝓜 𝓒), respectively. We show that ℐ and 𝓒 are strongly 𝔠-algebrable and 𝓜 𝓒 is 𝔠-lineable. We also show that 𝓒 is a dense $𝒢_δ$-set in ℓ₁ and ℐ is a true $ℱ_σ$-set. Finally we show that ℐ is spaceable while 𝓒 is not.
6
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Large structures made of nowhere $L^{q}$ functions

51%
Głąb S., Kaufmann P., Pellegrini L.
Studia Mathematica
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2014
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tom 221
|
nr 1
13-34
EN
We say that a real-valued function f defined on a positive Borel measure space (X,μ) is nowhere q-integrable if, for each nonvoid open subset U of X, the restriction $f|_U$ is not in $L^{q}(U)$. When (X,μ) has some natural properties, we show that certain sets of functions defined in X which are p-integrable for some p's but nowhere q-integrable for some other q's (0 < p,q < ∞) admit a variety of large linear and algebraic structures within them. The presented results answer a question of Bernal-González, improve and complement recent spaceability and algebrability results of several authors and motivate new research directions in the field of spaceability.
7
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Erratum to "Algebraic and topological properties of some sets in ℓ₁" (Colloq. Math. 129 (2012), 75-85)

51%
Banakh T., Bartoszewicz A., Głąb S., Szymonik E.
Colloquium Mathematicum
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2014
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tom 135
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nr 2
295-298
8
Content available remote

Dichotomies for Lorentz spaces

51%
Głąb S., Strobin F., Yang C.
Open Mathematics
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2013
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tom 11
|
nr 7
1228-1242
EN
Assume that L p,q, $L^{p_1 ,q_1 } ,...,L^{p_n ,q_n } $ are Lorentz spaces. This article studies the question: what is the size of the set $E = \{ (f_1 ,...,f_n ) \in L^{p_{1,} q_1 } \times \cdots \times L^{p_n ,q_n } :f_1 \cdots f_n \in L^{p,q} \} $. We prove the following dichotomy: either $E = L^{p_1 ,q_1 } \times \cdots \times L^{p_n ,q_n } $ or E is σ-porous in $L^{p_1 ,q_1 } \times \cdots \times L^{p_n ,q_n } $, provided 1/p ≠ 1/p 1 + … + 1/p n. In general case we obtain that either $E = L^{p_1 ,q_1 } \times \cdots \times L^{p_n ,q_n } $ or E is meager. This is a generalization of the results for classical L p spaces.
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