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Uniformly bounded set-valued Nemytskij operators acting between generalized Hölder function spaces

100%
Matkowski J., Wróbel M.
Open Mathematics
|
2012
|
tom 10
|
nr 2
609-618
EN
We show that the generator of any uniformly bounded set-valued Nemytskij composition operator acting between generalized Hölder function metric spaces, with nonempty, bounded, closed, and convex values, is an affine function.
2
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Uniformly bounded composition operators in the banach space of bounded (p, k)-variation in the sense of Riesz-Popoviciu

61%
Armao F., Głazowska D., Rivas S., Rojas J.
Open Mathematics
|
2013
|
tom 11
|
nr 2
357-367
EN
We prove that if the composition operator F generated by a function f: [a, b] × ℝ → ℝ maps the space of bounded (p, k)-variation in the sense of Riesz-Popoviciu, p ≥ 1, k an integer, denoted by RV(p,k)[a, b], into itself and is uniformly bounded then RV(p,k)[a, b] satisfies the Matkowski condition.
3
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A generalization of Krasnosel’skii fixed point theorem for sums of compact and contractible maps with application

61%
Przeradzki B.
Open Mathematics
|
2012
|
tom 10
|
nr 6
2012-2018
EN
The existence of a fixed point for the sum of a generalized contraction and a compact map on a closed convex bounded set is proved. The result is applied to a kind of nonlinear integral equations.
4
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Superposition operator on the space of sequences almost converging to zero

52%
Alekhno E.
Open Mathematics
|
2012
|
tom 10
|
nr 2
619-645
EN
We study the superposition operator f on on the space ac 0 of sequences almost converging to zero. Conditions are derived for which f has a representation of the form f x = a+bx +g x, for all x ∈ ac 0 with a = f 0, b ∈ D(ac 0), g a superposition operator from ℓ∞ into I(ac 0), D(ac 0) = {z: zx ∈ ac 0 for all x ∈ ac 0}, and I(ac 0) the maximal ideal in ac 0. If f is generated by a function f of a real variable, then f is linear. We consider the conditions for which a bounded function f generates f acting on ac 0 and the conditions for which there exists a sequence y ∈ ac 0 such that y − f y ∈ ac 0. In terms of f, criteria for the boundedness, continuity, and sequential σ(ac 0ℓ1)-continuity of f on ac 0 are given. It is shown that the continuity of f is equivalent to the weak sequential continuity. Finally, properties of spaces D(ac 0) and I(ac 0) are studied, and in particular it is established that the inclusion I(ac 0) ⊕ {λe: λ ∈ ℝ} ⊂ D(ac 0) is proper, where e = (1, 1, …). By means of D(ac 0), a number of Banach-Mazur limit properties are derived.
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