# On automatic boundedness of Nemytskiĭ set-valued operators

Studia Mathematica (1995)

• Volume: 113, Issue: 1, page 65-72
• ISSN: 0039-3223

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## Abstract

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Let X, Y be two separable F-spaces. Let (Ω,Σ,μ) be a measure space with μ complete, non-atomic and σ-finite. Let ${N}_{F}$ be the Nemytskiĭ set-valued operator induced by a sup-measurable set-valued function $F:\Omega ×X\to {2}^{Y}$. It is shown that if ${N}_{F}$ maps a modular space $\left(N\left(L\left(\Omega ,\Sigma ,\mu ;X\right)\right),{\varrho }_{N,\mu }\right)$ into subsets of a modular space $\left(M\left(L\left(\Omega ,\Sigma ,\mu ;Y\right)\right),{\varrho }_{M,\mu }\right)$, then ${N}_{F}$ is automatically modular bounded, i.e. for each set K ⊂ N(L(Ω,Σ,μ;X)) such that ${r}_{K}=sup{\varrho }_{N,\mu }\left(x\right):x\in K<\infty$ we have $sup{\varrho }_{M,\mu }\left(y\right):y\in {N}_{F}\left(K\right)<\infty$.

## How to cite

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Rolewicz, S., and Song, Wen. "On automatic boundedness of Nemytskiĭ set-valued operators." Studia Mathematica 113.1 (1995): 65-72. <http://eudml.org/doc/216160>.

@article{Rolewicz1995,
abstract = {Let X, Y be two separable F-spaces. Let (Ω,Σ,μ) be a measure space with μ complete, non-atomic and σ-finite. Let $N_F$ be the Nemytskiĭ set-valued operator induced by a sup-measurable set-valued function $F:Ω × X → 2^\{Y\}$. It is shown that if $N_F$ maps a modular space $(N(L(Ω,Σ,μ;X)), ϱ_\{N,μ\})$ into subsets of a modular space $(M(L(Ω,Σ,μ;Y)),ϱ_\{M,μ\})$, then $N_F$ is automatically modular bounded, i.e. for each set K ⊂ N(L(Ω,Σ,μ;X)) such that $r_K = sup\{ϱ_\{N,μ\}(x) : x ∈ K\} < ∞$ we have $sup\{ϱ_\{M,μ\}(y): y ∈ N_F(K)\} < ∞$.},
author = {Rolewicz, S., Song, Wen},
journal = {Studia Mathematica},
keywords = {Nemytskiĭ set-valued operators; superposition measurable set-valued operators; automatic boundedness; modular spaces; separable Fréchet spaces; sup-measurable multifunction; superposition operator; bounded on bounded sets},
language = {eng},
number = {1},
pages = {65-72},
title = {On automatic boundedness of Nemytskiĭ set-valued operators},
url = {http://eudml.org/doc/216160},
volume = {113},
year = {1995},
}

TY - JOUR
AU - Rolewicz, S.
AU - Song, Wen
TI - On automatic boundedness of Nemytskiĭ set-valued operators
JO - Studia Mathematica
PY - 1995
VL - 113
IS - 1
SP - 65
EP - 72
AB - Let X, Y be two separable F-spaces. Let (Ω,Σ,μ) be a measure space with μ complete, non-atomic and σ-finite. Let $N_F$ be the Nemytskiĭ set-valued operator induced by a sup-measurable set-valued function $F:Ω × X → 2^{Y}$. It is shown that if $N_F$ maps a modular space $(N(L(Ω,Σ,μ;X)), ϱ_{N,μ})$ into subsets of a modular space $(M(L(Ω,Σ,μ;Y)),ϱ_{M,μ})$, then $N_F$ is automatically modular bounded, i.e. for each set K ⊂ N(L(Ω,Σ,μ;X)) such that $r_K = sup{ϱ_{N,μ}(x) : x ∈ K} < ∞$ we have $sup{ϱ_{M,μ}(y): y ∈ N_F(K)} < ∞$.
LA - eng
KW - Nemytskiĭ set-valued operators; superposition measurable set-valued operators; automatic boundedness; modular spaces; separable Fréchet spaces; sup-measurable multifunction; superposition operator; bounded on bounded sets
UR - http://eudml.org/doc/216160
ER -

## References

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9. [9] H. Nakano, Topology and Linear Topological Spaces, Maruzen, Tokyo, 1951.
10. [10] V. Niemytzki [V. Nemytskiĭ], Sur les équations intégrales non linéaires, C. R. Acad. Sci. Paris 196 (1933), 836-838. Zbl0006.20901
11. [11] V. Niemytzki [V. Nemytskiĭ], Théorèmes d'existence et d'unicité des solutions de quelques équations intégrales non-linéaires, Mat. Sb. 41 (1934), 421-438. Zbl0011.02603
12. [12] T. Pruszko, Topological degree methods in multi-valued boundary value problems, Nonlinear Anal. 5 (1981), 959-973. Zbl0478.34017
13. [13] S. Rolewicz, Metric Linear Spaces, Reidel and PWN, 1985.
14. [14] W. Song, Multivalued superposition operators in ${L}^{p}\left(\Omega ,X\right)$, preprint.

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