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

Janusz Matkowski; Małgorzata Wróbel

Open Mathematics (2012)

- Volume: 10, Issue: 2, page 609-618
- ISSN: 2391-5455

## Access Full Article

top## Abstract

top## How to cite

topJanusz Matkowski, and Małgorzata Wróbel. "Uniformly bounded set-valued Nemytskij operators acting between generalized Hölder function spaces." Open Mathematics 10.2 (2012): 609-618. <http://eudml.org/doc/269761>.

@article{JanuszMatkowski2012,

abstract = {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.},

author = {Janusz Matkowski, Małgorzata Wróbel},

journal = {Open Mathematics},

keywords = {Nemytskij composition operator; Uniformly bounded operator; Set-valued function; Generalized Hölder function metric space; uniformly bounded operator; set-valued function; generalized Hölder function metric space},

language = {eng},

number = {2},

pages = {609-618},

title = {Uniformly bounded set-valued Nemytskij operators acting between generalized Hölder function spaces},

url = {http://eudml.org/doc/269761},

volume = {10},

year = {2012},

}

TY - JOUR

AU - Janusz Matkowski

AU - Małgorzata Wróbel

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

JO - Open Mathematics

PY - 2012

VL - 10

IS - 2

SP - 609

EP - 618

AB - 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.

LA - eng

KW - Nemytskij composition operator; Uniformly bounded operator; Set-valued function; Generalized Hölder function metric space; uniformly bounded operator; set-valued function; generalized Hölder function metric space

UR - http://eudml.org/doc/269761

ER -

## References

top- [1] Appell J., Zabrejko P.P., Nonlinear Superposition Operators, Cambridge Tracts in Math., 95, Cambridge University Press, Cambridge, 1990 Zbl0701.47041
- [2] Azócar A., Guerrero J.A., Matkowski J., Merentes N., Uniformly continuous set-valued composition operators in the spaces of functions of bounded variation in the sense of Wiener, Opuscula Math., 2010, 30(1), 53–60 Zbl1216.47090
- [3] Chistyakov V.V., Lipschitzian superposition operators between spaces of functions of bounded generalized variation with weight, J. Appl. Anal., 2000, 6(2), 173–186 http://dx.doi.org/10.1515/JAA.2000.173 Zbl0997.47051
- [4] Guerrero J.A., Leiva H., Matkowski J., Merentes N., Uniformly continuous composition operators in the space of bounded φ-variation functions, Nonlinear Anal., 2010, 72(6), 3119–3123 http://dx.doi.org/10.1016/j.na.2009.11.051 Zbl1225.47078
- [5] Ludew J.J., On Lipschitzian operators of substitution generated by set-valued functions, Opuscula Math., 2007, 27(1), 13–24 Zbl1160.47047
- [6] Mainka E., On uniformly continuous Nemytskij operators generated by set-valued functions, Aequationes Math., 2010, 79(3), 293–306 http://dx.doi.org/10.1007/s00010-010-0023-4 Zbl1227.47031
- [7] Matkowski J., Functional equations and Nemytskii operators, Funkcial. Ekvac., 1982, 25(2), 127–132 Zbl0504.39008
- [8] Matkowski J., Lipschitzian composition operators in some function spaces, Nonlinear Anal., 1997, 30(2), 719–726 http://dx.doi.org/10.1016/S0362-546X(96)00287-8 Zbl0894.47052
- [9] Matkowski J., Remarks on Lipschitzian mappings and some fixed point theorems, Banach J. Math. Anal., 2007, 2(1), 237–244 Zbl1146.47034
- [10] Matkowski J., Uniformly continuous superposition operators in the spaces of differentiable functions and absolutely continuous functions, In: Inequalities and Applications, Noszvaj, September 9–15, 2007, Internat. Ser. Numer. Math., 157, Birkhäuser, Basel, 2009, 155–166 http://dx.doi.org/10.1007/978-3-7643-8773-0_15
- [11] Matkowski J., Uniformly continuous superposition operators in the Banach space of Hölder functions, J. Math. Anal. Appl., 2009, 359(1), 56–61 http://dx.doi.org/10.1016/j.jmaa.2009.05.020 Zbl1173.47043
- [12] Matkowski J., Uniformly continuous superposition operators in the space of bounded variation functions, Math. Nachr., 2010, 283(7), 1060–1064 Zbl1235.47052
- [13] Matkowski J., Uniformly bounded composition operators between general Lipschitz function normed spaces, Topol. Methods Nonlinear Anal., 2011, 38(2), 395–406 Zbl1272.47070
- [14] Matkowski J., Miś J., On a characterization of Lipschitzian operators of substitution in the space BV〈a, b〉, Math. Nachr., 1984, 117, 155–159 http://dx.doi.org/10.1002/mana.3211170111 Zbl0566.47033
- [15] Matkowski J., Wróbel M., Uniformly bounded Nemytskij operators generated by set-valued functions between generalized Hölder function spaces, Discuss. Math. Differ. Incl. Control Optim., 2011, 31(2), 183–198 Zbl1264.47070
- [16] Smajdor A., Smajdor W., Jensen equation and Nemytskiı operator for set-valued functions, Rad. Math., 1989, 5(2), 311–320 Zbl0696.47057
- [17] Smajdor W., Note on Jensen and Pexider functional equations, Demonstratio Math., 1999, 32(2), 363–376 Zbl0938.39026

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.