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.
@article{bwmeta1.element.doi-10_2478_s11533-012-0002-1,
author = {Janusz Matkowski and Ma\l gorzata Wr\'obel},
title = {Uniformly bounded set-valued Nemytskij operators acting between generalized H\"older function spaces},
journal = {Open Mathematics},
volume = {10},
year = {2012},
pages = {609-618},
zbl = {1253.47039},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0002-1}
}
Janusz Matkowski; Małgorzata Wróbel. Uniformly bounded set-valued Nemytskij operators acting between generalized Hölder function spaces. Open Mathematics, Tome 10 (2012) pp. 609-618. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0002-1/
[1] Appell J., Zabrejko P.P., Nonlinear Superposition Operators, Cambridge Tracts in Math., 95, Cambridge University Press, Cambridge, 1990 | Zbl 0701.47041
[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 | Zbl 1216.47090
[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 | Zbl 0997.47051
[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 | Zbl 1225.47078
[5] Ludew J.J., On Lipschitzian operators of substitution generated by set-valued functions, Opuscula Math., 2007, 27(1), 13–24 | Zbl 1160.47047
[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 | Zbl 1227.47031
[7] Matkowski J., Functional equations and Nemytskii operators, Funkcial. Ekvac., 1982, 25(2), 127–132 | Zbl 0504.39008
[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 | Zbl 0894.47052
[9] Matkowski J., Remarks on Lipschitzian mappings and some fixed point theorems, Banach J. Math. Anal., 2007, 2(1), 237–244 | Zbl 1146.47034
[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 | Zbl 1173.47043
[12] Matkowski J., Uniformly continuous superposition operators in the space of bounded variation functions, Math. Nachr., 2010, 283(7), 1060–1064 | Zbl 1235.47052
[13] Matkowski J., Uniformly bounded composition operators between general Lipschitz function normed spaces, Topol. Methods Nonlinear Anal., 2011, 38(2), 395–406 | Zbl 1272.47070
[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 | Zbl 0566.47033
[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 | Zbl 1264.47070
[16] Smajdor A., Smajdor W., Jensen equation and Nemytskiı operator for set-valued functions, Rad. Math., 1989, 5(2), 311–320 | Zbl 0696.47057
[17] Smajdor W., Note on Jensen and Pexider functional equations, Demonstratio Math., 1999, 32(2), 363–376 | Zbl 0938.39026