Uniformly bounded Nemytskij operators generated by set-valued functions between generalized Hölder function spaces

Janusz Matkowski; Małgorzata Wróbel

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (2011)

  • Volume: 31, Issue: 2, page 183-198
  • ISSN: 1509-9407

Abstract

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We prove that the generator of any uniformly bounded set-valued Nemytskij operator acting between generalized Hölder function metric spaces, with nonempty compact and convex values is an affine function with respect to the function variable.

How to cite

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Janusz Matkowski, and Małgorzata Wróbel. "Uniformly bounded Nemytskij operators generated by set-valued functions between generalized Hölder function spaces." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 31.2 (2011): 183-198. <http://eudml.org/doc/271174>.

@article{JanuszMatkowski2011,
abstract = {We prove that the generator of any uniformly bounded set-valued Nemytskij operator acting between generalized Hölder function metric spaces, with nonempty compact and convex values is an affine function with respect to the function variable.},
author = {Janusz Matkowski, Małgorzata Wróbel},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
keywords = {Nemytskij composition operator; uniformly bounded operator; set-valued function; generalized Hölder function metric space; superposition operator},
language = {eng},
number = {2},
pages = {183-198},
title = {Uniformly bounded Nemytskij operators generated by set-valued functions between generalized Hölder function spaces},
url = {http://eudml.org/doc/271174},
volume = {31},
year = {2011},
}

TY - JOUR
AU - Janusz Matkowski
AU - Małgorzata Wróbel
TI - Uniformly bounded Nemytskij operators generated by set-valued functions between generalized Hölder function spaces
JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization
PY - 2011
VL - 31
IS - 2
SP - 183
EP - 198
AB - We prove that the generator of any uniformly bounded set-valued Nemytskij operator acting between generalized Hölder function metric spaces, with nonempty compact and convex values is an affine function with respect to the function variable.
LA - eng
KW - Nemytskij composition operator; uniformly bounded operator; set-valued function; generalized Hölder function metric space; superposition operator
UR - http://eudml.org/doc/271174
ER -

References

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  1. [1] J. Appell and P.P. Zabrejko, Nonlinear Superposition Operators, Cambridge University Press, 1990. doi:10.1017/CBO9780511897450 Zbl0701.47041
  2. [2] A. Azócar, J.A. Guerrero, J. Matkowski and N. Merentes, Uniformly continuous set-valued composition operators in the space of continuous functions of bounded variation in the sense of Wiener, Opuscula Math. 30 (2010), 53-60. Zbl1216.47090
  3. [3] V.V. Chistyakov, Lipschitzian superposition operators between spaces of functions of bounded generalized variation with weight, J. Appl. Anal. 6 (2000), 173-186. doi:10.1515/JAA.2000.173 
  4. [4] J.A. Guerrero, H. Leiva, J. Matkowski and N. Merentes, Uniformly continuous composition operators in the space of bounded φ-variation functions, Nonlinear Anal. 72 (2010), 3119-3123. doi:10.1016/j.na.2009.11.051 Zbl1225.47078
  5. [5] J.J. Ludew, On Lipschitzian operators of substitution generated by set-valued functions, Opuscula Math. 27 (1) (2007), 13-24. Zbl1160.47047
  6. [6] J. Matkowski, Functional equations and Nemytskij operators, Funkc. Ekvacioj Ser. Int. 25 (1982), 127-132. 
  7. [7] J. Matkowski, Lipschitzian composition operators in some function spaces, Nonlin. Anal. Theory Meth. Appl. 30 (2) (1997), 719-726. doi:10.1016/S0362-546X(96)00287-8 
  8. [8] J. Matkowski, Remarks on Lipschitzian mappings and some fixed point theorems, Banach J. Math. Anal. 2 (2007), 237-244 (electronic), www.math-analysis.org. Zbl1146.47034
  9. [9] J. Matkowski, Uniformly continuous superposition operators in the spaces of differentiable functions and absolutely continuous functions, Internat. Ser. Numer. Math. 157 (2008), 155-155. Zbl1266.47082
  10. [10] J. Matkowski, Uniformly continuous superposition operators in the space of Hölder functions, J. Math. Anal. Appl. 359 (2009), 56-61. doi:10.1016/j.jmaa.2009.05.020 Zbl1173.47043
  11. [11] J. Matkowski, Uniformly continuous superposition operators in the spaces of bounded variation functions, Math. Nachr. 283 (7) (2010), 1060-1064. Zbl1235.47052
  12. [12] J. Matkowski, Uniformly bounded composition operators between general Lipschitz function normed spaces, (accepted), Top. Math. Nonl. Anal. Zbl1272.47070
  13. [13] J. Matkowski and J. Miś, On a charakterization of Lipschitzian operators of substitution in the space BV[a,b], Math. Nachr. 117 (1984), 155-159. doi:10.1002/mana.3211170111 Zbl0566.47033
  14. [14] K. Nikodem, K-convex and K-concave set-valued functions, Zeszyty Naukowe Politechniki Łódzkiej, Mat. 559, Rozprawy Naukowe 114, 1989. 
  15. [15] A. Smajdor and W. Smajdor, Jensen equation and Nemytskii operator for set-valued functions, Rad. Math. 5 (1989), 311-320. Zbl0696.47057
  16. [16] W. Smajdor, Note on Jensen and Pexider functional equations, Demonstratio Math. 32 (1999), 363-376. Zbl0938.39026

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