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On a generalization of Krasnoselskii's theorem

Part of: Nonlinear operators and their properties Existence theories Miscellaneous applications of operator theory

Published online by Cambridge University Press:  01 August 2017

Dariusz Idczak  and
Andrzej Rogowski
Dariusz Idczak
Affiliation:
University of Łódź, S.Banacha 22, 90-238 Łódź, Poland e-mail: idczak@math.uni.lodz.pl, arogow@math.uni.lodz.pl
Andrzej Rogowski
Affiliation:
University of Łódź, S.Banacha 22, 90-238 Łódź, Poland e-mail: idczak@math.uni.lodz.pl, arogow@math.uni.lodz.pl
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Abstract

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In this paper we prove a generalization of the well known theorem of Krasnoselskii on the superposition operator in which the domain of Nemytskii's operator is a product space. We also give an application of this result.

Keywords

Nemytskii's operatorvariational methodscontinuous differentiability

MSC classification

Secondary: 49J50: Fréchet and Gateaux differentiability 47H30: Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.) 47N10: Applications in optimization, convex analysis, mathematical programming, economics
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 72 , Issue 3 , June 2002 , pp. 389 - 394
Copyright
Copyright © Australian Mathematical Society 2002

References

[1] Appell, J. and Zabrejko, P. P., ‘Boundedness properties of the superposition operator’. Bull. Polish Acad. Sci. Math. 37 (1989), 363–377.Google Scholar
[2] Appell, J. and Zabrejko, P. P., ‘Continuity properties of the superposition operator’, J. Austral. Math. Soc. Ser. A 47 (1989), 186–210.CrossRefGoogle Scholar
[3] Brézis, H.. Analyse fonctionelle. Theorie et applications (Masson, Paris, 1983).Google Scholar
[4] Ioffe, A. and Tikhomirov, V., Theory of extremal problems (North-Holland, Amsterdam, 1979).Google Scholar
[5] Krasnoselskii, M. K., Topological methods in the theory of nonlinear integral equations (Pergamon Press, New York, 1964).Google Scholar
[6] Krasnoselskii, M. K., Zabreiko, P. P., Pustylnik, J. I. and Sobolevskii, P. J., Integral operators in spaces of summable functions (Noordhoff, Leyden, 1976).CrossRefGoogle Scholar
[7] Mawhin, J. and Willem, P., Critical point theory and Hamiltonian systems (Springer, New York, 1989).CrossRefGoogle Scholar
[8] Struwe, M., Variational methods (Springer, Berlin, 1990).CrossRefGoogle Scholar
[9] Vainberg, M. M., Variational method and method of monotone operators in the theory of nonlinear equations (Halsted Press, New York, 1973).Google Scholar
[10] Willem, M., Minimax theorems (Birkhauser, Basel, 1996).CrossRefGoogle Scholar