We consider a neutral type operator differential inclusion and apply the topological degree theory for condensing multivalued maps to justify the question of existence of its periodic solution. By using the averaging method, we apply the abstract result to an inclusion with a small parameter. As example, we consider a delay control system with the distributed control.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1152,
author = {Mikhail Kamenskii and Valeri Obukhovskii and Jen-Chih Yao},
title = {On some topological methods in theory of neutral type operator differential inclusions with applications to control systems},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
volume = {33},
year = {2013},
pages = {193-204},
zbl = {1300.34146},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1152}
}
Mikhail Kamenskii; Valeri Obukhovskii; Jen-Chih Yao. On some topological methods in theory of neutral type operator differential inclusions with applications to control systems. Discussiones Mathematicae, Differential Inclusions, Control and Optimization, Tome 33 (2013) pp. 193-204. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1152/
[000] [1] Yu.G. Borisovich, B.D. Gelman, A.D. Myshkis and V.V. Obukhovskii, Topological methods in the theory of fixed points of multivalued mappings. (Russian) Uspekhi Mat. Nauk 35 (1980), 59-126. English translation: Russian Math. Surveys 35 (1980), 65-143. doi: 10.1070/RM1980v035n01ABEH001548
[001] [2] Yu.G. Borisovich, B.D. Gelman, A.D. Myshkis and V.V. Obukhovskii, Introduction to the Theory of Multivalued Maps and Differential Inclusions, (Russian) Second edition, Librokom, Moscow, 2011.
[002] [3] L. Górniewicz, Topological Fixed Point Theory of Multivalued Mappings, 2nd edition, Topological Fixed Point Theory and Its Applications, 4. Springer, Dordrecht, 2006. | Zbl 1107.55001
[003] [4] S. Hu and N.S. Papageorgiou, Handbook of Multivalued Analysis, Vol. I. Theory, Mathematics and its Applications, 419. Kluwer Academic Publishers, Dordrecht, 1997. | Zbl 0887.47001
[004] [5] M. Kamenskii, V. Obukhovskii and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, de Gruyter Series in Nonlinear Analysis and Applications, 7. Walter de Gruyter & Co., Berlin, 2001. doi: 10.1515/9783110870893 | Zbl 0988.34001
[005] [6] M.A. Krasnosel'skii and P.P. Zabreiko, Geometrical Methods of Nonlinear Analysis, A Series of Comprehensive Studies in Mathematics, 263, Springer-Verlag, Berlin-Heidelberg-New York-Tokio, 1984.
[006] [7] M.A. Krasnoselskii, P.P. Zabreiko, E.I. Pustyl'nik and P.E. Sobolevskii, Integral Operators in Spaces of Summable Functions, Noordhoff International Publishing, Leyden, 1976.