Abstract
We obtain the existence of a periodic solution for differential inclusions with lower semi-continuous right-hand sides on non-compact Lie groups. The construction is based on the combination of the method of guiding functions and the technique of integral operators with parallel translation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data availability
Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.
Notes
Notice that it does not follow from the homeomorphism of U to an open ball.
References
Krasnoselskii, M. A.; Perov, A. I.: On a certain principle of existence of bounded, periodic and almost periodic solutions of systems of ordinary differential equations. (Russian) Dokl. Akad. Nauk SSSR. 123, 1958, 235–238.
Krasnosel’skii, M. A.: Topological Methods in the Theory of Nonlinear Integral Equations, New York, Macmillan, 1964.
Krasnoselskii, M.A.: The Operator of Translation Along the Trajectories of Differential Equations, Providence, RI, AMS, 1968.
Krasnosel’skii, M. A. and Zabreiko, P. P.: Geometrical Methods of Nonlinear Analysis (Grundlehren der Mathematischen Wissenschaften, 263), Berlin, Springer-Verlag, 1984.
Obukhovskii, V.; Zecca, P.; Loi N.V. and Kornev, S. Method of Guiding Functions in Problems of Nonlinear Analysis. Lecture Notes in Mathematics, 2076. Springer, Heidelberg, 2013.
Gliklikh Yu., Kornev S. and Obukhovskii V.: Guiding potentials and periodic solutions of differential equations on manifolds. Global and Stochastic Analysis.- 2019.- Vol. 6.- No. 1.- P. 1 – 6.
Gliklikh Yu.E.: Global Analysis in Mathematical Physics. Geometric and Stochastic Methods. New York: Springer-Verlag,1997.
Gliklikh Yu. E.: Global and stochastic analysis with applications to mathematical physics, London, Springer-Verlag, 2011.
Obukhovskii V., Gel’man D.: Multivalued Maps and Differential Inclusions. Elements of Theory and Applications, World Scientific Publishing Co., Hackensack, NJ, 2020
Hirsch, M.W.: Differential Topology. (Graduate Texts in Mathematics, 33), New York, Springer-Verlag, 1994.
Borisovich, Ju. G. and Gliklih, Ju. E.: Fixed points of mappings of Banach manifolds and some applications. Nonlinear Anal., Vol. 4, no. 1 (1980) 165–192.
Funding
The research of S. Kornev and V. Obukhovskii was supported by the Ministry of Education of the Russian Federation within the framework of the state task in the field of science (topic number FZGF-0640-2020-0009).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Gliklikh, Y.E., Kornev, S.V. & Obukhovskii, V.V. INTEGRAL OPERATORS, GUIDING POTENTIALS AND PERIODIC SOLUTIONS OF DIFFERENTIAL INCLUSIONS ON NON-COMPACT LIE GROUPS. J Math Sci 266, 667–674 (2022). https://doi.org/10.1007/s10958-022-06004-6
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-022-06004-6
Keywords
- Lie groups
- Differential inclusions
- Guiding potentials
- Integral operators with parallel translation
- Periodic solution