Topological approach to random diferential inclusions

Abstract

In the present paper random multivalued admissible operators are considered. First for such operators we shall formulate the following topological results: Schauder-type Fixed Point Theorems, Leray?Schauder Alternative, Granas Continuation Method and Topological Degree. Next these problems will be transformed to the existence problems, periodic problems and implicit problems for random di?erentuial inclusions. Let us remark that this paper constitute a summary and complement of the following earlier papers: [2], [3], [5], [6], [10], [11], [14] and [15]. This work can be considered as an advanced survey with some new results: mainly concerning the theory of random di?erential inclusions. We believe that this paper will be useful for mathematiciants and students intrested in topological methods of nonconvex analysis.

Keywords

• random operators,
• multivalued mappings,
• fixed points
• topological essentiality
• topological degree
• differential inclusions

References

1. 1] J. Andres, L. Górniewicz, Topological Fixed Point Principles for Boundary Value Problems, Kluwer, 2003.
2. [2] J. Andres, L. Górniewicz, Random topological degree and random di?erential inclusions, Topol. Methods Nonlinear Anal. 40 (2012), 337-358.
3. [3] J. Andres , L. Górniewicz, Implicit diferential inclusions with acyclic right-hand sides an essential fixed point approach, Dyn. Syst. 26 (2017), 237-258.
4. [4] T.D. Benavides, G.L. Acedo, H.K. Xu, Random fixed points of set valued mappings, Proc. Amer. Math. Soc. 124 (1996), 431-438.
5. [5] R. Bielawski, L. Górniewicz, Some applications of the Leray-Schauder alternative to diferential equations, NATO ASI Series Ser. C Math. Phys. Ser., Vol. 173, edited by S.P. Singh, 187-194.
6. [6] R. Bielawski, L. Górniewicz, A fixed point approach to di?erential equations, Lecture Notes in Math., Vol. 1411, Springer, Berlin, 1989, 9-14.
7. [7] R. Bielawski, L. Górniewicz, S. Plaskacz, Topological approach to diferential inclusions on closed subsets of R n , Dynamics Reported, 1 New Series, Springer, 1991, 225-250.
8. [8] F.S. De Blasi, L. Górniewicz, G. Pianigiani, Topological degree and periodic solutions of di?erential inclusions, Nonlinear Anal. 3 (1999), 217-245.

9. [9] F.S. De Blasi, J. Myjak, On the solution sets for diferential inclusions, Bull. Polon. Math. Soc. 12 (1985), 17-23.
10. [10] M. Frigon, L. Górniewicz, T. Kaczyski, Diferential inclusions and implicit equations on closed subset of euclidean n- dimensional space, Proceedings of the First WCNA, Tampa, Florida, W. de Gruyter, Berlin 1996, 1197-1206.
11. [11] G. Gabor, L. Górniewicz, M. ‘losarski, Generalized topological essentiality and coincidence points of multivalued mappings, Set-Valued Anal. 17 (2009), 1-19.
12. [12] L. Górniewicz, Topological structure of solution sets: current results, Arch. Math. 36 (2000), 343-382.
13. [13] L. Górniewicz, Topological Fixed Point Theory for Multivalued Mappings, Springer, 2006 (second edition).
14. [14] L. Górniewicz, O. Górniewicz, Topological essentiality of random w-admissible operators, Discuss. Math. 39 (2019), 123- 134. [15] L. Górniewicz, M. Slosarski, Topological essentiality and diferential inclusions, Bull. Aust. Math. Soc. 45 (1992), 177-193.
15. [16] A. Granas, The theory of compact vector fields and some of its applications, Dissertationes Math. 30 (1962), 1-93.
16. [17] A. Granas, J. Dugundji, Fixed Point Theory, Springer, 2003.
17. [18] N.S. Papageorgiou, Random fixed points and random diferential inclusions, Int. J. Math. Sci. 3 (1988), 551-560.
18. [19] L.E. Rybinski, Random fixed points and viable random solutions of functional diferential equations, J. Math. Anal. 142 (1989), 53-61.
19. [20] J.L. Strand, Random ordinary diferential equations, J. Diferential Equations 7 (1970), 538-553.