The Method of Generalized Integral Guiding Function in the Periodic Problem of Differential Inclusions
In the present paper we consider new methods for solving the periodic problem for a nonlinear system governed by a differential inclusion of the following form:
x' (t) ∈ F(t, x(t)).
In the first part of the article we assume that the multivalued map F : R × Rn ⊸ Rn has convex compact values, satisfies the upper Caratheodory conditions, sublinear growth condition and T -periodic in the first argument. Under the above assumptions the closed multivalued superposition operator PF : C([0, T] Rn) → P(L1([0, T] Rn)) assigning to each function x(·) the set of all integrable selections of the multifunction F(t, x(t)) is well defined. In the second part of the article we assume that the multivalued map F : R × Rn ⊸ Rn is regular with compact values satisfying the T -periodicity condition in the first argument. Notice that the class of regular multimaps is broad enough. It includes, in particular, bounded almost lower semicontinuous multimaps with compact values. In both cases for the study of the periodic problem the generalized integral guiding function method is applied. An essential development of the concept of the guiding function is the fact that the basic condition is assumed to hold, firstly, in an integral form secondly, in the domain defined by the guiding function and at last, not necessarily for all integrable selections of the superposition multioperator. Application of the coincidence degree theory and the multivalued maps theory allows to establish the solvability of the periodic problem.
1. Borisovich Yu.G., Gel’man B.D., Myshkis A.D., Obukhovskii V.V. Vvedenie v teoriyu mnogoznachnykh otobrageniy i differentsial’nykh vklyucheniy [Introduction to the theory of multivalued maps and differential inclusions]. Moscow, Librokom, 2011. 226 p.
2. Kornev S.V., Obukhovskii V.V. On some versions of the theory of topological degree for nonconvex-valued multimap (in Russian) [O nekotorykh variantakh teorii topologicheskoy stepeni dlya nevypukloznachnykh multiotobrazhenij]. Trudymatematicheskogo fakulteta, Voronezh State University, Voronezh, 2004, vol. 8, pp. 56-74.
3. Kornev S.V., Obukhovskii V.V. On localization of the guiding function method in the periodic problem of differential inclusions (in Russian) [O lokalizatsii metoda napravlyayushchikh funktsij v zadache o periodicheskikh resheniyakhdifferentsial’nykh vklyuchenij]. Izv. vuzov. Matematika, 2009, no. 5, pp. 23-32.
4. Kornev S.V. Nonsmooth integral guiding functions in the problem of forced oscillations (in Russian) [Negladkie integral’nye napravlyaushchie funktsii v zadache o vynugdennykh kolebaniyakh]. Avtomatika i telemehanika, 2015, no. 9,pp. 31-43.
5. Krasnosel’skii M.A. Operator sdviga po traektoriyam differentsial’nykh uravnenii [The Operator of Translation along the Trajectories of Differential Equations]. Moscow, Nauka, 1966. 332 p.
6. Krasnosel’skii M.A., Perov A.I. On existence principle for bounded, periodic and almost periodic solutions to the systems of ordinary differential equations (in Russian) [Ob odnom printsipe sushchestvovaniya ogranichennykh, periodicheskikh i pochti periodicheskikh reshenij u sistem obyknovennykh differentsial’nykh uravnenij]. Dokl. Akad. Nauk SSSR, 1958, vol. 123, no 2, pp. 235-238.
7. Finogenko I.A. On differential equations with discontinuous right-hand side (in Russian) [O differentsial’nykh uravnenijkh s razryvnoy pravoy chast’yu]. Izv. Irkut. gos. un-ta. Ser. Math., 2010, vol. 3, no 2, pp. 88-102.
8. Bressan A. Upper and lower semicontinuous differential inclusions: A unified approach. In: H. Sussmann (Ed.), Nonlinear Controllability and Optimal Control. New York, Dekker, 1990, pp. 21-31.
9. Bressan A. Differential inclusions without convexity: A survey of directionally continuous selections. Proceedings of the First World Congress of Nonlinear Analyst.. Tampa, Florida, 1992, Lakshmikantham, V., Ed., Walter de Gruyter,1996, pp. 2081-2088.
10. Fonda A. Guiding functions and periodic solutions to functional differential equations. Proc. Amer. Math. Soc., 1987, vol. 99, no 1, pp. 79-85.
11. G´orniewicz L. Topological Fixed Point Theory of Multivalued Mappings. Berlin, Springer, 2006. 556 p.
12. Kamenskii M., Obukhovskii V., Zecca P. Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces. Berlin-New York, Walter de Gruyter, 2001. 231 p.
13. Kisielewicz M. Differential inclusions and optimal control, Kluwer, Dordrecht: PWN Polish Scientific Publishers, Warsaw, 1991.
14. Kornev S., Obukhovskii V., Yao J.C. On asymptotics of solutions for a class of functional differential inclusions. Discussiones Mathematicae. Differential Inclusions, Control and Optimization, 2014, vol. 34, issue 2, pp. 219-227.
15. Mawhin J.L. Topological degree methods in nonlinear boundary value problems. CBMS Regional Conf. Ser. in Math., Amer. Math. Soc. Providence, R.I., 1977, no 40.
16. Mawhin J., Ward James R. Jr. Guiding-like functions for periodic or bounded solutions of ordinary differential equations. Discrete and continuous dynamical systems, 2002, vol. 8, no 1, pp. 39-54.
17. Obukhovskii V., Zecca P., Loi N.V., Kornev S. Method of guiding functions in problems of nonlinear analysis. Lecture Notes in Math. V. 2076. Berlin, Springer, 2013. 177 p.
18. Pruszko T. A coincidence degree for L-compact convex-valued mappings and its application to the Picard problem for orientor fields. Bull. Acad. pol. sci. S´er. sci math., 1979, vol. 27, no 11–12, pp. 895–902.
19. Pruszko T. Topological degree methods in multi-valued boundary value problems. Nonlinear Anal.: Theory, Meth. and Appl., 1981, vol. 5, no 9, pp. 959-970.
20. Tarafdar E., Teo S.K. On the existence of solutions of the equation Lx ∈ Nx and a coincidence degree theory. J. Austral. Math. Soc., 1979, vol. A28, no 2, pp. 139-173.
21. Tolstonogov A. Differential inclusions in a Banach space. Kluwer Academic Publishers, 2000. 302 p.