# Method of Nonsmooth Integral Guiding Functions in Periodic Solutions Problem for Inclusions with Causal Multioperators

S.V. KornevAs it is known, differential inclusions are very useful mathematical tools to describe nonlinear control systems with feedback, automatic control systems, discontinuous systems, impulse response and other objects of modern engineering, mechanics, physics. In the present paper the new method to solving the problem of periodic oscillations of controlled systems described by a differential inclusion with a causal multioperator is introduced. Firstly differential equations with causal operator, or Volterra type equations where considered by L. Tonelli and A.N. Tikhonov. A.N. Tikhonov used them as the model in study of some thermal conductivity problems, in particular the problem of body coding when there is radiation from its surface. At first we consider the case when the multioperator is closed and convex-valued. Then the case of a non-convex-valued and lower semicontinuous right-hand part is considered. As the main research tool of the problem in both cases a modified method of the classical guiding function is applied. Namely, the method of nonsmooth integral guiding function is considered. Application of topological degree theory and this method allows to establish the solvability of periodic problem in each of the cases.Full text

- Keywords
- inclusion; causal multioperator; periodic solutions; nonsmooth integral guiding function; topological degree.
- References
- 1. Tonelli L. Sulle equazioni funzionali di Volterra. Bulletin of Calcutta Mathematical Society, 1930, vol. 20, P. 31-48.

2. Tihonov A.N. On Functional Equations of Volterra Type and Their Applications to Some Problems of Mathematical Physic. Bulletin of the Moscow State University. Section A. Series: Mathematics and Mechanics, 1938, vol. 1, issue 8, pp. 1-25. (in Russian)

3. Corduneanu, C. Functional Equations with Causal Operators. Stability and Control: Theory, Methods and Applications. London: Taylor and Francis, 2002.

4. Drici Z., McRae F.A., Devi J. Vasundhara Differential Equations with Causal Operators in a Banach Space. Nonlinear Analysis, 2005, vol. 62, no. 2, pp. 301-313.

5. Drici Z., McRae F.A., Vasundhara D.J. Monotone Iterative Technique for Periodic Boundary Value Problems with Causal Operators. Nonlinear Analysis, 2006, vol. 64, no. 6, pp. 1271-1277.

6. Jankowski T. Boundary Value Problems with Causal Operators. Nonlinear Analysis, 2008, vol. 68, no. 12, pp. 3625-3632.

7. Lupulescu V. Causal Functional Differential Equations in Banach Spaces. Nonlinear Analysis, 2008, vol. 69, no. 12, pp. 4787-4795.

8. Burlakov E.O., Zhukovskiy E.S. The Continuous Dependence of Solutions to Volterra Equations with Locally Contracting Operators on Parameters. Russian Mathematics (Izvestiya VUZ. Matematika), 2010, no. 8, pp. 12-23. DOI: 10.3103/S1066369X10080025

9. Zhukovskiy E.S., Zhukovskiy T.V., Alves M.J. Correctness of Equations with Generalized Volterra Maps of Metric Spaces. Tambov University Reports. Series: Natural and Technical Sciences, 2010, vol. 15, issue 6, pp. 1669-1672. (in Russian)

10. Obukhovskii V., Zecca P. On Certain Classes of Functional Inclusions with Causal Operators in Banach Spaces. Nonlinear Analysis, 2011, vol. 74, no. 8, pp. 2765-2777.

11. Krasnosel'skii M.A. Operator sdviga po traektoriyam differentsial'nykh uravneniy [The Operator of Translation along the Trajectories of Differential Equations]. Moscow, Nauka, 1966. 332 p.

12. Krasnosel'skii M.A., Perov A.I. On Existence Principle for Bounded, Periodic and Almost Periodic Solutions to the Systems of Ordinary Differential Equations. Dokl. Akad. Nauk SSSR, 1958, vol. 123, no. 2, pp. 235-238.(in Russian)

13. Krasnosel'skii M.A., Zabreiko P.P. Geometricheskie metody nelineynogo analiza [Geometrical Methods of Nonlinear Analysis]. Moscow, Nauka, 1975. 512 p.

14. Mawhin J. Topological Degree Methods in Nonlinear Boundary Value Problems. CBMS Regional Conference Series in Mathematics, 40. American Mathematical Society. Providence, 1979. DOI: 10.1090/cbms/040

15. Mawhin J., Ward J.R. 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.

16. 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.

17. G'orniewicz L. Topological Fixed Point Theory of Multivalued Mappings. Berlin: Springer, 2006.

18. Fonda A. Guiding Functions and Periodic Solutions to Functional Differential Equations. Proceedings of the American Mathematical Society, 1987, vol. 99, no 1, pp. 79-85.

19. Kornev S., Obukhovskii V. On Some Developments of the Method of Integral Guiding Functions. Functional Differential Equations, 2005, vol. 12, no. 3-4, pp. 303-310.

20. Loi N.V., Obukhovskii V., Zecca P. On the Global Bifurcation of Periodic Solutions of Differential Inclusions in Hilbert Spaces. Nonlinear Analysis, 2013, vol. 76, pp. 80-92.

21. 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.

22.Kornev S.V., Obukhovskii V.V. On Asymptotic Behavior of Solutions of Differential Inclusions and the Method of Guiding Functions. Differential Equations, 2015, vol. 51, no. 6, pp. 711-716. DOI: 10.1134/S0012266115060014

23. 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.

24. Deimling K. Multivalued Differential Equations. Berlin; N.Y.: Walter de Gruyter, 1992. DOI: 10.1515/9783110874228

25. Kamenskii M., Obukhovskii V., Zecca P. Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces. Berlin; N.Y.: Walter de Gruyter, 2001. DOI: 10.1515/9783110870893

26. Fryszkowski A. Fixed Point Theory for Decomposable Sets. Dordrecht: Kluwer AP, 2004. DOI: 10.1007/1-4020-2499-1

27.Clark F. Optimization and Nonsmooth Analysis. Wiley, 1983. 308 p.