Abstract
In this paper we consider the problem on the existence of forced oscillations in nonlinear objects governed by differential inclusions. We propose certain modifications of the methods of generalized and integral guiding functions.
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References
M. A. Krasnosel’skii, Translation Operator for Trajectories of Differential Equations (Nauka, Moscow, 1966) [in Russian].
M. A. Krasnosel’skii and P. P. Zabreiko, Geometric Methods of Nonlinear Analysis (Nauka, Moscow, 1975) [in Russian].
M. A. Krasnosel’skii and A. I. Perov, “On an Existence Principle for Bounded, Periodic, and Almost Periodic Solutions of Systems of Ordinary Differential Equations,” Sov. Phys. Dokl. 12(2), 235–238 (1958).
Yu. G. Borisovich, B. D. Gel’man, A. D. Myshkis, and V. V. Obukhovskii, Introduction to the Theory of Multi-Valued Maps and Differential Inclusions (KomKniga, Moscow, 2005) [in Russian].
K. Deimling, Multivalued Differential Equations (Walter de Gruyter, Berlin-New York, 1992).
L. Górniewicz, Topological Fixed Point Theory of Multivalued Mappings (Kluwer Acad. Publ., Dordrecht-Boston-London, 1999).
F. S. De Blasi, L. Górniewicz, and G. Pianigiani, “Topological Degree and Periodic Solutions of Differential Inclusions,” Nonlinear Anal. 37, 217–245 (1999).
J. Mawhin, J. R. Ward jr., “Guiding-Like Functions for Periodic or Bounded Solutions of Ordinary Differential Equations,” Discrete and Continuous Dynamical Systems 8(1), 39–54 (2002).
J. Mawhin and H. B. Thompson, “Periodic or Bounded Solutions of Caratheodory Systems of Ordinary Differential Equations,” J. Dyn. Diff. Eq. 15(2–3), 327–334 (2003).
M. Filippakis, L. Gasinski, and N. S. Papageorgiou, “Nonsmooth Generalized Guiding Functions for Periodic Differential Inclusions,” Nonlinear Diff. Eq. Appl. 13(1), 43–66 (2006).
M. Kamenskii, V. Obukhovskii, and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces (Walter de Gruyter, Berlin-New York, 2001).
J. L. Mawhin, “Topological Degree Methods in Nonlinear Boundary Value Problems,” in CBMS Regional Conf. Ser. Math. (Amer.Math. Soc. Providence, R. I, 1977), No. 40.
T. Pruszko, “A Coincidence Degree for L-compact Convex-Valued Mappings and its Application to the Picard Problem for Orientor Fields,” Bull. Acad. Polon. Sci. Sér. Sci. Math. 27(11–12), 895–902 (1979).
T. Pruszko, “Topological Degree Methods in Multi-Valued Boundary Value Problems,” Nonlinear Anal.: Theory,Meth. and Appl. 5(9), 959–973 (1981).
E. Tarafdar and S. K. Teo, “On the Existence of Solutions of the Equation ℒx ∈ Nx and a Coincidence Degree Theory,” J. Austral.Math. Soc. A28(2), 139–173 (1979).
S.V. Kornev and V. V. Obukhovskii, “On Some Versions of Topological Degree Theory for Nonconvex-Valued Multimaps,” in Sbornik trudov Matem. fakult. Voronezhsk. Univ. (Voronezh, 2004), No. 8, pp. 56–74.
F. Clarke, Optimization and Nonsmooth Analysis (JohnWiley & Sons, New York, 1983; Nauka, Moscow, 1988).
N. A. Bobylev, S.V. Emel’yanov, and S. K. Korovin, Geometric Methods in Variational Problems (Magistr, Moscow, 1998) [in Russian].
V. F. Dem’yanov and L. V. Vasil’ev, Nondifferential Optimization (Nauka, Moscow, 1981) [in Russian].
S. V. Emel’yanov, S. K. Korovin, N. A. Bobylev, and A. V. Bulatov, Homotopies of Extremal Problems (Nauka, Moscow, 2001) [in Russian].
A. Fonda, “Guiding Functions and Periodic Solutions to Functional Differential Equations,” Proc. Amer. Math. Soc. 99(1), 79–85 (1987).
S. Kornev and V. Obukhovskii, “On Some Developments of the Method of Integral Guiding Functions,” Functional Diff. Eq. 12(3–4), 303–310 (2005).
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Original Russian Text © S.V. Kornev and V.V. Obukhovskii, 2009, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2009, No. 5, pp. 23–32.
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Kornev, S.V., Obukhovskii, V.V. Localization of the method of guiding functions in the problem about periodic solutions of differential inclusions. Russ Math. 53, 19–27 (2009). https://doi.org/10.3103/S1066369X0905003X
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DOI: https://doi.org/10.3103/S1066369X0905003X