Abstract
In this paper, applying the method of guiding functions, we consider the existence and global bifurcation problems for periodic solutions to first order operator-differential inclusions whose multivalued parts are not necessarily convex-valued.
Similar content being viewed by others
References
Barbu V.: Nonlinear Semigroups and Differential Equations in Banach Spaces. Noordhoff International Publishing, Leyden (1976)
Borisovich, Yu.G., Gelman, B.D., Myshkis, A.D., Obukhovskii, V.V.: Topological methods in the theory of fixed points of multivalued mappings (in Russian) Uspekhi Mat. Nauk 35. 1(211), 59–126 (1980); Translation: Russian Math. Surveys 35, 65–143 (1980)
Borisovich Y.G., Gelman B.D., Myshkis A.D., Obukhovskii V.V.: Introduction to the Theory of Multivalued Maps and Differential Inclusions (in Russian), 2nd edn. Librokom, Moscow (2011)
Borsuk K.: Theory of Retracts. Monografie Mat. 44. PWN, Warszawa (1967)
Clarke, F.H.: Optimization and nonsmooth analysis, 2nd edn. In: Classics in Applied Mathematics 5. Society for Industrial and Applied Mathematics (SIAM), Philadelphia
Denkowski Z., Migórski S., Papageorgiou N.S.: An Introduction to Nonlinear Analysis: Theory. Kluwer Academic Publishers, Boston (2003)
Dem’yanov, V.F., Vasil’ev, L.V.: Nondifferentiable optimization. Nauka, Moscow (1981) (in Russian); Translation: Translation Series in Mathematics and Engineering. Optimization Software, Publications Division, New York (1985)
Fonda A.: Guiding functions and periodic solutions to functional differential equations. Proc. Am. Math. Soc. 99(1), 79–85 (1987)
Gabor D., Kryszewski W.: A global bifurcation index for set-valued perturbations of Fredholm operators. Nonlinear Anal. 73(8), 2714–2736 (2010)
Gaines R.E., Mawhin J.L.: Coincidence degree and nonlinear differential equations. In: Lecture Notes in Mathematics, vol. 568. Springer, Berlin–New York (1977)
Górniewicz L.: Topological Fixed Point Theory of Multivalued Mappings, 2nd edn. Springer, Dordrecht (2006)
Górniewicz L., Granas A., Kryszewski W.: On the homotopy method in the fixed point index theory of multi-valued mappings of compact absolute neighborhood retracts. J. Math. Anal. Appl. 161(2), 457–473 (1991)
Hyman D.M.: On decreasing sequences of compact absolute retracts. Fundam. Math. 64, 91–97 (1969)
Kornev S.V., Obukhovskii V.V.: On some versions of the topological degree theory for nonconvex-valued multimaps(in Russian). Trudy Mat. Fac. Voronezh Univ. (N.S.) 8, 56–74 (2004)
Kornev S., Obukhovskii V.: On some developments of the method of integral guiding functions. Funct. Differ. Equat. 12(3–4), 303–310 (2005)
Kamenskii M., Obukhovskii V., Zecca P.: Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces. de Gruyter Series in Nonlinear Analysis and Applications 7. Walter de Gruyter, Berlin–New York (2001)
Krasnosel’skii M.A., Perov A.I.: On a certain priciple of existence of bounded periodic and almost periodic solutions of systems of ordinary differential equations (in Russian). Dokl. Akad. Nauk SSSR. 123(2), 235–238 (1958)
Krasnosel’skii, M.A.: The Operator of Translation along the Trajectories of Differential Equations (in Russian), Nauka, Moscow (1966); Translation: Translations of Mathematical Monographs, vol. 19. American Mathematical Society, Providence (1968)
Kryszewski W.: Homotopy Properties of Set-valued Mappings. Univ. N. Copernicus Publishing, Torun (1997)
Lacher R.C.: Cell-like mappings and their generalizations. Bull. Am. Math. Soc. 83(4), 495–552 (1977)
Loi, N.V., Obukhovskii, V.V.: On application of the method of guiding functions to problem of bifurcation of periodic solutions of differential inclusions (in Russian). Vestnik Ross. Univ. Druzh. Nar. Ser. Matem. Inform. Phys. 4, 14–24 (2009)
Myshkis A.D.: Generalizations of the theorem on a fixed point of a dynamical system inside of a closed trajectory (in Russian). Mat. Sb. 34(3), 525–540 (1954)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Obukhovskii, V., Loi, N.V. & Kornev, S. Existence and Global Bifurcation of Solutions for a Class of Operator-Differential Inclusions. Differ Equ Dyn Syst 20, 285–300 (2012). https://doi.org/10.1007/s12591-012-0133-7
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12591-012-0133-7