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Method of Guiding Functions in Finite-Dimensional Spaces

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Method of Guiding Functions in Problems of Nonlinear Analysis

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2076))

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Abstract

In this section we present the guiding functions method for studying the periodic problem for a differential inclusion in a finite-dimensional space.

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References

  1. S. Aizicovici, N.H. Pavel, Anti-periodic solutions to a class of nonlinear differential equations in Hilbert space. J. Funct. Anal. 99(2), 387–408 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  2. J.C. Alexander, P.M. Fitzpatrick, Global bifurcation for solutions of equations involving several parameter multivalued condensing mappings, in Proceedings of the Fixed Point Theory (Sherbrooke, QC, 1980), ed. by E. Fadell, G. Fournier. Springer Lecture Notes, vol. 886, pp. 1–19

    Google Scholar 

  3. V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces (Noordhoff International Publishing, Leyden, 1976)

    Book  MATH  Google Scholar 

  4. N.A. Bobylev, V.S. Klimov, Methods of Nonlinear Analysis in Nonsmooth Optimization Problems (Russian) (Nauka, Moscow, 1992)

    Google Scholar 

  5. Yu.G. Borisovich, B.D. Gelman, A.D. Myshkis, V.V. Obukhovskii, Topological methods in the theory of fixed points of multivalued mappings. Uspekhi Mat. Nauk (Russian) 35(1)(211), 59–126 (1980); English translation: Russ. Math. Surv. 35, 65–143 (1980)

    Google Scholar 

  6. Yu.G. Borisovich, B.D. Gelman, A.D. Myshkis, V.V. Obukhovskii, Introduction to the Theory of Multivalued Maps and Differential Inclusions, 2nd edn. (Librokom, Moscow, 2011) (in Russian)

    Google Scholar 

  7. H.L. Chen, Anti-periodic wavelets. J. Comput. Math. 14(1), 32–39 (1996)

    MATH  Google Scholar 

  8. F.H. Clarke, in Optimization and Nonsmooth Analysis, 2nd edn. Classics in Applied Mathematics, vol. 5 (Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1990)

    Google Scholar 

  9. Yu. Chena, D. O’Regan, R.P. Agarwal, Anti-periodic solutions for evolution equations associated with monotone type mappings. Appl. Math. Lett. 23(11), 1320–1325 (2010)

    Article  MathSciNet  Google Scholar 

  10. J.-F. Couchouron, R. Precup, Anti-periodic solutions for second order differential inclusions. Electron. J. Differ. Equat. 2004(124), 1–17 (2004)

    MathSciNet  Google Scholar 

  11. K. Deimling, Nonlinear Functional Analysis (Springer, Berlin, 1985)

    Book  MATH  Google Scholar 

  12. K. Deimling, Multivalued Differential Equations, in De Gruyter Series in Nonlinear Analysis and Applications, vol. 1 (Walter de Gruyter, Berlin, 1992)

    Book  Google Scholar 

  13. V.F. Dem’yanov, L.V. Vasil’ev, Nondifferentiable Optimization (Nauka, Moscow, 1981) (in Russian); English translation: Translation Series in Mathematics and Engineering (Optimization Software, Inc., Publications Division, New York, 1985)

    Google Scholar 

  14. Z. Denkowski, S. Migòrski, N.S. Papageorgiou, An Introduction to Nonlinear Analysis: Theory (Kluwer, Boston, 2003)

    Google Scholar 

  15. S. Domachowski, J. Gulgowski, A global bifurcation theorem for convex-valued differential inclusions. Z. Anal. Anwendungen 23(2), 275–292 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  16. J. Eisner, M. Kuçera, M. Väth, Degree and global bifurcation for elliptic equations with multivalued unilateral conditions. Nonlinear Anal. 64(8), 1710–1736 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  17. M. Feçkan, Bifurcation from homoclinic to periodic solutions in ordinary differential equations with multivalued perturbations. J. Differ. Equat. 130, 415–450 (1996)

    Article  MATH  Google Scholar 

  18. M. Feçkan, Bifurcation of periodic solutions in differential inclusions. Appl. Math. 42(5), 369–393 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  19. M. Feçkan, Bifurcation from homoclinic to periodic solutions in singularly perturbed direrential inclusions. Proc. R. Soc. Edinb. 127A, 727–753 (1997)

    Article  Google Scholar 

  20. M. Feçkan, in Topological Degree Approach to Bifurcation Problems. Topological Fixed Point Theory and Its Applications, vol. 5 (Springer, New York, 2008)

    Google Scholar 

  21. M. Feçkan, Bifurcation of periodic solutions in forced ordinary differential inclusions. Differ. Equat. Appl. 4(1), 459–472 (2009)

    Google Scholar 

  22. A. Fonda, Guiding functions and periodic solutions to functional differential equations. Proc. Am. Math. Soc. 99(1), 79–85 (1987)

    MathSciNet  MATH  Google Scholar 

  23. D. Gabor, W. Kryszewski, A global bifurcation index for set-valued perturbations of Fredholm operators. Nonlinear Anal. 73(8), 2714–2736 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  24. D. Gabor, W. Kryszewski, Alexander invariant for perturbations of Fredholm operators. Nonlinear Anal. 74(18), 6911–6932 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  25. L. Górniewicz, in Topological Fixed Point Theory of Multivalued Mappings, 2nd edn. Topological Fixed Point Theory and Its Applications, vol. 4 (Springer, Dordrecht, 2006)

    Google Scholar 

  26. L. Górniewicz, W. Kryszewski, Bifurcation invariants for acyclic mappings. Rep. Math. Phys. 31(2), 217–239 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  27. J. Gulgowski, A global bifurcation theorem with applications to nonlinear Picard problems. Nonlinear Anal. 41, 787–801 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  28. Ph. Hartman, in Ordinary Differential Equations. Corrected reprint of the second (1982) edition (Birkhauser, Boston). Classics in Applied Mathematics, vol. 38 (Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 2002)

    Google Scholar 

  29. J. Ize, Bifurcation theory for Fredholm operators. Mem. Am. Math. Soc. 7(174), viii + 128 pp (1976)

    Google Scholar 

  30. J. Ize, Topological bifurcation, in Topological Nonlinear Analysis: Degree, Singularity and Variations, ed. by M. Matzeu, A. Vignoli. Progress in Nonlinear Differential Equations and Their Applications, vol. 15 (Birkhäuser, Boston, 1995), pp. 341–463

    Google Scholar 

  31. M. Kamenskii, V. Obukhovskii, P. Zecca, in Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces. de Gruyter Series in Nonlinear Analysis and Applications, vol. 7 (Walter de Gruyter, Berlin, 2001)

    Google Scholar 

  32. I.-S. Kim, Yu.-H. Kim, A global bifurcation for nonlinear inclusions. Nonlinear Anal. 68(1), 343–348 (2008)

    Article  Google Scholar 

  33. S. Kornev, V. Obukhovskii, On some developments of the method of integral guiding functions. Funct. Differ. Equat. 12(3–4), 303–310 (2005)

    MathSciNet  MATH  Google Scholar 

  34. S.V. Kornev, V.V. Obukhovskii, Nonsmooth guiding potentials in problems of forced oscillations, Avtomat. i Telemekh. (1), 3–10 (2007) (in Russian); English translation: Autom. Remote Control 68(1), 1–8 (2007)

    Google Scholar 

  35. S.V. Kornev, V.V. Obukhovskii, Localization of the method of guiding functions in the problem about periodic solutions of differential inclusions. Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika (5), 23–32 (2009) (in Russian). English translation: Russ. Math. 53(5), 19–27 (2009)

    Google Scholar 

  36. M.A. Krasnosel’skii, Topological Methods in the Theory of Nonlinear Integral Equations (Gostekhizdat, Moscow, 1956) (in Russian); English translation (A Pergamon Press Book The Macmillan Co., New York, 1964)

    Google Scholar 

  37. M.A. Krasnosel’skii, The Operator of Translation Along the Trajectories of Differential Equations (Nauka, Moscow, 1966) (in Russian); English translation: Translations of Mathematical Monographs, vol. 19 (American Mathematical Society, Providence, 1968)

    Google Scholar 

  38. M.A. Krasnosel’skii, P.P. Zabreiko, Geometrical Methods of Nonlinear Analysis (Nauka, Moscow, 1975); English translation: Grundlehren der Mathematischen Wissenschaften, vol. 263 (Springer, Berlin, 1984)

    Google Scholar 

  39. W. Kryszewski, Homotopy Properties of Set-Valued Mappings (Univ. N. Copernicus Publishing, Torun, 1997)

    MATH  Google Scholar 

  40. N.V. Loi, Application of the method of integral guiding functions to bifurcation problems of periodic solutions of differential inclusions. Tambov Univ. Rep. Ser. Nat. Tech. Sci. 14(4), 738–741 (2009) (in Russian)

    Google Scholar 

  41. N.V. Loi, Guiding functions and global bifurcation of periodic solutions of functional differential inclusions with infinite delay. Topol. Meth. Nolinear Anal. 40, 359–370 (2012)

    Google Scholar 

  42. N.V. Loi, V.V. Obukhovskii, On application of the method of guiding functions to bifurcation problem of periodic solutions of differential inclusions. Vestnik Ross. Univ. Dr. Narod. (Russian) 4, 14–27 (2009)

    Google Scholar 

  43. N.V. Loi, V. Obukhovskii, On the global bifurcation for solutions of linear fredholm inclusions with convex-valued perturbations. Fixed Point Theor. 10(2), 289–303 (2009)

    MathSciNet  MATH  Google Scholar 

  44. N.V. Loi, V. Obukhovskii, On global bifurcation of periodic solutions for functional differential inclusions. Funct. Differ. Equat. 17(1–2), 157–168 (2010)

    MathSciNet  MATH  Google Scholar 

  45. L. Nirengerg, in Topics in Nonlinear Functional Analysis. Revised Reprint of the 1974 Original. Courant Lecture Notes in Mathematics, vol. 6, New York University, Courant Institute of Mathematical Sciences, New York (American Mathematical Society, Providence, 2001)

    Google Scholar 

  46. V. Obukhovskii, N.V. Loi, S. Kornev, Existence and global bifurcation of solutions for a class of operator-differential inclusions. Differ. Equat. Dyn. Syst. 20(3), 285–300 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  47. V. Obukhovskii, P. Zecca and V. Zvyagin, On some generalizations of the Landesman-Laser theorem. Fixed Point Theory 8(1), 69–85 (2007).

    MathSciNet  MATH  Google Scholar 

  48. S. Pinsky, U. Trittmann, Anti-periodic boundary conditions in supersymmetric discrete light cone quantization. Phys. Rev. D 62, 087701 (2000)

    Article  MathSciNet  Google Scholar 

  49. P. Rabinowitz, Some global results for nonlinear eigenvalue problems. J. Funct. Anal. 7, 487–513 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  50. J. Shao, Anti-periodic solutions for shunting inhibitory cellular neural networks with time-varying delays. Phys. Lett. A 372(30), 5011–5016 (2008)

    Article  MATH  Google Scholar 

  51. M. Väth, New beams of global bifurcation points for a reaction-diffusion system with inequalities or inclusions. J. Differ. Equat. 247(11), 3040–3069 (2009)

    Article  MATH  Google Scholar 

  52. T. Yoshizawa, in Stability Theory by Liapunov’s Second Method. Publications of the Mathematical Society of Japan, vol. 9 (The Mathematical Society of Japan, Tokyo, 1966)

    Google Scholar 

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Authors and Affiliations

  1. Department of Physics and Mathematics, Voronezh State Pedagogical University, Voronezh, Russia

    Valeri Obukhovskii & Sergei Kornev

  2. Dipartimento di Matematica e Informatica “U Dini”, Università di Firenze, Firenze, Italy

    Pietro Zecca

  3. Faculty of Fundamental Science, PetroVietNam University, Ba Ria, Vietnam

    Nguyen Van Loi

Authors
  1. Valeri Obukhovskii
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  2. Pietro Zecca
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  3. Nguyen Van Loi
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  4. Sergei Kornev
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Obukhovskii, V., Zecca, P., Van Loi, N., Kornev, S. (2013). Method of Guiding Functions in Finite-Dimensional Spaces. In: Method of Guiding Functions in Problems of Nonlinear Analysis. Lecture Notes in Mathematics, vol 2076. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-37070-0_2

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