Geodesic
Conley type index and Hamiltonian inclusions
Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 49 (2010) no. 1, pp. 33-47 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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This paper is based mainly on the joint paper with W. Kryszewski [Dzedzej, Z., Kryszewski, W.: Conley type index applied to Hamiltonian inclusions. J. Math. Anal. Appl. 347 (2008), 96–112.], where cohomological Conley type index for multivalued flows has been applied to prove the existence of nontrivial periodic solutions for asymptotically linear Hamiltonian inclusions. Some proofs and additional remarks concerning definition of the index and special cases are given.
This paper is based mainly on the joint paper with W. Kryszewski [Dzedzej, Z., Kryszewski, W.: Conley type index applied to Hamiltonian inclusions. J. Math. Anal. Appl. 347 (2008), 96–112.], where cohomological Conley type index for multivalued flows has been applied to prove the existence of nontrivial periodic solutions for asymptotically linear Hamiltonian inclusions. Some proofs and additional remarks concerning definition of the index and special cases are given.
Classification : 34A60, 37B30, 37J45
Keywords: Conley index; multivalued dynamical system; Hamiltonian inclusion 
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Dzedzej, Zdzisław. Conley type index and Hamiltonian inclusions. Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 49 (2010) no. 1, pp. 33-47. http://geodesic.mathdoc.fr/item/AUPO_2010_49_1_a3/

[1] Amann, H., Zehnder, E.: Periodic solutions of asymptotically linear Hamiltonian systems. Manuscripta Math. 32 (1980), 149–189. | DOI | MR | Zbl

[2] Bobylev, N. A., Bulatov, A. V., Kuznetsov, Yu. O.: An approximation scheme for defining the Conley index of isolated critical points. Diff. Equ. 40 (2004), 1539–1544. | MR | Zbl

[3] Chang, K. C., Liu, J. Q., Liu, M. J.: Nontrivial periodic solutions for strong resonance Hamiltonian systems. Ann. Inst. Henri Poincare 14 (1997), 103–117. | DOI | MR | Zbl

[4] Clarke, F. H.: Periodic solutions to Hamiltonian inclusions. J. Diff. Eq. 40 (1981), 1–6. | DOI | MR | Zbl

[5] Clarke, F. H.: Optimization and Nonsmooth Analysis. Wiley, New York, 1983. | MR | Zbl

[6] Conley, C.: Isolated Invariant Sets and the Morse Index. CBMS 38 Amer. Math. Society, Providence, Rhode Island, 1978. | MR | Zbl

[7] Dzedzej, Z.: On the Conley index in Hilbert spaces – a multivalued case. Banach Center Publ. 77 Warszawa, 2007, 61–68. | MR | Zbl

[8] Dzedzej, Z., Gabor, G.: On homotopy Conley index for multivalued flows in Hilbert spaces. to appear.

[9] Dzedzej, Z., Kryszewski, W.: Conley type index applied to Hamiltonian inclusions. J. Math. Anal. Appl. 347 (2008), 96–112. | DOI | MR | Zbl

[10] Emelyanov, S. V., Korovin, S. K., Bobylev, N. A., Bulatov, A. V.: Homotopy of Extremal Problems. W. de Gruyter, Berlin, 2007. | MR | Zbl

[11] Fei, G. H.: Maslov-type index and periodic solution of asymptotically linear Hamiltonian systems which are resonant at infinity. J. Differential Equations 121 (1995), 121–133. | DOI | MR | Zbl

[12] Gabor, G.: Homotopy index for multivalued flows on sleek sets. Set-valued Analysis 13 (2005), 125–149. | DOI | MR | Zbl

[13] Gȩba, K., Izydorek, M., Pruszko, A.: The Conley index in Hilbert spaces and its applications. Studia Math. 134 (1999), 217–233. | MR

[14] Górniewicz, L.: Topological Fixed Point Theory of Multivalued Mappings. Kluwer, Dordrecht–Boston–London, 1999.

[15] Han, Z. Q.: Computations of cohomology groups and nontrivial periodic solutions of Hamiltonian systems. J. Math. Anal. Appl. 330 (2007), 259–275. | DOI | MR | Zbl

[16] Izydorek, M.: A cohomological index in Hilbert spaces and applications to strongly indefinite problems. J. Differential Equations 170 (2001), 22–50. | DOI | MR

[17] Izydorek, M., Rybakowski, K.: On the Conley index in the absence of uniqueness. Fund. Math. 171 (2002), 31–52. | DOI | MR

[18] Kryszewski, W., Szulkin, A.: An infinite dimensional Morse theory with applications. Trans. Amer. Math. Soc. 349 (1997), 3181–3234. | DOI | MR | Zbl

[19] Kunze, M., Küpper, T., Li, Y.: On Conley index theory for non-smooth dynamical systems. Differential Integral Equations 13 (2000), 479–502. | MR

[20] Mrozek, M.: A cohomological index of Conley type for multivalued admissible flows. J. Differential Equations 84 (1990), 15–51. | DOI | MR

[21] Rybakowski, K.: The Homotopy Index and Partial Differential Equations. Springer, Berlin, 1987. | MR | Zbl

[22] Szulkin, A.: Cohomology and Morse theory for strongly indefinite functionals. Math. Z. 209 (1992), 375–418. | DOI | MR | Zbl

[23] Szulkin, A., Zhou, W.: Infinite dimensional cohomology groups and periodic solutions of asymptotically linear Hamiltonian systems. J. Differential Equations 174 (2001), 369–391. | DOI | MR