| Title:
|
Conley type index and Hamiltonian inclusions (English) |
| Author:
|
Dzedzej, Zdzisław |
| Language:
|
English |
| Journal:
|
Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica |
| ISSN:
|
0231-9721 |
| Volume:
|
49 |
| Issue:
|
1 |
| Year:
|
2010 |
| Pages:
|
33-47 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
Conley index |
| Keyword:
|
multivalued dynamical system |
| Keyword:
|
Hamiltonian inclusion |
| MSC:
|
34A60 |
| MSC:
|
37B30 |
| MSC:
|
37J45 |
| idZBL:
|
Zbl 1229.37011 |
| idMR:
|
MR2797521 |
| Note:
|
Supported by the Ministry of sciences and higher education, Poland, under grant N N 201 394037 (English) |
| . |
| Date available:
|
2010-09-13T06:54:00Z |
| Last updated:
|
2013-09-18 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140735 |
| . |
| Reference:
|
[1] Amann, H., Zehnder, E.: Periodic solutions of asymptotically linear Hamiltonian systems.Manuscripta Math. 32 (1980), 149–189. Zbl 0443.70019, MR 0592715, 10.1007/BF01298187 |
| Reference:
|
[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. Zbl 1084.37015, MR 2200861 |
| Reference:
|
[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. Zbl 0881.34061, MR 1437190, 10.1016/S0294-1449(97)80150-3 |
| Reference:
|
[4] Clarke, F. H.: Periodic solutions to Hamiltonian inclusions.J. Diff. Eq. 40 (1981), 1–6. Zbl 0461.34030, MR 0614215, 10.1016/0022-0396(81)90007-3 |
| Reference:
|
[5] Clarke, F. H.: Optimization and Nonsmooth Analysis.Wiley, New York, 1983. Zbl 0582.49001, MR 0709590 |
| Reference:
|
[6] Conley, C.: Isolated Invariant Sets and the Morse Index.CBMS 38 Amer. Math. Society, Providence, Rhode Island, 1978. Zbl 0397.34056, MR 0511133 |
| Reference:
|
[7] Dzedzej, Z.: On the Conley index in Hilbert spaces – a multivalued case.Banach Center Publ. 77 Warszawa, 2007, 61–68. Zbl 1114.37013, MR 2338575 |
| Reference:
|
[8] Dzedzej, Z., Gabor, G.: On homotopy Conley index for multivalued flows in Hilbert spaces.to appear. |
| Reference:
|
[9] Dzedzej, Z., Kryszewski, W.: Conley type index applied to Hamiltonian inclusions.J. Math. Anal. Appl. 347 (2008), 96–112. Zbl 1149.37008, MR 2433828, 10.1016/j.jmaa.2008.05.081 |
| Reference:
|
[10] Emelyanov, S. V., Korovin, S. K., Bobylev, N. A., Bulatov, A. V.: Homotopy of Extremal Problems.W. de Gruyter, Berlin, 2007. Zbl 1141.49001, MR 2352609 |
| Reference:
|
[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. Zbl 0831.34046, MR 1348538, 10.1006/jdeq.1995.1124 |
| Reference:
|
[12] Gabor, G.: Homotopy index for multivalued flows on sleek sets.Set-valued Analysis 13 (2005), 125–149. Zbl 1096.54016, MR 2148132, 10.1007/s11228-004-4109-9 |
| Reference:
|
[13] Gȩba, K., Izydorek, M., Pruszko, A.: The Conley index in Hilbert spaces and its applications.Studia Math. 134 (1999), 217–233. MR 1685454 |
| Reference:
|
[14] Górniewicz, L.: Topological Fixed Point Theory of Multivalued Mappings.Kluwer, Dordrecht–Boston–London, 1999. |
| Reference:
|
[15] Han, Z. Q.: Computations of cohomology groups and nontrivial periodic solutions of Hamiltonian systems.J. Math. Anal. Appl. 330 (2007), 259–275. Zbl 1115.37056, MR 2302921, 10.1016/j.jmaa.2006.07.047 |
| Reference:
|
[16] Izydorek, M.: A cohomological index in Hilbert spaces and applications to strongly indefinite problems.J. Differential Equations 170 (2001), 22–50. MR 1813098, 10.1006/jdeq.2000.3818 |
| Reference:
|
[17] Izydorek, M., Rybakowski, K.: On the Conley index in the absence of uniqueness.Fund. Math. 171 (2002), 31–52. MR 1879882, 10.4064/fm171-1-2 |
| Reference:
|
[18] Kryszewski, W., Szulkin, A.: An infinite dimensional Morse theory with applications.Trans. Amer. Math. Soc. 349 (1997), 3181–3234. Zbl 0892.58015, MR 1422612, 10.1090/S0002-9947-97-01963-6 |
| Reference:
|
[19] Kunze, M., Küpper, T., Li, Y.: On Conley index theory for non-smooth dynamical systems.Differential Integral Equations 13 (2000), 479–502. MR 1750037 |
| Reference:
|
[20] Mrozek, M.: A cohomological index of Conley type for multivalued admissible flows.J. Differential Equations 84 (1990), 15–51. MR 1042657, 10.1016/0022-0396(90)90125-9 |
| Reference:
|
[21] Rybakowski, K.: The Homotopy Index and Partial Differential Equations.Springer, Berlin, 1987. Zbl 0628.58006, MR 0910097 |
| Reference:
|
[22] Szulkin, A.: Cohomology and Morse theory for strongly indefinite functionals.Math. Z. 209 (1992), 375–418. Zbl 0735.58012, MR 1152264, 10.1007/BF02570842 |
| Reference:
|
[23] Szulkin, A., Zhou, W.: Infinite dimensional cohomology groups and periodic solutions of asymptotically linear Hamiltonian systems.J. Differential Equations 174 (2001), 369–391. MR 1846740, 10.1006/jdeq.2000.3942 |
| . |
