|
Bourgin-Yang Type Theorem and its application to  -equivariant Hamiltonian systems
Author(s):
Marek
Izydorek
Journal:
Trans. Amer. Math. Soc.
351
(1999),
2807-2831.
MSC (1991):
Primary 58E05, 55M20;
Secondary 34C25, 34C35
Posted:
February 24, 1999
Retrieve article in:
PDF DVI PostScript
This article is available free of charge
Abstract |
References |
Similar articles |
Additional information
Abstract:
We will be concerned with the existence of multiple periodic solutions of asymptotically linear Hamiltonian systems with the presence of  -action. To that purpose we prove a new version of the Bourgin-Yang theorem. Using the notion of the crossing number we also introduce a new definition of the Morse index for indefinite functionals.
References:
- [AZ1]
- H. Amann and E. Zehnder, Nontrivial solutions for a class of nonresonance problems and applications to nonlinear differential equations, Ann. Scuola Norm. Sup. Pisa (4) 7 (1980), 539-603. MR 82b:47077
- [AZ2]
- H. Amann and E. Zehnder, Periodic solutions of asymptotically linear Hamiltonian systems, Manuscr. Math. 32 (1980), 149-189. MR 82i:58026
- [AR]
- A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349-381. MR 51:6412
- [B1]
- V. Benci, A new approach to the Morse-Conley theory and some applications, Ann. Mat. Pura Appl. (4) 158 (1991), 231-305. MR 92k:58043
- [B2]
- V. Benci, Introduction to Morse Theory, a New Approach, Birkhäuser, Boston, 1995. MR 96d:58019
- [B3]
- V. Benci, On critical point theory for indefinite functionals in the presence of symmetries, Trans. AMS, 274 (1982), 533-572. MR 84c:58014
- [Ch]
- K.C. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problems, Birkhäuser, Boston, 1993. MR 94e:58023
- [Ch1]
- K.C. Chang, Solutions of asymptotically linear operator equations via Morse theory, Comm. Pure Appl. Math. 34 (1981), 693-712. MR 82m:58015
- [C1]
- D.C. Clark, A variant of Lijusternik-Schnirelmann theory, Indiana Univ. Math. J. 22 (1972), 65-74. MR 45:5836
- [C]
- C.C. Conley, Isolated Invariant Sets and the Morse Index, CBMS 38, Amer. Math. Soc., Providence, R.I., 1978. MR 80c:58009
- [CZ]
- C.C. Conley and E. Zehnder, Morse type index theory for flows and periodic solutions for Hamiltonian equations, Comm. Pure Appl. Math. 37 (1984), 207-253. MR 86b:58021
- [DG]
- J. Dugundji and A. Granas, Fixed Point Theory, I, PWN-Polish Sci. Publ., Warsaw, 1982. MR 83j:54038
- [D]
- J. Dugundji, An extension of Tietze's theorem, Pacific Math. J. 1 (1951), 353-367. MR 13:373c
- [E]
- I. Ekeland, Index theory for periodic solutions of convex Hamiltonian systems, Proc. AMS Summer Institute on Nonlinear Functional Analysis, Berkeley, (1983). Proc of Symp. in Pure Math. 45 (1986), I, 395-423. MR 87j:58023
- [F]
- G. Fei, Maslov-type index and periodic solution of asymptotically linear Hamiltonian systems which are resonant at infinity, J. Differential Equations 121 (1995), 121-133. MR 96f:58127
- [F1]
- G. Fei, Relative Morse index and its application to Hamiltonian systems in the presence of symmetries, J. Differential Equations 122 (1995), 302-315. MR 96i:58028
- [G]
- A. Granas, An extension to functional spaces of Borsuk-Ulam theorem on antipodes, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astr. Phys. 10 (1962), 81-86. MR 25:2417
- [LL]
- S. Li and J.Q. Liu, Morse theory and asymptotic linear Hamiltonian systems, J. Differential Equations 79 (1988), 53-73. MR 90d:58041
- [L1]
- Y. Long, The Index Theory of Hamiltonian Systems with Applications, Science Press, Beijing, (1993).
- [L2]
- Y. Long, Maslov type index, degenerate critical points and asymptotically linear Hamiltonian systems, Science in China, Series A, 33, (1990), 1409-1419. MR 92d:58171
- [LZ]
- Y. Long and E. Zehnder, Morse theory for forced oscillations of asymptotically linear Hamiltonian systems, Stochastic Process, Physics and Geometry, S. Albeverio et al., eds., World Scientific, 1990, 528-563. MR 92j:58019
- [MW]
- J. Mawhin and M. Willem, Critical point theory and Hamiltonian systems, Berlin Heidelberg New York, Springer-Verlag, 1989. MR 90e:58016
- [R]
- P.H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Reg. Conf. Ser. Math., vol. 35, Providence, R.I., Amer. Math. Soc., 1986. MR 87j:58024
- [S]
- A. Szulkin, Cohomology and Morse theory for strongly indefinite functionals, Math. Z. 209 (1992), 375-418. MR 93c:58046
- [Y]
- C.T. Yang, On theorems of Borsuk-Ulam, Kakutani-Yamabe-Yujobo and Dyson, I, Ann. of Math. 60 (1954), 262-282. MR 16:502d
Similar Articles:
Retrieve articles in Transactions of the American Mathematical Society
with MSC
(1991):
58E05, 55M20,
34C25, 34C35
Retrieve articles in all Journals with MSC
(1991):
58E05, 55M20,
34C25, 34C35
Additional Information:
Marek
Izydorek
Affiliation:
Department of Technical Physics and Applied Mathematics, Technical University of Gdansk, 80-952 Gdansk, ul. Gabriela Narutowicza 11/12, Poland
Email:
izydorek@mifgate.gda.pl
DOI:
10.1090/S0002-9947-99-02144-3
PII:
S 0002-9947(99)02144-3
Keywords:
Morse index,
critical points,
periodic solutions,
Hamiltonian systems,
$Z_2$--genus of space
Received by editor(s):
January 9, 1996
Received by editor(s) in revised form:
March 7, 1997
Posted:
February 24, 1999
Copyright of article:
Copyright
1999,
American Mathematical Society
|
Comments: webmaster@ams.org