Bourgin-Yang Type Theorem and its application

to -equivariant Hamiltonian systems

Author:
Marek Izydorek

Journal:
Trans. Amer. Math. Soc. **351** (1999), 2807-2831

MSC (1991):
Primary 58E05, 55M20; Secondary 34C25, 34C35

DOI:
https://doi.org/10.1090/S0002-9947-99-02144-3

Published electronically:
February 24, 1999

MathSciNet review:
1467470

Full-text PDF Free Access

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.

**[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**

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 Gdańsk, 80-952 Gdańsk, ul. Gabriela Narutowicza 11/12, Poland

Email:
izydorek@mifgate.gda.pl

DOI:
https://doi.org/10.1090/S0002-9947-99-02144-3

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

Published electronically:
February 24, 1999

Article copyright:
© Copyright 1999
American Mathematical Society