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Transactions of the American Mathematical Society

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Bourgin-Yang Type Theorem and its application
to $Z_2$-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
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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 $Z_2$-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.


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  • [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

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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

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