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1999 | 134 | 3 | 217-233
Tytuł artykułu

The Conley index in Hilbert spaces and its applications

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We present a generalization of the classical Conley index defined for flows on locally compact spaces to flows on an infinite-dimensional real Hilbert space H generated by vector fields of the form f: H → H, f(x) = Lx + K(x), where L: H → H is a bounded linear operator satisfying some technical assumptions and K is a completely continuous perturbation. Simple examples are presented to show how this new invariant can be applied in searching critical points of strongly indefinite functionals having asymptotically linear gradient.
Słowa kluczowe
Czasopismo
Rocznik
Tom
134
Numer
3
Strony
217-233
Opis fizyczny
Daty
wydano
1999
otrzymano
1997-11-17
poprawiono
1998-10-02
Twórcy
autor
  • Institute of Mathematics, Polish Academy of Sciences, 18 Abrahama St., 81-825 Sopot, Poland., kgeba@ksinet.univ.gda.pl
autor
autor
  • Institute of Mathematics, Polish Academy of Sciences, 18 Abrahama St., 81-825 Sopot, Poland., amp@ksinet.univ.gda.pl
Bibliografia
  • [1] H. Amann and E. Zehnder, Periodic solutions of asymptotically linear Hamiltonian systems, Manuscripta Math. 32 (1980), 149-189.
  • [2] V. Benci, A new approach to the Morse-Conley theory and some applications, Ann. Mat. Pura Appl. (4) 158 (1991), 231-305.
  • [3] K. C. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problems, Birkhäuser, Boston, 1993.
  • [4] K. C. Chang, S. P. Wu and S. J. Li, Multiple periodic solutions for an asymptotically linear wave equation, Indiana Univ. Math. J. 31 (1982), 721-731.
  • [5] C. C. Conley, Isolated Invariant Sets and the Morse Index, CBMS Regional Conf. Ser. in Math. 38, Amer. Math. Soc., Providence, R.I., 1978.
  • [6] 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.
  • [7] K. Deimling, Ordinary Differential Equations in Banach Spaces, Lecture Notes in Math. 596, Springer, 1977.
  • [8] J. Dugundji and A. Granas, Fixed Point Theory, Vol. I, Monograf. Mat. 61, PWN-Polish Sci. Publ., 1982.
  • [9] S. Li and J. Q. Liu, Morse theory and asymptotic linear Hamiltonian systems, J. Differential Equations 79 (1988), 53-73.
  • [10] Y. Long, The Index Theory of Hamiltonian Systems with Applications, Science Press, Beijing, 1993.
  • [11] J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Springer, 1989.
  • [12] K. Mischaikow, Conley index theory, in: Dynamical Systems, R. Johnson (ed.), Lecture Notes in Math. 1609, Springer, 1995, 119-207.
  • [13] P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conf. Ser. in Math. 35, Amer. Math. Soc. Providence, R.I., 1986.
  • [14] P. H. Rabinowitz, Free vibrations for a semilinear wave equation, Comm. Pure Appl. Math. 31 (1978), 31-68.
  • [15] K. Rybakowski, The Homotopy Index and Partial Differential Equations, Universitext, Springer, 1987.
  • [16] D. Salamon, Connected simple systems and the Conley index of isolated invariant sets, Trans. Amer. Math. Soc. 291 (1985), 1-41.
  • [17] A. Szulkin, Cohomology and Morse theory for strongly indefinite functionals, Math. Z. 209 (1992), 375-418.
  • [18] A. Szulkin, Index theories for indefinite functionals and applications, in: Topological and Variational Methods for Nonlinear Boundary Value Problems (Cholin, 1995), Pitman Res. Notes Math. Ser. 365, Longman, 1997, 89-121.
  • [19] C. Troestler and M. Willem, Nontrivial solution of a semilinear Schrödinger equation, Comm. Partial Differential Equations 21 (1996), 1431-1449.
  • [20] G. W. Whitehead, Recent Advances in Homotopy Theory, CMBS Regional Conf. Ser. in Math. 5, Amer. Math. Soc., Providence, R.I., 1970.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-smv134i3p217bwm
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