The Conley index in Hilbert spaces and its applications

• Autorzy: K Gęba , M. Izydorek , A. Pruszko
• 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.

Kategorie tematyczne

• 58E05: Abstract critical point theory (Morse theory, Ljusternik-Schnirelman (Lyusternik-Shnirel\cprime man) theory, etc.)
• 34C25: Periodic solutions

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1999
• Tom: 134
• Numer: 3
• Strony: 217-233

Daty

wydano

1999

otrzymano

1997-11-17

poprawiono

1998-10-02

Twórcy

autor

K Gęba

• Institute of Mathematics, Polish Academy of Sciences, 18 Abrahama St., 81-825 Sopot, Poland.

autor

M. Izydorek

• Institute of Mathematics, Polish Academy of Sciences, 18 Abrahama St., 81-825 Sopot, Poland.

autor

A. Pruszko

• Institute of Mathematics, Polish Academy of Sciences, 18 Abrahama St., 81-825 Sopot, Poland.

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.