The Conley index in Hilbert spaces and its applications
Gęba, K. ; Izydorek, M. ; Pruszko, A.
Studia Mathematica, Tome 133 (1999), p. 217-233 / Harvested from The Polish Digital Mathematics Library
Access to full text
Full (PDF)

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.

Publié le : 1999-01-01
EUDML-ID : urn:eudml:doc:216635 
@article{bwmeta1.element.bwnjournal-article-smv134i3p217bwm,
     author = {K. G\k eba and M. Izydorek and A. Pruszko},
     title = {The Conley index in Hilbert spaces and its applications},
     journal = {Studia Mathematica},
     volume = {133},
     year = {1999},
     pages = {217-233},
     zbl = {0927.58004},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv134i3p217bwm}
}

Gęba, K.; Izydorek, M.; Pruszko, A. The Conley index in Hilbert spaces and its applications. Studia Mathematica, Tome 133 (1999) pp. 217-233. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv134i3p217bwm/

[00000] [1] H. Amann and E. Zehnder, Periodic solutions of asymptotically linear Hamiltonian systems, Manuscripta Math. 32 (1980), 149-189. | Zbl 0443.70019

[00001] [2] V. Benci, A new approach to the Morse-Conley theory and some applications, Ann. Mat. Pura Appl. (4) 158 (1991), 231-305. | Zbl 0778.58011

[00002] [3] K. C. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problems, Birkhäuser, Boston, 1993. | Zbl 0779.58005

[00003] [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. | Zbl 0465.35007

[00004] [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.

[00005] [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. | Zbl 0559.58019

[00006] [7] K. Deimling, Ordinary Differential Equations in Banach Spaces, Lecture Notes in Math. 596, Springer, 1977.

[00007] [8] J. Dugundji and A. Granas, Fixed Point Theory, Vol. I, Monograf. Mat. 61, PWN-Polish Sci. Publ., 1982.

[00008] [9] S. Li and J. Q. Liu, Morse theory and asymptotic linear Hamiltonian systems, J. Differential Equations 79 (1988), 53-73. | Zbl 0672.34037

[00009] [10] Y. Long, The Index Theory of Hamiltonian Systems with Applications, Science Press, Beijing, 1993.

[00010] [11] J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Springer, 1989.

[00011] [12] K. Mischaikow, Conley index theory, in: Dynamical Systems, R. Johnson (ed.), Lecture Notes in Math. 1609, Springer, 1995, 119-207. | Zbl 0847.58062

[00012] [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. | Zbl 0609.58002

[00013] [14] P. H. Rabinowitz, Free vibrations for a semilinear wave equation, Comm. Pure Appl. Math. 31 (1978), 31-68. | Zbl 0341.35051

[00014] [15] K. Rybakowski, The Homotopy Index and Partial Differential Equations, Universitext, Springer, 1987. | Zbl 0628.58006

[00015] [16] D. Salamon, Connected simple systems and the Conley index of isolated invariant sets, Trans. Amer. Math. Soc. 291 (1985), 1-41. | Zbl 0573.58020

[00016] [17] A. Szulkin, Cohomology and Morse theory for strongly indefinite functionals, Math. Z. 209 (1992), 375-418. | Zbl 0735.58012

[00017] [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. | Zbl 0918.58017

[00018] [19] C. Troestler and M. Willem, Nontrivial solution of a semilinear Schrödinger equation, Comm. Partial Differential Equations 21 (1996), 1431-1449. | Zbl 0864.35036

[00019] [20] G. W. Whitehead, Recent Advances in Homotopy Theory, CMBS Regional Conf. Ser. in Math. 5, Amer. Math. Soc., Providence, R.I., 1970. | Zbl 0217.48601