×

An approximation scheme for defining the Conley index of isolated critical points. (English. Russian original) Zbl 1084.37015

Differ. Equ. 40, No. 11, 1539-1544 (2004); translation from Differ. Uravn. 40, No. 11, 1462-1467 (2004).
Let \(f:H\to\mathbb{R}\) be a \(C^1\)-functional on a separable Hilbert space \(H\) with \(\nabla f:H\to H\) locally Lipschitz continuous and satisfying the following compactness condition: if \(x_n\rightharpoonup x_*\), and if \(\lim\sup_{n\to\infty}(\nabla f(x_n),x_n-x_*)\leq 0\) then \(x_n\to x_*\). The authors define the Conley index of an isolated critical point as follows:
Suppose \(0\in H\) is the only critical point of \(f\) in the ball \(B(0,r)\) of radius \(r\) around \(0\). Choose any sequence \(H_1\subset H_2\subset\ldots\subset H\) of finite-dimensional subspaces with \(\bigcup_n H_n\) dense in \(H\). Then for \(n\) large, \(B(0,r)\cap H_n\) is an isolating neighborhood of the negative gradient flow \(\varphi_n\) on \(H_n\) associated to the restriction \(f| H_n\). Let \(S_n:=\text{{inv}}(B(0,r)\cap H_n, \varphi_n)\) be the invariant set and \(h(S_n, \varphi _n)\) the usual finite-dimensional Conly index. The authors show that \(h(S_n, \varphi_n)\) is independent of \(n\) for \(n\) large, and independent of the choice of the sequence \(H_n\), \(n\geq1\). This is then, by definition, the Conley index of \(0\) as an isolated critical point of \(f\).
Related and more general versions of the Conley index are due to K. P. Rybakowski [The homotopy index and partial differential equations, Universitext, Berlin etc.: Springer-Verlag (1987; Zbl 0628.58006)], V. Benci [Prog. Nonlinear Differ. Equ. Appl. 15, 37–177 (1995; Zbl 0823.58008)], and K. Geba, M. Izydorek and A. Pruszko [Stud. Math. 134, 217–233 (1999; Zbl 0927.58004)].

MSC:

37B30 Index theory for dynamical systems, Morse-Conley indices
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Conley, C., Reg. Conf. Ser. Math., Providence: AMS, 1978.
[2] Wazewski, T., Ann. Soc. Polon. Math., 1947, vol. 20, pp. 279-313.
[3] Mischaikow, R. and Mrozek, M., Conley Index Theory. Handbook of Dynamical Systems, Amsterdam, 2002, vol. 2, pp. 393-460. · Zbl 1035.37010
[4] Rybakowski, K.P., The Homotopy Index and Partial Differential Equations, Berlin, 1987. · Zbl 0628.58006
[5] Plotnikov, P.I., Izv. Akad. Nauk SSSR. Ser. Mat., 1991, vol. 55, no. 2, pp. 339-366.
[6] Emel?yanov, S.V., Korovin, S.K., Bobylev, N.A., and Bulatov, A.V., Gomotopii ekstremal?nykh zadach (Homotopies of Extremal Problems), Moscow, 2001.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.