## 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

### Citations:

Zbl 0628.58006; Zbl 0823.58008; Zbl 0927.58004
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### References:

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