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Deformation properties for continuous functionals and critical point theory. (English) Zbl 0789.58021
Let $$X$$ be a complete metric space and $$f: X \to \mathbb{R}$$ be continuous. The authors define the notion of weak slope $$| df| (u) \in [0,\infty]$$, $$u\in X$$, which corresponds to $$\| df(u)\|$$ if $$X$$ and $$f$$ are of class $$C^ 1$$. The definition is based on the existence of certain deformations of neighborhoods of $$u$$ along which $$f$$ decreases. Using these local deformations the authors prove a version of the deformation lemma. If $$f$$ satisfies the Palais-Smale condition abstract critical point theorems follow in a standard way. No use is made of Ekeland’s variational principle. Under additional assumptions the theory can be generalized to lower semi-continuous functions.

##### MSC:
 5.8e+06 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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