Ngô Quốc Anh

November 1, 2007

Saddle Point Theorem

Filed under: Nghiên Cứu Khoa Học — Ngô Quốc Anh @ 20:34

Let X = Y \oplus Z be a Banach space with Z closed in X and \dim Y < \infty. For \varrho >0 define

\displaystyle\mathcal M: = \left\{ {u \in Y: \left\| u \right\| \leqslant \varrho} \right\} {\text{ and }} \mathcal M_0 : = \left\{ {u \in Y: \left\| u \right\| = \varrho} \right\}.

Let F \in C^1(X, \mathbb R) be such that

\displaystyle\mathop {\inf }\limits_{u \in Z} F\left( u \right) > \mathop {\max }\limits_{u \in \mathcal M_0 } F\left( u \right).

If F satisfies (PS)_c condition with

\displaystyle c = \mathop {\inf }\limits_{\gamma \in \Gamma } \mathop {\max }\limits_{u \in \mathcal M} F\left( {\gamma \left( u \right)} \right)


\displaystyle \Gamma : = \left\{ {\gamma \in C\left( {\mathcal M,X} \right): \gamma \left| {_{\mathcal M_0 } = I} \right.} \right\}

then c is a critical point of F.

1 Comment »

  1. Thanks!,

    Comment by Hcezdphj — December 13, 2008 @ 23:37

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