Ngô Quốc Anh

November 1, 2007

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

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

where

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

then $c$ is a critical point of $F$.