mountain pass theorem
Let $X$ a real Banach space^{} and $F\in {C}^{1}(X,\mathbb{R})$. Consider $K$ a compact^{} metric space, and ${K}^{*}\subset K$ a closed nonempty subset of $K$. If ${p}^{*}:{K}^{*}\to X$ is a continuous mapping, set
$$\mathcal{P}=\{p\in C(K,X);p={p}^{*}\text{on}{K}^{*}\}.$$ 
Define
$$c=\underset{p\in \mathcal{P}}{inf}\underset{t\in K}{\mathrm{max}}F(p(t)).$$ 
Assume that
$$c>\underset{t\in {K}^{*}}{\mathrm{max}}F({p}^{*}(t)).$$  (1) 
Then there exists a sequence $({x}_{n})$ in $X$ such that

(i)
$\underset{n\to \mathrm{\infty}}{lim}F({x}_{n})=c$;

(ii)
$\underset{n\to \mathrm{\infty}}{lim}\parallel {F}^{\prime}({x}_{n})\parallel =0$.
The name of this theorem is a consequence of a simplified visualization for the objects from theorem. If we consider the set ${K}^{*}=\{A,B\}$, where $A$ and $B$ are two villages, $\mathcal{P}$ is the set of all the routes from $A$ to $B$, and $F(x)$ represents the altitude of point $x$; then the assumption^{} (1) is equivalent^{} to say that the villages $A$ and $B$ are separated with a mountains chain. So, the conclusion^{} of the theorem tell us that exists a route between the villages with a minimal^{} altitude. With other words exists a “mountain pass” .
Title  mountain pass theorem 

Canonical name  MountainPassTheorem 
Date of creation  20130322 15:19:19 
Last modified on  20130322 15:19:19 
Owner  ncrom (8997) 
Last modified by  ncrom (8997) 
Numerical id  8 
Author  ncrom (8997) 
Entry type  Theorem 
Classification  msc 49J40 