# 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 $p^{*}:K^{*}\rightarrow X$ is a continuous mapping, set

 $\mathcal{P}=\{p\in C(K,\,X);\;p=p^{*}\;\textrm{on }K^{*}\}.$

Define

 $c=\inf_{p\in\mathcal{P}}\max_{t\in K}F(p(t)).$

Assume that

 $c>\max_{t\in K^{*}}F(p^{*}(t)).$ (1)

Then there exists a sequence $(x_{n})$ in $X$ such that

1. (i)

$\lim\limits_{n\rightarrow\infty}F(x_{n})=c$;

2. (ii)

$\lim\limits_{n\rightarrow\infty}\|F^{\prime}(x_{n})\|=0$.

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 MountainPassTheorem 2013-03-22 15:19:19 2013-03-22 15:19:19 ncrom (8997) ncrom (8997) 8 ncrom (8997) Theorem msc 49J40