mountain pass theorem

Let X a real Banach spaceMathworldPlanetmath and FC1(X,). Consider K a compactPlanetmathPlanetmath metric space, and K*K a closed nonempty subset of K. If p*:K*X is a continuous mapping, set

𝒫={pC(K,X);p=p*on K*}.



Assume that

c>maxtK*F(p*(t)). (1)

Then there exists a sequence (xn) in X such that

  1. (i)


  2. (ii)


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, 𝒫 is the set of all the routes from A to B, and F(x) represents the altitude of point x; then the assumptionPlanetmathPlanetmath (1) is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to say that the villages A and B are separated with a mountains chain. So, the conclusionMathworldPlanetmath of the theorem tell us that exists a route between the villages with a minimalPlanetmathPlanetmath altitude. With other words exists a “mountain pass” .

Title mountain pass theorem
Canonical name MountainPassTheorem
Date of creation 2013-03-22 15:19:19
Last modified on 2013-03-22 15:19:19
Owner ncrom (8997)
Last modified by ncrom (8997)
Numerical id 8
Author ncrom (8997)
Entry type Theorem
Classification msc 49J40