The mountain pass theorem on subsystems of second order arithmetic

Abstract

The main goal of this work is to formalize the Mountain Pass Theorem of Ambrosetti and Rabinowitz within the formal subsystem of second order arithmetic known as ACA0. We develop some Analysis within this system to have access to the space of continuous functions from [0, 1] into a separable Banach space and from there built formalized proofs of the basic ingredients of the Mountain Pass Theorem: The deformation lemma and the minimax principle that proves the theorem itself.