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1. Let ∈ C 1 (X ; R). Then, admits a pseudo-gradient vector field on X˜ . Proof. Take u˜ in X˜ , then there exists w ∈ X such that w =1 Set v = 3 2 ˜ w > (u), and 2 3 ˜ . (u) ˜ · w, then (u) ˜ , v <2 (u) ˜ v > ˜ 2. 1) ˜ The family {N (u)}u∈ X˜ is obviously an open cover of X˜ . Since hold for all u in N (u). X˜ is a metric space and hence paracompact, there exists an open cover {Ni }i∈I that is locally finite and is a refinement of {N (u)}u∈ X˜ . 1) holds for some u = u i in each Ni . Ni ⊂ N (u) Set for all u in X˜ ρi (u) = dist (u, X \ Ni ) and v(u) = i∈I ρi (u) ui .

He uses properties of -uniformly convex functionals. 4 below exist in the literature although probably not in the generality below. A. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. , 14, 349–381 (1973) In this chapter, we present perhaps the first MPT. It is a finite dimensional version of the MPT due to Courant (1950) [279, pp. 223–226]. It contains all the ingredients of the MPT in a finite dimensional setting. That way, we hope to avoid technicalities and make it easy to understand the basic ideas in a first contact.

Then w(t, S) ⊂ Sδ for any t ∈ [0, δ]. , u)) is decreasing. Let u ∈ c+ε ∩ S. If there is some t ∈ [0, δ[ such that (w(t, u)) < c − ε, then (w(δ, u)) < c − ε and w(δ, u) ∈ c−ε ∩ Sδ . Otherwise, for all t ∈ [0, δ], c−ε ≤ (w(t, u)) ≤ And then w(t, u) ∈ A1 for all t ∈ [0, δ [. (w(0, u)) = (u) ≤ c + ε. 2) and the definitions of ϕ and v, we deduce that (w(δ, u)) = δ (u) + 0 = δ (u) + d dt (w(t, u)) dt (w(t, u)), ϕ(w(t, u)) dt 0 = δ (u) + (w(t, u)), v(w(t, u))/ v(w(t, u)) dt 0 δ ≤c+ε− (w(t, u)) 2 / v(w(t, u)) dt 0 1 δ (w(t, u)) dt 2 0 δ 4ε ≤c+ε− = c − ε.

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