The mountain pass theorem: variants, generalizations, and by Youssef Jabri

By Youssef Jabri

Joussef Jabri provides min-max equipment via a accomplished learn of the several faces of the distinguished Mountain go Theorem (MPT) of Ambrosetti and Rabinowitz. Jabri clarifies the extensions and variations of the MPT in a whole and unified means and covers typical subject matters: the classical and twin MPT; second-order details from playstation sequences; symmetry and topological index idea; perturbations from symmetry; convexity and extra. He additionally covers the non-smooth MPT; the geometrically restricted MPT; numerical ways to the MPT; or even extra unique variations. A bibliography and precise index also are incorporated.

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For ε = δ, we get that x is the minimal point that appears in c, that is (x) < (x) + dist (x, x), ∀x = x. Otherwise, all y ∈ T x satisfy y = x. 8), we get the contradiction (y) ≤ (x) − dist (x, y) and (x) < (y) + dist (x, y). Proof of Ekeland’s principle using Caristi’s fixed point theorem. Suppose by contradiction that no point in X satisfies c. Then, for each x ∈ X , the set T x = y ∈ X ; (x) ≥ (y) + (ε/δ) dist (x, y), y = x is not empty. 8) is satisfied, so there must exist some point x ∈ X 32 3 Obtaining “Almost Critical Points” – Variational Principle such that x ∈ T x: a contradiction with the definition of T x which was not supposed to contain x.

7 (Deformation Lemma, Shafrir). Let be a C 1 -functional defined on a Banach space X and let A ⊂ X be a closed set (possibly empty). Then, there exists a continuous deformation η(t, u) : [0, 1] × X → X satisfying i. η(0, u) = u for all u ∈ X , ii. η(t, u) = u for all t ∈ [0, 1] if u ∈ A or if (u) = 0, iii. (η(t, u)) ≤ (u) for all t ∈ [0, 1] and all u ∈ X , iv. (η(t, u)) < (u) for all t ∈ (0, 1] if u ∈ A and (u) = 0. This result is used to get some additional information in the MPT when the inf max value c is attained by some the maximum of some competing path γ ∈ .

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 .

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