By Hille E., Tamarkin J. D.

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Function bounded from below on a complete metric space (X, dist ). Given x ∈ X with (x) > inf (X ), let dist (x, y) ≤ (x) − (y) for some y = x. Then, (z) = inf (X ) for some z ∈ X . It is also equivalent to the drop theorem of Daneˇs related to the theory of normal solvability of nonlinear equations [293], to the flower petal theorem of Penot [712], and to the Caristi fixed point theorem, also called the Kirk-Caristi fixed point theorem [182]. 9 (The Drop Theorem, Daneˇs). Given two closed nonempty sets A, B in a Banach space, with B bounded and convex and dist (A, B) > 0, there exists a point a in A such that there is no other point between a and B, that is, D(a, B) ∩ A = {a}, where D(x, B) = clco[{x} ∪ B], and where clco refers to the closure of the convex hull; this set is called a “drop” because of its geometry.

Such spaces were known as “subreflexive” spaces. The name is due to Phelps [720] who conjectured in this paper that every Banach space is subreflexive. In the proof of this result appears a certain convex cone in E, associated with a partial ordering, to which a transfinite argument is applied (Zorn’s lemma). c. functionals by Ekeland in the original proof of his variational principle. More precisely, for s ∈ R, consider the closed convex cone with nonempty interior, C(s) = (u, a) ∈ X × R; a + s u ≤ 0 .

Proof. 2) with S = {u} and c = infx∈X (x), if by contradiction, for all u ∈ −1 ([c, c + 2ε]) ∩ S2δ , we had 4ε (v) ≥ . δ Then, η(1, v) would be in c−ε ∩ Sδ , which is impossible since c−ε = ∅. ” When combining the quantitative deformation lemma with the compactness condition (PS)c , we obtain the useful result known widely in literature as the deformation lemma. 4 (Standard Deformation Lemma). Let c ∈ R and consider : X → R a C 1 -functional satisfying (PS)c . If c is a regular value of , then for every ε sufficiently small there exists η ∈ C([0, 1] × X ; X ) such that Comments and Additional Notes i.