The strong Ekeland variational principle, the strong drop theorem and applications (Q1104544)

Modified versions of Ekeland's variational principle [\textit{I. Ekeland}, ibid. 47, 324-353 (1974; Zbl 0286.49015)], of the drop theorem, of a lemma of \textit{R. R. Phelps} [Adv. Math. 13, 1-19 (1974; Zbl 0284.46015)] and of Penot's lower petal theorem [\textit{J.-P. Penot}, Nonlinear Anal., Theory Methods Appl. 10, 813-822 (1986; Zbl 0612.49011)] are proved, stating in addition, that every minimizing sequence is convergent. This is connected with the Tihonov and Hadamard well-posedness of minimization problems. All these versions are found to be equivalent to each other and to the corresponding original ones. As applications some generic results concerning well-posedness of minimization problems and single-valuedness and upper-semicontinuity of the metric projection are proved. As a corollary we have that almost all (in the Baire sense) sublinear and continuous functionals in a separable Banach space are Fréchet differentiable almost everywhere. Also the metric projection to a closed non-empty subset of a separable Bn of biped locomotion.