PARAMETRIC EKELAND'S VARIATIONAL PRINCIPLE AND APPLICATIONS FOR MIN-MAX PROBLEMS

Ekeland's variational principle is now a classical method for investigations of many non-linear problems in various areas in mathematics (see for instance [El], [E2] for a survey). In this paper we establish a parametric analogue of this principle (called here basic perturbation theorem), which states that the minimum point in the Ekeland's variational principle can be chosen to depend continuously on a parameter. We show throe applications of this parametric Ekeland's variational principle to min-max problems. The first one is an analogue of Ekeland's variational principle to min-max problems (see Theorem 2). The second one is a very simple proof of Ky Fan's inequality. The third one is the following "perturbed " version of Von-Neumann min-max theorem: under similar assumptions as in the Von-Neumann min-max theorem, but without, compactness of the one of the subsets, we can perturb the functions in such a way, that the perturb function has a saddle point. The perturbation is of the same type as in the Ekeland variational principle, but with respect to the two variables, The proof of this parametric Ekeland's variational principle uses Michael's selectio