We investigate a general question about the size and regularity of data solutions in implicit function problems with loss regularity. First, we give heuristic explanation fact that optimal found by Ekeland Sere their recent non-quadratic version Nash-Moser theorem can also be recovered, for large class nonlinear problems, quadratic schemes. Then prove this observation applies to singular pertur...

In this paper we prove a generalized version of the Ekeland variational principle, which is a common generalization of Zhong variational principle and Borwein Preiss Variational principle. Therefore in a particular case, from this variational principle we get a Zhong type variational principle, and a Borwein-Preiss variational principle. As a consequence, we obtain a Caristi type fixed point th...

The applicant develops new tools of variational analysis; in particular, two new versions of the Ekeland variational principle and subdifferential variational principle for set-valued mappings, as well as new constructions of extremal set systems. They play a significant role to studying an important class of constrained optimization problems called Multiobjective Optimization Problems with Equ...

Let X be a Banach space and Φ : X → R of class C1. We are interested in finding critical points for the restriction of Φ to the manifold M = {u ∈ X : G(u) = 1}, where G : X → R is a C1 function having 1 as a regular value. A point u ∈M is a critical point of the restriction of Φ to M if and only if dΦ(u)|TuM = 0 (see the definition in Section 2). Our purpose is to prove two general minimax prin...

This article concerns with the problem ?div(jruj m?2 ru) = hu q + u m ?1 ; in R N : Using the Ekeland Variational Principle and the Mountain Pass Theorem, we show the existence of > 0 such that there are at least two non-negative solutions for each in (0;).

Using the mountain pass theorem combined with the Ekeland variational principle, we obtain at least two distinct, non-trivial weak solutions for a class of p(x)-Kirchhoff type equations with combined nonlinearities. We also show that the similar results can be obtained in the case when the domain has cylindrical symmetry.

We prove a stability of weakly almost conformal mappings in W 1,p(Ω;Rn) for p not too far below the dimension n by studying the W 1,pquasiconvex hull of the set Cn of conformal matrices. The study is based on coercivity estimates from the nonlinear Hodge decompositions and reverse Hölder inequalities from the Ekeland variational principle.