Nonlinear systems with a partial Nash type equilibrium

"In this paper xed point arguments and a critical point technique are combined leading to hybrid existence results for a system of three operator equations where only two of the equations have a variational structure. The components of the solution which are associated to the equations having a variational form represent a Nash-type equilibrium of the corresponding energy functionals. The result is achieved by an iterative scheme based on Ekeland's variational principle."

Small perturbations of elliptic problems with variable growth in RN

In this paper, we study the existence of at least two non-trivial solutions for a class of p ( x )-Laplacian equations with perturbation in the whole space. Using Ekeland’s variational principle and the mountain pass theorem, under appropriate assumptions, we prove the existence of two solutions for the equations.

New Generalized Ekeland’s Variational Principle, Critical Point Theorems and Common Fuzzy Fixed Point Theorems Induced by Lin-Du’s Abstract Maximal Element Principle

In this paper, by applying the abstract maximal element principle of Lin and Du, we present some new existence theorems related with critical point theorem, maximal element theorem, generalized Ekeland’s variational principle and common (fuzzy) fixed point theorem for essential distances.

Ekeland’s variational principle in weak and strong systems of arithmetic

We analyze Ekeland’s variational principle in the context of reverse mathematics. We find that that the full variational principle is equivalent to $$\Pi ^1_1\text{- }\mathsf {CA}_0$$ Π 1 1 - CA 0 , a strong theory of second-order arithmetic, while natural restrictions (e.g. to compact spaces or to continuous functions) yield statements equivalent to weak König’s lemma ($$\mathsf {WKL}_0$$ WKL 0 ) and to arithmetical comprehension ($$\mathsf {ACA}_0$$ ACA 0 ). We also find that the localized version of Ekeland’s variational principle is equivalent to $$\Pi ^1_1\text{- }\mathsf {CA}_0$$ Π 1 1 - CA 0 , even when restricted to continuous functions. This is a rare example of a statement about continuous functions having great logical strength.