## A remark on the lower semicontinuity assumption in the Ekeland variational principle

What happens to the conclusion of the Ekeland variational principle (briefly, EVP) if a considered function $f:X\to \R\cup\{+\infty\}$ is lower semicontinuous not on a whole metric space $X$ but only on its domain? We provide a straightforward proof showing that it still holds but only for $\epsilon$ varying in some interval $]0,\beta-\inf_Xf[$, where $\beta$ … Read more

## An induction theorem and nonlinear regularity models

A general nonlinear regularity model for a set-valued mapping $F:X\times\R_+\rightrightarrows Y$, where $X$ and $Y$ are metric spaces, is considered using special iteration procedures, going back to Banach, Schauder, Lusternik and Graves. Namely, we revise the \emph{induction theorem} from Khanh, \emph{J. Math. Anal. Appl.}, 118 (1986) and employ it to obtain basic estimates for studying … Read more

## THE EKELAND VARIATIONAL PRINCIPLE FOR HENIG PROPER MINIMIZERS AND SUPER MINIMIZERS

In this paper we consider, for the first time, approximate Henig proper minimizers and approximate super minimizers of a set-valued map F with values in a partially ordered vector space and formulate two versions of the Ekeland variational principle for these points involving coderivatives in the senses of Ioffe, Clarke and Mordukhovich. As applications we … Read more