## Preprint 2010-09

## Fabián Flores-Bazán, Cesar Gutierrez, Vicente Novo:

### A Brezis-Browder principle on partially ordered spaces and related ordering theorems

### Abstract:

In this work, the existence of strong minimal points in quasi ordered spaces is studied. Through a simple extension of Brezis-Browder principle to partially ordered spaces, a very general strong minimal point existence theorem is proved. This theorem together with a generic quasi order and a new strong approximate solution notion are used to obtain two strong solution existence theorems and three general Ekeland variational principles in optimization problems where the objective space is quasi ordered. Then these results are applied to prove strong minimal point existence results and generalizations of Bishop-Phelps lemma in linear spaces, and Ekeland variational principles in set-valued optimization problems through a set solution criterion.

This preprint gave rise to the following definitive publication(s):

**Fabián FLORES-BAZáN, Cesar GUTIERREZ, Vicente NOVO: ***A Brezis-Browder principle on partially ordered spaces and related ordering theorems*. Journal of Mathematical Analysis and Applications, vol. 375, 1, pp. 245-260, (2011).