General theorems of the Knaster-Kuratowski- Mazurkiewicz type and applications to the existence study in optimization

ABSTRACT

We establish theorems of the Knaster-Kuratowski-Mazurkiewicz (KKM) type on topological spaces with topologically-based generalized convex structures and apply them to the existence study for variational relation problems, variational inequalities, minimax inequalities and saddle points, and to alternative theorems. Our first main KKM-type theorem is a necessary and sufficient condition, not only a sufficient one as usual. The results improve and extend corresponding contributions in the literature in many aspects.

Keywords:

• Theorems of the KKM type
• general KKM maps
• alternative theorems
• minimax inequalities
• variational relation problems
• variational inequalities

Acknowledgments

A part of this paper was completed during a scientific stay of the authors at Vietnam Institute for Advanced Study in Mathematics (VIASM), whose support and hospitality are gratefully acknowledged. We are deeply indebted to the two referees for their valuable remarks and suggestions on the prevous version of the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

Our work was supported by Vietnam National University-Hochiminh City, under grant number B2018-28-02.