A unified study of existence theorems in topologically based settings and applications in optimization

ABSTRACT

A unified study of necessary and sufficient conditions for the existence of intersection points and other important points in mathematical analysis is presented. The results are established in topologically based settings and shown to have equivalent formulations for various kinds of important points. For set-valued maps from a set to a topological space, instead of a convexity structure on this set, we use two general structures. The first structure is based on continuous single-valued maps from simpleces to the aforementioned set in order to extend the ideas of Knaster–Kuratowski–Mazurkiewicz and Fan for convex hulls. The second one is based on single-valued maps from the unit interval of the real line to the considered set and extends the classical connectedness properties. The general existence theorems of the above types are crucial tools in studies of the solution existence in optimization and other areas of applied mathematics. Here we choose minimax problems for applications of our general theorems. Our results are new, or improve or include as special cases many known results.

Keywords:

• Necessary and sufficient conditions for existence
• intersection points
• maximal elements
• sectional points
• minimax equalities
• saddle points
• KKM structure
• connectedness structure

Mathematics Subject Classification (2010):

• 54H25
• 54C60
• 49J35
• 54C55
• 47H04
• 54C65
• 49K35

Acknowledgments

This work was supported by Vietnam National University, Ho Chi Minh City, under the grant B2021-28-02. A part of the work was completed during a stay of the authors at the Vietnam Institute for Advanced Study in Mathematics (VIASM). The authors would like to thank VIASM for its hospitality and support. They are grateful to the anonymous referees for their very helpful and detailed remarks and suggestions.

Disclosure statement

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

Additional information

Funding

This work was supported by the Vietnam National University-Ho Chi Minh City, under the grant B2021-28-02.