Fixed Points, Continuous Selections, and Existence of Solution of Optimization-Related Problems

ABSTRACT

We propose a definition of a generalized type of Knaster–Kuratowski–Mazurkiewicz (KKM) mappings, called a weak T-KKM mapping, and a corresponding weak KKM property. A new extension of the Fan–Glicksberg fixed-point theorem is established. Sufficient conditions for the existence of a continuous selection, a fixed point of a composition, and a coincidence point are also provided. Then, we use the obtained results to study the existence of solutions to various optimization-related problems. Discussions and detailed examples are included as well to compare our results with existing ones and to explain their advantages in many situations.

KEYWORDS:

• Coincidence points
• continuous selections
• equilibria
• existence of solutions
• fixed points
• GFC-spaces
• quasivariational inclusions
• variational relations

MATHEMATICS SUBJECT CLASSIFICATION:

• 90C26
• 90C48
• 47J22
• 49J40

Acknowledgments

We are grateful to Dr. Nguyen Hong Quan for his valuable remarks and discussions.