Generalized KKM theorem with applications to generalized minimax inequalities and generalized equilibrium problems. (English) Zbl 1121.49005

Zeng, Lu-Chuan; Wu, Soon-Yi; Yao, Jen-Chih

Taiwanese J. Math. 10, No. 6, 1497-1514 (2006).

Summary: A generalized version of the famous KKM theorem is obtained by using the concept of generalized KKM mappings introduced by S. Chang and Y. Zhang [J. Math. Anal. Appl., No. 1, 208–223 (1991; Zbl 0739.47026)]. By employing our generalized KKM theorem, we obtain a generalized minimax inequality which includes several existing ones as special cases. Further, by applying our generalized minimax inequality we establish an existence result for the saddle-point problem under general setting. Finally, we also derive some existence results for generalized equilibrium problems and generalized variational inequalities.

MSC:

49J30	Existence of optimal solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.)
47H10	Fixed-point theorems
49J35	Existence of solutions for minimax problems

Keywords:

transfer closed-valued mappings; \(\gamma\)-transfer lower semi-continuous functions; generalized variational inequalities; generalized KKM mappings

Citations:

Zbl 0739.47026