COMMENTS ON HOU JICHENG'S "ON SOME KKM TYPE THEOREMS" Park, Se-Hie;
In a paper by Hou Jicheng, On some KKM type theorems, Advaces in Mathematics 36 (2007), no. 1, 86-88, the author claimed that some previous KKM type theorems are false by giving a counterexample. In the present paper, we show that the counterexample does not work and, consequently, the results are correct. Moreover, we claim that the artificial concept like transfer compactly closed-valued maps can be destroyed. Finally, we introduce a theorem generalizing the main target of Hou.
KKM type theorems;admissible class of multimaps;transfer compactly closedness;
X.-P. Ding, New H-KKM theorems and their applications to geometric property, coincidence
theorems, minimax inequality and maximal elements, Indian J. Pure Appl. Math.
26 (1995), no. 1, 1–19.
J. Hou, On some KKM type theorems, Adv. Math. (China) 36 (2007), no. 1, 86–88.
E. M. Kalmoun and H. Rihai, Topological KKM theorems and generalized vector equilibria
on G-convex spaces with applications, Proc. Amer. Math. Soc. 129 (2001), no. 5, 1335–1348
S. Park, Comments on generalized R-KKM type theorems, Comm. Korean Math. Soc.
25 (2010), no. 2, 303–311.
S. Park, Fixed point theorems in the new era of the KKM theory, Fixed Point Theory
and Its Applications (Proc. ICFPTA-2009), Yokohama Publ., to appear.
S. Park, General KKM theorems for abstract convex spaces, J. Inform. Math. Sci. 1
(2009), no. 1, 1–13.
S. Park and H. Kim, Generalizations of the KKM type theorems on generalized convex
spaces, Indian J. Pure Appl. Math. 29 (1998), no. 2, 121–132.