TWO-FUNCTION MINIMAX THEOREM INVOLVING LEVEL SETS WITH PARACONVEXITY

Gu-Sheng Tang (P. R. China) and Cao-Zong Cheng (P. R. China)

Received December 2, 2007

References:

[1] Cao-Zong Cheng and Bor-luh Lin, Nonlinear two function minimax theorems, Minimax Theory and Applications, B. Ricceri and S. Simonsw, eds., Nonconvex Optim. Appl., Vol. 26, Kluwer Acad. Publ. Dordrecht, 1998, pp. 1-20. [2] Ky Fan, Sur un théorèmme minimax, C. R. Acad. Sci. Paris 259 (1964), 3925-3928. [3] Ky Fan, A generalization of Tychonoff’s fixed point theorem, Math. Ann. 142 (1961), 305-310. [4] I. Joo and G. Kassy, Convexity, minimax theorems and their applications, Annal. Univ. Sci. Budapest, Sect. Math. 38 (1995), 71-93. [5] M. Lassonde, On the use of KKM multifunctions in fixed point theory and related topics, J. Math. Anal. Appl. 97 (1983), 151-201. [6] E. Michael, Continuous, I, Ann. Math. 63(2) (1956), 361-382. [7] D. Repovš and P. V. Semenov, Continuous Selections of Multivalued Mappings, Math. Appl., Vol. 455, Kluwer Academic, Dordrech, 1998. [8] P. V. Semenov, Functionally paraconvex sets, Mat. Zametki 54 (1993), 74-81 (in Russion), English transl. in Math. Notes 54 (1993), 1236-1240. [9] D. Repovš and P. V. Semenov, A minimax theorem for functions with possibly nonconnected intersections of sublevel sets, J. Math. Anal. Appl. 314 (2006), 537-545. [10] D. Repovš and P. V. Semenov, On functions of nonconvexity for graphs of continuous functions, J. Math. Anal. Appl. 196 (1995), 1021-1029. [11] M. Sion, On general minimax theorem, Pacific J. Math. 8 (1958), 171-178. [12] Wen-tsun Wu, A remark on the fundamental theorem in the theory of games, Sci. Record (N.S.) 3 (1959), 229-233.

Keywords and phrases: minimax theorems, paraconvexity, nonconvex function.