Previous |  Up |  Next

Article

Keywords:
acyclic map; convex space; matching theorem; coincidence theorem
Summary:
We obtain generalizations of the Fan's matching theorem for an open (or closed) covering related to an admissible map. Each of these is restated as a KKM theorem. Finally, applications concerning coincidence theorems and section results are given.
References:
[1] Allen G.: Variational inequalities, complementary problems and duality theorems. J. Math. Anal. Appl. 58 (1977), 1-10. MR 0513305
[2] Balaj M.: A variant of a fixed point theorem of Browder and some applications. Math. Montisnigri 9 (1998), 5-13. MR 1657668 | Zbl 0999.47045
[3] Ben-El-Mechaiekh H., Deguire P., Granas A.: Points fixes et coincidences pour les applications multivoque, $I$. C.R. Acad. Sci. Paris 295 (1982), 337-340; II, 381-384.
[4] Browder F.E.: The fixed point theory of multi-valued mappings in topological vector spaces. Math. Ann. 177 (1968), 283-301. MR 0229101 | Zbl 0176.45204
[5] Chang S.Y.: A generalization of KKM principle and its applications. Soochow J. Math. 15 (1989), 7-17. MR 1025967 | Zbl 0747.47033
[6] Ding X.-P., Tan K.-K.: Matching theorems, fixed point theorems, and minimax inequalities without convexity. J. Austral. Math. Soc. (Ser. A) 49 (1990), 111-128. MR 1054086 | Zbl 0709.47053
[7] Fan K.: A generalization of Tychonoff's fixed point theorem. Math. Ann 142 (1961), 305-310. MR 0131268 | Zbl 0093.36701
[8] Fan K.: A minimax inequality and applications. in ``Inequalities III'', O Shisha (ed.), Academic Press, New York, 1972, pp.103-113. MR 0341029 | Zbl 0302.49019
[9] Fan K.: Fixed-point and related theorems for non-compact convex sets. in ``Game Theory and Related Topics'', O. Moeschlin and D. Pallaschke (eds.), North-Holland, Amsterdam, 1979, pp.151-156. MR 0556363 | Zbl 0432.54040
[10] Fan K.: Some properties of convex sets related to fixed point theorems. Math. Ann. 266 (1984), 519-537. MR 0735533 | Zbl 0515.47029
[11] Gorniewicz L.: A Lefschetz-type fixed point theorem. Fund. Math. 88 (1975), 103-115. MR 0391062 | Zbl 0306.55007
[12] Gorniewicz L.: Homological methods in fixed point theory of multi-valued maps. Dissertationes Math. 129 (1976), 1-71. MR 0394637 | Zbl 0324.55002
[13] Ha C.-W.: Minimax and fixed point theorems. Math. Ann. 248 (1980), 73-77. MR 0569411 | Zbl 0413.47042
[14] Kim W.K.: Some applications of the Kakutani fixed point theorem. J. Math. Anal. Appl. 121 (1987), 119-122. MR 0869523 | Zbl 0612.54055
[15] Lassonde M.: On the use of KKM multifunctions in fixed point theory and related topics. J. Math. Anal. Appl. 97 (1983), 151-201. MR 0721236 | Zbl 0527.47037
[16] Lassonde M.: Sur le principle KKM. C.R. Acad. Sci. Paris 310 (1990), 573-576. MR 1050134
[17] Lin T.-C.: Convex sets, fixed points, variational and minimax inequalities. Bull. Austral. Math. Soc. (Ser. A) 34 (1986), 107-117. MR 0847978 | Zbl 0597.47038
[18] Park S.: Generalizations of Ky Fan's matching theorems and their applications. J. Math. Anal. Appl. 141 (1989), 164-176. MR 1004591 | Zbl 0681.47028
[19] Park S.: Convex spaces and KKM families of subsets. Bull. Korean Math. Soc. 27 (1990), 11-14. MR 1060811 | Zbl 0746.47036
[20] Park S.: Generalizations of Ky Fan's matching theorems and their applications, $II$. J. Korean Math. Soc. 28 (1991), 275-283. MR 1127832 | Zbl 0813.47063
[21] Park S.: Some coincidence theorems on acyclic multifunctions and applications to KKM theory. in ``Fixed Point Theory and Applications'' (K.-K. Tan ed.), World Scientific, Publishing River Edge, NY, 1992, pp.248-277. MR 1190044
[22] Park S.: Foundations of the KKM theory via coincidences of composites of upper semicontinuous maps. J. Korean Math. Soc. 31 (1994), 493-519. MR 1297433 | Zbl 0829.49002
[23] Park S.: Ninety years of the Browder fixed point theorem. Vietnam J. Math. 27 (1999), 193-232. MR 1811334
[24] Park S., Kim H.: Coincidences of composites of u.s.c. maps on $H$-spaces and applications. J. Korean Math. Soc. 32 (1995), 251-264. MR 1338994 | Zbl 0868.54015
[25] Sehgal V.M., Singh S.P., Whitfield J.H.M.: KKM-maps and fixed point theorems. Indian J. Math. 32 (1990), 289-296. MR 1088610
[26] Shih M.-H.: Covering properties of convex sets. Bull. London Math. Soc. 18 ((1986)), 57-59. MR 0841369 | Zbl 0579.52004
[27] Shih M.-H., Tan K.-K.: Covering theorems of convex sets related to fixed-point theorems. in ``Nonlinear and Convex Analysis-Proc. in Honor of Ky Fan'' (B-L. Lin and S. Simons, eds.), Marcel Dekker, New York, 1987, pp.235-244. MR 0892795 | Zbl 0637.47029
[28] Shih M.-H., Tan K.K.: A geometric property of convex sets with applications to minimax type inequalities and fixed point theorems. J. Austral. Math. Soc. 45 (1988), 169-183. MR 0951575 | Zbl 0664.52001
[29] Shioji N.: A further generalization of the Knaster-Kuratowski-Mazurkiewicz theorems. Proc. Amer. Math. Soc. 111 (1991), 187-195. MR 1045601
[30] Takahashi W.: Nonlinear variational inequalities and fixed point theorems. J. Math. Soc. Japan 28 (1976), 168-181. MR 0399979 | Zbl 0314.47032
[31] Tarafdar E.: On nonlinear variational inequalities. Proc. Amer. Math. Soc. 67 (1977), 95-98. MR 0467408 | Zbl 0369.47029
[32] Tarafdar E.: On minimax principles and sets with convex sections. Publ. Math. Debrecen 29 (1982), 219-226. MR 0678897 | Zbl 0536.47038
Partner of
EuDML logo