On various multimap classes in the KKM theory and their applications

Year 2018, Volume: 2 Issue: 2, 88 - 105, 30.06.2018

Sehei Park

https://doi.org/10.31197/atnaa.433869

https://izlik.org/JA57DT93WZ

Abstract

Fixed point theory of convex-valued multimaps are closely related to the KKM theory from the beginning. In the last twenty-five years, we introduced the acyclic multimap class, the admissible multimap class  A_c^\kappa , the better admissible class  B and the KKM

admissible classes KC, KO in the frame of the KKM theory. Our aim in this review is to collect the basic properties of our

multimap classes and some mutual relations among them in general topological spaces or our abstract convex spaces. We add some new

remarks and further comments to improve many of those results, and introduce some recent applications of our multimap classes. 

Keywords

Abstract convex space , (Partial) KKM space , Multimap classes , Fixed point , Admissible multimap class $\frak A_c^\kappa$ , Better admissible class $\frak B$ , KKM admissible classes

References

• [1] P. Agarwal and D. O’Regan, An essential map theory for A κ c and PK maps, Top. Meth. Nonlinear Anal., J. of the Juliusz Schauder Center, 21 (2003) 375–386. [2] P. Agarwal and D. O’Regan, Fixed point theory for multimaps defined on admissible subsets of topological vector spaces, Commentationes Mathematicae, XLIV(1) (2004) 1–10. [3] R. P. Agarwal and D. O’Regan, Fixed point theory for admissible type maps with applications, Fixed Point Theory Appl. (2009), Article ID 439176, 22 pages. doi:10.1155/2009/439176. [4] R. P. Agarwal, D. O’Regan and S. Park, Fixed point theory for multimaps in extension type spaces, J. Korean Math. Soc. 39 (2002) 579–591. [5] R. P. Agarwal, D. O’Regan and M.-A. Taoudi, Fixed Point theorems for general classes of maps acting on topological vector spaces, Asian-European J. Math. 4 (2011) 373–387. [6] A. Amini, M. Fakhar and J. Zafarani, Fixed point theorems for the class S-KKM mappings in abstract convex spaces, Nonlinear Anal. 66 (2007) 14–21. [7] M. Balaj and L.-J. Lin, Equivalent forms of a generalized KKM theorem and their applications, Nonlinear Anal. 73 (2010) 673–682. [8] F. E. Browder, The fixed point theory of multi-valued mappings in topological vector spaces, Math. Ann. 177 (1968) 283–301. [9] K. Fan, A generalization of Tychonoff’s fixed point theorem, Math. Ann. 142 (1961) 305–310. [10] C. J. Himmelberg, Fixed points of compact multifunctions, J. Math. Anal. Appl. 38 (1972) 205–207. [11] S. Kakutani, A generalization of Brouwer’s fixed-point theorem, Duke Math. J. 8 (1941) 457–459. [12] H. Kim, KKM property, S-KKM property and fixed point theorems, Nonlinear Anal. 63 (2005) e1877–e1884 [13] H. Kim and S. Park, Remarks on the KKM property for open-valued multimaps on generalized convex spaces, J. Korean Math. Soc. 42 (2005) 101–110. [14] W. Kulpa and A. Szymanski, Some remarks on Park’s abstract convex spaces, Top. Meth. Nonlinear Anal. 44(2) (2014) 369–379. [15] H. Lu and Q. Hu, A collectively fixed point theorem in abstract convex spaces and its applications, J. Function Spaces Appl. Vol. 2013, Article ID 517469, 10 pages. http://dx.doi.org/10.1155/2013 /517469. [16] H. Lu and J. Zhang, A section theorem with applications to coincidence theorems and minimax inequalities in FWC-spaces, Comp. Math. Appl. 64 (2012) 570–588. [17] D. T. Luc, E. Sarabi, and A. Soubeyran, Existence of solutions in variational relation problems without convexity, J. Math. Anal. Appl. 364 (2010) 544–555. [18] D. O’Regan, Coincidence theory for multimaps, Appl. Math. Comp. 219 (2012) 2026–2034. [19] D. O’Regan, A unified theory for homotopy principles for multimaps, Appl. Anal. 92 (2013) 1944–1958. [20] D. O’Regan and J. Perán, Fixed points for better admissible multifunctions on proximity spaces, J. Math. Anal. Appl. 380 (2011) 882–887. [21] D. O’Regan and N. Shahzad, Krasnoselskii’s fixed point theorem for general classes of maps, Advances in Fixed Point Theory 2 (2012) 248-257. [22] S. Park, Fixed point theory of multifunctions in topological vector spaces, J. Korean Math. Soc. 29 (1992) 191–208. [23] S. Park, Cyclic coincidence theorems for acyclic multifunctions on convex spaces, J. Korean Math. Soc. 29 (1992) 333–339 [24] S. Park Some coincidence theorems on acyclic multifunctions and applications to KKM theory, Fixed Point Theory and Applications (K.-K ˙ Tan, ed.), pp. 248–277, World Scientific Publ., River Edge, NJ, 1992. [25] S. Park, Fixed point theory of multifunctions in topological vector spaces, II, J. Korean Math. Soc. 30 (1993) 413–431. [26] S. Park, Foundations of the KKM theory via coincidences of composites of admissible u.s.c. maps, J. Korean Math. Soc. 31 (1994) 493-516. [27] S. Park, Acyclic maps, minimax inequalities, and fixed points, Nonlinear Anal., TMA 24 (1995) 1549–1554. [28] S. Park, Fixed points of acyclic maps on topological vector spaces, World Congress of Nonlinear Analysts ’92 (V ˙ Lakshimikantham, ed.), pp.2171-2177, Walter de Gruyter, Berlin-New York, 1996. [29] S. Park, Coincidence theorems for the better admissible multimaps and their applications, Nonlinear Anal. 30 (1997) 4183– 4191. [30] S. Park, A unified fixed point theory of multimaps on topological vector spaces, J. Korean Math. Soc. 35 (1998), 803–829. Corrections, ibid. 36 (1999) 829–832. [31] S. Park, Ninety years of the Brouwer fixed point theorem, Vietnam J. Math. 27 (1999) 187–222. [32] S. Park, Fixed points of generalized upper hemicontinuous maps, Revisited, Acta Math. Viet 27 (2002) 141–150. [33] S. Park, Coincidence, almost fixed point, and minimax theorems on generalized convex spaces, J. Nonlinear Convex Anal. 4 (2003) 151–164. [34] S. Park, The KKM principle implies many fixed point theorems, Topology Appl. 135 (2004) 197–206. [35] S. Park, Fixed points of multimaps in the better admissible class, J. Nonlinear Convex Anal. 5 (2004) 369–377. [36] S. Park, Fixed point theorems on KC-maps in abstract convex spaces, Nonlinear Anal. Forum 11(2) (2006) 117–127. [37] S. Park, Fixed point theorems for better admissible multimaps on almost convex sets, J. Math. Anal. Appl. 329(1) (2007) 690–702. [38] S. Park, A unified fixed point theory in generalized convex spaces, Acta Math. Sinica, English Ser. 23 (2007) 1509–1536. [39] S. Park, Elements of the KKM theory on abstract convex spaces, J. Korean Math. Soc. 45 (2008), 1–27.
• [40] S. Park, Comments on fixed point and coincidence theorems for families of multimaps, PanAmerican Math. J. 18 (2008) 21–34. [41] S. Park, Generalizations of the Himmelberg fixed point theorem, Fixed Point Theory and Its Applications (Proc. ICFPTA- 2007), 123–132, Yokohama Publ., 2008. [42] S. Park, Fixed point theory of multimaps in abstract convex uniform spaces, Nonlinear Anal. 71 (2009), 2468–2480. [43] S. Park, Applications of fixed point theorems for acyclic maps – A review, Vietnam J. Math. 37 (2009) 419–441. [44] S. Park, A unified approach to KC-maps in the KKM theory, Nonlinear Anal. Forum 14 (2009) 1–14. [45] S. Park, A brief history of the KKM theory, RIMS Kôkyûroku, Kyoto Univ. 1643 (2009) 1–16. [46] S. Park, The KKM principle in abstract convex spaces: Equivalent formulations and applications, Nonlinear Anal. 73 (2010) 1028–1042. [47] S. Park, Remarks on fixed points of generalized upper hemicontinuous maps, Comm. Appl. Nonlinear Anal. 18(3) (2011) 71–78. [48] S. Park, Applications of the KKM theory to fixed point theory, J. Nat. Acad. Sci., ROK, Nat. Sci. Ser. 50 (2011) 1–49. [49] S. Park, Applications of some basic theorems in the KKM theory, Fixed Point Theory Appl. vol.2011:98. doi:10.1186/1687- 1812-2011-98. [50] S. Park, Continuous selection theorems in generalized convex spaces: Revisited, Nonlinear Anal. Forum 16 (2011) 21–33. [51] S. Park, Abstract convex spaces, KKM spaces, and φ A -spaces, Nonlinear Anal. Forum 17 (2012) 1–10. [52] S. Park, A genesis of general KKM theorems for abstract convex spaces, J. Nonlinear Anal. Optim. 2 (2011) 133–146. [53] S. Park, Applications of multimap classes in abstract convex spaces, J. Nat. Acad. Sci., ROK, Nat. Sci. Ser. 51(2) (2012) 1–27. [54] S. Park, The Fan-Browder alternatives on abstract spaces: Generalizations and applications, J. Nat. Acad. Sci., ROK, Nat. Sci. Ser. 52(2) (2013) 1–55. [55] S. Park, A genesis of general KKM theorems for abstract convex spaces: Revisited, J. Nonlinear Anal. Optim. 4(2) (2013) 127–132. [56] S. Park, Evolution of the Fan-Browder type alternatives, Nonlinear Analysis and Convex Analysis (NACA 2013, Hirosaki), pp.401–418, Yokohama Publ., Yokohama, 2016. [57] S. Park, A unification of generalized Fan-Browder type alternatives, J. Nonlinear Convex Anal. 17(1) (2016) 1–15. [58] S. Park, Recent applications of some analytical fixed point theorems, Nonlinear Analysis and Convex Analysis (NACA 2015, Chiang Rai), pp.259–273. Yokohama Publ., Yokohama, 2016. [59] S. Park, On multimap classes in the KKM theory, RIMS Kôkyûroku, Kyoto Univ., to appear. [60] S. Park and H. Kim, Admissible classes of multifunctions on generalized convex spaces, Proc. Coll. Natur. Sci., SNU 18 (1993) 1-21. [61] S. Park and H. Kim, Coincidence theorems for admissible multifunctions on generalized convex spaces, J. Math. Anal. Appl. 197 (1996) 173–187. [62] S. Park and H. Kim, Foundations of the KKM theory on generalized convex spaces, J. Math. Anal. Appl. 209 (1997) 551–571. [63] S. Park, S.P. Singh and B. Watson, Some fixed point theorems for composites of acyclic maps, Proc. Amer. Math. Soc. 121 (1994), 1151–1158. [64] N. Shahzad, Fixed point and approximation results for multimaps in S-KKM class, Nonlinear Anal. 56 (2004) 905–918. [65] M.-g. Yang and N.-j. Huang, Coincidence theorems for noncompact KC-maps in abstract convex spaces with applications, Bull. Korean Math. Soc. 49 (2012) 1147–1161. [66] M.-g. Yang, J.-p. Xu and N.-j. Huang, Systems of generalized quasivariational inclusion problems with applications in LΓ-spaces, Fixed Point Theory Appl. Vol. 2011, Article ID 561573, 12 pages. doi:10.1155/2011/561573. [67] O. Agratini, I.A. Rus, Iterates of some bivariate approximation process via weakly Picard operators, Nonlinear Anal. Forum, 8 (2003), 159-168.