Improving some KKM theoretic results

Abstract

The purpose of this article is to introduce some works on the KKM theory which can be improved according to our theory on abstract convex spaces. In Section 2, we introduce some basic things on our abstract convex spaces as a preliminary. Section 3 deals with the basic results and subjects of our study on the KKM theory. In Section 4, we introduce several works that appeared since 2010. Most of these works are chosen on the basis that they can be improved by following our theory. Actually, we introduce abstracts of each work and add some comments showing how to improve them.

Keywords

• Abstract convex space,KKM theorem,KKM space,mapping classes $\mathfrak{KC}$,$\mathfrak{KO}$

Kaynakça

1. [1] Agarwal, R. P., Balaj, M. and O’Regan, D. Intersection theorems with applications in optimization, J. Optim. Theory Appl. https://doi.org/10.1007/s10957-018-1331-4
2. [2] Chaipunya, P. and Kumam, P. Topological aspects of circular metric spaces and some observations on the KKM property towards quasi-equilibrium problems, J. Ineq. Appl. 2013, 2013:343
3. [3] Chebbi, S., Gourdel, P. and Hammami, H. A generalization of Fan’s matching theorem, J. Fixed Point Theory Appl. 9 (2011) 117–124. DOI 10.1007/s11784-010-0022-z
4. [4] Darzi, R., Delavar, M.R., and Roohi, M. Fixed point theorems in minimal generalized convex spaces, Filomat 25(4) (2011) 165–176. DOI: 10.2298/FIL1104165D
5. [5] Espinoza, L. G. Some KKM type, intersection and minimax theorems in spaces with abstract convexities, Bolet. Asoc. Matem. Venezolana, XIX(2) (2012) 129–140.
6. [6] Fakhar, M., Lotfipour, M. and Zafarani, J. On the Brezis Nirenberg Stampacchia-type theorems and their applications, J. Glob. Optim. DOI 10.1007/s10898-012-9965-5
7. [7] Horvath, C. D. A note on metric spaces with continuous midpoints, Annal. Acad. Rumanian Scientists, Ser. Math. Appl. 1(2) (2009) 252–288.
8. [8] Hussain, N. and Shah, M. H. KKM mappings in cone b-metric spaces, Comp. Math. Appl. 62 (2011) 1677–1684.

9. [9] Jafari, S., Farajzadeh, A. P., Moradi, S. and Khanh, P. Q. Existence results for ϕ-quasimonotone equilibrium problems in convex metric spaces, Optimization 66(3) (2017) 293–310. http://dx.doi.org/10.1080/02331934.2016.1274989
10. [10] Khamsi, M. A., Latif, A. and Al-Sulami, H. KKM and Ky Fan theorems in modular function spaces, Fixed Point Theory Appl. 2011, 2011:57
11. [11] Le, T., Van, C.L., Pham, N.-S., and Săglam, C. Sperner lemma, fixed point theorems, and the existence of equilibrium, MPRA Paper No.100084 (May 3, 2020) https://mpra.ub.uni-muenchen.de/100084/
12. [12] Lu, H., Hu, Q. and Miao, Y. A maximal element theorem in FWC-spaces and its applications, Sci. World J. 2014, Article ID 890696, 18pp. http://dx.doi.org/10.1155/2014/890696
13. [13] Lu, H., Lan, D., Hu, Q. and Yuan, G. Fixed point theorems in CAT(0) spaces with applications, J. Ineq. Appl. 2014, 2014:320
14. [14] Lu, H. and Zhang, J. A section theorem with applications to coincidence theorems and minimax inequalities in FWC-spaces, Comp. Math. Appl. 64 (2012) 579–588.
15. [15] Mitrović, Z. D. On a coupled fixed point problem in topological vector spaces, Math. Comp. Model. 57 (2013) 2388–2392.
16. [16] Mitrović, Z. D., Hussain, A., de la Sen, M., and Radenovi, S. On best approximations for set-valued mappings in G-convex spaces, MPDA, 2020.
17. [17] Park, S. 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.
18. [18] Park, S. Ninety years of the Brouwer fixed point theorem, Vietnam J. Math. 27 (1999), 187–222.
19. [19] 19 Park, S. Applications of the KKM principle on abstract convex minimal spaces, Nonlinear Funct. Anal. Appl. 13(2) (2008) 179–191.
20. [20] Park, S. Comments on generalized R-KKM type theorems, Commun. Korean Math. Soc. 25(2) (2010) 303–311. DOI 10.4134/CKMS.2010.25.2.303
21. [21] Park, S. The KKM principle in abstract convex spaces: Equivalent formulations and applications, Nonlinear Anal. 73 (2010) 1028–1042.
22. [22] Park, S. New generalizations of basic theorems in the KKM theory, Nonlinear Anal. 74 (2011) 3000–3010.
23. [23] Park, S. Applications of multimap classes in abstract convex spaces, J. Nat. Acad. Sci., ROK, Nat. Sci. Ser. 51(2) (2012) 1–27.
24. [24] Park, S. A genesis of general KKM theorems for abstract convex spaces: Revisited, J. Nonlinear Anal. Optim. 4(2) (2013) 127–132.
25. [25] Park, S. Remarks on the KKM theory of abstract convex minimal spaces, Nonlinear Funct. Anal. Appl. 18(3) (2013) 383–395.
26. [26] Park, S. Comments on the FWC-spaces of H. Lu and J. Zhang, Nonlinear Anal. Forum 18 (2013) 33–38.
27. [27] Park, S. A unified approach to generalized KKM maps, J. Nat. Acad. Sci., ROK, Nat. Sci. Ser. 55(1) (2016) 1–20.
28. [28] Park, S. A history of the KKM Theory, J. Nat. Acad. Sci., ROK, Nat. Sci. Ser. 56(2) (2017) 1–51.
29. [29] Park, S. Extending the realm of Horvath spaces, J. Nonlinear Convex Anal. 20(8) (2019) 1609–1621.
30. [30] Plubtieng, S. and Sitthithakerngkiet, K. Existence result of generalized vector quasiequilibrium problems in locally G-convex spaces, Fixed Point Theory Appl. 2011, Article ID 967515, 13pp. doi:10.1155/2011/967515
31. [31] Shabanian, S. and Vaezpour, S. M. A minimax inequality and its applications to fixed point theorems in CAT(0) spaces, Fixed Point Theory Appl. 2011, 2011:61
32. [32] Shabanian, S. and Vaezpour, S. M. The KKM theorem in modular spaces and applications to minimax inequalities, Bull. Malays. Math. Sci. Soc. 39 (2016) 921–931. DOI 10.1007/s40840-015-0192-3
33. [33] Simić, S. A note on Stone’s, Baire’s, Ky Fan’s and Dugundji’s theorem in tvs-cone metric spaces, Appl. Math. Lett. 24 (2011) 999–1022.