# COLLECTIVE FIXED POINTS FOR GENERALIZED CONDENSING MAPS IN ABSTRACT CONVEX UNIFORM SPACES

• Kim, Hoonjoo (Department of Mathematics Education Sehan University)
• Accepted : 2020.09.22
• Published : 2021.03.15

#### Abstract

In this paper, we present a fixed point theorem for a family of generalized condensing multimaps which have ranges of the Zima-Hadžić type in Hausdorff KKM uniform spaces. It extends Himmelberg et al. type fixed point theorem. As applications, we obtain some new collective fixed point theorems for various type generalized condensing multimaps in abstract convex uniform spaces.

#### References

1. R.P. Agarwal and D. O'Regan, Fixed points, morphisms, countably condensing pairs and index theory, Nonlinear Funct. Anal. Appl., 7(2) (2002), 241-253.
2. A. Amini-Harandi, A.P. Farajzadeh, D. O'Regan and R.P. Agarwal, Fixed point theorems for condensing multimaps on abstract convex uniform spaces, Nonlinear Funct. Anal. Appl., 14 (2009), 109-120.
3. T.-H. Chang, Y.-Y. Huang, J.-C. Jeng and K.-H. Kuo, On S-KKM property and related topics, J. Math. Anal. Appl., 229 (1999), 212-227. https://doi.org/10.1006/jmaa.1998.6154
4. C.J. Himmelberg, J. R. Porter and F.S. Van Vleck, Fixed point theorems for condensing multifunctions, Proc. Amer. Math. Soc., 23 (1969), 635-641. https://doi.org/10.1090/S0002-9939-1969-0246175-1
5. Y.-Y. Huang, J.-C. Jeng and T.-Y. Kuo, Fixed point theorems for condensing maps in S-KKM class, Int. J. Math. Anal., 2 (2008), 1031-1044.
6. Y.-Y. Huang, T.-Y. Kuo and J.-C. Jeng, Fixed point theorems for condensing multimaps on locally G-convex spaces, Nonlinear Anal., 67 (2007), 1522-1531. https://doi.org/10.1016/j.na.2006.07.034
7. H. Kim, Fixed points for generalized condensing maps in abstract convex uniform spaces, Int. J. Math. Anal., 8 (2014), 2899-2908. https://doi.org/10.12988/ijma.2014.411362
8. H. Kim, Maximal elements of condensing maps on abstract convex spaces with applications, J. Nonlinear convex Anal., 20 (2019), 123-131.
9. S. Park, Fixed point theorems in locally G-convex spaces, Nonlinear Anal., 48 (2002), 869-879. https://doi.org/10.1016/S0362-546X(00)00220-0
10. S. Park, Elements of the KKM theory on abstract convex spaces, J. Korean Math. Soc., 45 (2008), 1-27. https://doi.org/10.4134/JKMS.2008.45.1.001
11. S. Park, Equilibrium existence theorems in KKM spaces, Nonlinear Anal., 69 (2008), 4352-4364. https://doi.org/10.1016/j.na.2007.10.058
12. S. Park, Fixed point theory of multimaps in abstract convex uniform spaces, Nonlinear Anal., 71 (2009), 2468-2480. https://doi.org/10.1016/j.na.2009.01.081
13. S. Park, The KKM principle in abstract convex spaces: equivalent formulations and applications, Nonlinear Anal., 73 (2010), 1028-1042. https://doi.org/10.1016/j.na.2010.04.029
14. S. Park and H. Kim, Admissible classes of multifunctions on generalized convex spaces, Proc. Coll. Natur. Sci., Seoul Nat. Univ., 18 (1993), 1-21.
15. S. Park and H. Kim, Coincidence theorems for admissible multifunctions on generalized convex spaces, J. Math. Anal. Appl., 197 (1996), 173-187. https://doi.org/10.1006/jmaa.1996.0014
16. S. Park and H. Kim, Foundations of the KKM theory on generalized convex spaces, J. Math. Anal. Appl., 209 (1997), 551-571. https://doi.org/10.1006/jmaa.1997.5388
17. W.V. Petryshyn and P.M. Fitzpatrick, Fixed-point theorems for multivalued noncompact inward maps, J. Math. Anal. Appl., 46 (1974), 756-767. https://doi.org/10.1016/0022-247x(74)90271-6
18. Y-L. Wu, C.-H. Huang and L.-J. Chu, An extension of Mehta theorem with applications, J. Math. Anal. Appl., 391 (2012), 489-495. https://doi.org/10.1016/j.jmaa.2012.02.027