Year 2019, Volume 2 , Issue 2, Pages 71 - 82 2019-08-30

Generalizations of Hyperconvex Metric Spaces

Sehei Park [1]


Since Khamsi found a KKM theorem for hyperconvex metric spaces in 1996, there have appeared a large number of works on them related to the KKM theory.  In our previous review [34], we followed the various stages of developments of the  KKM theory of hyperconvex metric spaces. In fact, we introduced abstracts of articles on such theory and gave comments or generalizations of the results there if necessary. We noted that many results in those articles are consequences of  (partial) KKM space theory developed by ourselves from 2006. The present survey is a continuation of [34] and aims to collect further generalizations of hyperconvex metric spaces related to the KKM theory.

convex space, Generalizations of Hyperconvex Metric Spaces, abstract convex space; Horvath space, metric type space, metric spaces with continuous midpoints, global nonpositive curvature (NPC)
  • [1] A. Amini, M. Fakhar, and J. Zafarani, KKM mappings in metric spaces, Nonlinear Anal. 60 (2005),1045--1052.
  • [2] N. Aronszajn and P. Panitchpakdi, Extensions of uniformly continuous transformations and hyper-convex metric spaces, Paci c J. Math. 6 (1956), 405--439.
  • [3] M. Balaj, E. D. Jorquera, and M. A. Khamsi, Common fi xed points of set-valued mappings inhyperconvex metric spaces, J. Fixed Point Theory Appl. 20:2.
  • [4] C. Bardaro and R. Ceppitelli, Some further generalizations of Knaster-Kuratowski-Mazurkiewicztheorem and minimax inequalities, J. Math. Anal. Appl. 132 (1988), 484--490.
  • [5] M. H. El Bansami and H. Riahi, Ky Fan Principle in ℵ0-spaces and some applications, J. NonlinearConvex Anal. 11(2) (2012), 229--241.
  • [6] R. Espinola and G. Lopez, Extension of compact mappings and ℵ0-hyperconvexity, Nonlinear Anal.TMA, 49 (2002), 1127{1135.
  • [7] C. Horvath, Contractibility and generalized convexity, J. Math. Anal. Appl. 156 (1991), 341-357.
  • [8] C. D. Horvath, Extension and selection theorems in topological spaces with a generalized convexitystructure, Ann. Fac. Sci. Toulouse 2 (1993), 253--269.
  • [9] C. D. Horvath, A note on metric spaces with continuous midpoints, Annal. Acad. Romanian Scientists, Ser. Math. Appl. 1(2) (2009), 252--288.
  • [10] N. Hussain and M. H. Shah, KKM mappings in cone b-metric spaces, Comput. Math. Appl. 62(2011), 1677--1684.
  • [11] M. A. Khamsi, KKM and Ky Fan theorems in hyperconvex metric spaces, J. Math. Anal. Appl. 204(1996), 298--306.
  • [12] M.A. Khamsi and N. Hussain, KKM mappings in metric type spaces, Nonlinear Anal. 73 (2010),3123{3129. doi:10.1016/j.na.2010.06.084.
  • [13] J.-H. Kim and S. Park, Comments on some fi xed point theorems in hyperconvex metric spaces, J.Math. Anal. Appl. 291 (2004), 154--164.
  • [14] Y. Kimura and Y. Kishi, Equilibrium problems and their resolvents in Hadamard spaces. Presentedat RIMS2018.
  • [15] W. A. Kirk and B. Panyanak, Best approximation in R-trees, Numer. Funct. Anal. Optimiz. 28(5-6)(2007), 681{690. Erratum, ibid. 30(3-4) (2009), 409.
  • [16] D. T. Luc, E. Sarabi, and A. Soubeyran, Existence of solutions in variational relation problemswithout convexity, J. Math. Anal. Appl. 364 (2010), 544--555
  • .[17] C. P. Niculescu and L. Roventa, Fan's inequality in geodesic spaces, Appl Math Lett. 22 (2009),1529{1533. doi:10.1016/j. aml.2009.03.020
  • [18] S. Park, The Schauder type and other fixed point theorems in hyperconvex spaces, Nonlinear Anal.Forum 3 (1998), 1--12. MR 99e:54036.
  • [19] S. Park, Fixed point theorems in hyperconvex metric spaces, Nonlinear Anal. 37 (1999), 467--472.
  • [20] S. Park and B. Sims, Remarks on xed point theorems on hyperconvex spaces, Nonlinear Funct.Anal. Appl. 5 (2000), 51--64.
  • [21] S. Park, On generalizations of the KKM principle on abstract convex spaces, Nonlinear Anal. Forum11 (2006), 67--77.[22] S. Park, Fixed point theorems on KC-maps in abstract convex spaces, Nonlinear Anal. Forum, 11(2)(2006), 117--127.[23] S. Park, Remarks on KC-maps and KO-maps in abstract convex spaces, Nonlinear Anal. Forum12(1) (2007), 29--40.
  • [24] S. Park, Examples of KC-maps and KO-maps on abstract convex spaces, Soochow J. Math. 33(3)(2007), 477--486.
  • [25] S. Park, Various subclasses of abstract convex spaces for the KKM theory, Proc. Nat. Inst. Math.Sci. 2(4) (2007), 35--47.
  • [26] S. Park, Elements of the KKM theory on abstract convex spaces, J. Korean Math. Soc. 45(1) (2008),1--27.
  • [27] S. Park, New foundations of the KKM theory, J. Nonlinear Convex. Anal. 9(3) (2008), 331--350.[28] S. Park, Remarks on the partial KKM principle, Nonlinear Anal. Forum 14 (2009), 51{62.
  • [29] S. Park, Comments on the KKM theory on hyperconvex metric spaces, Tamkang J. Math. 41(1)(2010), 1--14.
  • 30] Park, S. Generalizations of the Nash equilibrium theorem in the KKM theory, Takahashi Legacy, Fixed Point Theory Appl. vol. 2010, Article ID 234706, 23pp. doi:10.1155 /2010/234706.
  • [31] S. Park, The KKM principle in abstract convex spaces: Equivalent formulations and applications,Nonlinear Anal. 73 (2010), 1028{1042.15
  • [32] S. Park, A genesis of general KKM theorems for abstract convex spaces: Revisited, J. NonlinearAnal. Optim. 4(1) (2013), 127--132.
  • [33] S. Park, Review of recent studies on the KKM theory, II, Nonlinear Funct. Anal. Appl. 19(1) (2014)143--155.
  • [34] S. Park, Evolution of the KKM theory of hyperconvex spaces, J. Nat. Acad. Sci., ROK, Nat. Sci.Ser. 54(2) (2015), 1--28
  • [35] S. Park, Comments on the KKM theory of metric type spaces, Linear and Nonlinear Anal. 2(1)(2016), 39--45.[36] S. Park, Generalizations of Khamsi's KKM and fi xed point theorems on hyperconvex metric spaces,Nonlinear Anal. Forum 22(2) (2017), 7--15.
  • [37] S. Park, A history of the KKM Theory, J. Nat. Acad. Sci., ROK, Nat. Sci. Ser. 56(2) (2017), 1--51.
  • [38] S. Park, Various examples of the KKM spaces, presented at IWNAO2018.
  • 39] S. Park, A panoramic view of the KKM theory on abstract convex spaces, J. Nat. Acad. Sci., ROK,Nat. Sci. Ser. 57(2) (2018), 1--46.
  • [40] S. Park, From Hadamard manifolds to Horvath spaces, J. Nat. Acad. Sci., ROK, Nat. Sci. Ser. 58(1)(2019), 1--36.
  • [41] S. Park, Extending the realm of Horvath spaces, J. Nonlinear Convex Anal. (Takahashi Issue)
  • [42] S. Shabanian and S. M. Vaezpour, The KKM theorem in modular spaces and applications to minimaxinequalities, Bull. Malays. Math. Sci. Soc. 39 (2016) 921--931. DOI10.1007/s40840-015-0192-3
  • [43] D. Turkoglu, M. Abuloha, and T. Abdeljawad, KKM mappings in cone metric spaces and some xed point theorems, Nonlinear Anal. 72 (2010), 348--353. doi:10.1016/j.na.2009.06.058.
Primary Language en
Subjects Mathematics
Journal Section Articles
Authors

Author: Sehei Park (Primary Author)
Institution: The National Academy of Sciences, Republic of Korea
Country: South Korea


Dates

Publication Date : August 30, 2019

Bibtex @research article { rna573805, journal = {Results in Nonlinear Analysis}, issn = {}, eissn = {2636-7556}, address = {}, publisher = {Erdal KARAPINAR}, year = {2019}, volume = {2}, pages = {71 - 82}, doi = {}, title = {Generalizations of Hyperconvex Metric Spaces}, key = {cite}, author = {Park, Sehei} }
APA Park, S . (2019). Generalizations of Hyperconvex Metric Spaces. Results in Nonlinear Analysis , 2 (2) , 71-82 . Retrieved from https://dergipark.org.tr/en/pub/rna/issue/45041/573805
MLA Park, S . "Generalizations of Hyperconvex Metric Spaces". Results in Nonlinear Analysis 2 (2019 ): 71-82 <https://dergipark.org.tr/en/pub/rna/issue/45041/573805>
Chicago Park, S . "Generalizations of Hyperconvex Metric Spaces". Results in Nonlinear Analysis 2 (2019 ): 71-82
RIS TY - JOUR T1 - Generalizations of Hyperconvex Metric Spaces AU - Sehei Park Y1 - 2019 PY - 2019 N1 - DO - T2 - Results in Nonlinear Analysis JF - Journal JO - JOR SP - 71 EP - 82 VL - 2 IS - 2 SN - -2636-7556 M3 - UR - Y2 - 2019 ER -
EndNote %0 Results in Nonlinear Analysis Generalizations of Hyperconvex Metric Spaces %A Sehei Park %T Generalizations of Hyperconvex Metric Spaces %D 2019 %J Results in Nonlinear Analysis %P -2636-7556 %V 2 %N 2 %R %U
ISNAD Park, Sehei . "Generalizations of Hyperconvex Metric Spaces". Results in Nonlinear Analysis 2 / 2 (August 2019): 71-82 .
AMA Park S . Generalizations of Hyperconvex Metric Spaces. RNA. 2019; 2(2): 71-82.
Vancouver Park S . Generalizations of Hyperconvex Metric Spaces. Results in Nonlinear Analysis. 2019; 2(2): 82-71.