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The rise and fall of MC-spaces

Yıl 2021, , 21 - 32, 31.03.2021
https://doi.org/10.53006/rna.852462

Öz

In 1994, Llinares introduced mc-spaces and began to study KKM theoretic results on them. Since 1998, he became an L-space theorist and repeated to claim that his mc-spaces generalize G-convex spaces without any justifications. Later he insisted that his mc-spaces are the same as L-spaces. Hence his study on mc-spaces is
useless now as the L-space case shown by our previous works. The present article is a continuation of our previous works on L-spaces and concerns with the rise and fall of mc-spaces. This paper will be an important record for the history of the KKM theory.

Kaynakça

  • [1] H. Ben-El-Mechaiekh, S. Chebbi, M. Florenzano, and J.V. Llinares, Abstract convexity and fixed points, J. Math. Anal. Appl. 222 (1998) 138–150.
  • [2] H. Ben-El-Mechaiekh, S. Chebbi, and M. Florenzano, A generalized KKMF principle, J. Math. Anal. Appl. 309 (2005) 583–590.
  • [3] G.L. Cain Jr. and L. Gonzalez, The Knaster-Kuratowski-Mazurkiewitz theorem and abstract convex spaces, J. Math. Anal. Appl. 338 (2008) 563–571.
  • [4] L. González, S. Kilmer, and J. Rebaza, From a KKM theorem to Nash equilibria in L-spaes, Top. Appl. 155 (2007) 165–170.
  • [5] C. Horvath, Contractibility and generalized convexity, J. Math. Anal. Appl. 156 (1991) 341-357.
  • [6] C.D. Horvath and J.V. Llinares Ciscar, Maximal elements and fixed points for binary relations on topological ordered spaces, J. Math. Econom. 25 (1996) 291–306.
  • [7] W. Kulpa and A. Szymanski, Applications of general infimum principles to fixed-point theory and game theory, Set-valued Anal. 16 (2008) 375–398.
  • [8] J.V. LLinares, Abstract convexity. Fixed points and applications, Ph.D. Thesis. Univ. de Alicante, Spain. 1994.
  • [9] J.V. LLinares, Existence of maximal elements in a binary relation relaxing the convexity condition, WP-AD 95-10, Jan. 1995. Appeared ResearchGate on 15 May 2014.
  • [10] J.V. Llinares, Unified treatment of the problem of existence of maximal elements in bindary relations. A characterization, J. Math. Econom. 29 (1998) 285–302.
  • [11] J.V. LLinares, Existence of equilibrium in generalized games with non-convex strategy spaces, Working Paper CEPREMAP Couverture Orange no.98-01, Janvier 1998.
  • [12] J.V. LLinares, Abstract convexity, some relations and applications, Working Paper CEPREMAP Couverture Orange no.98- 03, Mars 1998.
  • [13] J.V. LLinares, Existence of equilibrium in generalized games with abstract convexity structure, J. Optim. Theory Appl. bf 105(1) (2000) 149–160.
  • [14] J.V. LLinares, Abstract convexity. Some relations and applications, Optimization 51(6) (2002) 797–818.
  • [15] H. Lu, A section theorem in topological ordered spaces and its applications to the existence of Pareto equilibria for multi- objective games, 2009 Inter. Joint Conf. Artificial Intelligence, IEEE, DOI 10.1109/JCAI.2009.14 3
  • [16] 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.
  • [17] Q. Luo, KKM and Nash equilibria type theorems in topological ordered spaces, J. Math. Anal. Appl. 264 (2001) 262–269. doi:10.1006/jmaa.2001.7624
  • [18] S. Park, Another five episodes related to generalized convex spaces, Nonlinear Funct. Anal. Appl. 3 (1998), 1–12.
  • [19] S. Park, On generalizations of the KKM principle on abstract convex spaces, Nonlinear Anal. Forum 11 (2006), 67–77.
  • [20] S. Park, Remarks on KC-maps and KO-maps in abstract convex spaces, Nonlinear Anal. Forum 12(1) (2007) 29–40.
  • [21] S. Park, Examples of KC-maps and KO-maps on abstract convex spaces, Soochow J. Math. 33(3) (2007) 477–486.
  • [22] S. Park, Elements of the KKM theory on abstract convex spaces, J. Korean Math. Soc. 45(1) (2008) 1–27.
  • [23] S. Park, New foundations of the KKM theory, J. Nonlinear Convex Anal. 9(3) (2008a), 331–350.
  • [24] S. Park, Comments on the KKM theory on ? A -spaces, PanAmerican Math. J. 18(2) (2008b), 61–71.
  • [25] S. Park, Generalized convex spaces, L-spaces, and FC-spaces, J. Global Optim. 45(2) (2009) 203–210.
  • [26] S. Park, The KKM principle in abstract convex spaces: Equivalent formulations and applications, Nonlinear Anal. 73 (2010) 1028–1042.
  • [27] S. Park, The rise and decline of generalized convex spaces, Nonlinear Anal. Forum 15 (2010) 1–12.
  • [28] S. Park, Comments on abstract convexity structures on topological spaces, Nonlinear Anal. 72 (2010) 549–554.
  • [29] S. Park, Several episodes in recent studies on the KKM theory, Nonlinear Anal. Forum 15 (2010) 13–26.
  • [30] S. Park, New generalizations of basic theorems in the KKM theory, Nonlinear Anal. 74 (2011) 3000–3010.
  • [31] S. Park, Remarks on simplicial spaces and L* -spaces of Kulpa and Szymanski, Comm. Appl. Nonlinear Anal. 19(1) (2012) 59–69.
  • [32] S. Park, A review of the KKM theory on pi A -spaces or GFC-spaces, Advances in Fixed Point Theory 3(2) (2013) 353–382.
  • [33] S. Park, A unified approach to generalized KKM maps, J. Nat. Acad. Sci., ROK, Nat. Sci. Ser. 55(1) (2016) 1–20.
  • [34] S. Park, The rise and fall of L-spaces, Adv. Th. Nonlinear Anal. Appl. 4(3) (2020) 152–166.
  • [35] S. Park, The rise and fall of L-spaces, II, Adv. Th. Nonlinear Anal. Appl. 5(1) (2021) 1–15.
  • [36] S. Park, Revisit to Generalized KKM maps, to appear.
  • [37] S. Park and H. Kim, Admissible classes of multifunctions on generalized convex spaces, Proc. Coll. Natur. Sci. SNU 18 (1993) 1–21.
  • [38] M.C. Sanchez, J.-V. Llinares, and B. Subiza, A KKM-result and an application for binary and non-binary choice functions, Economic Theory 21 (2003) 185–193.
Yıl 2021, , 21 - 32, 31.03.2021
https://doi.org/10.53006/rna.852462

Öz

Kaynakça

  • [1] H. Ben-El-Mechaiekh, S. Chebbi, M. Florenzano, and J.V. Llinares, Abstract convexity and fixed points, J. Math. Anal. Appl. 222 (1998) 138–150.
  • [2] H. Ben-El-Mechaiekh, S. Chebbi, and M. Florenzano, A generalized KKMF principle, J. Math. Anal. Appl. 309 (2005) 583–590.
  • [3] G.L. Cain Jr. and L. Gonzalez, The Knaster-Kuratowski-Mazurkiewitz theorem and abstract convex spaces, J. Math. Anal. Appl. 338 (2008) 563–571.
  • [4] L. González, S. Kilmer, and J. Rebaza, From a KKM theorem to Nash equilibria in L-spaes, Top. Appl. 155 (2007) 165–170.
  • [5] C. Horvath, Contractibility and generalized convexity, J. Math. Anal. Appl. 156 (1991) 341-357.
  • [6] C.D. Horvath and J.V. Llinares Ciscar, Maximal elements and fixed points for binary relations on topological ordered spaces, J. Math. Econom. 25 (1996) 291–306.
  • [7] W. Kulpa and A. Szymanski, Applications of general infimum principles to fixed-point theory and game theory, Set-valued Anal. 16 (2008) 375–398.
  • [8] J.V. LLinares, Abstract convexity. Fixed points and applications, Ph.D. Thesis. Univ. de Alicante, Spain. 1994.
  • [9] J.V. LLinares, Existence of maximal elements in a binary relation relaxing the convexity condition, WP-AD 95-10, Jan. 1995. Appeared ResearchGate on 15 May 2014.
  • [10] J.V. Llinares, Unified treatment of the problem of existence of maximal elements in bindary relations. A characterization, J. Math. Econom. 29 (1998) 285–302.
  • [11] J.V. LLinares, Existence of equilibrium in generalized games with non-convex strategy spaces, Working Paper CEPREMAP Couverture Orange no.98-01, Janvier 1998.
  • [12] J.V. LLinares, Abstract convexity, some relations and applications, Working Paper CEPREMAP Couverture Orange no.98- 03, Mars 1998.
  • [13] J.V. LLinares, Existence of equilibrium in generalized games with abstract convexity structure, J. Optim. Theory Appl. bf 105(1) (2000) 149–160.
  • [14] J.V. LLinares, Abstract convexity. Some relations and applications, Optimization 51(6) (2002) 797–818.
  • [15] H. Lu, A section theorem in topological ordered spaces and its applications to the existence of Pareto equilibria for multi- objective games, 2009 Inter. Joint Conf. Artificial Intelligence, IEEE, DOI 10.1109/JCAI.2009.14 3
  • [16] 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.
  • [17] Q. Luo, KKM and Nash equilibria type theorems in topological ordered spaces, J. Math. Anal. Appl. 264 (2001) 262–269. doi:10.1006/jmaa.2001.7624
  • [18] S. Park, Another five episodes related to generalized convex spaces, Nonlinear Funct. Anal. Appl. 3 (1998), 1–12.
  • [19] S. Park, On generalizations of the KKM principle on abstract convex spaces, Nonlinear Anal. Forum 11 (2006), 67–77.
  • [20] S. Park, Remarks on KC-maps and KO-maps in abstract convex spaces, Nonlinear Anal. Forum 12(1) (2007) 29–40.
  • [21] S. Park, Examples of KC-maps and KO-maps on abstract convex spaces, Soochow J. Math. 33(3) (2007) 477–486.
  • [22] S. Park, Elements of the KKM theory on abstract convex spaces, J. Korean Math. Soc. 45(1) (2008) 1–27.
  • [23] S. Park, New foundations of the KKM theory, J. Nonlinear Convex Anal. 9(3) (2008a), 331–350.
  • [24] S. Park, Comments on the KKM theory on ? A -spaces, PanAmerican Math. J. 18(2) (2008b), 61–71.
  • [25] S. Park, Generalized convex spaces, L-spaces, and FC-spaces, J. Global Optim. 45(2) (2009) 203–210.
  • [26] S. Park, The KKM principle in abstract convex spaces: Equivalent formulations and applications, Nonlinear Anal. 73 (2010) 1028–1042.
  • [27] S. Park, The rise and decline of generalized convex spaces, Nonlinear Anal. Forum 15 (2010) 1–12.
  • [28] S. Park, Comments on abstract convexity structures on topological spaces, Nonlinear Anal. 72 (2010) 549–554.
  • [29] S. Park, Several episodes in recent studies on the KKM theory, Nonlinear Anal. Forum 15 (2010) 13–26.
  • [30] S. Park, New generalizations of basic theorems in the KKM theory, Nonlinear Anal. 74 (2011) 3000–3010.
  • [31] S. Park, Remarks on simplicial spaces and L* -spaces of Kulpa and Szymanski, Comm. Appl. Nonlinear Anal. 19(1) (2012) 59–69.
  • [32] S. Park, A review of the KKM theory on pi A -spaces or GFC-spaces, Advances in Fixed Point Theory 3(2) (2013) 353–382.
  • [33] S. Park, A unified approach to generalized KKM maps, J. Nat. Acad. Sci., ROK, Nat. Sci. Ser. 55(1) (2016) 1–20.
  • [34] S. Park, The rise and fall of L-spaces, Adv. Th. Nonlinear Anal. Appl. 4(3) (2020) 152–166.
  • [35] S. Park, The rise and fall of L-spaces, II, Adv. Th. Nonlinear Anal. Appl. 5(1) (2021) 1–15.
  • [36] S. Park, Revisit to Generalized KKM maps, to appear.
  • [37] S. Park and H. Kim, Admissible classes of multifunctions on generalized convex spaces, Proc. Coll. Natur. Sci. SNU 18 (1993) 1–21.
  • [38] M.C. Sanchez, J.-V. Llinares, and B. Subiza, A KKM-result and an application for binary and non-binary choice functions, Economic Theory 21 (2003) 185–193.
Toplam 38 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Articles
Yazarlar

Sehie Park

Yayımlanma Tarihi 31 Mart 2021
Yayımlandığı Sayı Yıl 2021

Kaynak Göster

APA Park, S. (2021). The rise and fall of MC-spaces. Results in Nonlinear Analysis, 4(1), 21-32. https://doi.org/10.53006/rna.852462
AMA Park S. The rise and fall of MC-spaces. RNA. Mart 2021;4(1):21-32. doi:10.53006/rna.852462
Chicago Park, Sehie. “The Rise and Fall of MC-Spaces”. Results in Nonlinear Analysis 4, sy. 1 (Mart 2021): 21-32. https://doi.org/10.53006/rna.852462.
EndNote Park S (01 Mart 2021) The rise and fall of MC-spaces. Results in Nonlinear Analysis 4 1 21–32.
IEEE S. Park, “The rise and fall of MC-spaces”, RNA, c. 4, sy. 1, ss. 21–32, 2021, doi: 10.53006/rna.852462.
ISNAD Park, Sehie. “The Rise and Fall of MC-Spaces”. Results in Nonlinear Analysis 4/1 (Mart 2021), 21-32. https://doi.org/10.53006/rna.852462.
JAMA Park S. The rise and fall of MC-spaces. RNA. 2021;4:21–32.
MLA Park, Sehie. “The Rise and Fall of MC-Spaces”. Results in Nonlinear Analysis, c. 4, sy. 1, 2021, ss. 21-32, doi:10.53006/rna.852462.
Vancouver Park S. The rise and fall of MC-spaces. RNA. 2021;4(1):21-32.