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KKM implies the Brouwer fixed point theorem: Another proof

Yıl 2020, Cilt: 3 Sayı: 1, 12 - 17, 31.03.2020

Öz

It is well-known that the Brouwer fixed point theorem (BFPT), the weak Sperner combinatorial lemma, and the
Knaster-Kuratowski-Mazurkiewicz (KKM) theorem are mutually equivalent and have scores of equivalent formulations and several thousand applications. It is well-known that KKM deduced the BFPT from Sperner Lemma. In this article, we recall some KKM theoretic results implying the BFPT.

Kaynakça

  • [1] L.E.J. Brouwer, Über Abbildung von Mannigfaltigkeiten, Math. Ann. 71 (1912) 97–115.
  • [2] C.-M. Chen and T.-H. Chang, Some results for the family 2– g KKM(X,Y ) and the ?-mapping in hyperconvex metric spaces, Nonlinear Anal. 69 (2008) 2533–2540.
  • [3] K. Fan, A generalization of Tychonoff’s fixed point theorem, Math. Annalen 142 (1961) 305–310.
  • [4] B. Halpern, Fixed-point theorems for outward maps, Doctoral Thesis, U.C.L.A. 1965.
  • [5] M.A. Khamsi, KKM and Ky Fan theorems in hyperconvex metric spaces, J. Math. Anal. Appl. 204 (1996) 298–306.
  • [6] 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.
  • [7] B. Knaster, K. Kuratowski und S. Mazurkiewicz, Ein Beweis des Fixpunktsatzes für n-Dimensionale Simplexe, Fund. Math. 14 (1929) 132–137.
  • [8] R.D. Mauldin, The Scottish Book: Mathematics from the Scottish Café, Birkhäuser, Boston-Basel-Stuttgart, 1981.
  • [9] S. Park, A generalization of the Brouwer fixed point theorem, Bull. Korean Math. Soc. 28 (1991) 33–37.
  • [10] S. Park, Elements of the KKM theory on abstract convex spaces, J. Korean Math. Soc. 45(1) (2008) 1–27.
  • [11] S. Park, New foundations of the KKM theory, J. Nonlinear Convex. Anal. 9(3) (2008) 331–350.
  • [12] S. Park, The KKM principle in abstract convex spaces — Equivalent formulations and applications, Nonlinear Analysis 73 (2010) 1028–1042.
  • [13] S. Park, Fixed point theorems in the new era of the KKM theory, Fixed Point Theory and Its Applications (Proc. ICFPTA-2009), 145–159, Yokohama Publ., 2010.
  • [14] S. Park, Comments on the KKM theory of metric type spaces, Linear and Nonlinear Anal. 2(1) (2016) 39–45.
  • [15] S. Park, A history of the KKM Theory, J. Nat. Acad. Sci., ROK, Nat. Sci. Ser. 56(2) (2017) 1–51.
  • [16] S. Park, On further examples of partial KKM spaces, J. Fixed Point Theory 2019:10 (18 June, 2019), 1–18.
  • [17] S. Park, Extending the realm of Horvath spaces, J. Nonlinear Convex Anal. 20(8) (2019) 1609–1621.
  • [18] S. Park, KKM implies the Brouwer fixed point theorem: Revisited, to appear.
  • [19] S. Park and K.S. Jeong, A proof of the Sperner lemma from the Brouwer fixed point theorem, Nonlinear Anal. Forum 8 (2003), 65–67.
  • [20] E. Sperner, Neuer Beweis für die Invarianz der Dimensionszahl und des Gebietes, Abh. Math. Seminar Univ. Hamburg 6 (1928) 265–272.
Yıl 2020, Cilt: 3 Sayı: 1, 12 - 17, 31.03.2020

Öz

Kaynakça

  • [1] L.E.J. Brouwer, Über Abbildung von Mannigfaltigkeiten, Math. Ann. 71 (1912) 97–115.
  • [2] C.-M. Chen and T.-H. Chang, Some results for the family 2– g KKM(X,Y ) and the ?-mapping in hyperconvex metric spaces, Nonlinear Anal. 69 (2008) 2533–2540.
  • [3] K. Fan, A generalization of Tychonoff’s fixed point theorem, Math. Annalen 142 (1961) 305–310.
  • [4] B. Halpern, Fixed-point theorems for outward maps, Doctoral Thesis, U.C.L.A. 1965.
  • [5] M.A. Khamsi, KKM and Ky Fan theorems in hyperconvex metric spaces, J. Math. Anal. Appl. 204 (1996) 298–306.
  • [6] 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.
  • [7] B. Knaster, K. Kuratowski und S. Mazurkiewicz, Ein Beweis des Fixpunktsatzes für n-Dimensionale Simplexe, Fund. Math. 14 (1929) 132–137.
  • [8] R.D. Mauldin, The Scottish Book: Mathematics from the Scottish Café, Birkhäuser, Boston-Basel-Stuttgart, 1981.
  • [9] S. Park, A generalization of the Brouwer fixed point theorem, Bull. Korean Math. Soc. 28 (1991) 33–37.
  • [10] S. Park, Elements of the KKM theory on abstract convex spaces, J. Korean Math. Soc. 45(1) (2008) 1–27.
  • [11] S. Park, New foundations of the KKM theory, J. Nonlinear Convex. Anal. 9(3) (2008) 331–350.
  • [12] S. Park, The KKM principle in abstract convex spaces — Equivalent formulations and applications, Nonlinear Analysis 73 (2010) 1028–1042.
  • [13] S. Park, Fixed point theorems in the new era of the KKM theory, Fixed Point Theory and Its Applications (Proc. ICFPTA-2009), 145–159, Yokohama Publ., 2010.
  • [14] S. Park, Comments on the KKM theory of metric type spaces, Linear and Nonlinear Anal. 2(1) (2016) 39–45.
  • [15] S. Park, A history of the KKM Theory, J. Nat. Acad. Sci., ROK, Nat. Sci. Ser. 56(2) (2017) 1–51.
  • [16] S. Park, On further examples of partial KKM spaces, J. Fixed Point Theory 2019:10 (18 June, 2019), 1–18.
  • [17] S. Park, Extending the realm of Horvath spaces, J. Nonlinear Convex Anal. 20(8) (2019) 1609–1621.
  • [18] S. Park, KKM implies the Brouwer fixed point theorem: Revisited, to appear.
  • [19] S. Park and K.S. Jeong, A proof of the Sperner lemma from the Brouwer fixed point theorem, Nonlinear Anal. Forum 8 (2003), 65–67.
  • [20] E. Sperner, Neuer Beweis für die Invarianz der Dimensionszahl und des Gebietes, Abh. Math. Seminar Univ. Hamburg 6 (1928) 265–272.
Toplam 20 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Articles
Yazarlar

Sehie Park

Yayımlanma Tarihi 31 Mart 2020
Yayımlandığı Sayı Yıl 2020 Cilt: 3 Sayı: 1

Kaynak Göster

APA Park, S. (2020). KKM implies the Brouwer fixed point theorem: Another proof. Results in Nonlinear Analysis, 3(1), 12-17.
AMA Park S. KKM implies the Brouwer fixed point theorem: Another proof. RNA. Mart 2020;3(1):12-17.
Chicago Park, Sehie. “KKM Implies the Brouwer Fixed Point Theorem: Another Proof”. Results in Nonlinear Analysis 3, sy. 1 (Mart 2020): 12-17.
EndNote Park S (01 Mart 2020) KKM implies the Brouwer fixed point theorem: Another proof. Results in Nonlinear Analysis 3 1 12–17.
IEEE S. Park, “KKM implies the Brouwer fixed point theorem: Another proof”, RNA, c. 3, sy. 1, ss. 12–17, 2020.
ISNAD Park, Sehie. “KKM Implies the Brouwer Fixed Point Theorem: Another Proof”. Results in Nonlinear Analysis 3/1 (Mart 2020), 12-17.
JAMA Park S. KKM implies the Brouwer fixed point theorem: Another proof. RNA. 2020;3:12–17.
MLA Park, Sehie. “KKM Implies the Brouwer Fixed Point Theorem: Another Proof”. Results in Nonlinear Analysis, c. 3, sy. 1, 2020, ss. 12-17.
Vancouver Park S. KKM implies the Brouwer fixed point theorem: Another proof. RNA. 2020;3(1):12-7.