Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



An odd theorem

Author: B. Curtis Eaves
Journal: Proc. Amer. Math. Soc. 26 (1970), 509-513
MSC: Primary 90.60
MathSciNet review: 0270757
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ C$ be a bounded convex polyhedral set and let $ f:C \to C$ be continuous and piecewise linear. Using notions from complementary pivot theory, it is shown that if each fixed point of $ f$ lies interior to some piece of linearity, then $ f$ has an odd number of fixed points. In addition, an algorithm is given for computing a fixed point of $ f$.

References [Enhancements On Off] (What's this?)

  • [1] Richard W. Cottle and George B. Dantzig, Complementary pivot theory of mathematical programming, Linear Algebra and Appl. 1 (1968), no. 1, 103–125. MR 0226929
  • [2] B. Curtis Eaves, Linear complementarity problem in mathematical programming, Doctoral Dissertation, Stanford University, Stanford, Calif., 1969.
  • [3] C. E. Lemke, Bimatrix equilibrium points and mathematical programming, Management Sci. 11 (1964/1965), 681–689. MR 0189823
  • [4] C. E. Lemke and J. T. Howson Jr., Equilibrium points of bimatrix games, J. Soc. Indust. Appl. Math. 12 (1964), 413–423. MR 0173556
  • [5] K. G. Murty, On the number of solutions to the complementary quadratic programming problem, Doctoral Dissertation, Engineering Science, University of California, Berkeley, Calif., 1968.
  • [6] T. Hansen, On the approximation of a competitive equilibrium, Ph.D. Dissertation, Yale University, New Haven, Conn., 1968.
  • [7] Herbert E. Scarf, The core of an 𝑁 person game, Econometrica 35 (1967), 50–69. MR 0234735
  • [8] Herbert Scarf, The approximation of fixed points of a continuous mapping, SIAM J. Appl. Math. 15 (1967), 1328–1343. MR 0242483
  • [9] Daniel I. A. Cohen, On the Sperner lemma, J. Combinatorial Theory 2 (1967), 585–587. MR 0214047
  • [10] Harold W. Kuhn, Simplicial approximation of fixed points, Proc. Nat. Acad. Sci. U.S.A. 61 (1968), 1238–1242. MR 0488010
  • [11] H. W. Kuhn, Some combinatorial lemmas in topology, IBM J. Res. Develop. 4 (1960), 508–524. MR 0124038
  • [12] L. S. Pontryagin, Osnovy Kombinatornoĭ Topologii, OGIZ, Moscow-Leningrad,], 1947 (Russian). MR 0033516

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 90.60

Retrieve articles in all journals with MSC: 90.60

Additional Information

Keywords: Odd number, fixed points, piecewise linear, simplicial, complementary pivots, Sperner's Lemma, pseudomanifold
Article copyright: © Copyright 1970 American Mathematical Society