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)



A short proof and a generalization of Miranda's existence theorem

Author: Michael N. Vrahatis
Journal: Proc. Amer. Math. Soc. 107 (1989), 701-703
MSC: Primary 54H25; Secondary 47H10, 55M20, 58C30, 65H10
MathSciNet review: 993760
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Miranda gave in [5] an equivalent formulation of the famous Brouwer fixed point theorem. We give a short proof of Miranda's existence theorem and then using the results obtained in this proof we give a generalization of a well-known variant of Bolzano's existence theorem. Finally, we give a generalization of Miranda's theorem.

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

  • [1] L. E. J. Brouwer, Über Abbildungen von Mannigfaltigkeiten, Math. Ann. 71 (1912), 97-115. MR 1511644
  • [2] J. Cronin, Fixed points and topological degree in nonlinear analysis, Math. Surveys no. 11, Amer. Math. Soc., Providence, R.I., 1964. MR 0164101 (29:1400)
  • [3] V. I. Istraţescu, Fixed point theory, Mathematics and its applications, V. 7, D. Reidel Publishing Co., Dordecht, Holland, 1981. MR 620639 (83c:54065)
  • [4] J. B. Kioustelidis, Algorithmic error estimation for approximate solutions of nonlinear systems of equations, Computing 19 (1978), 313-320. MR 0474779 (57:14412)
  • [5] C. Miranda, Un'osservazione su un teorema di Brouwer, Bol. Un. Mat. Ital., 3 (1940), 5-7. MR 0004775 (3:60b)
  • [6] R. E. Moore and J. B. Kioustelidis, A simple test for accuracy of approximate solutions to nonlinear (or linear) systems, SIAM J. Numer. Anal. 17 (1980), 521-529. MR 584727 (83e:65085)
  • [7] M. Nagumo, A theory of degree of mapping based on infinitesimal analysis, Amer. J. Math. 73 (1951), 485-496. MR 0042696 (13:150a)
  • [8] J. M. Ortega and W. C. Rheinboldt, Iterative solution of nonlinear equations in several variables, Academic Press, New York, 1970. MR 0273810 (42:8686)
  • [9] J. T. Schwartz, Nonlinear functional analysis, Gordon and Breach, New York, 1969. MR 0433481 (55:6457)
  • [10] F. Stenger, Computing the topological degree of a mapping in $ {R^n}$, Numer. Math. 25 (1975), 23-38. MR 0394639 (52:15440)
  • [11] S. Zuhe and A. Neumaier, A note on Moore's interval test for zeros of nonlinear systems, Computing 40 (1988), 85-90. MR 938167 (89e:65052)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 54H25, 47H10, 55M20, 58C30, 65H10

Retrieve articles in all journals with MSC: 54H25, 47H10, 55M20, 58C30, 65H10

Additional Information

Keywords: Fixed points, Brouwer fixed point theorem, Miranda existence theorem, Bolzano existence theorem, systems of nonlinear algebraic or transcendental equations
Article copyright: © Copyright 1989 American Mathematical Society

American Mathematical Society