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

Vrahatis, Michael N.

doi:10.1090/S0002-9939-1989-0993760-8

A short proof and a generalization of Miranda’s existence theorem
HTML articles powered by AMS MathViewer

by Michael N. Vrahatis PDF

Proc. Amer. Math. Soc. 107 (1989), 701-703 Request permission

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

L. E. J. Brouwer, Über Abbildung von Mannigfaltigkeiten, Math. Ann. 71 (1911), no. 1, 97–115 (German). MR 1511644, DOI 10.1007/BF01456931
Jane Cronin, Fixed points and topological degree in nonlinear analysis, Mathematical Surveys, No. 11, American Mathematical Society, Providence, R.I., 1964. MR 0164101
Vasile I. Istrăţescu, Fixed point theory, Mathematics and its Applications, vol. 7, D. Reidel Publishing Co., Dordrecht-Boston, Mass., 1981. An introduction; With a preface by Michiel Hazewinkel. MR 620639, DOI 10.1007/978-94-009-8177-5
J. B. Kioustelidis, Algorithmic error estimation for approximate solutions of nonlinear systems of equations, Computing 19 (1977/78), no. 4, 313–320 (English, with German summary). MR 474779, DOI 10.1007/BF02252029
Carlo Miranda, Un’osservazione su un teorema di Brouwer, Boll. Un. Mat. Ital. (2) 3 (1940), 5–7 (Italian). MR 0004775
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), no. 4, 521–529. MR 584727, DOI 10.1137/0717044
Mitio Nagumo, A theory of degree of mapping based on infinitesimal analysis, Amer. J. Math. 73 (1951), 485–496. MR 42696, DOI 10.2307/2372303
J. M. Ortega and W. C. Rheinboldt, Iterative solution of nonlinear equations in several variables, Academic Press, New York-London, 1970. MR 0273810
J. T. Schwartz, Nonlinear functional analysis, Notes on Mathematics and its Applications, Gordon and Breach Science Publishers, New York-London-Paris, 1969. Notes by H. Fattorini, R. Nirenberg and H. Porta, with an additional chapter by Hermann Karcher. MR 0433481
Frank Stenger, Computing the topological degree of a mapping in $\textbf {R}^{n}$, Numer. Math. 25 (1975/76), no. 1, 23–38. MR 394639, DOI 10.1007/BF01419526
Zu He Shen and A. Neumaier, A note on Moore’s interval test for zeros of nonlinear systems, Computing 40 (1988), no. 1, 85–90 (English, with German summary). MR 938167, DOI 10.1007/BF02242192