Constructive Proof of Tychonoff’s Fixed Point Theorem for Sequentially Locally Non-Constant Functions

We present a constructive proof of Tychonoff’s fixed point theorem in a locally convex space for uniformly continuous and sequentially locally non-constant functions.

• Yasuhito Tanaka

• sequentially locally non-constant functions
• Tychonoff’s fixed point theorem
• constructive mathematics.

[1] J. Berger, D. Bridges, and P. Schuster. The fan theorem and unique
existence of maxima. Journal of Symbolic Logic, 71:713–720, 2006.
[2] J. Berger and H. Ishihara. Brouwer’s fan theorem and unique existence
in constructive analysis. Mathematical Logic Quarterly, 51(4):360–364,
2005.
[3] D. Bridges and F. Richman. Varieties of Constructive Mathematics.
Cambridge University Press, 1987.
[4] D. Bridges and L. Vˆıt¸˘a. Techniques of Constructive Mathematics.
Springer, 2006.
[5] V. I. Istrˇat¸escu. Fixed Point Theory. D. Reidel Publishing Company,
1981.
[6] R. B. Kellogg, T. Y. Li, and J. Yorke. A constructive proof of
Brouwer fixed-point theorem and computational results. SIAM Journal
on Numerical Analysis, 13:473–483, 1976.
[7] Y. Tanaka. Constructive proof of Brouwer’s fixed point theorem for sequentially
locally non-constant functions. http://arxiv.org/abs/1103.1776,
2011.
[8] Y. Tanaka. On constructive versions of tychonoff’s and schauder’s fixed
point theorems. Applied Mathematics E-Notes, 11:125–132, 2011.
[9] D. van Dalen. Brouwer’s ε-fixed point from Sperner’s lemma. Theoretical
Computer Science, 412(28):3140–3144, June 2011.
[10] W. Veldman. Brouwer’s approximate fixed point theorem is equivalent to
Brouwer’s fan theorem. In S. Lindstr¨om, E. Palmgren, K. Segerberg, and
V. Stoltenberg-Hansen, editors, Logicism, Intuitionism and Formalism.
Springer, 2009.