World Academy of Science, Engineering and Technology
Published August 28, 2012 | Version 11175
Journal article Open

A Constructive Proof of the General Brouwer Fixed Point Theorem and Related Computational Results in General Non-Convex sets

Creators

Description

In this paper, by introducing twice continuously differentiable mappings, we develop an interior path following following method, which enables us to give a constructive proof of the general Brouwer fixed point theorem and thus to solve fixed point problems in a class of non-convex sets. Under suitable conditions, a smooth path can be proven to exist. This can lead to an implementable globally convergent algorithm. Several numerical examples are given to illustrate the results of this paper.

Files

11175.pdf

Files (188.0 kB)

Name Size Download all
md5:4303ce521cd0023cb04acab5bda8719a
188.0 kB Preview Download

Additional details

References

  • B. Bollobas, W. Fulton, A. Katok, F. Kirwan, P. Sarnak, Fixed Point Theory and Applications. Lodon, England: Cambridge University Press, 2004.
  • Y. X. Gao, Y. Li, J. Zhang, Invariant tori of nonlinear Schr¨odinger equation, Journal of Differential Equations 246 (2009) 3296-3331.
  • S. Heikkila, K. Reffett, Fixed point theorems and their applications to theory of Nash equilibria, Nonlinear Anal. 64 (2006) 1415-1436.
  • S. Karamardian, Fixed Points: Algorithms and Applications. New York, America: Academic Press, 1977.
  • L. J. Lin, Z. T. Yu, Fixed point theorems and equilibrium problems, Nonlinear Anal. 43 (2001) 987-999.
  • S. Park, Fixed points and quasi-equilibrium problems, Math. Comput. Modelling 32 (2000) 1297-1303.
  • S. Robinson, Analysis and Computation of Fixed Points. New York, America: Academic Press, 1980.
  • M. J. Todd, Improving the convergence of fixed point algorithms, Math. Program. 7 (1978) 151-179.
  • K. B. Kellogg, T. Y. Li, J. A. Yorke, A constructive proof of the Brouwer fixed-point theorem and computational results, SIAM J. Numer. Anal. 13 (1976) 473-483. [10] J. C. Alexander, J. A. Yorke, The homotopy continuation method: numerically implementable topological procedure, Trans. Amer. Math. Soc. 242 (1978) 271-284. [11] E.L. Allgower, K. Georg, Introduction to Numerical Continuation Methods. Philadelphia, America: SIAM Society for Industried and Applied Mathematics, 2003. [12] C. B. Carcia, W. I. Zangwill, An approach to homotopy and degree theory, Math. Oper. Res 4 (1979) 390-405. [13] S. N. Chow, J. Mallet-Paret, J. A. Yorke, Finding zeros of maps: homotopy methods that are constructive with probability one, Math. Comput. 32 (1978) 887-899. [14] Y. Li, Z. H. Lin, A constructive proof of the Pincare-Birkhoff theorem, Trans.Amer. Math. Soc. 347 (1995) 2111-2126. [15] B. Yu, Z.H. Lin, Homotopy method for a class of nonconvex Brouwer fixed-point problems, Appl. Math. Comput. 74 (1996) 65-77.