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Homotopy method for a class of nonconvex Brouwer fixed-point problems. (English) Zbl 0840.65038

The paper gives a homotopy proof of the existence of fixed points of \(C^2\) maps \(F : \Omega \to \Omega\) where \(\Omega \subset \mathbb{R}^n\). The set \(\Omega\) is defined via \(\Omega = \{y \in \mathbb{R}^n : g_i(x) \leq 0, i =1,2,\dots,n\}\) where the \(g_i\) are assumed to be \(C^3\) functions. The assumptions are: (i) the interior of \(\Omega\) is nonempty (ii) regularity of the constraints, i.e. \(\nabla g_i(x)\) is of full rank if \(x \in \partial \Omega\) and \(g_i(x) = 0\), (iii) a “normal cone” condition on \(\partial \Omega\).
A homotopy map is constructed consisting of two components. One component is a linear convex homotopy between a trivial map and the Lagrangian \(F(x) - x + \nabla g(x)y\), and the other component is a global Newton homotopy. The parametrized version of Sard’s theorem is applied to conclude the existence of a path of regular zero points of the homotopy which emanates from a trivial zero point of the homotopy map. It is shown that the path must stay in a bounded “cylinder” and reach a level of the homotopy parameter where a fixed point of \(F\) must lie.
The proof is in the spirit of the paper of S. N. Chow, J. Mallet-Paret and J. A. Yorke [Math. Comput. 32, 887-899 (1978; Zbl 0398.65029)].

MSC:

65H20 Global methods, including homotopy approaches to the numerical solution of nonlinear equations
65H10 Numerical computation of solutions to systems of equations
55M20 Fixed points and coincidences in algebraic topology
55P99 Homotopy theory
54H25 Fixed-point and coincidence theorems (topological aspects)

Citations:

Zbl 0398.65029
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Full Text: DOI

References:

[1] Hirsch, M. W., Differential Topology (1976), Springer-Verlag: Springer-Verlag Berlin · Zbl 0121.18004
[2] Kellogg, R. B.; Li, T. Y.; Yorke, J. A., A constructive proof of the Brouwer fixed-point theorem and computational results, SIAM J. Num. Anal., 13, 473-483 (1976) · Zbl 0355.65037
[3] Chow, S. N.; Mallet-Paret, J.; Yorke, J. A., Finding zeros of maps: homotopy methods that are constructive with probability one, Math. Comput., 32, 887-899 (1978) · Zbl 0398.65029
[4] Naber, G. L., Topological Methods in Euclidean Space (1980), Cambridge Univ. Press: Cambridge Univ. Press London · Zbl 0437.55001
[5] Allgower, E. L.; Georg, K., Numerical Continuation Methods: An Introduction (1990), Springer-Verlag: Springer-Verlag Berlin · Zbl 0717.65030
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