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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)].

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)].

Reviewer: E.L.Allgower (Fort Collins)

### 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) |

### Keywords:

nonconvex Brouwer fixed point problems; homotopy methods; normal cone condition; global Newton homotopy; Sard’s theorem### Citations:

Zbl 0398.65029
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\textit{B. Yu} and \textit{Z. Lin}, Appl. Math. Comput. 74, No. 1, 65--77 (1996; Zbl 0840.65038)

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### 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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