Homotopy method for a class of nonconvex Brouwer fixed-point problems (Q1908258)

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 \textit{S. N. Chow, J. Mallet-Paret} and \textit{J. A. Yorke} [Math. Comput. 32, 887-899 (1978; Zbl 0398.65029)].