A topological generalization of the theorems of Gale-Nikaido and Mas Colell. (Q2711292)

Interconnections between analysis and topology are well-known [cf. D. H. Hyers et al., \textit{Topics in Nonlinear Analysis}, World Scientific (1997; Zbl 0878.47040)]. Exploiting these connections, the author generalizes several univalence theorems for continuous, open and discrete mappings in \(\mathbb{R}^n\).NEWLINENEWLINETheorem 1. Let \(K\subset {\mathbb R}^m\) be a compact and convex polyhedron, \(\dim K=m\), \(f\in C(K, {\mathbb R}^m)\) such that \(f\) is locally Lipschitzian on \({\partial K}\) and let \(L_1^p,\dots, L_{i_p}^p\) be the open faces of \(K\) of dimension \(p\), \(p=0,1,\dots, m-1.\) Suppose that \(f\in F(\text{int }K)\) and \(\varepsilon(f,\text{int }K)=1\) and for every face \(L\) of \(K\) of dimension \(p\), with \(1\leq p\leq m-1\), the map \(\pi_L\circ f|_L: L\to H\) is open, discrete and sense-preserving (here \(\pi_L: {\mathbb R}^m\to H\) is the projection on the minimal affine space containing \(L\), denoted by \(H\)). Then \(f\) is injective on \(K\).NEWLINENEWLINEFor the entire collection see [Zbl 0958.00011].