An extension of Brouwer's fixed point theorem allowing discontinuities (Q1876825)

Brouwer's fixed point theorem states that every continuous mapping \(f: \mathcal{B} \rightarrow \mathcal{B},\) where \(\mathcal{B}\) is a closed ball of \(\mathbb{R}^n,\) has a fixed point. This paper extends this famous theorem to the situation where discontinuities of the single-valued mapping \(f\) are permitted. In economics, a class of problems that takes the incompleteness of financial markets into account has recently emerged. The utility of the results in this paper is that it may be applied to explain discontinuites in economic situations arising from markets that are not complete. In particular, the authors consider an equilibrium existence problem in economics that involves a fixed point of mappings \(f(x,\text{span}~ V(x)),\) where \(V(x)\) is a matrix specifying the returns on financial assets.