A fixed point theorem for discontinuous functions
Let \(P\) be an non-empty polytope in the \(n\)-dimensional Euclidean space \(\mathbb{R}^n\) and \(f:P\to P\) be a function. A function \(f\) is locally gross direction preserving if for each \(x\in P\) for which \(f(x)\neq x\), there exists \(\delta>0\) such that for every \(y,z\in B(x,\delta)\cap P\), the function \(f\) satisfies \((f(y)-y)^T(f(z)-z)\geq 0\). This condition allows for various kinds od discontinuities and irregularities of the function \(f\). The main result of the present paper is the following theorem: Let \(P\) be an non-empty polytope in \(\mathbb{R}^n\) and the function \(f:P\to P\) satisfy the locally gross direction preserving property. Then \(f\) has a fixed point. Some applications of the result in the area of non-cooperative game theory are given.
scientific article; zbMATH DE number 5280662
scientific article