The index of Lyapunov stable fixed points in two dimensions (Q1340465)

Let \(U \subset R^ 2\) be an open and connected set, \(0 \in U\), \(f:U \to R^ 2\) be a continuous, one-to-one and orientation-preserving mapping, \(f(0) = 0\). If 0 is stable and isolated as a fixed point of \(f\), then the fixed point index is equal to one. Applications to differential equations are given.