The Global Injectivity of C² Maps in R²

Let F be a C² C 2 map from the real plane into itself whose Jacobian determinant is nonzero at every point. By the Inverse Function Theorem, such a map is locally injective. A classical and challenging problem is to determine conditions under which F is globally injective. In this work, using results from the qualitative theory of differential equations, we provide both a necessary and sufficient condition and a sufficient condition for the global injectivity of F .