Finding Domains of Invertibility for Smooth Functions by Means of Attraction Basins

The problem of finding a large domain in n, where a locally invertible smooth function is one-to-one, had been approached in different ways, none of them exhaustive. The present paper considers an auxiliary ordinary differential equation, built upon ƒ C1, which has an asymptotically stable equilibrium at the point around which we invert ƒ. On the basin of attraction of this point the function ƒ is proved to be one-to-one. A sufficient condition for injectivity on a compact ball is derived. The condition involves the values of ƒ and its Jacobian matrix Dƒ on the boundary ∂ . This criterion of invertibility is easily generalized to more general domains.