The problem of finding a large domain in $R^n$, 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 which has an asymptotically stable equilibrium at the point around which we invert the function. On the basin of attraction of this point the function is proved to be one-to-one. A sufficient condition for injectivity on a compact ball is derived. The condition involves the values of the function and its Jacobian matrix on the boundary of the compact ball. This criterion of invertibility is easily generalized to more general domains.
Finding domains of invertibility for smooth functions by means of attraction basins
ZAMPIERI, Gaetano
1993-01-01
Abstract
The problem of finding a large domain in $R^n$, 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 which has an asymptotically stable equilibrium at the point around which we invert the function. On the basin of attraction of this point the function is proved to be one-to-one. A sufficient condition for injectivity on a compact ball is derived. The condition involves the values of the function and its Jacobian matrix on the boundary of the compact ball. This criterion of invertibility is easily generalized to more general domains.
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