A global inversion theorem for functions with singular points
The article deals with some generalization of the classic Hadamard-Levi and Krasnosel'skii theorem about global homeomorphism of a smooth map \(T\) between Banach spaces \(X\) and \(Y\) whose derivatives \(T'(x)\) are invertible for all \(x_0 \in X\). More precisely, the author considers the case when the derivatives \(T'(x)\) are invertible everywhere besides \(x = x_0\), and for some \(n\), \(n \geq 1\), \[ T^{(k)}(x_0) = 0, \quad k = 1,\dots,2n, \] \[ \|f'(x)h\| \geq C_1\|x\|^{2n}\|h\|, \quad C_2\|x\|^{2n+1} \leq \|f(x)\| \leq C_3\|x\|^{2n+1} \quad x \in X,\ h \in X, \] where \(f(x) = ((2n+1)!)^{-1} T^{(2n+1)}\). The main result is the following: Theorem. The map \(T\) is a homeomorphism \(X\) onto \(Y\) provided that \[ \sup_{x \in X \setminus \{0\}} \|r'(x)\| \|x\|^{-2n} < C_1C_2C_3^{-2n/(2n+1)}, \] where \(r(x)\) is a Taylor reminder term of the map \(T\) at the point \(x_0\). At the end of the article, the author provides two applications of this result.
scientific article; zbMATH DE number 6807049
scientific article