Global inversion theorems via coercive functionals on metric spaces
Let \( X \) and \( Y \) be metric spaces. The author gives sufficient conditions for a local homeomorphism \( f :X\rightarrow Y \) to be a global one. Define the lower scalar derivative of \( f \) at \( u\in X \) by \( D^-_uf=\liminf_{v\to u}{d(f(u),f(v))\over d(u,v)} \). The main result then can be stated as follows. Assume that \( X \) is complete, that \( Y \) is contractible at a point \( y_0 \in f(X) \) by paths of finite length, and that for all bounded subset \( U \) of \( X \), one has \( \sup_{u \in U}(D^-_uf)^{-1}<\infty \). If there exists a coercive continuous function \( k: X\rightarrow {\mathbb R} \) and a constant \( \alpha>0 \) such that the condition \( \limsup_{v\to w}{| k(v)-k(w)|\over d(f(v),f(w))}\leq \alpha \) holds for any \( w\in X \), then \( f \) is a global homeomorphism. Here, a function \( k : X\rightarrow {\mathbb R} \) is said to be coercive if there is \( x_0\in X \) such that \(| k(x)| \to \infty\) when \( d(x_0,x)\to\infty \). Choosing various types of spaces and coercive functions, the author obtains several corollaries, recovering a result of \textit{G. Zampieri} [Nonlinear Anal., Theory Methods Appl. 19, No.10, 923--932 (1992; Zbl 0807.46044)], among others.
scientific article; zbMATH DE number 5144899
scientific article