The existence of positive solutions for the singular two-point boundary value problem

Abstract

Since the Hadamard Theorem, several metric and topological conditions have emerged in the literature to date, yielding global inverse theorems for functions in different settings. Relevant examples are the mappings between infinite-dimensional Banach-Finsler manifolds, which are the focus of this work. Emphasis is given to the nonlinear Fredholm operators of nonnegative index between Banach spaces. The results are based on good local behavior of $f$ at every $x$, namely, $f$ is a local homeomorphism or $f$ is locally equivalent to a projection. The general structure includes a condition that ensures a global property for the fibres of $f$, ideally expecting to conclude that $f$ is a global diffeomorphism or equivalent to a global projection. A review of these results and some relationships between different criteria are shown. Also, a global version of the Graves Theorem is obtained for a suitable submersion $f$ with image in a Banach space: given $r>0$ and $x_0$ in the domain of $f$ we give a radius $\varrho(r)> 0$, closely related to the hypothesis of the Hadamard Theorem, such that $B_{\varrho}(f(x_0))\subset f(B_r(x_0))$.