A remark on the existence of solutions to nonlinear equations with degenerate mappings (Q2479883)

This paper is concerned with the nonlinear equation \(F(x)=0\), where \(F:X\to Y\) is a (sufficiently many times) Fréchet differentiable map between Banach spaces. The main emphasis is on the case when \(F\) is singular but \(p\)-regular. In this situation, the main result gives a sufficient condition for the existence of a solution. Two illustrative examples are provided: the first is about a particular system of algebraic equations in \({\mathbb R}^2\), and the second concerns the existence of solutions of a Dirichlet boundary value problem for a particular ODE in \({\mathbb R}\).