Residual growth control for general maps and an approximate inverse function result

The need to control the residual of a potentially nonlinear function $\mathcal{F}$ arises in several situations in mathematics. For example, computing the zeros of a given map, or the reduction of some cost function during an optimization process are such situations. In this note, we discuss the existence of a curve $t\mapsto x(t)$ in the domain of the nonlinear map $\mathcal{F}$ leading from some initial value $x_0$ to a value $u$ such that we are able to control the residual $\mathcal{F}(x(t))$ based on the value $\mathcal{F}(x_0)$. More precisely, we slightly extend an existing result from J.W. Neuberger by proving the existence of such a curve, assuming that the directional derivative of $\mathcal{F}$ can be represented by $x \mapsto \mathcal{A}(x)\mathcal{F}(x_0)$, where $\mathcal{A}$ is a suitable defined operator. The presented approach covers, in case of $\mathcal{A}(x) = -\mathsf{Id}$, some well known results from the theory of so-called continuous Newton methods. Moreover, based on the presented results, we discover an approximate inverse function result.