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 \(\mathcal {A}(x) = -\textsf{Id}\), 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.