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

The need to control the residual of a potentially nonlinear function F 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↦ x(t) t ↦ x ( t ) in the domain of the nonlinear map F F leading from some initial value x₀ x 0 to a value u such that we are able to control the residual F(x(t)) F ( x ( t ) ) based on the value F(x₀) 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 F F can be represented by x ↦ A(x)F(x₀) x ↦ A ( x ) F ( x 0 ) , where A A is a suitable defined operator. The presented approach covers, in case of A(x) = -Id A ( x ) = - 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.