Linearization and Lemma of Newton for operator functions

Abstract We study the action of the nonlinear mapping G [ z ] between real or complex Banach spaces in the vicinity of a given curve with respect to possible linearization, emerging patterns of level sets, as well as existing solutions of $$G[z]=0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo>[</mml:mo> <mml:mi>z</mml:mi> <mml:mo>]</mml:mo> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> . The results represent local generalizations of the standard implicit or inverse function theorem and of Newton’s Lemma, considering the order of approximation needed to obtain solutions of $$G[z]=0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo>[</mml:mo> <mml:mi>z</mml:mi> <mml:mo>]</mml:mo> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> . The main technical tool is given by Jordan chains with increasing rank, used to obtain an Ansatz, appropriate for transformation of the nonlinear system to its linear part. The family of linear mappings is restricted to the case of an isolated singularity. Geometrically, the Jordan chains define a generalized cone around the given curve, composed of approximate solutions of order 2 k with k denoting the maximal rank of Jordan chains needed to ensure k -surjectivity of the linear family. Along these lines, the zero set of G [ z ] in the cone is calculated immediately, agreeing up to the order of $$k-1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> with the given approximation. Hence, the results may also be interpreted as a version of Tougeron’s implicit function theorem in Banach spaces, essentially restricted to the arc case of a single variable. Finally, by considering a left shift of the Jordan chains, the Ansatz can be modified in a systematic way to obtain a sequence of refined versions of linearization theorems and Newton Lemmas in Banach spaces.