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We study the action of the nonlinear mapping G[z] between real or complexBanach spaces in the vicinity of a given curve with respect to possiblelinearization, emerging patterns of level sets, as well as existing solutionsof G[z]=0. The results represent local generalizations of the standard implicitor inverse function theorem and of Newton's Lemma, considering the order ofapproximation needed to obtain solutions of G[z]=0.  The main technical tool is given by Jordan chains with increasing rank, usedto obtain an Ansatz, appropriate for transformation of the nonlinear system toits linear part. The family of linear mappings is restricted to the case of anisolated singularity.  Geometrically, the Jordan chains define a generalized cone around the givencurve, composed of approximate solutions of order 2k with k denoting themaximal rank of Jordan chains needed to ensure k-surjectivity of the linearfamily. Along these lines, the zero set of G[z] in the cone is calculatedimmediately, agreeing up to the order of k-1 with the given approximation.Hence, the results may also be interpreted as a version of Tougeron's implicitfunction theorem or Hensel's Lemma in Banach spaces, essentially restricted tothe arc case of a single variable.  Finally, by considering a left shift of the Jordan chains, the Ansatz can bemodified in a systematic way to obtain a sequence of refined versions oflinearization theorems and Newton Lemmas in Banach spaces.

Linearization and Lemma of Newton for Operator functions