Iteration and functional expansions

In a recent paper on nonlocal expansions necessary and sufficient conditions are given under whichf−1 has a generalized power series expansion, whenf is an invertible locally Lipschitz map between certain general subsets of a complex Banach space. Here we establish the validity of a conceptually interesting algorithm for obtaining the expansion.Basically, we show that a certain contraction mapping iteration generates iteratesℐ1,ℐ2,... such that eachℐk yields all of the terms of the generalized power series expansion off−1 up to order (k + 1), assuming merely that the expansion off−1 exists. An earlier different result along related lines is mentioned.