Partial linearization for noninvertible mappings
The authors prove a Hartman-Grobman result for noninvertible mappings in a rather general situation: Let \({\mathcal U}\), \(\mathcal V\), \(\mathcal W\) be Banach spaces and \(F: {\mathcal U}\to {\mathcal U}\), \(G: {\mathcal V}\to {\mathcal V}\), \(H: {\mathcal W}\to {\mathcal W}\) bounded linear operators such that there is an \(a\in (0,1)\) with \(\sigma(F)\subset \{\lambda\in \mathbb{C}\mid|\lambda |< a\}\), \(\sigma(G)\subset \{\lambda\in \mathbb{C}\mid a< | \lambda|< 1\}\), and \(\sigma(H)\subset \{\lambda\in \mathbb{C}\mid |\lambda|> 1\}\). The Banach spaces are assumed to possess a smooth norm in the sense that there exists a real-valued \(C^ 1\)- function \(\lambda\) with bounded and uniformly Lipschitzian derivative which vanishes outside the unit ball and equals 1 on some ball around zero. They consider nonlinear operators on a neighbourhood \(\mathcal N\) of zero in \({\mathcal E}:= {\mathcal U}\times {\mathcal V}\times {\mathcal W}\), \(U: {\mathcal N}\to {\mathcal U}\), \(V: {\mathcal N}\to {\mathcal V}\), \(W: {\mathcal N}\to {\mathcal W}\) of the form \(U(u,v,w)= Fu+ f(u,v,w)\), \(V(u,v,w)= Gv+ g(u,v,w)\), \(W(u,v,w)= Hw+ h(u,v,w)\) where \(f\), \(g\), \(h\) are assumed to be partially differentiable with respect to \(u\), \(v\), \(w\) with partial derivatives which are bounded and uniformly Lipschitzian in \(u\), \(v\), \(w\). Moreover, \(f\), \(g\), \(h\) are assumed to be flat at zero (i.e., \(f\), \(g\), \(h\) and their first partial derivatives with respect to \(u\), \(v\), \(w\) take the value zero at zero). It is conjectured that under these conditions there is an open neighbourhood \({\mathcal N}_ 0\) of zero in \(\mathcal E\) and a continuous function \(f_ 0: {\mathcal N}_ 0\to {\mathcal U}\) with \(f_ 0(0,0,0)= 0\) such that \((U,V,W)\) is locally conjugate to \((F+ f_ 0,G,H)\) (i.e., there is a homeomorphism of neighbourhoods of zero in \(\mathcal E\) which carries \((U,V,W)\) to \((F+ f_ 0,G,H)\)). (For example, the classical Hartman-Grobman lemma states that the conjecture is true if \(F\) is invertible.) Using rather elaborate computations the authors prove that the conjecture is true if one assumes in addition that there are finitely many numbers \(a_ 1,\dots, a_ n\) in \((a,1)\) such that \(|\lambda|= a_ k\) for some \(k\in \{1,\dots, n\}\) whenever \(\lambda\in \sigma(G)\).
scientific article; zbMATH DE number 690937
scientific article