A global representation of all solutions to a nonlinear equation and its applications (Q1206132)

Let \(U\) be a Hilbert space and \(Z\) be a Banach space. The equation \[ h(u)=0\qquad h: U\to Z\tag{1} \] with continuous map \(h\) is considered. For affine \(h\), i.e. \(h(u)= Gu-z\), \(\forall u\in U\), for some \(G\in{\mathcal L}(U,Z)\) and some \(z\in Z\), with invertible \(GG^*\in{\mathcal L}(Z)\) all possible solutions of (1) can be represented in the form \(u=G^*(GG^*)^{-1} z+ (I-G^*(GG^*)^{-1} G)v\) with \(v\in U\) as a parameter [see \textit{J. P. Aubin}, Applied functional analysis (1979; Zbl 0424.46001)]. With the aid of this representation the authors obtain a nonlinear generalized inverse theorem and give applications to control systems with mixture constraints.