Resolving singular nonlinear equations
The authors develop a unified theory for the solution \(x_ 0=(z_ 0,\lambda_ 0)\) of an operator equation \(G(z,\lambda)=0\) where \(\lambda\in R\) q, dim N(G'(x\({}_ 0))=m+q\), and dim N(G'(x\({}_ 0)\) \(*)=m\). This formulation includes multiple bifurcations for multi- parameter problems \((q>0\), \(m>1)\), as well as singular operator equations \((q=0\), \(m>0)\). The problem is unfolded by embedding the operator G into a family of operators \(F=F(x,c)\) such that \(F(x,0)=G(x)\) and that with a suitable linear operator L the augmented map \[ H(x,w_ 1,...,x_{m+q},c)=(F(x,c),F_ x(x,c)w_ i,\quad Lw_ i-a_ i,\quad i=1,...,m+q) \] has an invertible derivative at \((x_ 0,...)\). This map H may be used to define quadratically convergent Newton methods. The finite dimensional case is studied in detail and then some discretizations of H and their computation are considered. Various examples are included and there is some discussion of determining \(m+q\) when, as is typical in applications, m is not known a priori.
scientific article; zbMATH DE number 4064381