On the convergence of generalized Newton methods and implicit functions (Q1195742)

The paper is concerned with Newton-like iterations \[ x_{n+1}(\lambda)= x_ n(\lambda)- A(x_ n(\lambda),\lambda)^{-1} F(x_ n(\lambda),\lambda),\quad n\geq 0 \] to solve an equation \(F(x,\lambda)=0\), where \(F\) acts between Banach spaces. Also, the parameter \(\lambda\) is Banach space valued. The main point is that \(A(x,\lambda)\) (an approximation to the derivative of \(F\), of course) is replaced step by step by a linear operator which is inductively constructed from \(A(x,\lambda)\). The paper consists of a theorem and its proof on the local convergence of the method. The author uses the technique of majorizing sequences. It is basicallly an inductive argument which, step by step, justifies the inductive definitions. No worked example is given.