A continuation procedure based on projected Newton steps
Consider the fixed point equation \(u=GKu\) with a linear operator K on a real Hilbert space into a real Banach space and a nonlinear operator G. Hammerstein integral equations can be transformed into this (a bit unusual) form. The author considers an imbedding T(v,t) for \(t\in [0,1]\) such that \(T(Kz,1)=GKz.\) A sequence of pairs \((z_ n,t_ n)\) is constructed such that the inequality \(\| z_{n+1}-T(Kz_{n+1},t_{n+1})\| \leq \| z_ n-T(Kz_ n,t_ n)\| +(t_{n+1}-t_ n)\alpha\) with a prescribed real constant \(\alpha >0\) holds. The construction is on the basis of projected Newton steps and it is proved that there exists a natural number N such that \(\| z_ N-GKz_ N\| \leq \alpha\) holds. It is shown that the differences \(t_{n+1}-t_ n\) are uniformly bounded below so that the interval [0,1] is eventually covered and the endpoint \(t=1\) is reached. A central assumption lies on the error term of the forward finite difference quotient for T(Kz,t) with respect to (z,t). It allows the author to predict the next \(t_{n+1}\) from \(t_ n\). A worked numerical example concludes the paper.
scientific article; zbMATH DE number 4139211
scientific article