On the application of Newton's and chord methods of bifurcation problems (Q1321864)

The paper is concerned with a simple bifurcation from the trivial solution at \(\lambda = \lambda_ 0\) of \(G(x,\lambda)=0\) where \(G:H \times \mathbb{R}\to H\) satisfies \(G(0, \lambda) \equiv 0\) with a Hilbert space \(H\). The aim is to compute the nontrivial solution for a given \(\lambda\) near \(\lambda_ 0\) and to overcome the problems by the singularity of the derivative \(D_ xG (0,\lambda_ 0)\). It is proved that the Newton and the chord method converge for \(\lambda\) sufficiently near \(\lambda_ 0\) to the nontrivial solution provided that the initial guess is chosen from an asymptotic analysis for simple bifurcation points.