Recurrence relations for a Newton-like method in Banach spaces (Q2370695)

Let \(F: \Omega \subset X \to Y\) be a twice differentiable nonlinear operator on an open convex subset \(\Omega\) of a Banach space \(X\) with values in a Banach space \(Y\). The authors study the iterative method \(y_n=x_n-F^{\prime}(x_n)^{-1} F(x_n)\), \(x_{n+1}=x_n-2(F^{\prime}(x_n)+F^{\prime}(y_n))^{-1} F(x_n)\) for solving the operator equation \(F(x)=0\). It is shown that \(\| x_n-x^*\| =O(\sum_{k=n}^{\infty}\gamma^{2^k})\), \(\gamma \in (0,1)\).