A multipoint Jarratt-Newton type approximation algorithm for solving nonlinear operator equations in Banach spaces (Q1902334)

This paper discusses properties of the iterative method \[ \begin{aligned} y_k & = x_k- F'(x_k)^{- 1} F(x_k),\\ H(x_k, y_k) & = F'(x_k)^{- 1}[F'(x_k+ \textstyle{{2\over 3}}(y_k- x_k))- F'(x_k)],\\ x_{k+ 1} & = y_k- \textstyle{{3\over 4}} H(x_k, y_k)[I- \textstyle{{3\over 2}} H(x_k, y_k)] (y_k- x_k)\end{aligned} \] for approximations of nonsingular solutions of the operator equation \(F(x)= 0\) in Banach spaces. It is shown that under appropriate conditions \(\{x_k, k= 0,1,\dots\}\) converges with the order of four to a nonsingular solution.