A quadratically convergent iteration method for computing zeros of operators satisfying autonomous differential equations

If the Fréchet derivative P' of the operator P in a Banach space X is Lipschitz continuous, satisfies an autonomous differential equation $P'(x) = f(P(x))$, and $f(0)$ has the bounded inverse $\Gamma$, then the iteration process

$${x_{n + 1}} = {x_n} - \Gamma P({x_n}),\quad n = 0,1,2, \ldots ,$$

is shown to be locally quadratically convergent to solutions $x = {x^\ast}$ of the equation $P(x) = 0$. If f is Lipschitz continuous and $\Gamma$ exists, then the global existence of ${x^\ast}$ is shown to follow if $P(x)$ is uniformly bounded by a sufficiently small constant. The replacement of the uniform boundedness of P by Lipschitz continuity gives a semilocal theorem for the existence of ${x^\ast}$ and the quadratic convergence of the sequence $\{ {x_n}\}$ to ${x^\ast}$.