A new convergence theorem for the Jarratt method in Banach space

In this study, we approximate a locally unique solution of a nonlinear equation in Banach space using the Jarratt method. Sufficient convergence conditions for this method have already been given by several authors, when the equation is defined on the real line, or complex plane [1–3], or in Banach space [1,4–7]. If a certain Newton-Kantorovich type hypothesis is satisfied, then the Jarratt method converges to a solution of the equation with order four. The verification of some of the earlier hypotheses is too difficult or too expensive. Here, using Lipschitz conditions on the second Fréchet-derivative of the operator involved, we provide a convergence theorem for the Jarratt method which uses conditions that are very easy to check (see the Example and Remark 4). Finally, a numerical example is provided to show that our results apply to solve a nonlinear equation, where others fail.