A new convergence theorem for Newton-like methods in Banach space and applications

In this study, we approximate a locally unique solution of a nonlinear operator equation in Banach space using Newton-like methods. A semilocal convergence theorem is given based on Lipschitz conditions on the second Frechet-derivative. Earlier results use conditions on the first Frechet-derivative of the operator involved only. This way our conditions are different from earlier ones. Hence, our results have theoretical and practical value. Numerical examples are also provided to show that our results apply to solve a nonlinear equation, where others fail.