A convergence theorem for Newton-like methods under generalized Chen-Yamamoto-type assumptions

摘要

In this study, we provide sufficient conditions for the convergence of a Newton-like method to a locally unique solution of a nonlinear equation with a nondifferentiable term in a Banach space setting. We introduce new conditions by assuming the existence of real functions of two variables that serve as upper bounds on the operators involved. Special choices of the real functions mentioned above lead to Chen-Yamamoto-type assumptions or the usual Lipschitz conditions associated with the Fréchet derivatives of the nonlinear operators involved. Using the majorant method, we show that if a certain scalar inequality has a minimum positive solution, then the abstract equation has a locally unique solution also. We also show that our upper bounds on the distances ‖xn+1 – xn‖ and ‖xn – x∗‖ improve on earlier ones found by Chen-Yamamoto, Dennis, Rheinboldt, and Potra et al. Finally, our results can apply to solve nonlinear integral equations of Uryson type.