A semilocal convergence theorem for Newton-type method under γ-condition in Banach space

A third-order Newton-type method was recently studied [M. T. Darvishi and A. Barati, Appl. Math. Comput. 187(2007), pp. 630-635] to solve systems of nonlinear equations. In this paper, we aim to study the semilocal convergence property of the method. Assuming the nonlinear operator F is twice differentiable and satisfies γ-condition, we establish a semilocal convergence theorem for the Newton-type method. We also present the error estimate. Furthermore, several examples are given to show the application of our results, with comparison to other semilocal convergence theorems.