Abstract
A Newton–Kantorovich convergence theorem of a modified Newton’s method having third order convergence is established under the gamma-condition in a Banach space to solve nonlinear equations. It is assumed that the nonlinear operator is twice Fréchet differentiable and satisfies the gamma-condition. We also present the error estimate to demonstrate the efficiency of our approach. A comparison of our numerical results with those obtained by other Newton–Kantorovich convergence theorems shows high accuracy of our results.