Newton–Kantorovich Convergence Theorem of a Modified Newton’s Method Under the Gamma‐Condition in a Banach Space

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.