A variant of the Newton-Kantorovich theorem for nonlinear integral equations of mixed Hammerstein type

We study nonlinear integral equations of mixed Hammerstein type using Newton's method as follows. We investigate the theoretical significance of Newton's method to draw conclusions about the existence and uniqueness of solutions of these equations. After that, we approximate the solutions of a particular nonlinear integral equation by Newton's method. For this, we use the majorant principle, which is based on the concept of majorizing sequence given by Kantorovich, and milder convergence conditions than those of Kantorovich. Actually, we prove a semilocal convergence theorem which is applicable to situations where Kantorovich's theorem is not.