From Kantorovich’s theory we present a semilocal convergence result for Newton’s method
which is based mainly on a modification of the condition required to the second derivative
of the operator involved. In particular, instead of requiring that the second derivative
is bounded, we demand that it is centered. As a consequence, we obtain a modification of
the starting points for Newton’s method. We illustrate this study with applications to
nonlinear integral equations of mixed Hammerstein type.