On the convergence of Newton's method under $\omega ^\star $-conditioned second derivative
We provide a new semilocal result for the quadratic convergence of Newton's method under $\omega ^\star $-conditioned second Fréchet derivative on a Banach space. This way we can handle equations where the usual Lipschitz-type conditions are not verifiable. An application involving nonlinear integral equations and two boundary value problems is provided. It turns out that a similar result using $\omega $-conditioned hypotheses can provide usable error estimates indicating only linear convergence for Newton's method.