On the comparison of a weak variant of the Newton–Kantorovich and Miranda theorems

We recently showed a semilocal convergence theorem that guarantees convergence of Newtonʹs method to a locally unique solution of a nonlinear equation under hypotheses weaker than those of the Newton–Kantorovich theorem. Here we first weaken Mirandaʹs theorem, which is a generalization of the intermediate value theorem. Then we show that operators satisfying the weakened Newton–Kantorovich conditions satisfy those of the weakened Mirandaʹs theorem.