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

Abstract

We recently showed a semilocal convergence theorem that guarantees convergence of Newton's method to a locally unique solution of a nonlinear equation under hypotheses weaker than those of the Newton–Kantorovich theorem. Here we first weaken Miranda's 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 Miranda's theorem.