Abstract
Let an operator with f′ Hölder continuous. An existence result on the solutions of the equation f(x) = 0 was obtained by Vertgeim (Vertgeim, B. A. (1956). On conditions for the applicability of Newton's method. Dokl. Akad. Nauk. SSSR 110:719–722 (in Russian); Vertgeim, B. A. (1960). On some methods of the approximate solution of nonlinear functional equations in banach spaces. Uspekhi Mat. Nauk. 12:166–169 (in Russian) [(1960). Engl. Transl.: Amer. Math. Soc. Transl. 16(2):378–382.]) by means of the convergence of the modified Newton–Kantorovich method. There is an unknown land (“terra incognita”), left by earlier results on this topic, between the convergence region for the Newton–Kantorovich method and the convergence region for the modified Newton-Kantorovich method. We survey the attempts in literature to cover this “terra incognita” and we give a further step improving all previous results.
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