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Let an operator with f′ Holder 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.

Nondifferentiable operator equations in spaces equipped with aK-norm — map taking values in the cone of an ordered Banach space — are studied. The majorant method is used to derive the convergence of a modified sequence of Newton-Kantorovich approximations.

The paper is concerned with the application of Kantorovich (quasi-) majorants to the study of the convergence of Newton-Kantorovich approximations of solutions of nonlinear operator equations in Banach spaces. The derivative of the nonlinear operator is supposed to satisfy only a rather weak continuity condition. By means of an auxiliary scalar equation, we obtain convergence results for the Newton- Kantorovich approximations, as well as various new error estimates. The abstract results are illustrated by applications to nonlinear integral equations of Uryson type in the spaces C and .

Some elementary results about Hammerstein integral equations with non-symmetric kernels are proved by means of Minty’s monotonicity principle.

Soit Ω un domaine borné de RN, et soit f:Ω × R → R une fonction satisfaisant à la condition de Carathéodory (i.e., f(s, ·) est continue pour presque tout s ∊ Ω, et f (·, u) est mesurable pour tout u ∊ R). Considérons l'opérateur de la superposition(1.1)(encore appelé opérateur de Nemyckii), engendré par la fonction f. Cet opérateur joue un grand rôle dans la théorie des équations intégrales, différentielles (ordinaires et aux dérivées partielles), et fonctionnelles-différentielles, où il est important de connaître les propriétés analytiques et topologiques de F dans certains espaces de fonctions mesurables, intégrables, continues, différentiables, analytiques etc., les propriétés les plus importantes étant : théorèmes de transfert, de continuité, de bornage, et de compacité. Par exemple, on connaît de nombreux résultats sur l'opérateur (1) dans les espaces de Lebesgue L (voir [10] pour une présentation assez complète); en effet, si l'opérateur (1) envoie une partie de L , d'intérieur non vide, dans L, alors, il est automatiquement continu et borné sur chaque boule.

Let f:B(x0,R)⊆X→Y be an operator, with X and Y Banach spaces, and f′ be Holder continuous with exponent θ. The convergence of the sequence of Newton–Kantorovich approximations xn=xn−1−f′(xn−1)−1f(xn−1),n∈N, is a classical tool to solve the equation f(x)=0. The convergence of xn is often reduced to the study of the majorizing sequence rn defined by r0=0,r1=a,rn+1=rn+bk(rn−rn−1)1+θ(1+θ)(1−bkrnθ),n∈N, with a,b,k parameters related to f and f′. In the paper [F. Cianciaruso, E. De Pascale, Estimates of majorizing sequences in the Newton–Kantorovich method, submitted for publication] we proved that, if ξ:=aθbk⩽1(1+θθ1−θ)1−θ(θ1+θ)θ, then the following estimates for rn hold rn⩽(bk)−1θ(1+θθ1−θ)1−θθ(1−1(1+θ)n),∀n∈N. In the present paper we give a stronger (at least asymptotically) estimates on rn under a weaker condition on ξ. The techniques employed in the paper are similar to the ones used in [F. Cianciaruso, E. De Pascale, Estimates of majorizing sequences in the Newton–Kantorovich method, submitted for publication]. Finally, we make a comparison with previous results.