Inexact Newton methods are constructed by combining Newton’s method with another iterative method that is used to solve the Newton equations inexactly. In this paper, we establish two semilocal convergence theorems for the inexact Newton methods. When these two theorems are specified to Newton’s method, we obtain a different Newton-Kantorovich theorem about Newton’s method. When the iterative m...

Given a sequence {xn}n=Q in a Banach space, it is well known that if there a sequence {r^J^—QSUch that Wxn+X — xn\\ < tn + x — tn and lim tn = t* < °°, then {xn}n=Q converges to some x* and the error bounds IIjc* — jc II < f* — tn hold. It is shown that certain stronger hypotheses imply sharper error bounds, ■**-*„« <—-—Jxn-Xn-X*11* * \'*1-'0|M" > »• Representative applications to infinite seri...

Inexact Newton methods are constructed by combining Newton’s method with another iterative method that is used to solve the Newton equations inexactly. In this paper, we establish two semilocal convergence theorems for the inexact Newton methods. When these two theorems are specified to Newton’s method, we obtain a different Newton-Kantorovich theorem about Newton’s method. When the iterative m...

This review surveys a significant set of recent ideas developed in the study of nonlinear Galerkin approximation. A significant role is played by the Krasnosel’skii Calculus, which represents a generalization of the classical inf-sup linear saddle point theory. A description of a proper extension of this calculus, and the relation to the inf-sup theory are part of this review. The general study...

Halley’s method is a famous iteration method for solving nonlinear equations F (X) = 0. Some Kantorovich-like theorems have been given, including extensions for general spaces. Quasi-Halley methods were proposed too. This paper uses the generalized inverse approach in order to obtain a robust generalized Halley method.

The R-Function Method (RFM) solution structure is a functional expression that satisfies all given boundary conditions exactly and contains some undetermined functional components. It is complete if there exists a choice of undetermined component that transform the solution structure into an exact solution. Such a structure was used by Kantorovich (Kantorovich and Krylov, 1958) and his students...

By looking at the Γ-convergence of some easy functionals in terms of underlying Young measures, we emphasize the connection of this issue with the Monge-Kantorovich mass transfer problem. After exploring this relationship, we study a typical example in periodic homogenization to realize the difference between the Monge-Kantorovich problem and the usual cell-problem. We also state a version of t...

2 Formulation of the mass transport problems 4 2.1 The original Monge-Kantorovich problem . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Guessing a good dual to the Monge-Kantorovich problem . . . . . . . . . . . . . 6 2.3 Properties of ”Extreme points of C” . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.4 Existence of a minimizer . . . . . . . . . . . . . . . . . . . . . . . . . . . ...