Steffensen Methods for Solving Generalized Equations

2000 Mathematics Subject Classification: 65G99, 65K10, 47H04.We provide a local convergence analysis for Steffensen's method in order to solve a generalized equation in a Banach space setting. Using well known fixed point theorems for set-valued maps [13] and Hölder type conditions introduced by us in [2] for nonlinear equations, we obtain the superlinear local convergence of Steffensen's method. Our results compare favorably with related ones obtained in [11]

Plastic Deformation Instabilities: Lambert Solutions of Mecking-Lücke Equation with Delay

The aim of this paper is the study of instabilities during plastic deformation at constant cross‐head velocity. The deformation is supposed to be controlled by the emission of dislocation loops. Under some hypothesis analogous to the Mecking‐Lücke relation, we derive a linear delay differential‐difference equation. The “retarded” time term appears as the phase shift between the time of loop nucleation and the time at which the mean strain is recorded. We show the existence of the solution of strain equation. We give an analytic approach of solution using Lambert functions. The stability is also investigated close to the stable solution using a linearization of the number of nucleated loops functions

On Newton's method using recurrent functions under hypotheses up to the second Fréchet derivative

We provide semilocal result for the convergence of Newton method to a locally unique solution of an equation in a Banach space setting using hypotheses up to the second Fréchet-derivatives and our new idea of recurrent functions. The advantages of such conditions over earlier ones in some cases are: finer bounds on the distances involved, and a better information on the location of the solution

A unifying theorem for Newton’s method on spaces with a convergence structure

AbstractWe present a semilocal convergence theorem for Newton’s method (NM) on spaces with a convergence structure. Using our new idea of recurrent functions, we provide a tighter analysis, with weaker hypotheses than before and with the same computational cost as for Argyros (1996, 1997, 1997, 2007) [1–3,5], Meyer (1984, 1987, 1992) [13–15]. Numerical examples are provided for solving equations in cases not covered before

Semilocal convergence conditions for the secant method, using recurrent functions

Using our new concept of recurrent functions, we present new sufficient convergence conditions for the secant method to a locally unique solution of a nonlinear equation in a Banach space. We combine Lipschitz and center-Lipschitz conditions on the divided difference operator to obtain the semilocal convergence analysis of the secant method. Our error bounds are tighter than earlier ones. Moreover, under our convergence hypotheses, we can expand the applicability of the secant method in cases not covered before [8], [9], [12]-[14], [16], [19]-[21]. Application and examples are also provided in this study

Enlarging the convergence ball of Newton's method on Lie groups

We present a local convergence analysis of Newton's method for approximating a zero of a mapping from a Lie group into its Lie algebra. Using more precise estimates than before [55, 56] and under the same computational cost, we obtain a larger convergence ball and more precise error bounds on the distances involved. Some examples are presented to further validate the theoretical results

Analyse de quelques équations différentielles à retard et EDP modélisant les instabilités de surfaces

Cette thèse est divisée en deux parties principales : La première partie concerne la déformation plastique d'un matériau contraint. Nous commençons cette partie par une introduction physique sur la dislocation et son rôle dans l'étude de la déformation plastique. Nous exposons ensuite deux types de modélisation de la déformation plastique ce qui nous conduit à deux équations différentielles à retard de Mecking-Lüke-Grilhé. Nous présentons une analyse mathématique complète des deux modèles linéaire et non linéaire. Nous terminons cette partie par des tests numériques et une comparaison des deux modèles. La deuxième partie de la thèse traite l'instabilité de Rayleigh-Plateau. Cette étude porte sur les instabilités de surface d'un pore cylindrique sans contraintes. Nous nous intéressons à une EDP parabolique non linéaire d'ordre quatre, obtenue à partir d'une équation d'évolution des films minces. Le résultat principal est l'existence globale de la solution et la convergence vers la valeur moyenne de la donnée initiale en temps long. L'étude théorique est aussi appuyée comme dans la première partie par une validation numérique.This thesis is divided into two main parts: The first part relates to the plastic deformation of a constrained material. We begin this part by physical introduction on the dislocation and its role in the study of plastic deformation. We also present two types modelling for the plastic deformation, which leads to two delayed differential equations of Mecking-Lücke-Grilhé. We present a complete mathematical analysis of linear and nonlinear models. We conclude this part by numerical tests and a comparison of the two models. The second part of the thesis treats the Rayleigh-Plateau instability. This study focuses on the surface instabilities of a cylindrical pore without constraints. We are interested in a nonlinear parabolic PDE of fourth order, obtained from an evolution equation model of thin films. The main result is the global existence of the solution and the convergence to the average value of the initial data in long time. Numerical validation of the theoretical results is also presented in this part.POITIERS-SCD-Bib. électronique (861949901) / SudocSudocFranceF

On the convergence of Steffensen-type methods using recurrent functions nonexpansive mappings

We introduce the new idea of recurrent functions to provide a new semilocal convergence analysis for Steffensen-type methods (STM) in a Banach space setting. It turns out that our sufficient convergence conditions are weaker, and the error bounds are tighter than in earlier studies in many interesting cases[1]-[5], [12], [14]-[17], [23], [24], [26]. Applications and numerical examples, involving a nonlinear integral equation of Chandrasekhar-type, and a differential equation are also provided in this study