Ioannis K. Argyros

Quadratic equations and applications to Chandrasekhar's and related equations

A new technique, using the contraction mapping theorem, for solving quadratic equations in Banach space is introduced. The results are then applied to solve Chandrasekhars integral equation and related equations without the usual positivity assumptions.

Weaker conditions for the convergence of Newton's method

Newtons method is often used for solving nonlinear equations. In this paper, we show that Newtons method converges under weaker convergence criteria than those given in earlier studies, such as Argyros (2004) [2, p. 387], Argyros and Hilout (2010)[11, p. 12], Argyros et al. (2011) [12, p. 26], Ortega and Rheinboldt (1970) [26, p. 421], Potra and Ptak (1984) [36, p. 22]. These new results are illustrated by several numerical examples, for which the older convergence criteria do not hold but for which our weaker convergence criteria are satisfied.

A note on the Halley method in Banach spaces

Abstract We provide a convergence theorem under the usual Ostrowski–Kantorovich conditions for an equivalent form of the Halley method in Banach spaces. The best possible upper error bound for the method is obtained. Some applications to the solution of quadratic operator equations are provided.

The theory and applications of iteration methods

The Convergence of Algorithmic Models. The Convergence of Iteration Sequences. Monotone Convergence. Comparison Theorems. The Convergence of Newton Methods and Their Variants. The Monotone Convergence of Newton Methods and Their Variants. References. Index.

The Jarratt method in Banach space setting

Abstract In this note, we extend the Jarratt method of order four into Banach spaces. We also establish a Kantorovich-type convergence theorem (see the References) and give an explicit expression for the error bound of the method (Jarratt, 1966, 1969).

A semilocal convergence analysis for directional Newton methods

A semilocal convergence analysis for directional Newton methods in n-variables is provided in this study. Using weaker hypotheses than in the elegant related work by Y. Levin and A. Ben-Israel and introducing the center-Lipschitz condition we provide under the same computational cost as in Levin and Ben-Israel a semilocal convergence analysis with the following advantages: weaker convergence conditions; larger convergence domain; finer error estimates on the distances involved, and an at least as precise information on the location of the zero of the function. A numerical example where our results apply to solve an equation but not the ones in Levin and Ben-Israel is also provided in this study.

A local convergence theorem for the super-halley method in a Banach space

Abstract A local convergence theorem for the super-Halley method is presented here to solve nonlinear equations in Banach space. The method is of order four for quadratic equations. Most authors (including the famous conjecture by Traub for functions of one variable) have shown that this method is of order three only. Some applications are also provided, where our results apply, but previous related results do not.

On the convergence of an optimal fourth-order family of methods and its dynamics

In this paper, we present the study of the semilocal and local convergence of an optimal fourth-order family of methods. Moreover, the dynamical behavior of this family of iterative methods applied to quadratic polynomials is studied. Some anomalies are found in this family be means of studying the dynamical behavior. Parameter spaces are shown and the study of the stability of all the fixed points is presented.

Computational methods in nonlinear analysis : efficient algorithms, fixed point theory and applications

Kantorovich Theory for Newton-Like Methods Holder Conditions and Newton-Type Methods Regular Smoothness Conditions for Iterative Methods Fixed Point Theory and Iterative Methods Mathematical Programming Fixed Point Theory for Set-Valued Mapping Special Convergence Conditions Recurrent Functions and Newton-Like Methods Recurrent Functions and Special Iterative Methods.

Approximation methods for common solutions of generalized equilibrium, systems of nonlinear variational inequalities and fixed point problems

In this paper, we introduce an iterative method for finding a common element of the set of solutions of the generalized equilibrium problems, the set of solutions for the systems of nonlinear variational inequalities problems and the set of fixed points of nonexpansive mappings in Hilbert spaces. Furthermore, we apply our main result to the set of fixed points of an infinite family of strict pseudo-contraction mappings. The results obtained in this paper are viewed as a refinement and improvement of the previously known results.