On an Application of Newton's Method to Nonlinear Operators with w-Conditioned Second Derivative

We present a new approach to study the convergence of Newton's method in Banach spaces, which relax the conditions appearing in the usual studies. The approach is based on the bound required for the second derivative of the operator involved. An application to a nonlinear integral equation is presented.     

Resolution of quadratic equations in banach spaces

A new convergence theorem is established for the super-Halley method. This method has, in general, order three, but when it is applied to quadratic equations, its order is four.     

Toward a unified theory for third R-order iterative methods for operators with unbounded second derivative

In this paper, we provide a semilocal convergence analysis for a family of Newton-like methods, which contains the best-known third-order iterative methods for solving a nonlinear equation F(x)=0 in Banach spaces. It is assumed that the operator F is twice Fréchet differentiable and F^″ satisfies a Lipschitz type condition but it is unbounded. By using majorant sequences, we provide sufficient convergence conditions to obtain cubic semilocal convergence. Results on existence and uniqueness of solutions, and error estimates are also given. Finally, a numerical example is provided.     

New iterations of <Emphasis Type="Italic">R</Emphasis>-order four with reduced computational cost

From a one-point iterative method of R-order at least three, we construct new two-point iterations to solve nonlinear equations in Banach spaces such that the computational cost is reduced, whereas the R-order of convergence is increased to at least four.      

On the R-order of the Halley method

An R-order bound for the Halley method is obtained in this work, where an analysis of the convergence of the method is also presented under mild differentiability conditions. To do this, a new technique is developed, where the involved operator must satisfy some recurrence relations.     

Third Order Iterative Methods Without Using Second Fréchet Derivative

A modification of classical third order methods is proposed. The main advantage of these methods is they do not need evaluate any second order Frechet derivative. A convergence theorem in Banach spaces is analyzed. Finally, some preliminary numerical results are presented.     

On the efficiency index of one-point iterative processes

In this paper, we analyze the index of efficiency of one-point iterative processes, which are in practice the most used methods to solve a nonlinear equation. We obtain the best situation for one-point iterative processes with cubic convergence: Chebyshev’s method, Halley’s method, the super-Halley method and many others classical iterative methods with order of convergence three. By means of a construction of particular multipoint iterations, we get to improve the best situation obtained for one-point methods. Moreover, these type of multipoint iterations, can be considered as quasi-one-point iterations, since they only depend on one initial approximation. Numerical examples are given and the computed results support this theory. Partly supported by the Ministry of Education and Science (MTM 2005-03091) and the University of La Rioja (ATUR-05/43).     

On Osgood theorem in Banach spaces

Let X be a real Banach space, ω : [0, +∞) → ? be an increasing continuous function such that ω(0) = 0 and ω(t + s) ≤ ω(t) + ω(s) for all t, s ∈ [0, +∞). According to the infinite dimensional analog of the Osgood theorem if ∫¹₀ (ω(t))^?1 dt = ∞, then for any (t₀, x₀) ∈ ?×X and any continuous map f : ?×X → X such that ∥f(t, x) – f(t, y)∥ ≤ ω(∥x – y∥) for all t ∈ ?, x, y ∈ X, the Cauchy problem (t) = f(t, x(t)), x(t₀) = x₀ has a unique solution in a neighborhood of t₀. We prove that if X has a complemented subspace with an unconditional Schauder basis and ∫¹₀ (ω(t))^?1 dt < ∞ then there exists a continuous map f : ? × X → X such that ∥f(t, x) – f(t, y)∥ ≤ ω(∥x – y∥) for all (t, x, y) ∈ ? × X × X and the Cauchy problem (t) = f(t, x(t)), x(t₀) = x₀ has no solutions in any interval of the real line.     

Second-Derivative-Free Variant of the Chebyshev Method for Nonlinear Equations

In this paper, we introduce a numerical method for nonlinear equations, based on the Chebyshev third-order method, in which the second-derivative operator is replaced by a finite difference between first derivatives. We prove a semilocal convergence theorem which guarantees local convergence with R-order three under conditions similar to those of the Newton-Kantorovich theorem, assuming the Lipschitz continuity of the second derivative. In a subsequent theorem, the latter condition is replaced by the weaker assumption of Lipschitz continuity of the first derivative.     

On Newton''s method under Hölder continuous derivative

A Mysovskii-type theorem for Newton's method under (k,p)-Hölder continuous derivative is considered in this paper. For the application studied by others, the new condition is weaker than ones in the literature. Also we prove that the optimal convergent order is p+1 for 0<p<1.