J.A. Ezquerro

Secant-like methods for solving nonlinear integral equations of the Hammerstein type

We consider a one-parametric family of secant-type iterations for solving nonlinear equations in Banach spaces. We establish a semilocal convergence result for these iterations by means of a technique based on a new system of recurrence relations. This result is then applied to obtain existence and uniqueness results for nonlinear integral equations of the Hammerstein type. We also present a numerical example where the solution of a particular Hammerstein integral equation is approximated by different secant-type methods.

An optimization of Chebyshev's method

From Chebyshevs method, new third-order multipoint iterations are constructed with their efficiency close to that of Newtons method and the same region of accessibility.

Avoiding the computation of the second Fre´chet-derivative in the convex acceleration of Newton's method

We introduce a new two-step method to approximate a solution of a nonlinear operator equation in a Banach space. An existence-uniqueness theorem and error estimates are provided for this iteration using Newton-Kantorovich-type assumptions and a technique based on a new system of recurrence relations. For a special choice of the parameter involved we use, our method is of fourth order.

Solving a special case of conservative problems by Secant-like methods

We study a class of Secant-like iterations for solving nonlinear equations in Banach spaces. A semilocal convergence result is obtained, where the first order divided difference of the nonlinear operator is Holder continuous. For that, we use a technique based on a new system of recurrence relations to obtain existence-uniqueness domains of the solution and a priori error bounds. These results are applied to solve a special case of conservative problems.

On the semilocal convergence of efficient Chebyshev-Secant-type methods

We introduce a three-step Chebyshev-Secant-type method (CSTM) with high efficiency index for solving nonlinear equations in a Banach space setting. We provide a semilocal convergence analysis for (CSTM) using recurrence relations. Numerical examples validating our theoretical results are also provided in this study.

A discretization scheme for some conservative problems

We approximate a locally unique solution of an equation in Banach spaces using a Newton-like method of R-order three. Then we apply this method to obtain an existence-uniqueness result for a basic conservative problem given by a nonlinear boundary-value problem. Next, by means of a discretization method, we approximate the solution of the conservative problem.

Majorizing sequences for Newton's method from initial value problems

The most restrictive condition used by Kantorovich for proving the semilocal convergence of Newtons method in Banach spaces is relaxed in this paper, providing we can guarantee the semilocal convergence in situations that Kantorovich cannot. To achieve this, we use Kantorovichs technique based on majorizing sequences, but our majorizing sequences are obtained differently, by solving initial value problems.

On Iterative Methods with Accelerated Convergence for Solving Systems of Nonlinear Equations

We present a modified method for solving nonlinear systems of equations with order of convergence higher than other competitive methods. We generalize also the efficiency index used in the one-dimensional case to several variables. Finally, we show some numerical examples, where the theoretical results obtained in this paper are applied.

A modification of the super-Halley method under mild differentiability conditions

A new two-point iteration of order three is introduced to approximate a solution of a nonlinear operator equation in Banach spaces. Under the same assumptions as for Newtons method, we provide a result on the existence of a unique solution for the nonlinear equation, which is based on a technique consisting of a new system of recurrence relations.

The application of an inverse-free Jarratt-type approximation to nonlinear integral equations of Hammerstein-type

Abstract We consider an inverse-free Jarratt-type approximation, whose order of convergence is four, for solving nonlinear equations. The convergence of this method is analysed under two different types of conditions. We use a new technique based on constructing a system of real sequences. Finally, this method is applied to the study of Hammersteins integral equations.