in this paper, two pairs of new inequalities are given, which decompose two hilbert-type inequalities.

We present a local convergence analysis for a family of Steffensen-type fourth-order methods in order to approximate a solution of a nonlinear equation. We use hypotheses up to the first derivative in contrast to earlier studies such as [1], [5]-[28] using hypotheses up to the fifth derivative. This way the applicability of these methods is extended under weaker hypotheses. Moreover the radius ...

It is attempted to present an efficient and free derivative class of Steffensen-like methods for solving nonlinear equations. To this end, firstly, we construct an optimal eighth-order three-step uniparameter without memory of iterative methods. Then the self-accelerator parameter is estimated using Newton's interpolation in such a way that it improves its convergence order from 8 to 12 without...

For iteration sequences otherwise converging linearly, the proposed one-point extrapolation method attains a convergence rate and efficiency of 1.618. This is accomplished by retaining an estimate of the linear coefficient from the previous step and using the estimate to extrapolate. For linear convergence problems, the classical Aitken-Steffensen 6 -process has an efficiency of just -J2, while...

A local and semi-local convergence is developed of a class iterative methods without derivatives for solving nonlinear Banach space valued operator equations under the classical Lipschitz conditions first-order divided differences. Special cases this method are well-known algorithms, in particular, Secant, Kurchatov, Steffensen as well Newton method. For analysis, we use technique recurrent fun...