Secant-like Method for Solving Generalized Equations

Abstract

Abstract. In [2], [3], Argyros introduced a new derivative-free quadratically convergent method for solving a nonlinear equation in Banach space. In this paper, we extend this method to generalized equations in order to approximate a locally unique solution. The method uses only divided differences operators of order one. Under some Lipschitz-type conditions on the first and second order divided differences operators and Lipschitz-like property of set-valued maps, an existence-convergence theorem and a radius of convergence are obtained. Our method has the following advantages: we extend the applicability of this method than all the previous ones [2]-[5], [7], and we do not need to evaluate any Fréchet derivative. We provide also an improvement on the radius of convergence for our algorithm, under some center-condition and less computational cost. Numerical examples are also provided.

Improved Convergence of Steffensen's Method for Approximating Fixed Points of Operators in Banach Space

“…We present a new local as well as a semilocal convergence analysis for Steffensen's method in order to locate fixed points of operators on a Banach space setting. Using more precise majorizing sequences we show under the same or less computational cost that our convergence criteria can be weaker than in earlier studies such as [1][2][3][4][5][6][7][8][9][10][11][12][13], [21,22]. Numerical examples are provided to illustrate the theoretical results.…”

An improved local convergence analysis for Newton–Steffensen-type method

“…In [11] we provide a superlinear local convergence for a Newton-Steffensen type method using weaker hypotheses than (7) and (8) (see [6][7][8]). A linear convergence of method (2) to x * is showed in [18] using conditions (5) and (6).…”

“…We reintroduce [11,18] the iterative method 0 ∈ F (x k ) + H (x k ) + ∇F (x k ) + [g 1 (x k ), g 2 (x k ); H ] (x k+1 − x k ) + G(x k+1 ), (2) where g i : D −→ X (i = 1, 2) is a continuous mapping and [x, y; H ] ∈ L(X) is a divided difference of order one satisfying …”