A Semilocal Convergence of a Secant–Type Method for Solving Generalized Equations

Abstract

In this paper we present a study of the existence and the convergence of a secant-type method for solving abstract generalized equations in Banach spaces. With different assumptions for divided differences, we obtain a procedure that have superlinear convergence. This study follows the recent results of semilocal convergence related to the resolution of nonlinear equations (see [11]).

A Fréchet derivative-free cubically convergent method for set-valued maps

“…Faster convergence to the solution than the corresponding ones in [8,11,12]; 2. Our method does not need the evaluation of any Fréchet derivative; 3.…”

“…In the case G = {0}, super-linear convergence results of an uniparametric Secant-type method for solving (1.1) are developed in [12] using Lipschitz and Hölder conditions on the first order divided differences operators. Using some ideas introduced by us in [6] for nonlinear equations, a Newton-like method is used in [8] for solving perturbed generalized equation under some condition on the second order divided difference operator.…”

“…Using some condition on the first order divided differences introduced for nonlinear equation in [7], and under Aubin's continuity of set-valued map G −1 around (−F(x ), x ), we provide a convergence analysis of method (1.5). Our approach has the following advantages: faster convergence to the solution than all the previous known ones [8,11,12], and our method does not need the evaluation of any Fréchet derivative.…”

Newton's methods for variational inclusions under conditioned Fréchet derivative

“…A similar condition to (1.4) on the first order divided difference is used in [13]- [16] to study the local convergence for the secant and Steffensentype methods. Some convergence analysis of (1.2) is presented in [2] using a condition on the mth (m ≥ 2) Fréchet derivative ∇ (m) f :…”

An Uniparametric Secant–Type Method for Nonsmooth Generalized Equations

“…We provide a local convergence analysis under ω-conditioned divided difference which generalizes the usual Lipschitz continuous and Hölder continuous conditions used in [14]. …”