On the solution of generalized equations and variational inequalities

Abstract

Uko and Argyros provided in [18] a Kantorovich-type theorem on the existence and uniqueness of the solution of a generalized equation of the form f (u)+g(u) ∋ 0, where f is a Fréchet-differentiable function, and g is a maximal monotone operator defined on a Hilbert space. The sufficient convergence conditions are weaker than the corresponding ones given in the literature for the Kantorovich theorem on a Hilbert space. However, the convergence was shown to be only linear.In this study, we show under the same conditions, the quadratic instead of the linear convergenve of the generalized Newton iteration involved. RESUMENUko y Argyros estudian en [18] un teorema tipo-Kantorovich en el existencia y unicidad de la solución de una ecuación generalizada de la forma f (u) + g(u) ∋ 0, donde f es una función Fréchet-diferenciable, y g es un operador monotono máximo definido en un espacio de Hilbert. Las condiciones de convergencia suficientes son más débiles que los correspondientemente dadas en la literatura para el teorema de Kantorovich en un espacio de Hilbert. Sin embargo, la convergencia ha demostrado ser sólo lineal.En este estudio, mostramos en las mismas condiciones, la ecuación cuadrática en lugar de la lineal convergente de la iteración generalizada de Newton involucradas.

Convergence conditions for secant-type methods

“…P r o o f. The proof as being identical to that of Theorem 1 in [6] is omitted. Note that in [6], we simply used sufficient convergence conditions different from the ones in Lemma 3.1.…”

“…Bosarge and Falb [7], Dennis [9], Potra [16], Argyros [1]- [5], Hernández et al [10] and others [11], [15], [18] have provided sufficient convergence conditions for the Secant method based on Lipschitz-type conditions on δF (see also relevant results in [6]- [9], [12], [14], [16], [17], [19]). The conditions usually associated with the semilocal convergence of the Secant method (1.2) are:…”

Secant-like Method for Solving Generalized Equations

“…Using Lipschitz and Hölder conditions on the first order divided differences operators, some convergence results of an uniparametric Secant-type method for solving (1.1) are developed in [5]. Using some ideas introduced by us in [3] for nonlinear equations, a Newton-like method is used in [4] for solving perturbed generalized equation under some condition on the second order divided difference operator.…”

“…Using some ideas introduced by us in [3] for nonlinear equations, a Newton-like method is used in [4] for solving perturbed generalized equation under some condition on the second order divided difference operator. A family of Steffensen-type methods is presented in [5], [6], [10] for solving (1.1) under ω-conditioned divided differences operator, where ω is a continuous nondecreasing function.…”

“…Our approach has the following advantages: we extend the applicability of this method than all the previous know ones [2]- [5], [7], and we do not need to evaluate any Fréchet derivative.…”