Positivity 10 (2006), 693–700
2006 Birkh¨auser Verlag Basel/Switzerland
1385-1292/040693-8, published online July 11, 2006
A Semilocal Convergence of a Secant–Type
Method for Solving Generalized Equations
Said Hilout and Alain Pi´etrus
Abstract. In this paper we present a study of the existence and the conver-
gence 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 equa-
tions (see ).
Mathematics Subject Classiﬁcations (2000). 47H04; 65K10.
Keywords. Set–valued mapping, generalized equation, super–linear conver-
gence, Aubin continuity, divided diﬀerence.
This paper is concerned with the problem of approximating a solution of the
“abstract” generalized equation
0 ∈ f(x)+G(x) (1)
where f is a continuous function from X into Y and G is a set–valued map from
X to the subsets of Y with closed graph and X, Y are two Banach spaces. Let
us recall that equation (1) is an abstract model for various problems, the reader
could be referred to [5, 6]. For solving (1), we consider the sequence
are given starting points
; α is ﬁxed in [0, 1[
0 ∈ f(x
; f] is a ﬁrst order divided diﬀerence of f on the points y
This operator will be deﬁned in section 2.
In , the authors consider a similar iterative method like (2) with α =0to
solve nonlinear equations (G ≡ 0), they prove a semilocal convergence result for
this method assuming existence of divided diﬀerences for f . Analogous results can