Positivity 8: 75–84, 2004.
© 2004 Kluwer Academic Publishers. Printed in the Netherlands.
Convergence of Iterative Schemes for Multivalued
and MUHAMMAD ASLAM NOOR
Univeristé Antilles Guyane, GRIMAAG, Département Scientiﬁque Interfacultaire, F-97200
Schoelcher, Martinique, France. E-mail: abdellatif.moudaﬁ@martinique.univ-ag.fr;
Mathematics, Etisalat College of Engineering, P.O. Box 980, Sharjah, United Arab Emirates.
(Received 25 February 2002; accepted 16 January 2003)
Abstract. Relying on the resolvent operator method and using Nadler’s theorem, we suggest and
analyze a class of iterative schemes for solving multivalued quasi-variational inclusions. In fact,
by considering problems involving composition of mutivalued operators and by replacing the usual
compactness condition by a weaker one, our result can be considered as an improvement and a
signiﬁcant extension of previously known results in this ﬁeld.
AMS Subject Classiﬁcation 2000: 49J40, 90C33
Key words: variational inclusions, convergence, monotone operators, pseudo-Lipschitz conditions
Quasi-variational inclusions are being used as mathematical programming models
to study a large number of equilibrium problems arising in ﬁnance, economics,
transportation, optimization , operations research and engineering sciences, see,
for example [3,6,7] and the references therein. They have been extended and gen-
eralized in different directions by using novel and innovative techniques and ideas,
both for their own sake and for their applications. In recent years, much attention
has been given to develop efﬁcient and implementable numerical methods includ-
ing the projection method and its variant forms, Wiener-Hopf (normal) equations,
linear approximation, auxiliary principle, proximal-point algorithm and descent
framework for solving variational inequalities and related optimization problems.
It is well known that the projection methods and its variant forms; and Wiener-
Hopf equation techniques cannot be used to suggest and analyze iterative methods
for solving quasi-variational inequalities due to the presence of the nonlinear term.
This fact motivated to develop another technique, which involves the use of the
resolvent operator associated with maximal monotone operator. Using this tech-
nique, one shows that the variational inclusions are equivalent to a ﬁxed point
problem. This alternative formulation was used to develop numerical methods for