Positivity 3: 201–213, 1999.
© 1999 Kluwer Academic Publishers. Printed in the Netherlands.
Random Completely Generalized Set-Valued
Implicit Quasi-Variational Inequalities
NAN JING HUANG and YEOL JE CHO
Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, People’s Republic of
Department of Mathematics, Gyeongsang National University, Chinju 660-701, Korea
(Received: 17 April 1998; accepted 2 September 1998)
Abstract. The purpose of this paper is to introduce a class of new random completely generalized
set-valued implicit quasi-variational inequalities, to construct new random iterative algorithms, and
to give some existence theorems of random solutions for this class of random completely generalized
set-valued implicit quasi-variational inequalities. We also prove the convergence of random iterative
sequences generated by the algorithms. Our results extend and improve the earlier and recent results.
Mathematics Subject Classiﬁcation (1991): 49A29, 60H25
Key words: Random completely generalized set-valued implicit quasi-variational inequality, random
representation theorem, random algorithm
Variational inequality theory has become a rich source of inspiration in pure and
applied mathematics. Variational inequalities not only have stimulated new re-
sults dealing with nonlinear partial differential equations, but also have been used
in a large variety of problems arising in mechanics, physics, optimization and
control, nonlinear programming, economics and transportation equilibrium and
engineering sciences, etc.
Quasi-variational inequalities are the extended form of variational inequalities
in which the constrained set depends upon the solutions. These were introduced
and studied by Bensoussan, Goursat and Lions . For more details, we may refer
to [1, 2, 4, 5, 11, 16–18], and the references therein.
Recently, the random variational inequality and random quasi-variational in-
equality problems have been introduced and studied by Chang , Chang and
Huang [7, 8], Chang and Zhu , Huang [12, 13], Huang and Cho , Huang,
Long and Cho , Noor and Elsanousi , Tan  and Yuan .
In this paper, we introduce and study a class of new random completely general-
ized set-valued implicit quasi-variational inequalities and construct some new ran-
dom iterative algorithms. We prove the existence of random solutions for this class
of random completely generalized set-valued implicit quasi-variational inequalities
without compactness and the convergence of the random iterative sequences gen-
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