Positivity 3: 201–213, 1999. © 1999 Kluwer Academic Publishers. Printed in the Netherlands. 201 Random Completely Generalized Set-Valued Implicit Quasi-Variational Inequalities NAN JING HUANG and YEOL JE CHO 1 Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, People’s Republic of China; 2 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 1. Introduction 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 [3]. 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 [5], Chang and Huang [7, 8], Chang and Zhu [9], Huang [12, 13], Huang and Cho [14], Huang, Long and Cho [15], Noor and Elsanousi [20], Tan [22] and Yuan [23]. 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- 82.tex; 22/07/1999; 15:41; p.1 ICPC: PIPS No.: 189314 (postkap:mathfam) v.1.15