find u e B(u) + [a, b] such that (A(u), v -u) >_ (f, v -u), V v โข B(u) + In, b]. (1.1) If a = (0,..., 0) T, b = (+c~,..., +c~) T, (1.1) reduces into quasi-complementarity problems. If a, b are finite vectors, B(x) = O, Vx โข R n, (1.1) turns into usual two-sided obstacle problem.
Quasi-variational inequalities arise in many scientific, engineering, or economic problems, see and the references therein. The implicit two-sided obstacle problem is one of the basic quasi-variational inequalities. The Schwarz algorithm is one of the most important methods for solving discrete problems of partial differential equations, see and the references therein. Some authors have studied the Schwarz algorithm for variational inequalities, see for examples. The main work of this paper is to extend the Schwarz algorithm to solve an important quasi-variational inequality--an implicit two-sided obstacle problem. We have constructed some algorithms of this kind and proved the monotonic convergence of the algorithms.
In Section 2, we give equivalent systems for (1.1). In Section 3, we define lower (upper) solutions of (1.1). Their properties are presented. In Sections 4 and 5, algorithms for the problem are proposed. The monotone convergence of these algorithms has been proved. In the last section, some numerical tests are given. 2. EQUIVALENT FORM Let P be a projection of R n --* In, b], i.e., P(x) = argmin{]lx-vH2: v โข [a,b]}.
The authors are grateful to the refere~ for their many helpful suggestions.