Stability of monotone variational inequalities with various applications.

*(English)*Zbl 0848.49006Giannessi, F. (ed.) et al., Variational inequalities and network equilibrium problems. Proceedings of a conference, Erice, Italy, June 19-25, 1994. New York, NY: Plenum, 123-142 (1995).

Let $K$ be a closed convex set in a real reflexive Banach space $V$ with dual space ${V}^{*}$. For given $f\in V$, find $u\in K$ such that

$$\varphi (u,\nu )\ge \langle f,\nu -u\rangle ,\phantom{\rule{2.em}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{all}\phantom{\rule{4.pt}{0ex}}\nu \in K,\phantom{\rule{2.em}{0ex}}\left(1\right)$$

where $\varphi :K\times K\to \mathbb{R}$ satisfies the appropriate convexity and monotonicity assumptions. This problem was introduced by *E. Blum* and *W. Oettli* [Math. Stud. 63, No. 1-4, 123-145 (1994)]. In this paper, the convergence and stability theory for variational inequality (1) is studied using the technique of Mosco and the montonicity of $\varphi $. Applications to distributed market equilibria with bound and obstacle $p$-harmonic boundary value problems are given.

Reviewer: M.A.Noor (Riyadh)