## Browder-Hartman-Stampacchia variational inequalities for multi-valued monotone operators.(English)Zbl 0671.47043

Let E be a reflexive Banach space, X a non-empty closed convex subset of E and $$T:X\to 2^{E'}\setminus \{\emptyset \}^ a$$multi-valued monotone map with T(x) weakly compact in $$E'$$. Suppose that T is upper semi-continuous from line segments in X to the weak topology of $$E'$$. The author proves that:
1) if there is $$x_ 0\in X$$ such that $$\lim_{\| y\| \to \infty} \inf_{w\in T(y)} Re<w,y-x_ 0>>0,$$ then there exists $$\hat y\in X$$ such that $$\sup_{x\in X} \inf_{w\in T(\hat y)} Re<w,\hat y-x>\leq 0.$$
If, in addition, T(ŷ) is convex, then there is $$\hat w\in T(\hat y)$$ with $$Re<\hat w,\hat y-x>\leq 0$$ for all $$x\in X$$. 2) If all T(x) are convex and there is $$x_ 0\in X$$ such that $$\lim_{\| y\| \to \infty} \inf_{w\in T(y)} Re<w,y-x_ 0>/\| y\| =\infty,$$ then for each $$w_ 0\in E'$$ there exist $$\hat y\in X$$ and $$\hat w\in T(\hat y)$$ such that $$Re<\hat w-w_ 0,\hat y-x>\leq 0$$ for all $$x\in X.$$
In the case when T is single-valued these results reduce to Browder- Hartman-Stampacchia’s theorem. Surjectivity for multi-valued monotone operators and properties of solution sets are also discussed.
Reviewer: R.Precup

### MSC:

 47H05 Monotone operators and generalizations 49J40 Variational inequalities
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