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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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