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


47H05 Monotone operators and generalizations
49J40 Variational inequalities
Full Text: DOI


[1] Aubin, J.-P, Applied functional analysis, (1979), Wiley-Interscience New York
[2] Baiocchi, C; Capelo, A, Variational and quasivariational inequalities: applications to free-boundary problems, (1984), Wiley-Interscience New York, translated by L. Jayakar · Zbl 0551.49007
[3] Brězis, H; Nirenberg, L; Stampacchia, G, A remark on Ky Fan’s minimax principle, Boll. un. mat. ital., 6, 293-300, (1972) · Zbl 0264.49013
[4] Browder, F.E, Nonlinear elliptic boundary problems, Bull. amer. math. soc., 69, 862-874, (1963) · Zbl 0127.31901
[5] Browder, F.E, Nonlinear elliptic problems, II, Bull. amer. math. soc., 70, 299-301, (1964) · Zbl 0131.09702
[6] Browder, F.E, Nonlinear elliptic boundary value problems, II, Trans. amer. math. soc., 117, 530-550, (1965) · Zbl 0127.31903
[7] Browder, F.E, Existence and uniqueness theorems for solutions of nonlinear boundary value problems, (), 24-29 · Zbl 0145.35302
[8] Browder, F.E, Nonlinear monotone operators and convex sets in Banach spaces, Bull. amer. math. soc., 71, 780-785, (1965) · Zbl 0138.39902
[9] Browder, F.E, Nonlinear operators and nonlinear equation of evolution in Banach spaces, (), Part 2 · Zbl 0176.45301
[10] Dugundji, J; Granas, A, KKM maps and variational inequlities, Ann. scuolo norm. sup. Pisa cl. sci., 5, 679-682, (1978) · Zbl 0396.47037
[11] Fan, K, A generalization of tychonoffs fixed point theorem, Math. ann., 142, 305-310, (1961) · Zbl 0093.36701
[12] Fan, K, A minimax inequality and applications, (), 103-113
[13] Fan, K, Some properties of convex sets related to fixed point theorems, Math. ann., 266, 519-537, (1984) · Zbl 0515.47029
[14] Hartman, P; Stampacchia, G, On some nonlinear elliptic differential functional equations, Acta math., 115, 271-310, (1966) · Zbl 0142.38102
[15] Knaster, B; Kuratowski, C; Mazukiewicz, S, Ein beweis des fixpunktsatzes für n dimensionale simplexe, Fund. math., 14, 132-137, (1929) · JFM 55.0972.01
[16] Kneser, H, Sur un théorème fondamental de la théorie des jeux, C. R. acad. sci. Paris, 234, 2418-2420, (1952) · Zbl 0046.12201
[17] Minty, G, Monotone (nonlinear) operators in Hilbert space, Duke math. J., 29, 341-346, (1962) · Zbl 0111.31202
[18] Minty, G, On a “monotonicity” method for the solution of nonlinear equations in Banach spaces, (), 1038-1041 · Zbl 0124.07303
[19] Mosco, U, Implicit variational problems and quasi variational inequalities, (), 83-156 · Zbl 0346.49003
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