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Remarks on some variational inequalities. (English) Zbl 0753.47035

First, a generalization of the variational type inequality of Juberg and Karamardian is obtained and it is applied to obtain strengthened versions of the Hartman-Stampacchia inequality and the Brouwer fixed point theorem. Next, fairly general versions of Browder’s variational inequality and its subsequent generalizations due to Brezis et al., Takahashi, Shih and Tan, Simons, and others are obtained. Finally, a variational inequality for non-real locally convex t.v.s. generalizing a result of Shih and Tan is given.
Let \(E\) be a real vector space, \(F\) a nonempty set, and \(\langle\cdot,\cdot\rangle: E\times F\to\mathbb{R}\) a real-valued function which is linear in the first variable in the sense: for each given \(y\in F\), \(\langle\cdot,y\rangle\) maps \(E\) linearly into \(\mathbb{R}\). The main result is the following:
Theorem 1. Let \(X\) be a convex space in \(E\), \(h:X\to\mathbb{R}\cup\{+\infty\}\) and \(f,g:X\to F\) functions satisfying
(i) \(\langle x-y,gy\rangle\leq\langle x-y,fy\rangle\) for \((x,y)\in X\times X\);
(ii) for each \(y\in X\), \(\{x\in X:\;\langle x-y,fy\rangle+h(y)<h(x)\}\) is convex or empty;
(iii) for each \(x\in X\), \(\{y\in X:\;\langle x-y,gy\rangle+h(y)>h(x)\}\) is compactly open; and
(iv) there exists a nonempty compact subset \(K\) of \(X\) and, for each finite subset \(N\) of \(X\), a compact convex subset \(L_ N\) of \(X\) containing \(N\) such that \(y\in L_ N\setminus K\) implies \(\langle x- y,gy\rangle+h(y)>h(x)\) for some \(x\in L_ N\).
Then there exists a \(y_ 0\in K\) such that \[ \langle x-y_ 0,gy_ 0\rangle+h(y_ 0)\leq h(x)\text{ for all }x\in X. \] Moreover, if \(h: E\to\mathbb{R}\cup\{+\infty\}\) is convex, then the inequality holds for all \(x\in I_ X(y_ 0)=\{x+r(u-x)\in E:\;u\in X,\;r>0\}\).
Reviewer: S.Park

MSC:

47H10 Fixed-point theorems
47J20 Variational and other types of inequalities involving nonlinear operators (general)
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