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Nonlinear complementarity problem and systems of convex inequalities

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LetH be a nonempty closed convex subset of a topological vector spaceE andF be a real-valued function onH × H Then, we prove that, under some conditions, there existsx 0εH such thatF(x 0,y)≥0 for ally ε H. Furthermore, we obtain a necessary and sufficient condition that a finite system of convex inequalities is irreducibly inconsistent.

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Communicated by S. Karamardian

This work was supported in part by the Matsunaga Research Grant. The author wishes to express his sincere thanks to Professor H. Umegaki for his invaluable suggestions and advice.

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Takahashi, W. Nonlinear complementarity problem and systems of convex inequalities. J Optim Theory Appl 24, 499–506 (1978).

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