Abstract
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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References
Browder, F. E.,The Fixed Point Theory of Multi-Valued Mappings in Topological Vector Spaces, Mathematische Annalen, Vol. 177, pp. 283–301, 1968.
Hartman, P., andStampacchia, G.,On Some Nonlinear Elliptic Differential Functional Equations, Acta Mathematica, Vol. 115, pp. 271–310, 1966.
Browder, F. E.,On a New Generalization of the Schauder Fixed Point Theorem, Mathematische Annalen, Vol. 174, pp. 285–290, 1967.
Fan, K.,Existence Theorems and Extreme Solutions for Inequalities Concerning Convex Functions or Linear Transformations, Mathematische Zeitschrift, Vol. 68, pp. 205–217, 1957.
Fan, K.,Applications of a Theorem Concerning Sets with Convex Sections, Mathematische Annalen, Vol. 163, pp. 189–203, 1966.
Karamardian, S.,Generalized Complementarity Problem, Journal of Optimization Theory and Applications, Vol. 8, pp. 161–168, 1971.
Moré, J. J.,Coercivity Conditions in Nonlinear Complementarity Problems, SIAM Review, Vol. 16, pp. 1–16, 1974.
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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). https://doi.org/10.1007/BF00932892
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DOI: https://doi.org/10.1007/BF00932892