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On vector variational inequalities. (English) Zbl 0848.49012
Summary: The concept of weakly $$C$$-pseudomonotone operator is introduced. By employing the Fan lemma, we establish several existence results. The new results extend and unify existence results of vector variational inequalities for monotone operators under a Banach space setting. In particular, existence results for the generalized vector complementarity problem with weakly $$C$$-pseudomonotone operators in a Banach space are obtained.

##### MSC:
 49J40 Variational inequalities 47H05 Monotone operators and generalizations
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##### References:
 [1] Giannessi, F.,Theorem of the Alternative, Quadratic Programs, and Complementarity Problems, Variational Inequalities and Complementarity Problems, Edited by R. W. Cottle, F. Giannessi, and J. L. Lions, Wiley, New York, New York, pp. 151–186, 1980. [2] Chen, G. Y., andYang, X. Q.,Vector Complementarity Problem and Its Equivalence with Weak Minimal Element in Ordered Spaces, Journal of Mathematical Analysis and Applications, Vol. 153, pp. 136–158, 1990. · Zbl 0719.90078 [3] Chen, G. Y., andCheng, G. M.,Vector Variational Inequalities and Vector Optimization, Lecture Notes in Economics and Mathematical Systems, Springer Verlag, Heidelberg, Germany, Vol. 258, 1987. [4] Chen, G. Y., andCraven, B. D.,A Vector Variational Inequality and Optimization over an Efficient Set, Zeitschrift für Operations Research, Vol. 3, pp. 1–12, 1990. · Zbl 0693.90091 [5] Chen, G. Y.,Existence of Solution for a Vector Variational Inequality: An Extension of the Hartmann-Stampacchia Theorem, Journal of Opimization Theory and Applications, Vol. 74, pp. 445–456, 1992. · Zbl 0795.49010 [6] Yang, X. Q.,Vector Variational Inequality and Its Duality, Nonlinear Analysis: Theory, Methods, and Analysis, Vol. 21, pp. 869–877, 1993. · Zbl 0809.49009 [7] Hartmann, G. J., andStampacchia, G.,On Some Nonlinear Elliptic Differential Functional Equations, Acta Mathematica, Vol. 115, pp. 271–310, 1966. · Zbl 0142.38102 [8] Fan, K.,A Generalization of Tychonoff’s Fixed-Point Theorem Mathematische Annalen, Vol. 142, pp. 305–310, 1961. · Zbl 0093.36701 [9] Knaster, B., Kurotowski, C., andMazukiewicz, S.,Ein Beweis des Fixpunktsatzes für N-Dimensionale Simplexe, Fundamental Mathematica, Vol. 14, pp. 132–137, 1929. · JFM 55.0972.01 [10] Karmardian, S.,Complementarity over Cones with Monotone and Pseudomonotone Maps, Journal of Optimization Theory and Applications, Vol. 18, pp. 445–454, 1976. · Zbl 0304.49026 [11] Lee, G. M., Kim, D. S., Lee, B. S., andCho, S. J.,Generalized Vector Variational Inequality and Fuzzy Extension, Applied Mathematics Letters, Vol. 6, pp. 47–51, 1993. · Zbl 0804.49004 [12] Minty, G.,Monotone Nonlinear Operators in Hilbert Space, Duke Mathematical Journal, Vol. 29, pp. 341–346, 1962. · Zbl 0111.31202 [13] Yao, J. C.,Variational Inequalities with Generalized Monotone Operators, Mathematics of Operations Research, Vol. 19, pp. 691–705, 1994. · Zbl 0813.49010 [14] Yu, L. P.,Multiple-Criteria Decision Making: Concepts, Techniques, and Extensions, Plenum Press, New York, New York, 1985. · Zbl 0643.90045 [15] Sawaragi, Y., Nakayama, H., andTanino, T.,Theory of Multiobjective Optimization, Academic Press, New York, New York, 1985. · Zbl 0566.90053 [16] Moré, J. J.,Coercivity Conditions in Nonlinear Complementarity Problems, SIAM Review, Vol. 16, pp. 1–16, 1974. · Zbl 0272.65041 [17] Browder, F. E.,Nonlinear Monotone Operators and Convex Sets in Banach Space, Bulletin of the American Mathematical Society, Vol. 71, pp. 780–785, 1965. · Zbl 0138.39902 [18] Opial, Z.,Nonexpansive Monotone Mapping in Banach Spaces, Technical Report 67-1, Department of Mathematics, Brown University, Providence, Rhode Island, 1967. · Zbl 0179.19902 [19] Stampacchia, G.,Variational Inequalities Theory and Applications of Monotone Operators, Edited by A. Ghizzetti, Edizioni Oderisi, Gubbio, Italy, 1969. · Zbl 0167.11501 [20] Karamardian, S.,The Nonlinear Complementarity Problem, with Applications, Parts 1 and 2, Journal of Optimization Theory and Applications, Vol. 4, pp. 87–98, 1969 and Vol. 4, pp. 167–181, 1969. · Zbl 0169.06901 [21] Karamardian, S.,The Complementarity Problem, Mathematical Programming, Vol. 2, pp. 107–129, 1972. · Zbl 0247.90058 [22] Rheinboldt, W. C.,On M-Functions and Their Application to Nonlinear Gauss-Seidel Iterations and Network Flows, Journal of Mathematical Analysis and Applications, Vol. 32, pp. 274–307, 1971. · Zbl 0206.46504 [23] Schaible, S., andYao, J. C.,On the Equivalence of Nonlinear Complementarity Problems, Mathematical Programming, 1995. [24] Yao, J. C.,Multi-Valued Variational Inequalities with K-Pseudomonotone Operators, Journal of Optimization Theory and Applications, Vol. 83, pp. 445–454, 1994. · Zbl 0812.47055 [25] Conway, J. B.,A Course in Functional Analysis, 2nd Edition, Springer Verlag New York, New York, 1990. · Zbl 0706.46003 [26] Karamardian, S.,Generalized Complementarity Problem, Journal of Optimization Theory and Applications, Vol. 8, pp. 161–168, 1971. · Zbl 0208.46301 [27] Théra, M.,Existence Results for the Nonlinear Complementarity Problem and Applications to Nonlinear Analysis, Journal of Mathematical Analysis and Applications, Vol. 154, pp. 572–584, 1991. · Zbl 0728.90088 [28] Harker, P. T., andPang, J. S.,Finite-Dimensional Variational Inequality and Nonlinear Complementarity Problems: A Survey of Theory, Algorithms, and Applications, Mathematical Programming, Vol. 48B, pp. 161–220, 1990. · Zbl 0734.90098 [29] Nanda, S.,Nonlinear Complementarity Problem of Mathematical Programming in Banach Space, Indian Journal of Pure Applied Mathematics, Vol. 18, pp. 215–218, 1987. · Zbl 0628.90083 [30] Isac, G., andThéra, M.,Complementarity Problem and the Existence of the Post-Critical Equilibrium State of a Thin Elastic Plate, Journal of Optimization Theory and Applications, Vol. 58, pp. 241–257, 1988. · Zbl 0631.49005 [31] Dash, A. T., andNanda, S.,A Complementarity Problem in Mathematical Programming in Banach Space, Journal of Mathematical Analysis and Applications, Vol. 98, pp. 328–331, 1984. · Zbl 0547.90099 [32] Borwein, J. M.,Generalized Linear Complementarity Problems Treated without Fixed-Point Theory, Journal of Optimization Theory and Applications, Vol. 43, pp. 445–454, 1984. · Zbl 0532.90089
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