Abstract
The purpose of this paper is to study the solvability for a class of generalized vector variational inequalities in reflexive Banach spaces. Utilizing the KKM-Fan lemma and the Nadler’s result, we prove the solvability results for this class of generalized vector variational inequalities for monotone vector multifuctions. On the other hand, we first introduce the concepts of complete semicontinuity and strong semicontinuity for vector multifunctions. Then we prove the solvability for this class of generalized vector variational inequalities without monotonicity assumption by using these concepts and by applying the Brouwer fixed point theorem. The results in this paper are extension and improvement of the corresponding results in Huang and Fang (2006).
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References
L. Brouwer, Zur Invarianz des n-dimensional Gebietes. Mathematische Annalen 71 (1912) 305-313
G.Y. Chen, Vector variational inequality and its applications for multiobjective optimization. Chinese Science Bulletin 34 (1989) 969-972
G.Y. Chen, Existence of solutions for a vector variational inequality: An extension of Hartman-Stampacchia theorem. Journal of Optimization Theory and Applications 74 (1992) 445-456
G.Y. Chen and X.Q. Yang, The vector complementarity problem and its equivalences with the weak minimal element. Journal of Mathematical Analysis and Applications 153 (1990) 136-158
K. Fan, A generalization of Tychonoff’s fixed-point theorem. Mathematische Annalen 142 (1961) 305-310
F. Giannessi, Theorems of alternative, quadratic programs and complementarity problems. In: R.W. Cottle, F. Giannessi and L.J.L. Lions (eds.) Variational Inequalities and Complementarity Problems.. New York: Wiley & Sons (1980) pp.
F. Giannessi, Vector Variational Inequalities and Vector Equilibria. Dordrechet, Holland: Kluwer Academic Publisher (2000).
F. Giannessi and A. Maugeri, Variational Inequalities and Network Equilibrium Problems. New York: Plenum Press (1995).
Huang, N.J. and Fang, Y.P. (2005), On vector variational inequalities in reflexive banach spaces, Journal of Global Optimization(to appear).
I.V. Konnov and J.C. Yao, On the generalized vector variational inequality problems. Journal of Mathematical Analysis and Applications 206 (1997) 42-58
T.C. Lai and J.C. Yao, Existence results for VVIP. Applied Mathematical Letters 9 (1996) 17-19
S.B. Nadler Jr, Multi-valued contraction mappings. Pacific Journal of Mathematics 30 (1969) 475-488
A.H. Siddiqi, Q.H. Ansari and A. Khaliq, On vector variational inequalities.. journal of Optimization Theory and Applications 84 (1995) 171-180
X.Q. Yang, Vector complementarity and minimal element problems. Journal of Optimization Theory and Applications 77 (1993) 483-495
X.Q. Yang, Vector variational inequality and vector pseudolinear optimization. Journal of Optimization Theory and Applications 95 (1997) 729-734
X.Q. Yang and C.J. GOH, On vector variational inequality application to vector traffic equilibria. Journal of Optimization Theory and Applications 95 (1997) 431-443
S.J. Yu and J.C. Yao, On vector variational inequalities. Journal of Optimization Theory and Applications 89 (1996) 749-769
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Zeng, LC., Yao, JC. Existence of Solutions of Generalized Vector Variational Inequalities in Reflexive Banach Spaces. J Glob Optim 36, 483–497 (2006). https://doi.org/10.1007/s10898-005-5509-6
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DOI: https://doi.org/10.1007/s10898-005-5509-6
Keywords
- Brouwer fixed point theorem
- Complete semicontinuity
- Generalized vector variational inequalities
- Hausdorff metric
- KKM-Fan lemma