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Strong vector variational inequalities in Banach spaces. (English) Zbl 1138.49300
Summary: In this work, we study some existence results for solutions for a class of strong vector variational inequalities (for short, SVVI) in Banach spaces. The solvability of the $SVVI$ without monotonicity is presented by using the fixed point theorems of Brouwer and Browder, respectively. The solvability of the SVVI with monotonicity is also proved by using the {\it Ky Fan} lemma [Math. Ann. 266, No. 4, 519--537 (1984; Zbl 0515.47029)]. Our results give a positive answer to an open problem proposed by {\it G. Chen} and {\it S.-H. Hou} [Nonconvex Optim. Appl. 38, 73--86 (2000; Zbl 1012.49007)].

##### MSC:
 49J40 Variational methods including variational inequalities 47H10 Fixed-point theorems for nonlinear operators on topological linear spaces 47J20 Inequalities involving nonlinear operators
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##### References:
 [1] Giannessi, F.: Theorems of alterative, quadratic programs and complementarity problems. Variational inequalities and complementarity problems, 151-186 (1980) [2] Lee, G. M.; Kim, D. S.; Lee, B. S.; Chen, G. Y.: Generalized vector variational inequality and its duality for set-valued maps. Appl. math. Lett. 11, 21-26 (1998) · Zbl 0940.49008 [3] N.J. Huang, Y.P. Fang, On vector variational inequalities in reflexive Banach spaces, J. Global Optim. (in press) · Zbl 1097.49009 [4] Kim, W. K.; Tan, K. K.: On generalized vector quasi-variational inequalities. Optimization 46, 185-198 (1999) · Zbl 0960.47037 [5] Chen, G. Y.: Existence of solutions for a vector variational inequality: an extension of hartman--stampacchia theorem. J. optim. Theory appl. 74, 445-456 (1992) · Zbl 0795.49010 [6] Chen, G. Y.; Yang, X. Q.: The vector complementarity problem and its equivalences with the weak minimal elements. J. math. Anal. appl. 153, No. 1, 136-158 (1990) · Zbl 0712.90083 [7] Chen, G. Y.; Hou, S. H.: Existence of solutions for vector variational inequalities. Vector variational inequalities and vector equilibria, 73-86 (2000) · Zbl 1012.49007 [8] Y.P. Fang, N.J. Huang, On the strong vector variational inequalities, Research Report, Department of Mathematics, Sichuan University, 2002 [9] Giannessi, F.: Vector variational inequalities and vector equilibria. (2000) · Zbl 0952.00009 [10] Konnov, I. V.; Yao, J. C.: On the generalized vector variational inequality problems. J. math. Anal. appl. 206, 42-58 (1997) · Zbl 0878.49006 [11] Lee, G. M.; Kim, D. S.; Lee, B. S.: Generalized vector variational inequality. Appl. math. Lett. 9, 39-42 (1996) · Zbl 0862.49014 [12] Siddiqi, A. H.; Ansari, Q. H.; Ahmad, R.: On vector variational inequalities. J. optim. Theory appl. 84, 171-180 (1995) · Zbl 0827.47050 [13] Yang, X. Q.: Vector variational inequality and its duality. Nonlinear anal. 95, 729-734 (1993) [14] Brouwer, L.: Zur invarianz des n-dimensional gebietes. Math. ann. 71, 305-313 (1912) · Zbl 42.0418.01 [15] Browder, F. E.: A new generalization of the Schauder fixed point theorem. Math. ann. 174, 285-290 (1967) · Zbl 0176.45203 [16] Fan, K.: Some properties of convex sets to fixed point theorem. Math. ann. 266, 519-537 (1984) · Zbl 0515.47029