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Inthispaper,westudysomeexistencetheoremsofsolutionsforvectorvariational inequality by using the generalized KKM theorem. Also, we investigate the properties of so- lution set of the Minty vector variational inequality in G–convex spaces. Finally, we prove the equivalence between a Browder fixed point theorem type and the vector variational in- equality in G-convex spaces.
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A. Bensussan, J. L. Lions, Applications des Inèquations variationnelles en Contrôle Stochas- tique, Dunod,Paris,(1987).
F. Browder, The fixed point theory of multi-valued mappings in topo-logical vector spaces, Math Ann, 177, 183301 (1968).
I. Chen and Sh. Zou and Y. Zhang, New existence theorems for vector equilibrium problems with set-valued mappings, J. Nonlinear Funct. Anal, 2019 (2019), Article ID4.
R. Ferrentino,Variational Inequalities and optimization problems. Appl. Math. Scl, 1, 2327- 2343 (2007).
F. Giannessi, theorems of alternative, quadratic programs and complementary problems, Variational Inequalities and complementarity problems and Applications, PP. 151-186. Wiley, chichester (1980).
F. Giannessi, Vector Variational Inequality and Vector Equilibrium. Kluwer Academic, Dordrecht, (2000).
C. D. Horvath, Some results on multivalued mappings and inequalities without convexity, in : B. L. Lin, S. Simons (Eds), Nonlinear and Convex Analysis, in: Lect. Notes Pure Appl. Math, Marcel Dekker, 1987, pp. 99-106.
C. D. Horvath, Contractibility and generalized convexity, J. Math. Anal. Appl, 156 (1991), 341-357.
B. Knaster and C. Kuratowski and S. Mazurkiewicz, Ein Beweis des Fixpunktsatsez fu ̈r n- dimensionale simplexe, Fund. Math. 14, (1929), 132137.
M. Lassonde, “On the use of KKM multifunctions in fixed point theory and related topics”, Journal of Mathematical Analysis and Applications, vol. 97, no. 1, 1983, (151-201).
I. I. Mai and Do Van Luu, Optimality conditions for weakly efficient solutions of vector vari- ational inequalities via convexificators, J. Nonlinear Var. Anal, 2 (2018), 347-389.
S. Park and H. Kim, Foundations of the KKM theory on generalized convex spaces, J. Math. Anal. App, 209 (1997), 551-571.
S. Park, Another five episodes related to generalized convex spaces, Nonlinear Funct. Anal. Appl, 3 (1998), 1-12.
S. Park, Fixed Point, Intersection Theorems, Variational Inequlities, and Equilibrium theorems, J. Math. & Math. Sci. Vol 24, No, 2 (2000).
S. Park and M. P. Chen, Unified approach to variational inequalities on compact convex sets, Nonlinear Anal, 33, (1998), 637-644.
S. Park, Elements of the KKM theory for generalized convex spaces, Korean J. comput. & Appl. Math, 7 (2000), 1-28.
X. Pin Ding, Generalized variational Inequalities and Equilibrium problems in Generalized convex spaces. J. computers and Mathematics with Applications, 38 (1999) 189-197
N. S. Parageorgiou, Nonsmooth analysis on partially ordered vector spaces. 1. Convex case, Pacific J. Math, 107 (1983), 403-458.
G. Stampacchia, Formes bihneares coercitives sur les ensembles convexes. C. R. Acad. Sci. Paris, Ser, 1. Math. 258, (1964).
K. Tan, K. G-KKM theorem, minimax inequalities and saddle points, Nonlinear Anal, 30 (1997), 4151-4160.
B. S. Thakar and S. Varghesa, Generalized Nonlinear Variational Inequalities. Novi. Sad. J. Math. Vol 44, No, 2, 2014, 29-40.
R. U. Verma, Nonlinear Variational and constrained hemivariational inequalities involving relaxed operators, Zeitschrift fur Ang. Math. and Mechanik, 77 (1997), 387-91.
J. Wang, The Existence of Solutions for general variational Inequality and applications in FC-Spaces. Jourral of Inequalities and Applications, (2012), 1-8.
X. Wu and Xian-Zhi Yuan, Nonlinear variational Inequalitoties and Implicit Variational Inequality of Ky Fan Type in H-Spaces, J. Computers and Mathematics with Applications, 38, (1999), 1-8.