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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 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 G. Chen and 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