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)].