Existence of vector mixed variational inequalities in Banach spaces

The purpose of this paper is to study the solvability for vector mixed variational inequalities (for short, VMVI) in Banach spaces. Utilizing Ky Fan’s Lemma and Nadler’s theorem, we derive the solvability for VMVIs with compositely monotone vector multifunctions. On the other hand, we first introduce the concepts of compositely complete semicontinuity and compositely strong semicontinuity for vector multifunctions. Then we prove the solvability for VMVIs without monotonicity assumption by using these concepts and by applying Brouwer’s fixed point theorem. The results presented in this paper are extensions and improvements of some earlier and recent results in the literature.