Generalized mixed variational-like inequalities with compositely pseudomonotone multifunctions

In this paper, we introduce and consider a class of generalized mixed variational-like inequalities with compositely pseudomonotone multifunctions in Banach spaces. By using the Brouwer's fixed point theorem and the Nadler's fixed point theorem, we prove the existence of solutions for generalized mixed variational-like inequalities with compositely relaxed η-α pseudomonotone multifunctions in reflexive Banach spaces. On the other hand, we also derive the solvability of generalized mixed variational-like inequalities with compositely relaxed η-α semi-pseudomonotone multifunctions in nonreflexive Banach spaces by virtue of the method of finite-dimensional successive approximation. The results presented in this paper extend and improve some earlier and recent results in the literature.