Generalized Vector Equilibrium-Like Problems without Pseudomonotonicity in Banach Spaces

Let and be real Banach spaces, a nonempty closed convex subset of , and a multifunction such that for each is a proper, closed and convex cone with , where denotes the interior of . Given the mappings , , and , we study the generalized vector equilibrium-like problem: find such that for all for some . By using the KKM technique and the well-known Nadler result, we prove some existence theorems of solutions for this class of generalized vector equilibrium-like problems. Furthermore, these existence theorems can be applied to derive some existence results of solutions for the generalized vector variational-like inequalities. It is worth pointing out that there are no assumptions of pseudomonotonicity in our existence results.