First, this paper introduces a family of generalized equi-KKM mappings and the concept of quasi-abstract convexity (concavity) defined by parameter multi-valued mappings in abstract convex spaces that satisfy the $H_0$-condition. Then, using the Brouwer fixed-point theorem, we prove a common fixed-point theorem for this family of generalized equi-KKM mappings. Finally, we introduce two classes of generalized abstract equilibrium problems and obtain two existence theorems for their solutions, as an application of the common fixed-point theorem.