On generalized implicit vector equilibrium problems in Banach spaces

Let X and Y be real Banach spaces, K be a nonempty convex subset of X, and C : K → 2^Y be a multifunction such that for each u ∈ K, C (u) is a proper, closed and convex cone with int C (u) ≠ 0{combining long solidus overlay}, where int C (u) denotes the interior of C (u). Given the mappings T : K → 2^{L (X, Y)}, A : L (X, Y) → L (X, Y), f₁ : L (X, Y) × K × K → Y, f₂ : K × K → Y, and g : K → K, we introduce and consider the generalized implicit vector equilibrium problem: Find u^* ∈ K such that for any v ∈ K, there is s^* ∈ T u^* satisfying f₁ (A s^*, v, g (u^*)) + f₂ (v, g (u^*)) ∉ - int C (u^*). By using the KKM technique and the well-known Nadler's result, we prove some existence theorems of solutions for this class of generalized implicit vector equilibrium problems. Our theorems extend and improve the corresponding results of several authors.