On a generalized vector equilibrium problem with bounds

(1) x0 ∈ K such that c1 ≤ f(x0, y) ≤ c2, for all y ∈ K, where K is a given set, f : K × K → R is a given function and c1 and c2 two real numbers such that c1 ≤ c2. If K is a nonempty closed subset in a locally convex semi-reflexive topological vector space, then the problem (1) is problem by Isac, Sehgal and Singh [16] (Open Problem 2). In [20], Li gave the answer of this problem by introducing and the using the concept of extremal subsets. In [8], Chadli, Chang and Yao derived some results in answering this problem by using a fixed point theorem due to Ansari and Yao [4] and the Ky Fan Lemma [10]. Al-Homidan and Ansari in [14] consider system of quasi-equilibrium problems with lower and upper bounds and establish the existence of their solution by using maximal elements theorems. Liya Fan studies the existence of solution of weighted quasi-equilibrium problems with lower and upper bounds by using maximal element theorems, a fixed point theorem of set-valued maps and the Fan-KKM theorem in [13]. In this paper we study generalized vector equilibrium problem with bounds and establish some existence theorems in Hausdorff topological vector spaces. Our results improve some recent results in the literature. Note that if c1 = 0, c2 = 1 and f(x, y) = e−h(x,y), where h : K ×K → R, the problem (1) is known as the scalar equilibrium problem