The connectedness of the solutions set for set-valued vector equilibrium problems under improvement sets

In this paper, we provide the connectedness of the sets of weak efficient solutions, Henig efficient solutions and Benson proper efficient solutions for set-valued vector equilibrium problems under improvement sets.


Introduction
In recent years, many scholars paid attention to developing concepts to unify various kinds of solution notions of vector optimization problems, for instance, efficiency, weak efficiency, proper efficiency and ε-efficiency. Chicoo et al. [1] putted forward a new concept of improvement set E and defined E-optimal solution in finite dimensional Euclidean space. E-optimal solution unifies some known concepts of exact and approximate solutions of vector optimization problems. Gutiérrez et al. [2] extended the notion of improvement set and E-optimal solution to a locally convex topological vector spaces. Much follow-up work about the improvement set E one finds in [3][4][5][6][7][8][9][10][11][12]. Chen et al. [13] introduced a new vector equilibrium problem based on improvement set E named the unified vector equilibrium problem (UVEP), linear scalarization characterizations of the efficient solutions, weak efficient solutions, Benson proper efficient solutions for (UVEP) were established, and some continuity results of parametric (UVEP) were obtained by applying scalarization method.
Vector equilibrium problems (shortly, VEP) provides a unified model of many significant problems (see [14][15][16]). An important topic about (VEP) is the connectedness of the solutions set. Lee et al. [17] and Cheng [18] discussed the path-connectedness and connectedness of weakly efficient solutions set for vector variational inequalities in finite dimensional Euclidean space, respectively. Applying the scalarization approaches, Gong [19] studied the connectedness of the sets of Henig efficient solutions and weakly efficient solutions for the vector Hartman-Stampacchia variational inequality in normed vector spaces (in short, n.v.s.). By employing scalarization results, Gong [20] investigated the connectedness and path-connectedness of sets of weak efficient solutions and various proper efficient solutions for (VEP)in locally convex spaces (in short, l.c.s.). By the density results, Gong and Yao [21] first showed the connectedness of efficient solutions set for (VEP) in l.c.s. Chen et al. [22] studied the connectedness and the compactness of weak efficient solutions set for set-valued vector equilibrium problems (shortly, SVEP) in n.v.s. Chen et al. [23] discussed the connectedness of the sets of -weak efficient solutions and -efficient solutions for (VEP) in l.c.s.
All the papers mentioned above, the hypotheses of compactness and monotonicity are essential in discussing the connectedness of the sets of various kinds of efficient solutions for (VEP). Han and Huang [24] studied the connectedness of the sets of (weakly) efficient solutions and various proper efficient solutions for the (GVEP) not using the conditions of compactness and monotonicity in n.v.s. Han and Huang [25] investigated the connectedness of the sets of weakly efficient solutions and ε-efficient solutions for the(SVEP) by using the scalarization results and the density results in n.v.s, respectively. The improvement set E is a tool to unify some exact and approximate solution notions, hence, it is very meaningful in establishing the connectedness of the solutions set for VEP based on improvement set.
Motivated by the work of [13,24,25], in this paper, by using the scalarization results, we study the connectedness of the sets of weakly efficient solutions, Henig efficient solutions and Benson proper efficient solutions for set-valued vector equilibrium problems under improvement sets. The main results unify and extend some exact and approximate cases.

Preliminaries
Throughout this paper, let X, Y be real locally convex Hausdorff topological vector spaces and let Z be a real vector topological space. Let Y * be the topological dual space of Y and let C be a pointed closed convex cone in Y with its topological interior int C = ∅.
Let Q be a nonempty subset of Y , denote the closure of Q by cl Q and the topological interior of Q by int Q. The cone hull of Q is defined by We say Q is solid if int Q = ∅.
The positive polar cone C * and the strict positive polar cone C of C are defined as

respectively.
A nonempty convex set B ⊂ C is said to be a base of C if It is clear that C = ∅ if and only if C has a base. Let B be a base of C, because of 0 Y / ∈ cl B, by the separation theorem of convex sets, there exists 0 = ϕ ∈ Y * such that Then For convenience, we denote by N(0) the family of neighborhoods of zero in Y . Assume that B is a base of C and write By the separation theorem of the convex sets (see [27]), we know B st = ∅.
Proof It follows directly from Definition 2.8, Definition 2.9 and Lemma 2.6. In this paper, we let R is the set of real numbers and R + = {r | r ≥ 0}, R ++ = {r | r > 0}. From now on, we presume that A is a nonempty subset of X, F : A×A ⇒ Y is a set-valued map.
We have the usual set-valued vector equilibrium problem (SVEP) of findingx ∈ A such that

Definition 2.14
(i) An element x ∈ A is called a weakly efficient solution of the (SVEP) (see [22]) if (ii) An element x ∈ A is called a Benson proper efficient solution of the (SVEP) (see [20]) if (iii) An element x ∈ A is called a C-Heing efficient solution of the (SVEP) (see [32] We consider the unified set-valued vector equilibrium problem (USVEP) through improvement set E of findingx ∈ A such that

Definition 2.17
An element x ∈ A is said to be a weakly efficient solution of the (USVEP) if Denote by We(F, A; E) the set of weakly efficient solutions of the (USVEP).

Definition 2.18
An element x ∈ A is said to be a Benson proper efficient solution of the (USVEP) if In what follows, we prove Letx ∈ Be(F, A; E), then This implies Therefore, Hence,x ∈ We(F, A; E).
then the weak efficiency of (USVEP) reduces to the weak efficiency of (SVEP).
(ii) If E = C \ {0 Y }, then the Benson proper efficiency and Heing efficiency of (USVEP) reduce to the Benson proper efficiency and of C-Heing efficiency of (SVEP), respectively.
Remark 3.2 By Theorem 2.12, the condition nearly E-subconvexlike in Theorem 3.1 is weaker than nearly C-convexlike and the convexity of E in Theorem 3.1 of [13]. So, compared with Theorem 3.1 in [13], Theorem 3.1 extends the model from vector-valued maps to set-valued maps under the weaker condition.
In the following, we show thatx ∈ Be(F, A; E).
Hence, there exist sequences {μ n } ⊂ R + , {c n } ⊂ C, {e n } ⊂ E and {y n } ⊂ A such that μ n (z n + e n + c n ) → d, ∀z n ∈ F(x, y n ).

Connectedness of the solutions set
In this section, we discuss the connectedness of We(F, A; E), He(F, A, B; E) and Be(F, A; E).
The proofs of (ii) and (iii) are similar to (i).

Conclusions
In this paper, under the assumption of nearly E-subconvexlikeness of the binary function in real locally convex Hausdorff topological vector spaces, we obtain the linear scalarization of weak efficient solutions, Benson proper efficient solutions, Heing efficient solutions for (USVEP). By means of the scalarization results, we investigate the connectedness of the sets of weak efficient solutions, Benson proper efficient solutions, Heing efficient solutions for (USVEP). However, the connectedness of efficient solutions sets of for (USVEP) have been not established, it may be of great interest for us to discuss.