Let
X be a real reflexive Banach space with its dual space
\(X^{*}\) and
\(K\subseteq X\) be a closed set. Let
\(F: K\times K\to R\) be a real-valued bifunction. The equilibrium problems (for short EP) is to find
\(\bar{x}\in K\) such that
$$(\mathrm{EP})\quad F(\bar{x},y)\geq0,\quad \forall y\in K. $$
The equilibrium problems play an important role in economics, finance, image reconstruction, ecology, transportation, network, and so on (see,
e.g., [
1‐
4]). Later, many researchers extended (EP) to the vector set-valued case in different ways; see [
5‐
7] and the references therein.
Let
\(F: K\times K\to2^{Y}\) be a set-valued mapping, where
Y is a real normed space with an ordered cone
C, that is, a pointed, closed, and convex cone. It is well known that weak vector set-valued equilibrium problems (for short WVSEP) include two basic types. The first type is to find
\(\bar {x}\in K\) such that
$$(\mathrm{FWVSEP})\quad F(\bar{x},y) \nsubseteq-\operatorname{int} C,\quad \forall y\in K. $$
The second type is to find
\(\bar{x}\in K\) such that
$$(\mathrm{SWVSEP})\quad F(\bar{x},y) \cap-\operatorname{int} C=\emptyset, \quad \forall y\in K, $$
where int
C denotes the interior of
C.
It is worth noting that strong vector set-valued equilibrium problems (for short SVSEP) include two basic types, too. The first type is to find
\(\bar{x}\in K\) such that
$$(\mathrm{FSVSEP})\quad F(\bar{x},y) \subseteq C, \quad \forall y\in K. $$
The second type is to find
\(\bar{x}\in K\) such that
$$(\mathrm{SSVSEP})\quad F(\bar{x},y)\cap C\neq\emptyset, \quad \forall y\in K. $$
The issues of nonemptiness and boundedness of the solution set are among the most interesting and important problems in the theory of (WVSEP), as they can guarantee the weak convergence of some solution algorithms [
8,
9] in infinite dimensional spaces. For (FWVSEP), based on dual formulations, Ansari
et al. [
6,
7] proved the existence theorems under generalized pseudomonotonicity conditions. For (SWVSEP), several necessary and/or sufficient conditions for the solution set to be nonempty and bounded were established in [
10,
11]. Furthermore, the semicontinuity and connectedness of (approximate) solution sets can be found in [
12‐
15] for weak vector set-valued equilibrium problems. On the other hand, if
\(\operatorname{int} C=\emptyset\), then (WVSEP) cannot be studied. It is well known that for the classical Banach spaces
\(l^{p}\),
\(L_{p}\), where
\(1 < p < +\infty\), the standard ordered cone has an empty interior [
16]. Thus, for
C-monotone-type (SVSEP), finding sufficient and/or necessary conditions for the nonemptiness and boundedness of the solution set is very important. To our knowledge, existence results proposed in [
17] can be considered as a pioneering work for (SVSEP). Characterizations of nonemptiness and boundedness of the solution set for strong vector equilibrium problems were derived in different spaces [
18,
19]. Recently, Long
et al. [
20] obtained the existence theorems for the generalized strong vector quasi-equilibrium problems by the Kakutani-Fan-Glicksberg fixed point theorem on compact sets. For (FSVSEP), on noncompact sets, Wang
et al. [
21] obtained some existence theorems by virtue of the Brouwer fixed point theorem in general real Hausdorff topological vector spaces. Since the characterizations of nonemptiness and boundedness of the solution set for strong vector equilibrium problems can be derived when
F is a single-valued map, it is natural to ask whether characterizations on nonemptiness and boundedness of the solution set for (SVSEP) can be obtained in the case that
F is multi-valued, which constitutes the motivation of this article. In this paper, we present equivalent characterizations on the nonemptiness and boundedness of the solution set for (SVSEP) by means of the asymptotic cone theory in which the decision space is a real reflexive Banach space. Then we apply the equivalent characterizations to establish the stability theorems for (SVSEP) on a noncompact set, when both the mapping and the constraint set are perturbed by different parameters.
The rest of the paper is organized as follows. In Section
2, we introduce some basic notations and preliminary results. In Section
3, under suitable conditions we investigate the equivalence between the nonemptiness and boundedness of the solution set and the asymptotic cone
\(R_{2}=\{0\}\) for (SSVSEP). Stability results are presented for (SVSEP) on a noncompact set, when both the mapping and the constraint set are perturbed by different parameters in Section
4. Our results generalize and extend some results of [
3‐
7,
13‐
15,
19‐
21] in some sense.