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Existencestability theorems for strong vector setvalued equilibrium problems in reflexive Banach spaces
Journal of Inequalities and Applications volumeÂ 2015, ArticleÂ number:Â 239 (2015)
Abstract
In this paper, we develop existence and stability theorems for strong vector setvalued equilibrium problems in reflexive Banach spaces. Based on asymptotic cone theory, we present the equivalent characterizations on the nonemptiness and boundedness of the solution set for strong vector setvalued equilibrium problems. Furthermore, stability results are established for strong vector setvalued equilibrium problems, when both the mapping and the constraint set are perturbed by different parameters.
1 Introduction
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 realvalued bifunction. The equilibrium problems (for short EP) is to find \(\bar{x}\in K\) such that
The equilibrium problems play an important role in economics, finance, image reconstruction, ecology, transportation, network, and so on (see, e.g., [1â€“4]). Later, many researchers extended (EP) to the vector setvalued case in different ways; see [5â€“7] and the references therein.
Let \(F: K\times K\to2^{Y}\) be a setvalued 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 setvalued equilibrium problems (for short WVSEP) include two basic types. The first type is to find \(\bar {x}\in K\) such that
The second type is to find \(\bar{x}\in K\) such that
where intC denotes the interior of C.
It is worth noting that strong vector setvalued equilibrium problems (for short SVSEP) include two basic types, too. The first type is to find \(\bar{x}\in K\) such that
The second type is to find \(\bar{x}\in K\) such that
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 [12â€“15] for weak vector setvalued 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 Cmonotonetype (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 quasiequilibrium problems by the KakutaniFanGlicksberg 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 singlevalued 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 multivalued, 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 [3â€“7, 13â€“15, 19â€“21] in some sense.
2 Notations and preliminaries
In this section, we introduce some basic notations and preliminary results.
Consider the following modeling: The first type strong vector setvalued equilibrium problems, abbreviated by (FSVSEP), is to find \(\bar{x}\in K\) such that
For (FSVSEP), its dual problem is to find \(\bar{x}\in K\) such that
We denote the solution set of (FSVSEP) and the solution set of (DFSVSEP) by \(SS_{K}\) and \(SS^{D}_{K}\), respectively.
The second type strong vector setvalued equilibrium problems, abbreviated by (SSVSEP), is to find \(\bar{x}\in K\) such that
For (SSVSEP), its dual problem is to find \(\bar{x}\in K\) such that
We denote the solution set of (SSVSEP) and the solution set of (DSSVSEP) by \(S_{K}\) and \(S^{D}_{K}\), respectively.
It is easy to see \(SS_{K}\subseteq S_{K}\) and \(SS^{D}_{K}\subseteq S^{D}_{K}\).
Definition 2.1
[22]
Let K be a nonempty, closed, and convex subset of a real reflexive Banach space X with dual space \(X^{*}\). The dual cone \(K^{*}\) of K is defined as
It is well known that
where â€˜intâ€™ means the interior of a set. The asymptotic cone \(K^{\infty}\) and the barrier cone \(\operatorname{barr}(K)\) of K are, respectively, defined by
and
where â‡€ stands for the weak convergence.
The asymptotic cone \(K^{\infty}\) has the following useful properties.
Lemma 2.1
[22]
Let \(K\subset X\) be nonempty and closed. Then the following conclusions hold:

(i)
\(K^{\infty}\) is a closed cone.

(ii)
If K is convex, then \(K^{\infty}=\{d\in X\mid x+td\in K, \forall t>0\}\), for all \(x\in K\).

(iii)
If K is a convex cone, then \(K^{\infty}=K\).
The following result can be found in Proposition 2.2 of [23] showing the property of K.
Lemma 2.2
Let K be a nonempty, closed, and convex subset of a real reflexive Banach space X. If \(\operatorname{barr}(K)\) has a nonempty interior, then there does not exist \(\{x_{n}\}\subseteq K\) with \(\x_{n}\\to+\infty\) such that the origin is a weak limit of \(\{\frac{x_{n}}{\x_{n}\}\}\). If K is a cone, then there does not exist \(\{d_{n}\}\subset K\) with each \(\d_{n}\=1\) such that \(d_{n}\rightharpoonup0\).
To obtain the characterization of the solution sets for (SSVSEP) and (FSVSEP), we recall generalized monotonicity and generalized convexity [24].
Definition 2.2
[24]
Let K be a nonempty, closed, and convex subset of X. A mapping \(F: K\times K\to2^{Y}\) is said to be

(i)
type I Cpseudomonotone if, for all \(x,y \in K\),
$$F(x,y)\subseteq C \quad \Rightarrow \quad F(y,x)\subseteqC; $$ 
(ii)
type II Cpseudomonotone if, for all \(x,y \in K\),
$$F(x,y)\cap C\neq\emptyset \quad \Rightarrow \quad F(y,x)\capC\neq\emptyset. $$
It is easy to verify that type I Cpseudomonotonicity implies type II Cpseudomonotonicity. However, the converse is not true.
Example 2.1
Let \(X=R^{1}\), \(K=[1,+\infty)\), \(Y=R^{2}\), \(C=R^{2}_{+}\). Let \(F: K\times K\to2^{Y}\) be defined by
Clearly, its dual is
For all \(x,y\in K\), we have
However, we cannot find \(x,y\in K\) such that
Thus, F is not type I Cpseudomonotone.
Definition 2.3
[24]
The mapping \(F: K\times K\to2^{Y}\) is said to be strongly

(i)
type I Cdiagonally quasiconvex in the first argument if, for any finite \(\{x_{1},x_{2},\ldots, x_{n}\}\subseteq K\) and any \(t_{i}\geq0\) with \(\sum^{n}_{i=1}t_{i}=1\), \(x=\sum^{n}_{i=1}t_{i}x_{i}\), and for some \(x_{i}\in K\), one has
$$F(x_{i},x)\subseteqC; $$ 
(ii)
type II Cdiagonally quasiconvex in the first argument if, for any finite \(\{x_{1},x_{2},\ldots, x_{n}\}\subseteq K\) and any \(t_{i}\geq0\) with \(\sum^{n}_{i=1}t_{i}=1\), \(x=\sum^{n}_{i=1}t_{i}x_{i}\), and for some \(x_{i}\in K\), one has
$$F(x_{i},x)\capC\neq\emptyset; $$ 
(iii)
Cconvex in the second argument if, for any \(y_{1},y_{2}\in K\) and \(t\in(0,1)\), one has
$$t F(x,y_{1})+ (1t)F(x,y_{2}) \subseteq F \bigl(x,ty_{1}+(1t)y_{2}\bigr)+C. $$
The following lemma is the wellknown KKM theorem, we refer the reader to Lemma 1 of [25].
Lemma 2.3
[25]
Let \(K\subseteq X\) be a nonempty convex of a topological vector space X and \(F : K\to2^{X}\) be a setvalued mapping from K into X satisfying the following properties:

(i)
F is a KKM mapping: For every finite subset A of E, \(\operatorname{co}(A)\subseteq\bigcup_{x\in A}F(x)\), where \(\operatorname{co}(\cdot)\) stands for the convex hull;

(ii)
\(F(x)\) is closed in X for every \(x\in K\);

(iii)
\(F(x_{0})\) is compact in X for some \(x_{0}\in K\).
Then \(\bigcap_{x\in K}F(x)\neq\emptyset\).
3 The nonemptiness and boundedness of the solution sets for (FSVSEP) and (SSVSEP)
In this section, we present the equivalent characterizations of the solution set for strong vector setvalued equilibrium problems to be nonempty and bounded based on asymptotic cone theory.
Theorem 3.1
Let K be a nonempty, closed, and convex subset of X with \(\operatorname{int} (\operatorname{barr}(K))\neq\emptyset\). Suppose that \(F :K\times K\to2^{Y} \) satisfies the following:

(i)
F is type II Cpseudomonotone and \(F(x,x)\subset C\capC\), \(\forall x \in K\);

(ii)
the set \(\{x\in K: F(x,y)\cap C\neq\emptyset\}\) is closed for any \(y\in K\) and F is strongly type II Cdiagonally quasiconvex in the first argument;

(iii)
the set \(\{y\in K: F(x,y)\capC\neq\emptyset\}\) is closed for any \(x\in K\) and F is Cconvex in the second argument.
Then the following statements are equivalent:

(I)
(SSVSEP) has a nonempty, convex, closed, and bounded solution set;

(II)
(DSSVSEP) has a nonempty, convex, closed, and bounded solution set;

(III)
\(R_{2}=\{d\in K_{\infty}: F(y,y+td)\capC\neq\emptyset, \forall y\in K, t>0\}=\{0\}\);

(IV)
there exists a bounded set \(D\subset K\) such that for every \(x\in K \backslash D\), there exists some \(y\in D\) such that
$$F(y,x)\capC=\emptyset. $$
Proof
(I) â‡” (II). Suppose that (SSVSEP) has a nonempty, convex, closed, and bounded solution set. By the type II Cpseudomonotonity of F, we obtain \(S_{K}\subseteq S^{D}_{K}\).
On the other hand, let us verify \(S^{D}_{K}\subseteq S_{K}\). Take any \(x^{*}\in S^{D}_{K}\), obviously
For every \(y \in K\), consider \(x_{t} = x^{*}+t (yx^{*})\), \(\forall t \in(0, 1)\). Clearly, \(x_{t} \in K\). The Cconvexity of \(F(x_{t},\cdot)\) implies that
Let us by contradiction show \(tF(x_{t},y)\cap C\neq\emptyset\). Suppose the contrary, then \(tF(x_{t},y) \subseteq Y\backslash C\). Consequently,
which contradicts (3.1). Noting that \(Y\backslash C\) is a cone, we deduce
By assumption (ii), letting \(t\to0\) in (3.3), one has
Hence, (SSVSEP) has a nonempty, convex, closed, and bounded solution set if and only if (DSSVSEP) has a nonempty, convex, closed, and bounded solution set.
(II) â‡’ (III). We know
For fixed \(y\in K\), one has
Since (DSSVSEP) has a nonempty, convex, closed, and bounded solution set, we have
On the other hand, one has
Thus, (III) holds.
(III) â‡’ (IV). If (III) does not hold, then there exists a sequence \(\{x_{n}\}\subset K\) such that \(n< \x_{n}\\) with
for \(y\in K\) with \(\y\\leq n\). For fixed \(y\in K\) and \(t>0\), without loss of generality, we may take a subsequence \(\{x_{n_{k}}\}\) of \(\{x_{n}\}\) such that
Noting that X is a reflexive Banach space and \(\operatorname{int}(\operatorname{barr}(K))\neq \emptyset\), from Lemma 2.2, we have \(d_{0}\neq0\). The Cconvexity of \(F(x,\cdot)\) implies
From \(F(y,y)\subseteq C\) and \(F(y,x_{n_{k}})\capC\neq\emptyset\), one has
By assumption (iii), letting \(t\to0\) in (3.4), we obtain
which contradicts (III). So (IV) holds.
(IV) â‡’ (II). Set \(M: K\times K\to2^{K}\) by
It is easy to verify that \(M(y)\) is a closed subset of K. We shall show that M is a KKM mapping. We may assume that D is a bounded, closed, and convex set (otherwise, consider the closed convex hull of D instead of D). Let \(\{y_{1},\ldots, y_{m}\}\) be finite number of points in K and let \(L= \operatorname{co}(D\cup \{y_{1},\ldots, y_{m}\})\). So, L is a weakly compact convex set, since X is a reflexive Banach space. Consider the setvalued mapping \(\overline{M(y)}\) defined by
Obviously, \(\overline{M(y)}\) is a weakly compact convex subset of L. We claim \(\overline{M(y)}\) has the finite intersection property.
Indeed, for all \(\{x_{1}, x_{2}\ldots, x_{p}\}\subset L\), \(\sum^{p}_{t_{i}}=1\), \(t_{i}\geq0\), and \(\bar{x}=\sum^{p}_{i=1}t_{i}x_{i}\), it follows from assumption (ii) that there exists \(x_{i}\in L\) such that
namely, \(\bar{x}\in\bigcup^{p}_{i=1}\overline{M(x_{i})}\). Hence, \(\overline{M(y)}\) has the finite intersection property. Noting that \(\overline{M(y)}\) has the finite intersection property and \(\overline{M(y)}\) is a weakly compact convex subset, we obtain, from Lemma 2.3
Let \(\bar{x}\in\bigcap_{y\in L}\overline{M(y)}\), we have
We assert that \(\bar{x}\in D\). Suppose to the contrary that if there exists some \(\bar{x}\in \overline{M(y)}\) but \(\bar{x}\notin D\), by the assumption (IV), one has
Consequently, \(\bar{x}\notin\overline{M(y)}\), which is a contradiction with (3.6). Thus,
For \(\bar{x}\in\bigcap_{y\in L}\overline{M(y)}\), by (3.7), we deduce \(\bar{x}\in\bigcap^{m}_{i=1}(M(y_{i})\cap D)\), which implies the collection \(\{M(y)\cap D: y\in K\}\) has the finite intersection property. Since for each \(y\in K\), \((M(y)\cap D)\) is weakly compact, it follows from Lemma 2.3 that \(\bigcap_{y\in K} (M(y)\cap D)\neq\emptyset\), which coincides with the solution set of (DSSVSEP). Notice that
From assumption (iii), it is easy to see that \(S^{D}_{K}\) is closed. Next, we shall show that \(S^{D}_{K}\) is convex. For all \(x_{1},x_{2}\in S^{D}_{K}\) and \(t\in[0,1]\), from the Cconvexity of \(F(y,\cdot)\), we have
It is easy to verify \(F(y, tx_{1}+(1t)x_{2})\capC\neq\emptyset\). Thus, (DSSVSEP) has a nonempty, convex, closed, and bounded solution set.â€ƒâ–¡
Remark 3.1
When F is a singlevalued and \(C=R^{1}_{+}\), a similar result to (III) can be obtained in [3] for equilibrium problems and in [26] for the variational inequality. Furthermore, if D is closed, convex, and bounded, the result (IV) was proved in [4] for equilibrium problems. It is worth to stress that Theorem 3.1 is a new result for (SSVSEP), since similar results cannot be found in [18, 19].
Now by the following example we illustrate Theorem 3.1.
Example 3.1
Let \(X=R^{1}\), \(K=[0,+\infty)\), \(Y=R^{2}\), \(C=R^{2}_{+}\). Let \(F: K\times K\to2^{Y}\) be defined by
Clearly, its dual problem is
It is easy to see that assumptions (i)(iii) are satisfied. We can verify that (I) â‡” (II), that is, \(S_{K}=S^{D}_{K}=\{0\}\); (III) holds, namely, \(R_{2}=\{0\}\); and if we take the bounded set \(\{0\}=D\subset K\), then (IV) is true.
Similar to the proof of Theorem 3.1, the following result holds for (FSVSEP).
Theorem 3.2
Let K be a nonempty, closed, and convex subset of X with \(\operatorname{int} \operatorname{barr}(K)\neq\emptyset\). Suppose that \(F :K\times K\to2^{Y} \) satisfies the following:

(i)
F is type I Cpseudomonotone and \(F(x,x)\subset C\capC\), \(\forall x \in K\);

(ii)
the set \(\{x\in K: F(x,y)\subseteq C\}\) is closed for any \(y\in K\) and F is strongly type I Cdiagonally quasiconvex in the first argument;

(iii)
the set \(\{y\in K: F(x,y)\subseteqC\}\) is closed for any \(x\in K\) and F is Cconvex in the second argument.
Then the following statements are equivalent:

(I)
(FSVSEP) has a nonempty, convex, closed, and bounded solution set;

(II)
(DFSVSEP) has a nonempty, convex, closed, and bounded solution set;

(III)
\(R_{1}=\{d\in K_{\infty}: F(y,y+td)\subseteqC, \forall y\in K, t>0\}=\{0\}\);

(IV)
there exists a bounded set \(D\subset K\) such that for every \(x\in K \backslash D\), there exists some \(y\in D\) such that
$$F(y,x)\nsubseteqC. $$
4 Stability analysis for (FSVSEP) and (SSVSEP)
In this section, we shall establish the stability theorems for (FSVSEP) and (SSVSEP) when the mapping F and the domain set K are simultaneously perturbed by different parameters. We take (SSVSEP) and (DSSVSEP) as examples to present the stability theorems.
First recall some important notions and results. Let \((Z_{1}, d_{1})\) and \((Z_{2}, d_{2})\) be two metric spaces. Let \(K(p)\) be perturbed by a parameter p, which varies over \((Z_{1}, d_{1})\), that is, \(K : Z_{1}\to2^{X}\) is a setvalued mapping with nonempty, closed, and convex values. Let F be perturbed by a parameter z, which varies over \((Z_{2}, d_{2})\), that is, \(F:Z_{2}\times K\times K\to2^{Y}\) is a parametric setvalued mapping.
Consider the perturbed second type strong vector setvalued equilibrium problems, denoted by (PSSVSEP), which consists in finding \(\bar{x}\in K(p)\) such that
Its dual problem is to find \(\bar{x}\in K(p)\) such that
We denote the solution set of (PSSVSEP) and solution set of (PDSSVSEP) by \(S_{K(p)}(z)\) and \(S^{D}_{K(p)}(z)\), respectively.
Definition 4.1
Let X and Y be topological spaces. A setvalued mapping \(T: X\to2^{Y}\) is said to be

(i)
upper semicontinuous at \(x_{0}\in X\) if and if only, for any neighborhood \(U(T(x_{0}))\) of \(T(x_{0})\), there exists a neighborhood \(U(x_{0})\) of \(x_{0}\) such that
$$T(x)\subset U\bigl(T(x_{0})\bigr),\quad \forall x\in U(x_{0}); $$ 
(ii)
lower semicontinuous at \(x_{0}\in X\) if and if only, for any \(y_{0}\in T(x_{0})\) and any neighborhood \(U(y_{0})\) of \(y_{0}\), there exists a neighborhood \(U(x_{0})\) of \(x_{0}\) such that
$$T(x)\cap U(y_{0})\neq\emptyset,\quad \forall x\in U(x_{0}); $$ 
(iii)
continuous at \(x_{0}\) if and if only it is both upper and lower semicontinuous at \(x_{0}\).
It is evident that T is lower semicontinuous at \(x_{0}\in Y\) if and if only, for any net \(\{x_{\beta}\}\) with \(x_{\beta}\to x_{0}\) and \(y_{0}\in T(x_{0})\), there exists a net \(\{y_{\beta}\} \) with \(y_{\beta}\in T(x_{\beta})\) such that \(y_{\beta}\to y_{0}\).
The following lemma will be useful for proving our results.
Lemma 4.1
[26]
Let \((Z, d)\) be a metric space and \(u_{0}\in Z\) be a given point. Let \(L: Z\to2^{X}\) be a setvalued mapping with nonempty values and be upper semicontinuous at \(u_{0}\). Then there exists a neighborhood U of \(u_{0}\) such that \(L(u)^{\infty}\subset L(u_{0})^{\infty}\) for all \(u\in U\).
Theorem 4.1
Let \((Z_{1}, d_{1})\), \((Z_{2}, d_{2})\) be metric spaces and let \(K :Z_{1}\to2^{X}\) be a setvalued mapping with nonempty, closed, and convex values. Suppose that

(i)
\(K(p)\) is continuous at \(p_{0}\) and \(\operatorname{int}(\operatorname{barr}K(p))\neq\emptyset\), \(\forall p\in Z_{1}\);

(ii)
\(F(z,x,x)\subseteq C\capC\), \(\forall x \in K(p)\), \(z\in Z_{2}\);

(iii)
the set \(\{(z,y)\in(Z_{2},K(p)): F(z,x,y)\capC\neq\emptyset \}\) is closed for any \(x\in K(p)\); F is Cconvex in the third argument for any \(x\in K(p)\) and \(z\in Z_{2}\), and F is strongly type II Cdiagonally quasiconvex in the second argument for any \(z\in Z_{2}\), \(y\in K(p)\);

(iv)
\(S^{D}_{K(p_{0})}(z_{0})\) is nonempty and bounded.
Then

(I)
there exists a neighborhood \(U_{1}\times U_{2}\) of \((p_{0},z_{0})\) such that (PDSSVSEP) has a nonempty and bounded solution set for all \((p,z)\in(U_{1}, U_{2})\);

(II)
\(\limsup_{(p,z)\to(p_{0},z_{0})} S^{D}_{K(p)}(z)\subseteq S^{D}_{K(p_{0})}(z_{0})\).
Proof
Since \(S^{D}_{K(p_{0})}(z_{0})\) is nonempty and bounded, from Theorem 3.1, we deduce
We claim that there exists a neighborhood \((U_{1},U_{2})\) of \((p_{0},z_{0})\) such that
Assume by contradiction that there exists \((p_{n},z_{n})\to(p_{0},z_{0})\) such that \(R_{2}(p_{n})\neq\{0\}\). Noting that K is lower semicontinuous at \(p_{0}\), for any \(y\in K(p_{0})\), we have \(y_{n}\in K(p_{n})\) such that \(y_{n}\to y\). Together with \(z_{n}\to z\), we have \((y_{n},z_{n})\to(y,z_{0})\). Thus, we can select a sequence \(\{d_{n}\}\) such that
with \(\d_{n}\= 1\) for all \(n=1,2,\ldots, n\). As X is reflexive, without loss of generality, we can assume that \(d_{n}\rightharpoonup d_{0}\). It follows from Lemma 2.2 that \(d_{0}\neq0\). We claim that \(d_{0}\in K(p_{0})^{\infty}\). Since K is upper semicontinuous at \(p_{0}\) and \(d_{n}\in K(p_{n})^{\infty}\), from Lemma 4.1, one has \(d_{n}\in K(p_{0})^{\infty}\), for all sufficiently large n. By the closure of \(K(p_{0})^{\infty}\), we have \(d_{0}\in K(p_{0})^{\infty}\). Noticing assumption (iii), taking the limit in (4.2), we have
which contradicts (4.1). Thus, the result (I) holds.
For the result (II), we need to prove that for any \((p,z)\to(p_{0},z_{0})\),
Let \(x\in\limsup_{(p,z)\to(p_{0},z_{0})} S^{D}_{K(p)}(z)\). Then there exists a sequence \(x_{n_{k}}\in S^{D}_{K(p_{n_{k}})}(z_{n_{k}})\) such that \(x_{n_{k}}\rightharpoonup x\) as \(k \in\infty\). Since K is upper semicontinuous at \(p_{0}\), we obtain
where B denotes the closed unit ball. This together with \(x_{n_{k}}\in K(p_{n_{k}})\) implies that
Since \(x_{n_{k}}\rightharpoonup x \) and \(K(p_{0})\) is closed and convex, one has \(x\in K(p_{0})\).
For any \(y\in K(p_{0})\), it follows from the lower semicontinuity of K at \(p_{0}\) that there exists \(y_{n}\in K(p_{n})\) with \(y_{n}\to y\). By the assumption \(x_{n_{k}} \in S^{D}_{K(p_{n_{k}})}(z_{n_{k}})\), we have
From assumption (iii), taking the limit in (4.3), one has
This yields \(x\in S^{D}_{K(p_{0})}(z_{0})\).â€ƒâ–¡
Remark 4.1

(i)
In Theorem 4.1, F is vector setvalued mapping, including the scalar variational inequality and scalar equilibrium problems. Thus, Theorem 4.1 generalizes and extends Theorem 4.1 of [23], Theorem 3.2 of [26], and Theorem 4.1 of [27] in some sense.

(ii)
We establish the stability theorem for (DSSVSEP) in Theorem 4.1, when the mapping F and the domain set K are simultaneously perturbed by different parameters. Theorem 4.1 of [23] and Theorem 4.1 of [27] only show that the mapping F or the domain set K is perturbed, respectively.
As a consequence of Theorem 3.1 and Theorem 4.1, the following results follow immediately.
Corollary 4.1
Let \((Z_{1}, d_{1})\), \((Z_{2}, d_{2})\) be metric spaces and let \(K :Z_{1}\to2^{X}\) be a setvalued mapping with nonempty, closed, and convex values. Suppose that

(i)
\(K(p)\) is continuous at \(p_{0}\) and \(\operatorname{int}(\operatorname{barr}K(p))\neq\emptyset\), \(\forall p\in Z_{1}\);

(ii)
\(F(z,x,x)\subseteq C\capC\), \(\forall x \in K(p)\), \(z\in Z_{2}\);

(iii)
the set \(\{(z,x)\in(Z_{2},K(p)): F(z,x,y)\capC\neq\emptyset \}\) is closed for any \(y\in K(p)\); \(F(z,\cdot,\cdot)\) is type II Cpseudomonotone for any \(z\in Z_{2}\);

(iv)
the set \(\{(z,y)\in(Z_{2},K(p)): F(z,x,y)\capC\neq\emptyset\} \) is closed for any \(x\in K(p)\); F is Cconvex in the third argument for any \(x\in K(p)\) and \(z\in Z_{2}\), and F is strongly type II Cdiagonally quasiconvex in the second argument for any \(z\in Z_{2}\), \(y\in K(p)\);

(v)
\(S_{K(p_{0})}(z_{0})\) is nonempty and bounded.
Then

(I)
there exists a neighborhood \(U_{1}\times U_{2}\) of \((p_{0},z_{0})\) such that (PSSVSEP) has a nonempty and bounded solution set for all \((p,z)\in(U_{1}, U_{2})\);

(II)
\(\limsup_{(p,z)\to(p_{0},z_{0})} S_{K(p)}(z)\subseteq S_{K(p_{0})}(z_{0})\).
Consider the perturbed the first type strong vector setvalued equilibrium problems, denoted by (PFSVSEP), which consists in finding \(\bar{x}\in K(p)\) such that
Its dual problem is to find \(\bar{x}\in K(p)\) such that
We denote the solution set of (PFSVSEP) and the solution set of (PDFSVSEP) by \(SS_{K(p)}(z)\) and \(SS^{D}_{K(p)}(z)\), respectively.
Similar to the proof of Theorem 4.1, we have the following stability results for (FSVSEP) and (DFSVSEP).
Theorem 4.2
Let \((Z_{1}, d_{1})\), \((Z_{2}, d_{2})\) be metric spaces and let \(K :Z_{1}\to2^{X}\) be a setvalued mapping with nonempty, closed, and convex values. Suppose that

(i)
\(K(p)\) is continuous at \(p_{0}\) and \(\operatorname{int}(\operatorname{barr}K(p))\neq\emptyset\), \(\forall p\in Z_{1}\);

(ii)
\(F(z,x,x)\subseteq C\capC\), \(\forall x \in K(p)\), \(z\in Z_{2}\);

(iii)
the set \(\{(z,y)\in(Z_{2},K(p)): F(z,x,y)\subseteqC\}\) is closed for any \(x\in K(p)\); F is Cconvex in the third argument for any \(x\in K(p)\), \(z\in Z_{2}\), and F is strongly type I Cdiagonally quasiconvex in the second argument for any \(z\in Z_{2}\), \(y\in K(p)\);

(iv)
\(SS^{D}_{K(p_{0})}(z_{0})\) is nonempty and bounded.
Then

(I)
there exists a neighborhood \(U_{1}\times U_{2}\) of \((p_{0},z_{0})\) such that (PDFSVSEP) has a nonempty and bounded solution set for all \((p,z)\in(U_{1}, U_{2})\);

(II)
\(\limsup_{(p,z)\to(p_{0},z_{0})} SS^{D}_{K(p)}(z)\subseteq SS^{D}_{K(p_{0})}(z_{0})\).
As a consequence of Theorem 3.2 and Theorem 4.2, the following results follow immediately.
Corollary 4.2
Let \((Z_{1}, d_{1})\), \((Z_{2}, d_{2})\) be metric spaces and let \(K :Z_{1}\to2^{X}\) be a setvalued mapping with nonempty, closed, and convex values. Suppose that

(i)
\(K(p)\) is continuous at \(p_{0}\) and \(\operatorname{int}(\operatorname{barr}K(p))\neq\emptyset\), \(\forall p\in Z_{1}\);

(ii)
\(F(z,x,x)\subseteq C\capC\), \(\forall x \in K(p)\), \(z\in Z_{2}\);

(iii)
the set \(\{(z,x)\in(Z_{2},K(p)): F(z,x,y)\subseteqC\}\) is closed for any \(y\in K(p)\); \(F(z,\cdot,\cdot)\) is type I Cpseudomonotone for any \(z\in Z_{2}\);

(iv)
the set \(\{(z,y)\in(Z_{2},K(p)): F(z,x,y)\subseteqC\}\) is closed for any \(x\in K(p)\); F is Cconvex in the third argument for any \(x\in K(p)\), \(z\in Z_{2}\), and F is strongly type I Cdiagonally quasiconvex in the second argument for any \(z\in Z_{2}\), \(y\in K(p)\);

(v)
\(SS_{K(p_{0})}(z_{0})\) is nonempty and bounded.
Then

(I)
there exists a neighborhood \(U_{1}\times U_{2}\) of \((p_{0},z_{0})\) such that (PFSVSEP) has a nonempty and bounded solution set for all \((p,z)\in(U_{1}, U_{2})\);

(II)
\(\limsup_{(p,z)\to(p_{0},z_{0})} S_{K(p)}(z)\subseteq S_{K(p_{0})}(z_{0})\).
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Acknowledgements
This research is supported by the Specialized Research Fund for the Doctoral Program of Higher Education of China (20113705110002, 20113705120004), China Postdoctoral Science Foundation funded project (2013M531566) and Promotive Research Fund for Young and Middleaged Scientists of Shandong Province (BS2012SF008), the Natural Science Foundation of China (61403228, 11171180, 11401438). The author would like to thank the reviewers for their careful reading, insightful comments, and constructive suggestions, which helped improve the presentation of the paper.
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Wang, G. Existencestability theorems for strong vector setvalued equilibrium problems in reflexive Banach spaces. J Inequal Appl 2015, 239 (2015). https://doi.org/10.1186/s136600150760y
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DOI: https://doi.org/10.1186/s136600150760y