Existence-stability theorems for strong vector set-valued equilibrium problems in reflexive Banach spaces
- Gang Wang^{1}Email author
https://doi.org/10.1186/s13660-015-0760-y
© Wang 2015
Received: 5 May 2015
Accepted: 11 July 2015
Published: 30 July 2015
Abstract
In this paper, we develop existence and stability theorems for strong vector set-valued 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 set-valued equilibrium problems. Furthermore, stability results are established for strong vector set-valued equilibrium problems, when both the mapping and the constraint set are perturbed by different parameters.
Keywords
MSC
1 Introduction
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.
2 Notations and preliminaries
In this section, we introduce some basic notations and preliminary results.
It is easy to see \(SS_{K}\subseteq S_{K}\) and \(SS^{D}_{K}\subseteq S^{D}_{K}\).
Definition 2.1
[22]
The asymptotic cone \(K^{\infty}\) has the following useful properties.
Lemma 2.1
[22]
- (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]
- (i)type I C-pseudomonotone if, for all \(x,y \in K\),$$F(x,y)\subseteq C \quad \Rightarrow \quad F(y,x)\subseteq-C; $$
- (ii)type II C-pseudomonotone if, for all \(x,y \in K\),$$F(x,y)\cap C\neq\emptyset \quad \Rightarrow \quad F(y,x)\cap-C\neq\emptyset. $$
It is easy to verify that type I C-pseudomonotonicity implies type II C-pseudomonotonicity. However, the converse is not true.
Example 2.1
Definition 2.3
[24]
- (i)type I C-diagonally quasi-convex 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$$F(x_{i},x)\subseteq-C; $$
- (ii)type II C-diagonally quasi-convex 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$$F(x_{i},x)\cap-C\neq\emptyset; $$
- (iii)C-convex in the second argument if, for any \(y_{1},y_{2}\in K\) and \(t\in(0,1)\), one has$$t F(x,y_{1})+ (1-t)F(x,y_{2}) \subseteq F \bigl(x,ty_{1}+(1-t)y_{2}\bigr)+C. $$
The following lemma is the well-known KKM theorem, we refer the reader to Lemma 1 of [25].
Lemma 2.3
[25]
- (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 set-valued equilibrium problems to be nonempty and bounded based on asymptotic cone theory.
Theorem 3.1
- (i)
F is type II C-pseudomonotone and \(F(x,x)\subset C\cap-C\), \(\forall x \in K\);
- (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 C-diagonally quasi-convex in the first argument;
- (iii)
the set \(\{y\in K: F(x,y)\cap-C\neq\emptyset\}\) is closed for any \(x\in K\) and F is C-convex in the second argument.
- (I)
(SSVSEP) has a nonempty, convex, closed, and bounded solution set;
- (II)
(DSSVSEP) has a nonempty, convex, closed, and bounded solution set;
- (III)
\(R_{2}=\{d\in K_{\infty}: F(y,y+td)\cap-C\neq\emptyset, \forall y\in K, t>0\}=\{0\}\);
- (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$$F(y,x)\cap-C=\emptyset. $$
Proof
(I) ⇔ (II). Suppose that (SSVSEP) has a nonempty, convex, closed, and bounded solution set. By the type II C-pseudomonotonity of F, we obtain \(S_{K}\subseteq S^{D}_{K}\).
Remark 3.1
When F is a single-valued 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
Similar to the proof of Theorem 3.1, the following result holds for (FSVSEP).
Theorem 3.2
- (i)
F is type I C-pseudomonotone and \(F(x,x)\subset C\cap-C\), \(\forall x \in K\);
- (ii)
the set \(\{x\in K: F(x,y)\subseteq C\}\) is closed for any \(y\in K\) and F is strongly type I C-diagonally quasi-convex in the first argument;
- (iii)
the set \(\{y\in K: F(x,y)\subseteq-C\}\) is closed for any \(x\in K\) and F is C-convex in the second argument.
- (I)
(FSVSEP) has a nonempty, convex, closed, and bounded solution set;
- (II)
(DFSVSEP) has a nonempty, convex, closed, and bounded solution set;
- (III)
\(R_{1}=\{d\in K_{\infty}: F(y,y+td)\subseteq-C, \forall y\in K, t>0\}=\{0\}\);
- (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$$F(y,x)\nsubseteq-C. $$
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 set-valued 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 set-valued mapping.
Definition 4.1
- (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 set-valued 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
- (i)
\(K(p)\) is continuous at \(p_{0}\) and \(\operatorname{int}(\operatorname{barr}K(p))\neq\emptyset\), \(\forall p\in Z_{1}\);
- (ii)
\(F(z,x,x)\subseteq C\cap-C\), \(\forall x \in K(p)\), \(z\in Z_{2}\);
- (iii)
the set \(\{(z,y)\in(Z_{2},K(p)): F(z,x,y)\cap-C\neq\emptyset \}\) is closed for any \(x\in K(p)\); F is C-convex in the third argument for any \(x\in K(p)\) and \(z\in Z_{2}\), and F is strongly type II C-diagonally quasi-convex 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.
- (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
Remark 4.1
- (i)
In Theorem 4.1, F is vector set-valued 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
- (i)
\(K(p)\) is continuous at \(p_{0}\) and \(\operatorname{int}(\operatorname{barr}K(p))\neq\emptyset\), \(\forall p\in Z_{1}\);
- (ii)
\(F(z,x,x)\subseteq C\cap-C\), \(\forall x \in K(p)\), \(z\in Z_{2}\);
- (iii)
the set \(\{(z,x)\in(Z_{2},K(p)): F(z,x,y)\cap-C\neq\emptyset \}\) is closed for any \(y\in K(p)\); \(F(z,\cdot,\cdot)\) is type II C-pseudomonotone for any \(z\in Z_{2}\);
- (iv)
the set \(\{(z,y)\in(Z_{2},K(p)): F(z,x,y)\cap-C\neq\emptyset\} \) is closed for any \(x\in K(p)\); F is C-convex in the third argument for any \(x\in K(p)\) and \(z\in Z_{2}\), and F is strongly type II C-diagonally quasi-convex 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.
- (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)
\(\limsup_{(p,z)\to(p_{0},z_{0})} S_{K(p)}(z)\subseteq S_{K(p_{0})}(z_{0})\).
Similar to the proof of Theorem 4.1, we have the following stability results for (FSVSEP) and (DFSVSEP).
Theorem 4.2
- (i)
\(K(p)\) is continuous at \(p_{0}\) and \(\operatorname{int}(\operatorname{barr}K(p))\neq\emptyset\), \(\forall p\in Z_{1}\);
- (ii)
\(F(z,x,x)\subseteq C\cap-C\), \(\forall x \in K(p)\), \(z\in Z_{2}\);
- (iii)
the set \(\{(z,y)\in(Z_{2},K(p)): F(z,x,y)\subseteq-C\}\) is closed for any \(x\in K(p)\); F is C-convex in the third argument for any \(x\in K(p)\), \(z\in Z_{2}\), and F is strongly type I C-diagonally quasi-convex 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.
- (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
- (i)
\(K(p)\) is continuous at \(p_{0}\) and \(\operatorname{int}(\operatorname{barr}K(p))\neq\emptyset\), \(\forall p\in Z_{1}\);
- (ii)
\(F(z,x,x)\subseteq C\cap-C\), \(\forall x \in K(p)\), \(z\in Z_{2}\);
- (iii)
the set \(\{(z,x)\in(Z_{2},K(p)): F(z,x,y)\subseteq-C\}\) is closed for any \(y\in K(p)\); \(F(z,\cdot,\cdot)\) is type I C-pseudomonotone for any \(z\in Z_{2}\);
- (iv)
the set \(\{(z,y)\in(Z_{2},K(p)): F(z,x,y)\subseteq-C\}\) is closed for any \(x\in K(p)\); F is C-convex in the third argument for any \(x\in K(p)\), \(z\in Z_{2}\), and F is strongly type I C-diagonally quasi-convex 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.
- (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)
\(\limsup_{(p,z)\to(p_{0},z_{0})} S_{K(p)}(z)\subseteq S_{K(p_{0})}(z_{0})\).
Declarations
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 Middle-aged 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.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
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