On a new generalized symmetric vector equilibrium problem
 Rahmatollah Lashkaripour^{1} and
 Ardeshir Karamian^{1}Email author
https://doi.org/10.1186/s136600171511z
© The Author(s) 2017
Received: 2 August 2017
Accepted: 11 September 2017
Published: 22 September 2017
Abstract
In this paper, a new form of the symmetric vector equilibrium problem is introduced and, by mixing properties of the nonlinear scalarization mapping and the maximal element lemma, an existence theorem for it is established. We show that Ky Fan’s lemma, as a usual technique for proving the existence results for equilibrium problems, implies the maximal element lemma, while it is useless for proving the main theorem of this paper. Our results can be viewed as an extension and improvement of the main results obtained by Farajzadeh (Filomat 29(9):20972105, 2015) and some corresponding results that appeared in this area by relaxing the lower semicontinuity, quasiconvexity on the mappings and being nontrivial of the dual cones. Finally, some examples are given to support the main results.
Keywords
MSC
1 Introduction and preliminaries
Existence results for vector equilibrium problems (in short, VEP) have been extensively studied in recent years. It is wellknown that VEP provides a general model of several classes of problems such as the vector variational inequality, the vector complementarity problems, the vector optimization problems, the vector saddle point (minimax) problems, the multiobjective game problems, and fixed point problems (see, e.g., [2–9] and the references therein).
The symmetric vector equilibrium problem (in short, SVEP) which is a generalization of the vector equilibrium problem has been studied by many authors. One of the important symmetric vector equilibrium problems is to investigate the existence theorems in order to guarantee its solution set is nonempty (see, e.g., [10–12] and the references therein).
Recently, Farajzadeh [1] established some existence results for a solution of SVEP whose proof strongly depends on nontriviality of the dual cones, lower semicontinuity and quasiconvexity of the mappings (see Theorem 3.4 in [1]). In this paper, we first introduce a general form of SVEP (called GSVEP), and then we relax the nontriviality of the cones, lower semicontinuity and quasiconvexity that appeared in Theorem 3.4 in [1] by ‘mixing’ the properties of the nonlinear scalarization mapping and the maximal element lemma (see Lemma 2.5 and Theorem 3.1). It is worth noting that the hypotheses of lower semicontinuity and quasiconvexity on the mappings are common assumptions in most of papers that have appeared in this area.
In the rest of this section we introduce our problem and recall some notations which are needed in the next sections. Throughout the paper, unless otherwise specified, we use the following notations.
Let \(I_{m} \) denote the finite index set \(\lbrace1,2,\ldots,m \rbrace\). For each \(i \in I_{m} \), \(X_{i} \) and \(Y_{i} \) stand for topological vector spaces (for short, t.v.s.) and \(C_{i}\) is a proper, closed and convex cone of \(Y_{i}\) with \(\operatorname{int} C_{i}\neq\emptyset\), where \(\operatorname{int} C_{i}\) denotes the topological interior of \(C_{i}\).
The symbols \(X=\prod_{i \in I_{m}} X_{i}\) and \(C=\prod_{i \in I_{m}} C_{i} \) denote the Cartesian product of \(X_{i}\) and \(C_{i}\), respectively. So, for each \(x \in X\) and \(c \in C\), we have \(x=(x_{i})_{i \in I_{m}}\) and \(c=(c_{i})_{i \in I_{m}}\), where \(x_{i}\in X_{i}\) and \(c_{i}\in C_{i}\). It is well known that X and C are respectively t.v.s. and proper, closed and convex cone with \(\operatorname{int} C=\prod_{i \in I_{m}} \operatorname{int} C_{i}\). The dual cone of \(C_{i}\) is denoted by \(C_{i}^{\ast}\) and defined by \(C_{i}^{\ast}=\lbrace f\in Y_{i}^{\ast} : f(c_{i}) \geq0, \forall c_{i}\in C_{i} \rbrace\), where \(Y_{i}^{\ast}\) is the topological dual space of \(Y_{i}\). For each \(i \in I_{m}\), let \(K_{i} \) be a nonempty, closed and convex subset of \(X_{i}\) and \(F_{i}:\prod_{j \in I_{m} } K_{j} \times K_{i} \to 2^{Y_{i} } \) be a setvalued mapping with nonempty values, where \(2^{Y_{i} }\) denotes the class of all subsets of \(Y_{i} \). Now, we are ready to introduce the following problem which we call a generalized symmetric vector equilibrium problem (in short, GSVEP):
Remark 1.1
 (i)
If we take \(I_{2}=\lbrace1,2\rbrace\), then GSVEP collapses to the symmetric vector equilibrium problem; see [1] and the references therein.
 (ii)If we take \(I_{2}=\lbrace1,2\rbrace\), \(C_{1} = C_{2}\);and$$f_{1}:\prod_{i \in I_{2} } K_{i} \to X_{1} $$are two singlevalued mappings, and$$f_{2}:\prod_{i \in I_{2} } K_{i} \to X_{2} $$and$$F_{1}(x, y, z)=\bigl\lbrace f_{1}(z,y)f_{1}(x, y)\bigr\rbrace ,\quad \forall \bigl((x, y), z\bigr) \in\prod _{i \in I_{2}}K_{i} \times K_{1} $$then (1.1) reduces to the symmetric vector equilibrium problem which was studied in [10].$$F_{2}(x, y, z)=\bigl\lbrace f_{2}(x,z)f_{2}(x, y)\bigr\rbrace ,\quad \forall \bigl((x, y), z\bigr) \in\prod _{i \in I_{2}}K_{i} \times K_{2}, $$
 (iii)If we take \(I_{2}=\lbrace1,2\rbrace\), \(f_{1}: K_{1}\times K_{1}\rightarrow2^{Y_{1}}\), \(F_{2}=\lbrace0_{Y_{2}} \rbrace\) andwhere \(((x , y), z)\in\prod_{i\in I_{2} } K_{i} \times K_{1}\).$$F_{1}(x , y, z)=\bigl\lbrace f_{1}(x , z)\bigr\rbrace , $$
Then (1.1) is the vector equilibrium problem which was introduced by Blum and Oettli [13]. For more details, we refer to [5, 14–16] and the references therein.
 (iv)
We can state the classical vector variational inequality problem which was introduced by Giannessi [15] in the form of GSVEP as follows:
Take \(I_{2}=\lbrace1,2\rbrace\), \(T: K_{1}\rightarrow L(X_{1},X_{2})\) and defineand$$F_{1}\bigl((x_{1},x_{2}),y_{1}\bigr) = T(x_{1}) (y_{1}x_{1}),\quad \forall \bigl((x_{1},x_{2}),y_{1}\bigr)\in\prod _{i \in I_{2} } K_{i} \times K_{1} $$where \(L(X_{1},X_{2})\) denotes the space of all continuous linear operators from \(X_{1}\) to \(X_{2}\).$$F_{2}=\lbrace0_{Y_{2}}\rbrace, $$  (v)Finally, if we take \(I_{1}=\lbrace1\rbrace\), \(C_{1}=[0, +\infty]\), \(Y_{1}=\mathbb{R}\), \(K_{1}\subseteq\mathbb{R}\), then (1.1) reduces to the scalar equilibrium problem forwhich was studied by many authors (see, for example, [17, 18] and the references therein).$$F_{1}: K_{1}\times K_{1}\rightarrow2^{\mathbb{R}}, $$
2 Some notes on nonlinear scalarization mapping, Ky Fan’s lemma and maximal element lemma
In this section, we introduce the nonlinear scalarization mapping and some of its important properties. Also, the maximal element lemma (i.e., Lemma 2.5) and the notion of KKM mapping and Ky Fan’s lemma (i.e., Lemma 2.4) are stated. Moreover, we show that Lemma 2.5 and Ky Fan’s lemma are not equivalent. It is a remarkable fact that we cannot apply Ky Fan’s lemma, which plays an important role in the study of the existence results of equilibrium problems (see, e.g., [1, 19] and the references therein), when we work with the scalarization mapping, while Lemma 2.5 is a useful tool for proving our existence results.
The nonlinear scalarization mapping that plays a key role in the paper was first introduced in [20] in order to study the vector optimization theory and vector equilibrium problems (see [14]).
Definition 2.1
The following lemma characterizes some of the important properties of the nonlinear scalarization mapping which are used in the sequel.
Lemma 2.2
 (i)
\(\xi_{e} (x) = \min \{r\in\mathbb{R}: rex\in C \}\).
 (ii)
\(\xi_{e} (x) \leq r \Longleftrightarrow rex\in C \).
 (iii)
\(\xi_{e} (x) < r \Longleftrightarrow rex\in \operatorname{int} C \).
 (iv)
\(\xi_{e} (x) = r \Longleftrightarrow x \in re\partial C\), where ∂C is the topological boundary of C.
 (v)
\(y_{2}y_{1}\in C \Longrightarrow\xi_{e}(y_{1}) \leq\xi_{e}(y_{2})\).
 (vi)
The mapping \(\xi_{e} \) is continuous, positively homogeneous and subadditive (that is, sublinear) on X.
For proving an existence result of the equilibrium problems, Ky Fan’s lemma plays a key role. Now we are going to state it. Before stating it, we also need the following definition.
Definition 2.3
[25]
Let K be a nonempty subset of the topological vector space X. A setvalued mapping \(T:K\rightarrow2^{X}\) is called a KKMmapping if, for every finite subset \(\lbrace x_{1},x_{2},\ldots,x_{n}\rbrace\) of K, \(\operatorname{conv}\lbrace x_{1},x_{2},\ldots,x_{n}\rbrace\) is contained in \(\bigcup_{i=1}^{n}T(x_{i})\), where conv denotes the convex hull.
Ky Fan in 1984 obtained the following result, which is known as Ky Fan’s lemma.
Lemma 2.4
Ky Fan1984 [25]
Note that if we have a multivalued mapping A on a set K and there exists an element \(x\in K\) such that \(A(x)\) is empty, then, in a way, such element x is called ‘maximal’. The existence of maximal elements for a multivalued mapping in topological vector spaces and its important applications to mathematical economies have been studied by many authors in both mathematics and economies, see, for example, [26–28] and the references therein. Moreover, the maximal element lemma plays a crucial role in establishing the existence of solutions for GSVEP. In the following, for the sake of readers, we prove it by Ky Fan’s lemma.
Lemma 2.5
 (i)
for each \(x \in K\), \(A(x)\) is convex,
 (ii)
for each \(x \in K\), \(x \notin A(x)\),
 (iii)
for each \(y \in K\), \(A^{1} (y)=\{x\in K : y\in A(x)\}\) is open in K,
 (iv)there exist a nonempty compact convex subset B of K and a nonempty compact subset N of K such that$$A(x) \cap B \neq\emptyset,\quad \forall x\in K\backslash N. $$
Proof
Remark 2.6
Moreover, it follows from condition (ii) that the setvalued mapping A is never a KKM mapping. Hence we cannot use Ky Fan’s lemma for mapping A.
Lemma 2.7
Then \(\bigcap_{x\in K}T(x)\neq\emptyset\).
Proof
The following example illustrates Lemma 2.7.
Example 2.8
Let \(X=\mathbb{R}\), \(K=[0,1]\) and define \(T:K\longrightarrow2^{X}\) by \(Tx=[x,1]\). It is easy to check that T satisfies all the conditions of Lemma 2.7 and \(\bigcap_{x\in K}T(x)=\{1\}\).
The following lemma provides a link between a vector ordering and a scalar ordering, which is a useful tool for reducing a vector problem to the scalar problem.
Lemma 2.9
Proof
To prove the main results of this paper, we need the following lemma.
Lemma 2.10
Proof
It is sufficient, for each \(i\in I_{m}\), we take \(B_{i}=F_{i}((x_{j}^{*} )_{j \in I_{m} },y_{i} ) \). Now, the result follows by Lemma 2.9. □
3 Existence results
Now, we are ready to present an existence result of a solution for GSVEP by using the scalarization method and the maximal element lemma, which one can consider as an extension of the wellknown results in this area from SVEP to GSVEP. In fact, some sufficient conditions to guarantee the existence of the solution for GSVEP, by relaxing the lower semicontinuity, quasiconvexity on the mappings and without assuming the nontriviality (that is, each cone contains a nonzero element) of the dual cones of the spaces, which in most of references are assumed to be nontrivial, are given. Moreover, in order to illustrate the main theorem of this section, some examples are provided. We note that Ky Fan’s lemma is useless for proving Theorem 3.1 (see Remark 2.6).
Theorem 3.1
 (i)
for all \((x_{j})_{j \in I_{m}}\in\prod_{j \in I_{m}} K_{j} \), \(F_{i} ((x_{j})_{j \in I_{m}},x_{i} ) \cap\operatorname{int} C_{i}=\emptyset\), \(\forall i\in I_{m}\);
 (ii)for all \((x_{j})_{j\in I_{m}} \in\prod_{j\in I_{m}}K_{j}\), the setis convex, where \(\varPhi=\underbrace{(\emptyset,\emptyset, \ldots,\emptyset)}_{m \textit{ elements}}\);$$\biggl\lbrace (y_{i})_{i\in I_{m}} \in\prod _{j \in I_{m}} K_{j}: \prod_{i\in I_{m}}F_{i} \bigl((x_{j})_{j\in I_{m}},y_{i}\bigr)\cap\prod _{i\in I_{m}}(\operatorname{int} C_{i})\neq{\varPhi} \biggr\rbrace , $$
 (iii)for all \((y_{i})_{i\in I_{m}} \in\prod_{j \in I_{m}} K_{j}\), the setis open;$$\biggl\lbrace (x_{j})_{j\in I_{m}} \in\prod _{j \in I_{m}} K_{j}: \prod_{i\in I_{m}}F_{i} \bigl((x_{j})_{j\in I_{m}},y_{i}\bigr)\cap\prod _{i\in I_{m}}(\operatorname{int} C_{i})\neq{\varPhi} \biggr\rbrace ,$$
 (iv)there exist a nonempty compact convex subset \(\prod_{j \in I_{m} } B_{j}\) of \(\prod_{j \in I_{m} } K_{j} \) and a nonempty compact subset \(\prod_{j \in I_{m} } N_{j}\) of \(\prod_{j \in I_{m} } K_{j}\) such that for eachthere exists \((y_{i} )_{i \in I_{m}} \in\prod_{j \in I_{m}} B_{j}\) satisfying$$(x_{j} )_{j \in I_{m}} \in\prod_{j \in I_{m} } K_{j}\smallsetminus\prod_{j \in I_{m} } N_{j}, $$$$P_{e} \biggl(\prod_{i \in I_{m}} F_{i} \bigl((x_{j} )_{j \in I_{m}},y_{i}\bigr)\biggr)\nsubseteq \mathbb{R}_{+}^{m}. $$
Proof
We claim that the mapping A satisfies all the conditions of Lemma 2.5.
First, applying Lemma 2.9 and condition (ii), it is clear that \(A((x_{j})_{j \in I_{m}} )\) is a convex set for any \((x_{j})_{j\in I_{m}}\in\prod_{j \in I_{m}}K_{j}\).
Finally, applying condition (iv), there exist a nonempty compact convex subset \(B=\prod_{j \in I_{m}} B_{j}\) of \(\prod_{j\in I_{m}}K_{j}\) and a nonempty compact subset \(N=\prod_{j \in I_{m} } N_{j}\) of \(\prod_{j \in I_{m} } K_{j} \) such that, for any \((x_{j} )_{j \in I_{m}} \in\prod_{j \in I_{m} } K_{j}\smallsetminus N\), \((y_{i} )_{i \in I_{m}} \in A((x_{j})_{j \in I_{m}} ) \cap B \).
 (iii)′:

for all \((y_{i})_{i\in I_{m}} \in\prod_{j \in I_{m}} K_{j}\), the setis open.$$\bigcup_{i=1}^{m}\biggl\lbrace (x_{j})_{j\in I_{m}} \in\prod_{j \in I_{m}} K_{j}: F_{i}\bigl((x_{j})_{j\in I_{m}},y_{i} \bigr)\cap\operatorname{int} C_{i}\neq \emptyset\biggr\rbrace ,$$
The following example satisfies all the assumptions of Theorem 3.1, while \(F_{1}\) is not lower semicontinuous at \((\frac{1}{2}, \frac{1}{2},1)\). Because, if we take the open set \(U=(1,\frac {3}{2})\), then \((F_{1}(\frac{1}{2}, \frac{1}{2},1)=[\frac {1}{2},\frac{3}{2}])\cap U\neq\emptyset\). Now, for each neighborhood V of \((\frac{1}{2}, \frac{1}{2},1)\), we have \(F_{1}(w_{1},w_{2},w_{3})\cap U=\emptyset\), where \((w_{1},w_{2},w_{3})\in V\) and \(w_{1}\in Q^{c}\), \(w_{1}<\frac{1}{2}\), \(1< w_{2}\). This means that \(F_{1}\) is not lower semicontinuous at \((\frac{1}{2}, \frac{1}{2},1)\). Hence it does not satisfy all the assumptions of Theorem 3.4 of [1]. Thus Theorem 3.4 of [1] does not work for the example.
Example 3.2
Condition (i) trivially holds.
Remark 3.3
Obviously, when \(I_{m}=\lbrace1,2\rbrace\), conditions (i) and (ii) in Theorem 3.1 are weaker than conditions (i) and (ii) in Theorem 3.4 in [1]. Moreover, the proof of Theorem 3.1 is different from the proof of Theorem 3.4 given in [1]. Furthermore, in Theorem 3.4 in [1] the proof is strongly dependent on the nontriviality of the dual cones, while it has been relaxed in Theorem 3.1. There are many cones whose duals are trivial, for instance, see the following example. Hence we cannot apply Theorem 3.4 in [1] for it, while Theorem 3.1 enables us to discuss the existence of solution for it.
Example 3.4
For \(0< p<1\), let \(L^{p}[0,1]\) denote the set of all measurable functions \(f:[0,1]\longrightarrow\mathbb{R}\) with \(\int_{0}^{1} \vert f(x) \vert ^{p}<\infty\). It can be shown that the topological dual of \(L^{p}[0,1]\), i.e.,\((L^{p}[0,1])^{\ast}\), equals zero. So there is no cone in \((L^{p}[0,1])^{\ast}\) containing a nonzero element.
4 Conclusion
In this paper, first a new form of the symmetric vector equilibrium problem is introduced and the relationship between the maximal element lemma and Ky Fan’s lemma is discussed. Then, by mixing the properties of the nonlinear scalarization mapping and the maximal element lemma, an existence theorem for the generalized symmetric vector equilibrium problem (GSVEP), by relaxing or weakening some assumptions, is established. Moreover, under some suitable assumptions, the convexity of the solution set of GSVEP is studied. Finally, some examples are given in order to support the main results. The main results of this note can be viewed as an extension and improvement of the main results obtained by Farajzadeh (Filomat 29(9):20972105, 2015) and some corresponding results that appeared in this area.
Declarations
Funding
The authors declare that there is no source of funding for this research.
Authors’ contributions
The authors contributed equally to the manuscript and read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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Authors’ Affiliations
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