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Connectedness of approximate solutions set for vector equilibrium problems in Hausdorff topological vector spaces
Fixed Point Theory and Applicationsvolume 2011, Article number: 36 (2011)
Abstract
In this paper, a scalarization result of εweak efficient solution for a vector equilibrium problem (VEP) is given. Using this scalarization result, the connectedness of εweak efficient and εefficient solutions sets for the VEPs are proved under some suitable conditions in real Hausdorff topological vector spaces. The main results presented in this paper improve and generalize some known results in the literature.
1 Introduction
Let K be a nonempty subset of a real Hausdorff topological vector space E, and f : K × K → R a bifunction such that f(x, x) ≥ 0 for all x ∈ K. Then, the scalar equilibrium problem consists in finding $\stackrel{\u0304}{x}\in K$ such that
It provides a unifying framework for many important problems, such as optimization problems, variational inequality problems, complementary problems, minimax inequality problems, Nash equilibrium problems, and fixed point problems, and has been widely applied to study problems arising in economics, mechanics, and engineering science (see [1]). On the other hand, several operations research problems are formulated with a multicriteria consideration. These are vector optimization problems, vector variational inequality and complementarity problems and vector equilibrium problems (VEPs). Recently, the VEP has received much attention by many authors because it provides a unified model including vector optimization problems, vector variational inequality problems, vector complementarity problems and vector saddle point problems as special cases (see, for example, [2–24] and the references therein).
It is well known that another important problem for VEPs is to study the topological properties of solutions set. Among its topological properties, the connectedness is of interest. Recently, Lee et al. [25], Cheng [26] have studied the connectedness of weak efficient solutions set for vector variational inequalities in finite dimensional Euclidean space. Gong [27–29] has studied the connectedness of the various solutions set for VEPs in infinite dimension space. Chen et al. [30] studied the connectedness and the compactness of the weak efficient solutions set for setvalued VEPs and the setvalued vector HartmanStampacchia variational inequalities in normed linear space. Gong and Yao [31] have studied the connectedness of the set of efficient solutions for generalized systems. Zhong et al. [32] have studied the connectedness and pathconnectedness of solutions set for symmetric VEPs. However, the connectedness of approximate solutions set for VEPs remained unstudied.
In this paper, we show a scalarization result of εweak efficient solution for a VEP. Using this scalarization result, we discuss the connectedness of εweak efficient and εefficient solutions sets for VEPs under some suitable conditions in real Hausdorff topological vector spaces. The main results presented in this paper generalize some known results due to Gong [27] and Gong and Yao [31].
2 Preliminaries
Throughout this paper, let X and Y be two real Hausdorff topological vector spaces and A a nonempty subset of X. Let F : A × A → Y be a mapping and C be a closed convex pointed cone in Y. The cone C induces a partial ordering in Y, defined by
Let
be the dual cone of C.
Denote the quasiinterior of C* by C^{#}, that is,
Let D be a nonempty subset of Y. The cone hull of D is defined as
A nonempty convex subset B of the convex cone C is called a base of C if
It is easy to see that C^{#} ≠ ∅ if and only if C has a base.
Throughout this paper, we always assume that intC ≠ ∅. Let e be a fixed point in intC and we set
Now, we give the concepts of εweak efficient solution, εefficient solution, and εf efficient solution for the VEP.
Definition 2.1. A vector x ∈ A satisfying
is called a εweak efficient solution to the VEP. Denote by V_{εW}(A, F) the set of all εweak efficient solutions to the VEP.
Definition 2.2. A vector x ∈ A satisfying
is called a εefficient solution to the VEP. Denote by V_{ ε } (A, F) the set of all εefficient solutions to the VEP.
Definition 2.3. Let f ∈ C*\{0}. A vector x ∈ A is called a εf efficient solution to the VEP if
Denote by V_{εf}(A, F) the set of all εf efficient solutions to the VEP.
Definition 2.4. [33] Let G be a setvalued map from a topological space W to another topological space Q. We say that G is

(i)
upper semicontinuous at x_{0} ∈ W if, for any neighborhood U(G(x_{0})) of G(x_{0}), there is a neighborhood U (x_{0}) of x_{0} such that G(x) ⊂ U (G(x_{0})) for all x ∈ U(x_{0});

(ii)
upper semicontinuous on W if it is upper semicontinuous at each x ∈ W;

(iii)
lower semicontinuous at x_{0} ∈ W if, for any net {x_{α} : α ∈ I} converging to x_{0} and for any y_{0} ∈ G(x_{0}), there exists a net y_{ α } ∈ G(x_{ α }) that converges to y_{0};

(iv)
lower semicontinuous on W if it is lower semicontinuous at each x ∈ W;

(v)
continuous on W if it is both upper semicontinuous and lower semicontinuous on W;

(vi)
closed if Graph(G) = {(x, y) : x ∈ W, y ∈ G(x)} is a closed subset in W × Q.
3 Scalarization
Lemma 3.1. Suppose F (x, A) + C is a convex set for each x ∈ A. Then,
Proof. We first prove that
In fact, letting ${x}_{0}\in {\bigcup}_{f\in {C}^{\prime}}\phantom{\rule{2.77695pt}{0ex}}{V}_{\in f}\left(A,\phantom{\rule{2.77695pt}{0ex}}F\right)$, then there exists f ∈ C',
We claim that x_{0} ∈ V_{εW}(A, F). If not, then there exists y_{0} ∈ A such that F (x_{0}, y_{0}) ∈ intC  εe. Thus, we have
and so
which is a contradiction to (3.1).
We next prove that
Let x ∈ V_{εW}(A, F). Then, F (x, A) ∩ (intC  εe) = ∅. Since C is a pionted convex cone, we have
By assumption, we know that F (x, A) + C is a convex set. Using the separation theorem for convex sets, there exists some g ∈ Y *\{0}, such that
From (3.2), we get g ∈ C*\{0} and so
where e ∈ intC with g (e) > 0. Letting $f=\frac{g}{g\left(e\right)}$, then f (F (x, y)) ≥ ε, for all y ∈ A and f (e) = 1. Thus, f ∈ C' and so
This completes the proof.
Remark 3.1 When ε = 0, it is easy to see that
Therefore, Lemma 3.1 generalizes Lemma 2.1 in [27].
4 Existence of the solutions
Definition 4.1. The bifunction F : A × A → Y is concavelike with respect to the first variable, if for t ∈ [0, 1], the following condition is satisfied: For x_{1}, x_{2} ∈ A, there exists x_{3} ∈ A, such that
The bifunction F : A×A → Y is convexlike with respect to the second variable, if for t ∈ [0, 1], the following condition is satisfied: for y_{1}, y_{2} ∈ A, there exists y_{3} ∈ A, such that
Theorem 4.1. Let A be a nonempty compact subset of X, and f ∈ C'. Assume that the following conditions are satisfied:

(i)
F (x, x) ∈ C  εe, for all x ∈ A;

(ii)
F : A × A → Y is concavelike with respect to the first variable and convexlike with respect to the second variable;

(iii)
For each fixed y ∈ A, the function x ↦ f (F (x, y)) is upper semicontinuous on A.
Then, V_{εf}(A, F) ≠ ∅.
Proof. Define the setvalued map G : A → 2^{A} by
By assumption, y ∈ G (y), for all y ∈ A, so G (y) ≠ ∅. By assumption, we can see that G (y) is a closed subset of A. Next, we prove that ∩{G (y) : y ∈ A} ≠ ∅. Since A is a compact, we need to show that $\bigcap _{i=1}^{n}G\left({y}_{i}\right)\ne \varnothing $ for any arbitrary chosen y_{1},..., y_{ n } in A. Suppose it is not true. Then, there exists a set B = {y_{1},..., y_{ n }} ⊂ A such that $\bigcap _{i=1}^{n}G\left({y}_{i}\right)\ne \varnothing $. Thus, for any x ∈ A, there exists y_{ i } ∈ B such that x ∉ G (y_{ i }). It follows that
and so there exists η_{ i } > 0 such that
Since x ↦ f (F (x, y)) is upper semicontinuous on A, we can choose η > 0 such that, for any x ∈ A, there exists y_{ i } ∈ B satisfying
Define g : A → R^{n} by
where x ∈ A. We get
Since f ∈ C' and F (x, y) is concavelike with respect to the first variable, we can see that, for t ∈ [0, 1], x_{1}, x_{2} ∈ A, there exists x_{3} ∈ A such that
This shows that $g\left(A\right)+{R}_{+}^{n}$ is a convex set. It follows from (4.1) that
By the separation theorem of convex sets (see, for example, [34]), we can find t_{1},..., t_{ n } ≥ 0 with $\sum _{i=1}^{n}{t}_{i}=1$ such that
It follows that
By assumption, there exists y ∈ A such that
Since f ∈ C', we have
So f (F (x, y)) ≤ ε  η < ε, for all x ∈ A. Setting x = y, it follows that
On the other hand, by the assumption,
This is a contradiction. Therefore, ∩{G (y) : y ∈ A} ≠ ∅, and so there exists x ∈ ∩{G (y) : y ∈ A}. This means that V_{εf}(A, F) ≠ ∅. This completes the proof.
5 Connectedness of the solutions set
In this section, we discuss the connected results of εweak efficient solutions set and εefficient solutions set.
Definition 5.1. Let A be a convex set. The bifunction F : A × A → Y is Cconcave with respect to the first variable, if for t ∈ [0, 1], x_{1}, x_{2} ∈ A,
It is clear that when A is a convex set, F : A×A → Y is Cconcave with respect to the first variable, then it is concavelike about the first variable.
Theorem 5.1 Let A be a nonempty compact convex subset of X, and f ∈ C'. Assume that the following conditions are satisfied.
(i) F (x, x) ∈ C  εe, for all x ∈ A;
(ii) F : A × A → Y is Cconcave with respect to the first variable and convexlike with respect to the second variable;
(iii) For each fixed y ∈ A, the function x ↦ f (F (x, y)) is upper semicontinuous on A;
(iv) D = {F (x, y) : x, y ∈ A} is a bounded set of Y.
Then, V_{εW}(A, F) is a connected set.
Proof. Define a setvalued mapping H : C' → 2^{A} by
By Theorem 4.1, we know that, for each f ∈ C', V_{εf}(A, F) ≠ ∅. It is easy to see that C' is a convex set and so is connected. Next, we prove that, for each f ∈ C', H (f) is a connected set. Let x_{1}, x_{2} ∈ H (f), we have x_{1}, x_{2} ∈ A, and
Because F : A ×A → Y is Cconcave with respect to the first variable, for each fixed y ∈ A, t ∈ [0, 1],
Hence
It follows from (5.1) that
Hence
and so H (f) is a convex set. Thus, it is a connected set.
Next, we show that H is upper semicontinuous on C'. Since A is compact, we only need to show that H is closed. Let {(f_{ α }, x_{ α }) : α ∈ I} ⊂ Graph (H) be a net such that (f_{ α }, x_{ α }) → (f, x), where f_{ α } → f means that {f_{ α }} converges to f with respect to the strong topology β (Y*, Y) in Y*. Since C' is a closed set and A is a compact set, we know that (f, x) ∈ C' × A. From x_{ α } ∈ H (f_{ α }) = V_{ε fα}(A, F), we have
For any δ > 0,
is a neighborhood of 0 with respect with to β (Y *, Y). Since f_{ α } → f, there exists α_{0} ∈ I such that f_{ α }  f ∈ U, for all α ≥ α_{0}. It follows that
Therefore, for any y ∈ A,
and so
Because x ↦ f (F (x, y)) is upper semicontinuous on A, then
From (5.2), (5.3) and (5.4), we have
Hence, x ∈ H (f) = V_{εf}(A, F). By Theorem 3.1 in [35], $\phantom{\rule{2.77695pt}{0ex}}{\bigcup}_{f\in {C}^{\prime}}\phantom{\rule{2.77695pt}{0ex}}{V}_{\epsilon f}\left(A,\phantom{\rule{2.77695pt}{0ex}}F\right)$ is a connected set. Because F : A × A → Y is convexlike with respect to the second variable, we have F (x, A) + C is a convex set, by Lemma 3.1,
is a connected set. This completes the proof.
Next, we give an example to illustrate Theorem 5.1.
Example 5.1 Let X = R, Y = R^{2}, $C={R}_{+}^{2}=\left\{\left({x}_{1},\phantom{\rule{2.77695pt}{0ex}}{x}_{2}\right):{x}_{1}\ge 0,\phantom{\rule{2.77695pt}{0ex}}{x}_{2}\ge 0\right\}$, ε = 2, e = (2, 3), and A = [0, 2]. Let
Then, F satisfies all conditions of Theorem 5.1. It is easy to see that ${V}_{\epsilon W}\left(A,\phantom{\rule{2.77695pt}{0ex}}F\right)=\left[0,\phantom{\rule{2.77695pt}{0ex}}\sqrt{2}\right]$. Clearly, V_{εW}(A, F) is a nonempty connected set.
Definition 5.2. The bifunction F : A ×A → Y is εC strictly monotone if, for any x, y ∈ A, x ≠ y,
Theorem 5.2 Suppose that all conditions of Theorem 5.1 are satisfied and F : A × A → Y is εC strictly monotone. Then, V_{εW}(A, F) is a path connected set.
Proof. Define the setvalued mapping H : C' → 2^{A} by
By Theorem 4.1, we know that, for each f ∈ C', V_{εf}(A, F) ≠ ∅. Furthermore, because F : A × A → Y is εC strictly monotone, it is easy to see that, for each f ∈ C', H (f) = V_{εf}(A, F) is a single point set. From the proof of Theorem 5.1, we know that H is upper semicontinuous on C' and so it is continuous on C'. Since C' is a convex set, so it is a path connected set. Hence,
is a path connected set. This completes the proof.
Next, we give an example to illustrate Theorem 5.2.
Example 5.2 Let X = R, Y = R^{2}, $C={R}_{+}^{2}=\left\{\left({x}_{1},\phantom{\rule{2.77695pt}{0ex}}{x}_{2}\right):{x}_{1}\ge 0,\phantom{\rule{2.77695pt}{0ex}}{x}_{2}\ge 0\right\}$, ε = 1, e = (1, 2), and A = [1, 1]. Let
Then, F satisfies all the conditions of Theorem 5.2. It is easy to see that V_{εW}(A, F) = {0}. Clearly, V_{εW}(A, F) is a nonempty path connected set.
Next, we give a lemma before we give the connectedness theorem of εefficient solutions set.
Lemma 5.1. Suppose that all conditions of Theorem 5.2 are satisfied, then
Proof. By Theorem 4.1, we know that, for each f ∈ C', V_{εf}(A, F) ≠ ∅. By definition, we have
From Lemma 3.1, we have
By (5.5) and (5.6), we have
Next, we prove that
Define the setvalued mapping: H : C' → 2^{A} by
From Theorem 5.2, we know that H (f) is a singlevalued mapping, and H is continuous on C'.
Let $x\in {\bigcup}_{f\in {C}^{\prime}}\phantom{\rule{2.77695pt}{0ex}}{V}_{\epsilon f}\left(A,\phantom{\rule{2.77695pt}{0ex}}F\right)$. Then, there exists f ∈ C', such that
Let g ∈ C", and set
Then, f_{ n } ∈ C^{#} and f_{ n } (e) = 1. Thus, f_{ n } ∈ C".
Next, we show that {f_{ n }} converges to f with respect to the topology β (Y*, Y). For any neighborhood of 0 with respect to β (Y*, Y), there exist bounded subsets B_{ i } ∈ Y (i = 1, 2,..., m) and δ > 0 such that
Since B_{ i } is bounded and g  f ∈ Y *, it is easy to see that (g  f) (B_{ i }) is bounded for i = 1, 2,..., m. This implies that there exists N such that
Hence, $\frac{1}{n}\left(gf\right)\in U$, that is f_{ n }  f ∈ U. Hence, {f_{ n }} converges to f with respect to β (Y*, Y).
Since H is continuous on f, we have H (f_{ n }) → H (f). Set {x_{ n }} = H (f_{ n }), then
Because {x} = H (f), we have x_{ n } → x. This implies that
Since $x\in {\bigcup}_{f\in {C}^{\prime}}\phantom{\rule{2.77695pt}{0ex}}{V}_{\epsilon f}\left(A,\phantom{\rule{2.77695pt}{0ex}}F\right)$ is arbitrary, we have
Therefore,
This completes the proof.
Theorem 5.3. Suppose that all the conditions of Theorem 5.2 are satisfied. Then, V_{ ε }(A, F) is a connected set.
Proof. By Lemma 5.1, we have
From Theorem 5.2, we can get
is connected set and so (5.7) implies that V_{ ε } (A, F) is a connected set. This completes the proof.
Remark 5.1 When ε = 0, we can get
Therefore, Theorem 5.3 generalizes Theorem 2.2 in [31].
Next, we give an example to illustrate Theorem 5.3.
Example 5.3 Let X = R, Y = R, C = R_{+}, ε = 1, e = 1, and A = [1, 2]. Let F (x, y) = x (y  x)  1 for all x, y ∈ A. Then, it is easy to check that all the conditions of Theorem 5.3 are satisfied and
Clearly, V_{ ε } (A, F) is a nonempty connected set.
Abbreviations
 VEP:

vector equilibrium problem.
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Acknowledgements
This work was supported by the National Natural Science Foundation of China (50874096, 70831005) and the Open Fund (PLN0904) of State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation (Southwest Petroleum University).
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Authors' contributions
BC carried out the study of connectedness of εweak efficient and ε efficient solutions sets for VEPs and drafted the manuscript. QYL participated in the design of the study. ZBL gave some examples to show the main results. NJH conceived of the study, and participated in its design and coordination. All authors read and approved the final manuscript.
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Keywords
 vector equilibrium problem
 scalarization method
 εweak efficient solution
 εefficient solution
 connectedness