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Existence of Solutions for Generalized Vector Quasi-Equilibrium Problems by Scalarization Method with Applications
The aim of this paper is to study generalized vector quasi-equilibrium problems (GVQEPs) by scalarization method in locally convex topological vector spaces. A general nonlinear scalarization function for set-valued mappings is introduced, its main properties are established, and some results on the existence of solutions of the GVQEPs are shown by utilizing the introduced scalarization function. Finally, a vector variational inclusion problem is discussed as an application of the results of GVQEPs.
Recently, various vector equilibrium problems were investigated by adopting many different methods, such as the scalarization method (e.g., [1, 2]), the recession method (e.g., ), and duality method (e.g., ).
The scalarization method is an important and efficacious tool of translating the vector problems into the scalar problems. In 1992, Chen  translated a vector variational inequality into a classical variational inequality by providing a kind of solution conceptions with variable domination structures. Gerth and Weidner  solved a vector optimization problem by introducing a scalarization function with a variable and Gong  dealt with vector equilibrium problems by using the scalarization function defined in . By constructing new nonlinear scalarization functions with two variables, Chen and Yang  discussed a vector variational inequality and Chen et al.  investigated a generalized vector quasi-equilibrium problem (GVQEP), respectively. In addition, the authors in [8, 9] studied the systems of vector equilibrium problems by the scalarization method since the gap functions, indeed established by the nonlinear scalarization function defined in , were adopted.
In this paper, we will discuss the GVQEPs by utilizing scalarization method. The essential preliminaries are listed in Section 2. On the basis of the works in [2, 6, 7], a general nonlinear scalarization function of a set-valued mapping is produced under a variable ordering structure and its main properties are discussed in Section 3. The results of properties for the general nonlinear scalarization function generalized the corresponding ones in . In Section 4, some results on the existence of solutions of the GVQEPs are proved by employing the scalarization function introduced in Section 3. The GVQEPs are different from the one in  and include the one in  as a special case. It is worth mentioning that the existence results of solutions for the GVQEPs extend the corresponding one in . Finally, a vector variational inclusion problem (VVIP) is given as an application of the GVQEPs in Section 5.
Suppose that and are topological vector spaces (TVSs). The subset is called a cone, if for all and . A cone is said to be proper, if . is called -closed  if is closed and -bonded  if for each neighborhood of zero in , there exists such that . A set-valued mapping is called strict, if for any . Throughout this paper, denotes by the set of the real numbers. Several notations are also listed as follows: where are nonempty, and . Incidentally, every TVS is Hausdorff (see ).
Let and be topological spaces and a nonempty subset. A function is said to be upper semicontinuous (usc for brevity) on , if is open for each ; to be lower semicontinuous (lsc for brevity), is open for each . In addition, some known notions of continuity and closeness for a set-valued mapping are given (see ). A set-valued mapping is said to be usc at , if for any neighborhood of , there exists a neighborhood of such that for all ; to be lsc at , if for any and any neighborhood of , there exists a neighborhood of such that for all ; to be usc (resp., lsc) on , if is usc (resp., lsc) at each ; to be continuous at (resp., on ), if is usc and lsc at (resp., on ); to be closed, if its graph is closed in .
Let and be real TVSs, a nonempty subset, and a convex cone in . is called, vector minimal point (resp., weakly vector minimal point) of , if (resp., ) for each . The set of vector minimal points (resp., weakly vector minimal points) of is denoted by (resp., .
Definition 1. Let be nonempty convex subset. is called generalized -quasiconvex, if for any , the set is convex (here, is regarded as a convex set).
The generalized -quasiconvexity and the -quasiconvexity introduced in  for the set-valued mappings are both generalizations of -quasiconvexity for the single-valued mapping introduced in , but they are indeed distinct. See the following example.
Example 2. Let , and and define as Then is generalized -quasiconvex but not -quasiconvex, while is just the reverse. Indeed, for any fixed , both the sets are convex. But the set is not convex and neither is the set
Lemma 3 (see ). Let and be topological spaces and a set-valued mapping. (1) If is usc with closed values, then is closed.(2) If is compact and is usc with compact values, then is compact.
Lemma 4 (see ). Let and be Hausdorff topological spaces and a nonempty compact set and let be a function and a set-valued mapping. If is continuous on and is continuous with compact values, then the marginal function is continuous and the marginal set-valued mapping is usc.
Lemma 5 (Kakutani, see [13, Theorem 13 in Section 4 Chapter 6]). Let be a nonempty compact and convex subset of a locally convex TVS . If is usc and is a nonempty, convex, and closed subset for any , then there exists such that .
3. A General Nonlinear Scalarization Function
From now on, unless otherwise specified, let , and be real TVSs and and nonempty subsets. Let be a set-valued mapping such that for any , is a proper, closed, and convex cone with .
In this section, suppose that is a strict mapping with compact values and is a vector-valued mapping with . Obviously, is -closed [11, Definition 3.1 and Proposition 3.3] for each and . Then, in view of [16, Lemma ], we can define a general nonlinear scalarization function of as follows.
Definition 6. The general nonlinear scalarization function , of is defined as
Remark 7. If and , then the general nonlinear scalarization function of becomes the nonlinear scalarization function defined in .
Example 8. Let , and and let define as respectively. We can see that .
Definition 9. Let be a cone, , and . is called monotone (resp., strictly monotone) with respect to (wrt for brevity) , if for any , implies that ).
Obviously, the strict monotonicity wrt implies the monotonicity wrt . Moreover, if , then the (strict) monotonicity wrt of is equivalent to the (strict) monotonicity under the general order structures. The following examples illuminate the relationship between the monotonicity wrt and the monotonicity in the normal sense when is not the identity mapping.
Example 10. Let ,, and and let Then implies that . Thus is strictly monotone wrt , while is not monotone in the normal sense.
Example 11. Let , and and let , and . Then is strictly monotone in the normal sense, but is not monotone wrt . As the case stands, is equivalent to and , which just results in .
Now some main properties of the general nonlinear scalarization function are established. First, according to [16, Proposition 3.1], we have the following.
Theorem 12. For each and , the following assertions hold.(1). (2).
Theorem 13. is strictly monotone wrt in the second variable.
Proof. Letting such that and , we have It is from this assertion that by Theorem 12 (1).
Theorem 14. Suppose that is continuous.(1) If and are usc on and , respectively, where then is usc on .(2) If is usc on and is lsc on , then is lsc on .
Proof. (1) It is sufficient to attest the fact that for each , the set
is closed. As a matter of fact, for any sequence in such that as , , that is, , by Theorem 12 (1). Obviously, is compact and so is by Lemma 3 (2). Then there exists such that and a subsequence of , such that as and
Obviously, is usc with closed values, which implies that is closed by Lemma 3 (1). Hence, . Similarly, is closed. Letting in (12) and applying the continuity of and the closeness of , we obtain that . Namely, . Thus , which is equivalent to by Theorem 12 (1). Therefore, and is closed.
(2) It's enough to argue that for each , the set is closed. In fact, for any sequence in such that as , by Theorem 12 (2), , equivalently, . For any , there exists such that as according to the equivalent definition of lower semicontinuity of (see , page 108). Also, Linking the continuity of with the closeness of inferred from Lemma 3 (1) and letting in (14), we get and . Consequently, Theorem 12 (2) leads to which amounts to . is closed.
If ( is not required to be compact) and , then Theorem 14 becomes [2, Theorem 2.1]. Actually, [2, Theorem 2.1] can be regarded as the case where is continuous in Theorem 14. In addition, Example 2.1 (resp., Example 2.2) in  shows that if (resp., ) is not usc, maybe the general nonlinear scalarization function fails to be lsc (resp., usc) under all the other assumptions. Now the following example demonstrates that the assumption of the upper semicontinuity (resp., lower semicontinuity) of is necessary in Theorem 14 (1) (resp., (2)) even if is continuous.
Example 15. Let , , , and , and let
(1) Define Evidently, is not usc on . After simply calculating, is not usc on due to the fact that is not closed.
(2) Consider the following mapping: Obviously, is not lsc on . Also, fails to be lsc on , where
4. Existence Results on Solutions of the GVQEPs
In this section, further suppose that , and are locally convex. Let be a vector-valued mapping. Now two GVQEPs are characterized as follows: GVQEP1: Seek and such that where , and are set-valued mappings. GVQEP2: Find and such that where , and are set-valued mappings.
It's worth noting that the GVQEP considered in  is just the special case of the GVQEP2 (when is single valued).
Lemma 16. Let be a vector-valued mapping such that for all and a set-valued mapping and define for each . If for each , is generalized -quasiconvex with compact values, then is -quasiconvex, where and is the general nonlinear scalarization function of .
Proof. It's sufficient to testify that is convex, where for each and . Indeed, for any , Clearly, for each , where is convex by reason of the generalized -quasiconvexity of and so for all . Hence, which is equal to by Theorem 12 (2). Thus, and is convex.
Now a result on existence of solutions of the GVQEP1 is verified by making use of the general nonlinear scalarization function defined in Section 3.
Theorem 17. Let and be compact and convex subsets and , and set-valued mappings. For each , define . Suppose that the following conditions are fulfilled:(a) has a continuous select ;(b)both and are usc on , where (c) is continuous, and are strict and continuous, and is strict and usc;(d)for each is generalized -quasiconvex;(e)for each , is compact and for each , both and are closed and convex.Then the GVQEP1 has a solution ; that is, there exist and such that Further assume that for each and . Then
Proof. Denote and where and , and define as
Obviously, is continuous on . The general nonlinear scalarization function of
is continuous on by Theorem 14. Set , where is the general nonlinear scalarization function of . Since
is continuous on by virtue of the continuity of and . Define as
Obviously, has compact values and is continuous. Define a set-valued mapping as
By Lemma 4, is usc. Let
Then . The upper semicontinuity of is obvious. Now we show that for each is convex and closed. Detailedly,(i) Define
Then for any , . Clearly, for all since is convex. In view of condition (d) and Lemma 16, is -quasiconvex, which deduces that for each , the set
is convex. Thus, . It follows from the definition of that , which results in and the convexity of .(ii) For each and for any sequence such that as ,
Since is continuous,
Thus and is closed.
Now consider a set-valued mapping prescribed as It's easy to check that is usc on and for each is convex and closed. By Lemma 5, exists a fixed point , that is to say, Hence, for each , . In other words, which deduces that , and by Theorem 13. The first conclusion is proved. Further assume that for each and . If the second conclusion is not true; namely, there exists such that , then This is absurd.
If mapping in Theorem 17 satisfies that for each , is open, then must exist a continuous selection by Browder Selection Theorem . Especially, when is single valued in Theorem 17, a result is stated as follows.
Corollary 18. Let and be compact and convex subsets, and set-valued mappings and a vector-valued mapping. For each , define . Suppose that the following conditions are in force:(a) has a continuous select ;(b) both and are usc on , where (c) and are continuous, is strict and continuous, and is strict and usc;(d) for each is -quasiconvex;(e) for each , both and are closed and convex.Then there exist and such that Further suppose that for each and . Then
Theorem 19. Let and be compact and convex subsets and , and set-valued mappings. For each , define . Suppose that the following conditions hold:(a) has a continuous select ;(b) both and are usc on , where (c) is continuous, and are strict and continuous, and is strict and usc;(d) for each , is generalized -quasiconvex;(e) for each , is compact and for each , both and are closed and convex.Then the GVQEP2 exists a solution ; namely, there exist and such that Furthermore, if for each and , then
Proof. Denoting , we see that is compact. Define and as and , respectively. Then is continuous on and for each , is generalized -quasiconvex according to condition (d). Since is strict and usc with compact values, so is . Replacing , and in Theorem 17 by , and , respectively, we see that these conclusions are true.
The result below follows from Theorem 19 immediately by further assuming that is single-valued.
Corollary 20 (see ). Let and be compact and convex subsets, and set-valued mappings, and a vector-valued mapping. For each , define . The following assumptions are in operation:(a) has a continuous select ;(b)both and are usc on , where (c) and are continuous, is strict and continuous, and is strict and usc;(d)for each , is -quasiconvex;(e)for each , both and are closed and convex.Then there exist and such that Additionally, on the assumption that for each and ,
5. An Application of GVQEP1: A VVIP
Let be a real -space ( is called an -space , if it is a TVS such that its topology is induced by a complete invariant metric), a real locally convex TVS, a nonempty subset and a proper, closed, and convex cone with and let be strict and continuous with compact values. A VVIP is described as follows: to find such that where The VVIP was investigated in [18, 19]. If is single valued, then the VVIP becomes a vector variational inequality problem, which was discussed in [20–22]. Note that is indeed locally convex.
Assume that is compact and convex and is a singleton for each .
(a) and (b) Since is a constant cone, has a continuous selection and both and are usc.
(c) Obviously, is strict, is continuous, is strict and continuous, and is strict and usc. In addition, is continuous with compact values.
In fact, for each fixed , letting , we see that is continuous with compact values. So is compact by the compactness of and Lemma 3 (2). Moreover, take and . Obviously, is continuous with compact values. is continuous in view of the bilinearity of itself and [11, Theorem 2.17]. Thus is continuous by [13, Theorem 1].
(d) For each , is generalized -quasiconvex by its linearity.
(e) The assertion that has compact values was verified in (c). Clearly, for each , both and are closed and convex.
Thus there exist , , and such that Since is a singleton for each , which implies that is a solution of VVIP.
Incidentally, the set-valued mapping satisfying the conditions above exists. For instance, let , and Define by Clearly, is strict and continuous with compact values. Moreover, for each .
The work was supported by both the Doctoral Fund of innovation of Beijing University of Technology (2012) and the 11th graduate students Technology Fund of Beijing University of Technology (No. ykj-2012-8236).
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