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The uniqueness and well-posedness of vector equilibrium problems with a representation theorem for the solution set
Fixed Point Theory and Applicationsvolume 2014, Article number: 115 (2014)
This paper aims to present some uniqueness and well-posedness results for vector equilibrium problems (for short, VEPs). We first construct a complete metric space M consisting of VEPs satisfying some conditions. Using the method of set-valued analysis, we prove that there exists a dense everywhere residual subset Q of M such that each VEP in Q has a unique solution. Moreover, we introduce and obtain the generalized Hadamard well-posedness and generic Hadamard well-posedness of VEPs by considering the perturbations of both vector-valued functions and feasible sets. As an application, we provide a representation theorem for the solution set to each VEP in M.
MSC:49K40, 90C31, 46B40, 47H04.
The vector equilibrium problem (for short, VEP) is a natural generalization of the equilibrium problem for the vector-valued function. It is well known that the vector equilibrium problem is a unified model of several fundamental mathematical problems, namely, the vector optimization problem, the vector variational inequality, the vector complementarity problem, the multiobjective game, the vector network equilibrium problem etc. Since the VEP was proposed at about 1997 lots of peoples have made many contributions to this problem and hundreds of papers have been published; see, e.g., the collection  and the monograph . However, works on the uniqueness of solutions to VEPs were hardly seen. The only work we can find about the uniqueness of solutions to VEPs is , in which Khanh and Tung established sufficient conditions for the local uniqueness of solutions to VEPs by using approximations as generalized derivatives under the assumption that the functions have first and second Fréchet derivatives. The reason why results on uniqueness are so few is due to the fact that except for a few types of mathematical problems, most of the mathematical problems cannot guarantee the uniqueness of the solution. Therefore, to consider the generic uniqueness of the solutions may be more suitable, which will answer the question how many problems there are in a class of problems having a unique solution. The generic uniqueness of solutions to VEPs is our main motivation in this paper.
As we know, several works have been achieved about the generic uniqueness of solutions to some optimization-related problems such as optimization problems [4, 5], two-person zero-sum continuous games , saddle point problems , large crowding games . Recently, some new results were obtained. Yu et al.  obtained the generic uniqueness of equilibrium points for a class of equilibrium problems. The results in  showed that most of the monotone equilibrium problems (in the sense of Baire category) have a unique equilibrium point and that each monotone equilibrium problem can be arbitrarily approached by a sequence of such equilibrium problems that each of them has a unique equilibrium point. Moreover, Peng et al.  provided a unified approach to the generic uniqueness and applied it to several nonlinear problems. However, the study of the generic uniqueness of solutions to VEPs has an essential difficulty: that the values of different vector-valued functions are incomparable. To overcome such a difficulty is one of the main tasks in this paper.
The stability of solutions to nonlinear problems is also an important topic. The notion of well-posedness is just one of the approaches to the stability. There have been several notions of well-posedness about optimization-related problems. We refer to [11–15] for more details. For well-posedness of equilibrium problems or vector equilibrium problems, there are some results. Fang et al.  investigated the well-posedness of equilibrium problems; Kimura et al.  studied the parametric well-posedness for vector equilibrium problems; Bianchi et al.  introduced and studied two types of well-posedness for vector equilibrium problems; Li and Li  investigated the Levitin-Polyak well-posedness of vector equilibrium problems with variable domination structures; Salamon  analyzed the Hadamard well-posedness of parametric vector equilibrium problems; Peng et al.  investigated several types of Levitin-Polyak well-posedness of generalized vector equilibrium problems. Most of these works considered the perturbation of the parameters in the vector-valued functions. Different from these works, we will not only consider the perturbation of objective functions but also consider the perturbation of feasible sets.
This paper aims to present some generic uniqueness and well-posedness results for VEPs. We consider both the perturbation of vector-valued functions and the perturbation of feasible sets. The paper is organized as follows. In Section 2, we recall some definitions and preliminaries. In Section 3, we investigate the uniqueness of solutions to VEPs. We first construct a complete metric space M consisting of VEPs satisfying some conditions. Then we prove that most of the VEPs (in the sense of Baire category) in M have a unique solution. In Section 4, the Hadamard well-posedness of VEPs is introduced and studied. The generalized Hadamard well-posedness and generic Hadamard well-posedness of VEPs are derived. In Section 5, applying the above results we provide an interesting representation theorem for the solution set of each VEP in M. Finally, we briefly conclude our results in Section 6.
Throughout this section, let H be a Hausdorff topological vector space and C be a nonempty closed, convex and pointed cone in H with , where intC denotes the topological interior of C. We note that (see ).
Let X be a nonempty set and be a vector-valued function. The so-called vector equilibrium problem (for short, VEP)  is to find such that
Definition 2.1 (see )
Let X be a nonempty subset of a Hausdorff topological vector space E and be a vector-valued function. f is said to be C-upper semi-continuous at iff for any open neighborhood V of 0 in H, there exists an open neighborhood U of x in X such that, for any ,
f is said to be C-upper semi-continuous on X iff f is C-upper semi-continuous at each ; and f is said to C-lower semi-continuous on X iff −f is C-upper semi-continuous on X.
Definition 2.2 Let X be a nonempty subset of a Hausdorff topological vector space E and be a vector-valued function. ϕ is said to be C-strictly-quasi-monotone on iff for any with ,
Example 2.3 Let , , and . Define
One can easily check that is C-upper semi-continuous on X but not C-lower semi-continuous at ; is both C-upper semi-continuous and C-lower semi-continuous on X; ϕ is both C-upper semi-continuous and C-lower semi-continuous on ; ϕ is C-strictly-quasi-monotone on ; and that are the only two solutions to .
To investigate the uniqueness of solutions to VEPs, we will use the way of set-valued analysis. So let us recall some definitions and lemmas about set-valued mappings; for more details see .
Definition 2.4 Let X, M be two topological spaces. Denote by the space of all nonempty subsets of X. Let be a set-valued mapping. Then (i) S is said to be upper (respectively, lower) semi-continuous at iff for each open set G in X with (respectively, ), there exists an open neighborhood O of u such that (respectively, ) for each ; (ii) S is said to be continuous at iff S is both upper semi-continuous and lower semi-continuous at u; (iii) S is said to be a usco mapping iff S is upper semi-continuous on M and is compact for each ; (iv) A subset Q of M is called residual iff it contains the intersection of countably many dense everywhere open subsets of M.
Let M be a Baire space, X be a metric space and be a usco mapping, then there exists a dense everywhere residual subset Q of M such that S is lower semi-continuous at each .
Remark 2.6 If there exists a dense everywhere residual subset Q of M such that, for each , a certain property P depending on u holds, then we say that the property P is generic on M. Since Q is a second category set, we may say that the property P holds for most of the points (in the sense of Baire category) in M. The research on generic properties (including generic existence, generic uniqueness, generic stability, generic well-posedness and so on) has attracted much attention; see, e.g., [4–7, 9, 22, 24, 28, 29] and the references therein.
Lemma 2.7 (see )
Let A and () all be nonempty compact subsets of a metric space X with in the Hausdorff distance topology, then the following statements hold:
is also nonempty compact subset of X;
If , , then .
3 Uniqueness of solutions to VEPs
In the rest of this paper, let X be a nonempty and closed subset of a complete metric space E, be a Banach space, and C be a nonempty, closed, convex, and pointed cone in H with . For any , denote by and . We emphasize that the open neighborhood V in Definition 2.1 can be replaced by in the case that H is a normed space.
Let a space M of VEPs be defined by
For any , define
where h is the Hausdorff distance on X.
Lemma 3.1 is a complete metric space.
Proof Clearly, ρ is a metric on M. We only need to show that is complete. Let be a Cauchy sequence of M, then for any , there exists a positive integer such that
Since H is a Banach space, for any , there exists such that and
Since X is complete, is also complete, where denotes the space of all nonempty compact subsets of X and is endowed with the Hausdorff distance h induced by the metric on X. Consequently, by , there exists such that . Next, we will prove .
Fix . Since is C-upper semi-continuous on , there exists a neighborhood of such that(3.2)
Thus, by (3.1) and (3.2), for any , we have
It follows that ϕ is C-upper semi-continuous on .
(ii) For any with , suppose . Since intC is open and , we have when n is big enough. It follows from the C-strictly-quasi-monotonicity of that . Since C is closed and , we obtain . Therefore, ϕ is C-strictly-quasi-monotone on .
(iii) For each , we have
Since for each , there exists such that(3.3)
Since A and () are all compact and , by Lemma 2.7(i), is also compact. Note that . Without loss of generality, suppose . By Lemma 2.7(ii), . We shall show that fulfills
Assume, by contradiction, that there exists such that . Since intC is open, there exists such that
For , by virtue of , there exist () such that . By (3.3), it follows from that
Since ϕ is C-upper semi-continuous on and , , there exists such that
From (3.1), we derive
By (3.4), (3.6), and (3.7), we obtain for all ,
which contradicts (3.5). Hence for all . Thus we have shown . Consequently, the inequality (3.1) and imply that . Therefore, is a complete metric space. □
Lemma 3.2 Let be C-upper semi-continuous on X, then the set is closed in X.
Proof Let with . We only need to prove . Assume, by contradiction, that , then . Note that is open, there exists such that . Since is C-upper semi-continuous at x and , there exists such that, for any , we have . But it follows from that , which is a contradiction. The proof is complete. □
For each , by the definition of M, must have at least one solution in A, i.e., such that for all . Denote by the set of all solutions to in A. Then the correspondence yields a set-valued mapping .
Lemma 3.3 is a usco mapping.
Proof For each , note that
Since ϕ is C-upper semi-continuous on , it is also C-upper semi-continuous on . Moreover, is also C-upper semi-continuous on A. By Lemma 3.2, for each , the set is closed in A. Thus is closed in A. Furthermore, is compact since A is compact.
Next, we will prove that S is upper semi-continuous on M. We assume, by contradiction, that there exists such that S is not upper semi-continuous at u, then there exists an open neighborhood G in X with such that, for each and each open neighborhood of u, there exist and but .
Since for each , we have . Then
It follows from that and
By Lemma 2.7(i), is compact due to the compactness of and A. Note that . Without loss of generality, we suppose that is convergent. Moreover, by Lemma 2.7(ii), the limit of belongs to A, i.e., . Meanwhile, and G is open, thus . Since , we have . Consequently, there exists such that . Note that is open; then there exists such that
Since and , there exists a sequence such that and . Since , there exists such that, for any ,
Moreover, ϕ is C-upper semi-continuous on as well as , , hence there exists such that, for any ,
By (3.10)-(3.12), we have for any ,
which is in contradiction with (3.9). Therefore, S must be upper semi-continuous on M. The proof is thus complete. □
Theorem 3.4 There exists a dense everywhere residual subset Q of M such that is a singleton for each , that is, has a unique solution in A.
Proof By Lemma 3.1, M is a complete metric space, so it is a Baire space. Since is a usco mapping (Lemma 3.3) and X is a metric subspace, by Lemma 2.5, there exists a dense everywhere residual subset Q of M such that S is lower semi-continuous at each .
Assume, by contradiction, that is not a singleton for some . Then there exist at least two points with . Consequently, there exist two open subsets U and V in X such that , and .
Define a function as follows:
where d is the metric on X. Note that g is continuous on X; for all ; if and only if ; for all .
Take . For each , let be defined by
For each , we will prove .
It is easy to check that is C-upper semi-continuous on ;
For any with , suppose . Then we can claim that . Otherwise . Note that , then
which is a contradiction. By the C-strictly-quasi-monotonicity of and , we get . Hence
That is, is C-strictly-quasi-monotone on .
From and , we derive
which implies .
Thus we have shown for each . Consequently, as .
Note that , then . Since S is lower semi-continuous at and , there exists a positive integer big enough such that . Take , then we have , and for any . Take (), then we get
Note that . If , then , which contradicts (3.13). Hence we have
Since , we have for any . Taking (), we get . It follows from the C-strictly-quasi-monotonicity of that , which is in contradiction with (3.14). Therefore, must be a singleton for each . □
When , , we get Corollary 3.5 as follows.
Corollary 3.5 Let
where f is called pseudo-monotone (see ) on iff for any with ,
Then there exists a dense everywhere residual subset of such that, for each , f has a unique equilibrium point in A.
we do not require the convexity of function ;
we do not require the convexity and linear structure of the set X;
we omit the requirement that for all ;
we replace the monotonicity of f by pseudo-monotonicity which is weaker than the former.
4 Well-posedness of VEPs
As is well known, the notions of well-posedness can be mainly divided into three groups, namely, Hadamard type, Tykhonov type and Levitin-Polyak type. Generally speaking, to consider Tykhonov well-posedness of a problem, one introduces the notion of ‘approximating sequence’ for the solution and requires some convergence of such sequences to a solution of the problem; while, Hadamard well-posedness of a problem means the continuous dependence of the solutions on the data or the parameter of the problem; as for Levitin-Polyak well-posedness, we mean the convergence of the approximating solution sequence to a solution of the problem with some constraints; for more details to see [11–15]. In this section, we will investigate the Hadamard well-posedness of VEPs.
Definition 4.1 Let . (1) The VEP associated with u is said to be generalized Hadamard well-posed iff for any and any , implies that has a subsequence converging to an element of ; (2) The VEP associated with u is said to be Hadamard well-posed iff (a singleton) and for any and any , implies that converges to x.
Theorem 4.2 For each , the VEP associated with u is generalized Hadamard well-posed.
Proof Let , , and . Note that because . According to Lemma 2.7(i), it follows from the compactness of and A that is compact. Since , there exists a convergent subsequence of . Moreover, by Lemma 2.7(ii), the limit point of belongs to A, i.e., . By Lemma 3.3, S is upper semi-continuous at u and is compact.
If , then there exists an open set O in X such that and . Since S is upper semi-continuous at u and , there is a positive integer N such that for all . From for all and , it follows that , which is a contradiction. Hence . The proof is complete. □
Theorem 4.3 There exists a dense everywhere residual subset Q of M such that, for each , the VEP associated with u is Hadamard well-posed, that is, VEPs in M are generic Hadamard well-posed.
Proof By Theorem 3.4, there exists a dense everywhere residual subset Q of M such that, for each , is a singleton.
Let and . Suppose , , and . We shall prove . If it is not true, then there exist an open neighborhood O of x and a subsequence such that . By Theorem 4.2, the VEP associated u is generalized Hadamard well-posed. Since , has a subsequence converging to x, which contradicts . □
5 A representation theorem of the solution set to VEPs
In this section, we use the limits of the solutions to VEPs, each of which has a unique solution, to provide an interesting representation of the solution set of each VEP in M, which in forms is very similar with the Clarke subdifferentials of the local Lipschitz functions.
By Theorem 3.4, . It is clear that P is the largest dense everywhere residual subset (ordered by the set inclusion) of M such that, for each , is a singleton and the VEP associated with u is Hadamard well-posed.
Theorem 5.1 For each ,
Proof The right-hand side of (5.1) means that we only consider such sequences and satisfying that ; converges to u; is the unique point of ; and that is convergent.
Since P is dense in M and , there exists such that converges to u. By the definition of P, has a unique point, denoted by . Note that , as well as and A are compact and , due to Lemma 2.7, or its subsequence converges to a point of A. Hence the right-hand side set of (5.1) is well-defined and nonempty.
First, suppose , , and . By Theorem 4.2, the VEP associated with u is generalized Hadamard well-posed. It follows from that . Hence .
Next, let . Define a function as follows:
where d is the metric on X. Note that g is continuous on X; for all ; if and only if .
Take . For each , define
Same as in the proof of Theorem 3.4, one can check that and for each , and that as . Moreover, we shall show that is a singleton, and hence for each . By way of contradiction, assume that is not a singleton for some . Then there exists with . It follows from that . By the C-strictly-quasi-monotonicity of ϕ, we get . Note that and . Then
But it follows from that , which is a contradiction. Thus for each .
Take for each , then we have , , , and . From the arbitrariness of , we derive
Combining the above two parts, we get the conclusion. □
In this paper, we considered a class of vector equilibrium problems. By considering the perturbations of vector-valued functions and feasible sets, we proved that each of the problems is generalized Hadamard well-posed, and that in the sense of Baire category, most of the problems have unique solution and are Hadamard well-posed. As an application, an interesting representation theorem for the solution set to each of the problems was provided.
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The first author was supported by the NSFC (11271098), the Science and Technology Foundation of Guizhou Province (20102133), and the Scientific Research Projects for the Introduced Talents of Guizhou University (201343). The third author was supported by the National Basic Research Program of China (2010CB732501) and the NSFC (71271021). The authors, therefore, acknowledge these supports.
The authors declare that they have no competing interests.
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.