Painleve-Kuratowski Convergences for the Solution Sets of Set-Valued Weak Vector Variational Inequalities
© Z. M. Fang et al. 2008
Received: 16 July 2008
Accepted: 10 December 2008
Published: 21 December 2008
Painleve-Kuratowski convergence of the solution sets is investigated for the perturbed set-valued weak vector variational inequalities with a sequence of mappings converging continuously. The closedness and Painleve-Kuratowski upper convergence of the solution sets are obtained. We also obtain Painleve-Kuratowski upper convergence when the sequence of mappings converges graphically. By virtue of a sequence of gap functions and a key assumption, Painleve-Kuratowski lower convergence of the solution sets is established. Some examples are given for the illustration of our results.
Since the concept of vector variational inequality (VVI) was introduced by Giannessi  in 1980, many important results on various kinds of vector variational inequality problems have been established, such as existence of solutions, relations with vector optimization, stability of solution set maps, gap function, and duality theories (see, e.g., [2–8] and the references cited therein).
The stability analysis of the solution set maps for the parametric (VVI) problem is of considerable interest amongst researchers in the area. Some results on the semicontinuity of the solution set maps for the parametric (VVI) problem with the parameter perturbed in the space of parameters are now available in the literature. In , Khanh and Luu proved the upper semicontinuity of the solution set map for two classes of parametric vector quasivariational inequalities. In , Li et al. established the upper semicontinuity property of the solution set map for a perturbed generalized vector quasivariational inequality problem and also obtained the lower semicontinuity property of the solution set map for a perturbed classical scalar variational inequality. In , Cheng and Zhu investigated the upper and lower semicontinuities of the solution set map for a parameterized weak vector variational inequality in a finite dimensional Euclidean space by using a scalarization method. In , Li and Chen obtained the closedness and upper semicontinuity of the solution set map for a parametric weak vector variational inequality under weaker conditions than those assumed in . Then, under a key assumption, they proved a lower semicontinuity result of the solution set map in a finite dimensional space by using a parametric gap function.
However, there are few investigations on the convergence of the sequence of mappings. In particular, almost no stability results are available for the perturbed (VVI) problem with the sequence of mappings converging continuously or graphically. It appears that the only relevant paper is , where Lignola and Morgan considered generalized variational inequality in a reflexive Banach space with a sequence of operators converging continuously and graphically and obtained the convergence of the solution sets under an assumption of pseudomonotonicity. Since the perturbed (VVI) problem with a sequence of mappings converging is different from the parametric (VVI) problem with the parameter perturbed in a space of parameters, these results do not apply to the parametric (VVI) problem with the parameter perturbed in a space of parameters. Thus, it is important to study Painleve-Kuratowski upper and lower convergences of the sequence of solution sets.
In passing, it is worth noting that some stability results are available for the vector optimization and vector equilibrium problems with a sequence of sets converging in the sense of Painleve-Kuratowski (see [11–13]). It is well known that the vector equilibrium problem is a generalization of (VVI) problem. However, if the results obtained for the vector equilibrium problem are to be applied to the (VVI) problem, the required assumptions are on the (VVI) problem as a whole. There is no information about the conditions that are required on the functions defining the (VVI) problem. Clearly, this is unsatisfactory. Our study of the stability properties for the perturbed (VVI) problem with a sequence of converging mappings is under appropriate assumptions on the function defining the (VVI) problem rather than on the (VVI) problem as a whole.
In this paper, we should establish Painleve-Kuratowski upper and lower convergences of the solution sets of the perturbed set-valued weak variational inequity (SWVVI) with a sequence of converging mappings in a Banach space. We first discuss Painleve-Kuratowski upper convergence and closedness of the solution sets. To obtain Painleve-Kuratowski lower convergence of the solution sets, we introduce a sequence of gap functions based on the nonlinear scalarization function introduced by Chen et al. in  and a key assumption imposed on the sequence of gap functions. Then, we obtain Painleve-Kuratowski lower convergence of the solution sets for .
The rest of the paper is organized an follows. In Section 2, we introduce problems (SWVVI) and , and recall some definitions and important properties of these problems. In Section 3, we investigate Painleve-Kuratowski upper convergence and the closedness of the solution sets. In Section 4, we introduce respective gap functions for problems (SWVVI) and and then establish Painleve-Kuratowski lower convergence of the solution sets under a key assumption.
Let and be two Banach spaces and let be the set of all linear continuous mappings from to . The value of a linear mapping at is denoted by . Let be a closed and convex cone with nonempty interior, that is, . We define the ordering relations as follows.
where is a nonempty subset and is a set-valued mapping.
where is a sequence of nonempty subsets.
Throughout this paper, we assume that and . The stability analysis is to investigate the behaviors of the solution sets and .
Now we recall some basic definitions and properties of problems (SWVVI) and . For each and a subset , let the open -neighborhood of be defined as . The notation denotes the open ball with center and radius .
where denotes the set of all positive integer numbers and is an integer in .
It is said that the sequence upper converges in the sense of Painleve-Kuratowski to if . Similarly, the sequence is said to lower converge in the sense of Painleve-Kuratowski to if .
Definition 2.2 (see ).
A set-valued mapping is outer semicontinuous (osc) at if with .
On the other hand, it is inner semicontinuous (isc) at if with .
The set-valued mapping is said to be continuous at , written as as if it is both outer semicontinuous and inner semicontinuous.
Definition 2.3 (see ).
If converges continuously to at every , then it is said that converges continuously to on .
Applying set convergence theory to the graphs of the mappings, we obtain the graphical convergence of the sequence of mappings.
Definition 2.4 (see ).
If the outer and inner limits of the mappings agree, it is said that their graphical limit, , exists. In this case, the notation is used, and the sequence of mappings is said to converge graphically to . Clearly, .
Proposition 2.5 (see ).
From Proposition 2.5 and Definition 2.3, the following proposition follows readily.
Definition 2.7 (see ).
Proposition 2.8 (see ).
For any fixed , , , and the nonlinear scalarization function defined by :
(i) is a continuous and convex function on ;
3. Painleve-Kuratowski Upper Convergence of the Solution Sets
In this section, our focus is on the Painleve-Kuratowski upper convergence and the closedness of the solution sets.
(iii) are uniformly bounded.
Then, , that is to say for any subsequence of solutions to , if , then is a solution to .
The proof is listed on contradiction arguments. On a contrary, suppose that but .
which is a contradiction to (3.4). This completed the proof.
Let and , where is a reflexive Banach space and is its dual. If we take , reduce to the generalized variational inequality problems with perturbed operators considered in [10, Section 3]. The convergence for the solution sets of was studied under the the pseudomonotonicity assumption in . Furthermore, if and are vector-valued mappings, then reduce to considered in [10, Section 2]. We also notice that the Painleve-Kuratowski upper convergence of the solution sets of is obtained under weaker assumptions than these assumed in [10, Proposition 2.1] for obtaining convergence of the solution sets.
From Proposition 2.5 and Theorem 3.1, we obtain readily the following corollary.
(iii) is uniformly bounded.
Let . Then, problems reduce to the generalized variational inequalities with perturbed operators considered in [10, Proposition 3.1] and the convergence was obtained under the assumption that the operators converge graphically.
(i) is osc on , that is, for all , ;
(ii) and are compact sets.
Then, is a compact set.
which contradicts with (3.9). Hence, and is a closed set. Next, it follows from and the compactness of that is a compact set. The proof is completed.
Similarly, we have the following result.
For any , suppose that
(ii) and are compact sets.
Then, is a compact set.
4. Painleve-Kuratowski Lower Convergence of the Solution Sets
Since , , , and are compact sets and is continuous, and are well defined.
- (i)for all ;
- (ii)for all
- (iii)if and only if
- (iv)if and only if
Clearly, (4.7) holds if and only if for any , , that is, . This proves that (iii) holds.
Similarly, we can show that (iv) holds.
The functions are called the gap functions for if properties (ii) and (iv) of Proposition 4.1 are satisfied.
(H g ):given the sequence for any , there exist an and an such that for all and for all .
Geometrically, the hypothesis means that given a sequence of mappings , we can find for any small positive number , a small positive number and a large-enough positive number such that for all , if a feasible point is away from the solution sets by distance of at least , then the values of all gap functions for is less than or equal to at least some " ."
To illustrate assumption , we give the following example.
For any given , we take and . Then, for all and for all , we have . Hence, assumption is valid.
(iii) is a compact set.
Then, for any , and sequence with and , there exists a subsequence of and such that for all .
Hence, the result holds.
Set and . We have the following lemma.
Then, if and only if for all , such that for all .
We assume , but there exists an such that for all there exists an satisfying . Then, there exists a sequence with , but . From Theorem 3.5, we note that is a compact set. Without loss of generality, we assume and . Thus, for any sequence satisfying with , we have . Letting , we get . Therefore, there does not exist any sequence satisfying . This is a contradiction to .
Conversely, suppose that for any such that for all . From Theorem 3.6, we note that is compact for all . Thus, for any , there exists such that for all . So, we have and . Therefore, the result of the lemma follows readily.
Now, we are in a position to state and prove our main result in the following theorem.
Suppose that assumption holds and that the following conditions are satisfied:
(iii) is a compact set;
From the compactness of , we can assume, without loss of generality, that . Then, there exists an such that . It is clear that for any positive integer . Since , there exist a sequence satisfying . Then, there exists an such that for all .
Consequently, by Proposition 2.8, we have , which shows that . This contradicts with . Therefore, our result follows readily.
Now, we explain the applicability of Theorem 4.5 through an example.
Consider Example 4.2. It follows from a direct computation that . It is easy to testify that assumption holds and so are conditions (i)–(v) of Theorem 4.5. Obviously, the solution sets of problem lower converge in the sense of Painleve-Kuratowski.
This research was partially supported by the National Natural Science Foundation of China (Grants no. 10871216 and no. 60574073).
- Giannessi F: Theorems of alternative, quadratic programs and complementarity problems. In Variational Inequalities and Complementarity Problems (Proc. Internat. School, Erice, 1978). Edited by: Cottle RW, Giannessi F, Lions J-L. John Wiley & Sons, Chichester, UK; 1980:151–186.Google Scholar
- Chen GY: Existence of solutions for a vector variational inequality: an extension of the Hartmann-Stampacchia theorem. Journal of Optimization Theory and Applications 1992,74(3):445–456. 10.1007/BF00940320MATHMathSciNetView ArticleGoogle Scholar
- Chen GY, Li SJ: Existence of solutions for a generalized vector quasivariational inequality. Journal of Optimization Theory and Applications 1996,90(2):321–334. 10.1007/BF02190001MATHMathSciNetView ArticleGoogle Scholar
- Khanh PQ, Luu LM: Upper semicontinuity of the solution set to parametric vector quasivariational inequalities. Journal of Global Optimization 2005,32(4):569–580. 10.1007/s10898-004-2694-7MATHMathSciNetView ArticleGoogle Scholar
- Lee JR, Park C, Shin DY: On the stability of generalized additive functional inequalities in Banach spaces. Journal of Inequalities and Applications 2008, 2008:-13.Google Scholar
- Li SJ, Chen CR: Stability of weak vector variational inequality. Nonlinear Analysis: Theory, Methods & Applications 2009,70(4):1528–1535. 10.1016/j.na.2008.02.032MATHMathSciNetView ArticleGoogle Scholar
- Li SJ, Chen GY, Teo KL: On the stability of generalized vector quasivariational inequality problems. Journal of Optimization Theory and Applications 2002,113(2):283–295. 10.1023/A:1014830925232MATHMathSciNetView ArticleGoogle Scholar
- Yang XQ: Vector variational inequality and its duality. Nonlinear Analysis: Theory, Methods & Applications 1993,21(11):869–877. 10.1016/0362-546X(93)90052-TMATHMathSciNetView ArticleGoogle Scholar
- Cheng YH, Zhu DL: Global stability results for the weak vector variational inequality. Journal of Global Optimization 2005,32(4):543–550. 10.1007/s10898-004-2692-9MATHMathSciNetView ArticleGoogle Scholar
- Lignola MB, Morgan J: Generalized variational inequalities with pseudomonotone operators under perturbations. Journal of Optimization Theory and Applications 1999,101(1):213–220. 10.1023/A:1021783313936MathSciNetView ArticleMATHGoogle Scholar
- Durea M: On the existence and stability of approximate solutions of perturbed vector equilibrium problems. Journal of Mathematical Analysis and Applications 2007,333(2):1165–1179. 10.1016/j.jmaa.2006.12.009MATHMathSciNetView ArticleGoogle Scholar
- Lucchetti RE, Miglierina E: Stability for convex vector optimization problems. Optimization 2004,53(5–6):517–528. 10.1080/02331930412331327166MATHMathSciNetView ArticleGoogle Scholar
- Miglierina E, Molho E: Scalarization and stability in vector optimization. Journal of Optimization Theory and Applications 2002,114(3):657–670. 10.1023/A:1016031214488MATHMathSciNetView ArticleGoogle Scholar
- Chen GY, Yang XQ, Yu H: A nonlinear scalarization function and generalized quasi-vector equilibrium problems. Journal of Global Optimization 2005,32(4):451–466. 10.1007/s10898-003-2683-2MATHMathSciNetView ArticleGoogle Scholar
- Rockafellar RT, Wets RJ-B: Variational Analysis, Fundamental Principles of Mathematical Sciences. Volume 317. Springer, Berlin, Germany; 1998:xiv+733.Google Scholar
- Chen GY, Huang X, Yang X: Vector Optimization: Set-Valued and Variational Analysis, Lecture Notes in Economics and Mathematical Systems. Volume 541. Springer, Berlin, Germany; 2005:x+306.MATHGoogle Scholar
- Zhao J: The lower semicontinuity of optimal solution sets. Journal of Mathematical Analysis and Applications 1997,207(1):240–254. 10.1006/jmaa.1997.5288MATHMathSciNetView ArticleGoogle Scholar
- Kien BT: On the lower semicontinuity of optimal solution sets. Optimization 2005,54(2):123–130. 10.1080/02331930412331330379MATHMathSciNetView ArticleGoogle Scholar
- Aubin J-P, Ekeland I: Applied Nonlinear Analysis, Pure and Applied Mathematics. John Wiley & Sons, New York, NY, USA; 1984:xi+518.MATHGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.