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.
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 .
Definition 2.2 (see ).
Definition 2.3 (see ).
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 ).
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.
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.
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.
Similarly, we have the following result.
4. Painleve-Kuratowski Lower Convergence of the Solution Sets
Similarly, we can show that (iv) holds.
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 " ."
Hence, the result holds.
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.
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 .
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).
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