An Approximation Theorem and Generic Convergence of Solutions of Inverse Quasivariational Inequality Problems

Liu, Ting; Jia, Wensheng

doi:https://doi.org/10.1155/2022/2539961

Journal of Function Spaces

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Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 2539961 | https://doi.org/10.1155/2022/2539961

An Approximation Theorem and Generic Convergence of Solutions of Inverse Quasivariational Inequality Problems

Ting Liu^1,2and Wensheng Jia^1,2

Academic Editor: Samuel Nicolay

Received31 May 2022

Accepted18 Aug 2022

Published05 Sept 2022

Abstract

In this paper, we mainly obtain an approximation theorem and generic convergence of solutions for inverse quasivariational inequality problems. First, we define the concept of the approximate solution to inverse quasivariational inequality problems under bounded rationality theory. Afterward, an approximation theorem that satisfies fairly mild assumptions is proved. Moreover, we establish a function space and discuss the convergence properties of solutions for inverse quasivariational inequality problems by the method of set-valued analysis. Finally, we prove that most of inverse quasivariational inequality problems are stable in the case of perturbation of the objective function. These results are new, which improve the corresponding outcomes of the recent literatures.

1. Introduction

Inverse quasivariational inequality (briefly, IQVI) was first proposed by Aussel et al. [1] in 2013. The specific format of this inequality is as follows:

Finding a vector such that , we have where represents two continuous mappings, respectively. denotes a set-valued mapping. For all , is a nonempty closed convex set in . In addition, represents the inner products and denotes the norms in . When the mapping is the identity mapping or represents a constant on , then the IQVI problem converts to a classical quasivariational inequality (briefly, QVI) problem or an inverse variational inequality (briefly, IVI) problem. In recent years, the IQVI model has attracted much attention. Many related results have been extensively investigated as follows: (i)For the IQVI problems, Aussel et al. obtained the global/local error bounds of problems by using different gap (merit) functions, namely the regularized, residual, and -gap function [1]. In addition, Han et al. proved the existence theorem for the solution of the IQVI problems [2]. Dey and Vetrivel first defined an approximate solution to IQVI problems. And based on the existence theorem of the IQVI problems in literature [2], they obtained the existence theorem of the approximate solution for the IQVI problems in a locally convex Hausdorff topological vector space [3](ii)For the inverse mixed quasivariational inequality problems, scholars explored several properties of problems in Hilbert space, such as generalized -projection operator, error bounds, existence and uniqueness outcomes, and gap functions [4, 5](iii)For the differential IQVI problems, Li et al. obtained some existence theorems for Carathéodory weak solutions of the problems. Besides, the convergence consequent on the Euler time-dependent scheme was proved [6](iv)For the mixed set-valued vector IQVI and the vector inverse mixed quasivariational inequality problems, three gap functions were provided, respectively. Using generalized -projection operator and three gap functions, scholars obtained error bounds of the generalized vector IQVI and the vector inverse mixed quasivariational inequality problems under the Lipschitz continuity and strong monotonicity of the underlying mappings [7–9]

It can be seen that the research on the model of the IQVI problem is relatively active and has a wide application. However, the uniqueness of the IQVI problem is still few. Hence, this is one of the motivations why we study the uniqueness of the IQVI problem.

On the other hand, the bounded rationality theory was proposed by Simon, and the core of this theory is the principle of satisfaction. The decision-maker is to seek the satisfactory solution rather than the optimal solution [10]. In 2001, Anderlini and Canning [11] established an abstract model for the study of bounded rationality. The model is a type of general games with abstract rational functions, which reflects the approximation of bounded rationality to full rationality. In 2007, Khanh and Luu [12] obtained semi(upper)continuous of the approximate solution sets and solution sets of parametric multivalued QVI in topological vector spaces. Chen et al. [13] first obtained a scalarization result of the -weakly efficient solution for a class of vector equilibrium problems under the Hausdorff topological vector space. Then, they proved the connectedness of the -efficient solution sets and -weakly efficient solution sets for this problem by applying the scalarization result. Subsequently, Han and Huang [14] provided the scalarization results and density theorems for the efficient and weakly efficient approximate solution sets of generalized vector equilibrium problems and established their connectedness. And these researchers obtained the upper (lower) semicontinuous of efficient and weakly efficient approximate solution mappings for parametric generalized vector equilibrium problems in which both the feasible regions and the objective mappings were simultaneously perturbed. Research on the bounded rationality theory has been increasing in recent years. In 2018, Qiu et al. [15] discussed an approximation theorem for equilibrium problems. At the same time, on the meaning of the Baire category, they obtained the generic convergence of the solution for the monotone equilibrium problem. Moreover, Qiu et al. discussed the applications of such approximation theorems to saddle point problems, optimization problems, and variational inequality problems. Especially, in 2020, under certain assumptions, Jia et al. [16] proved an approximation theorem and obtained several corollaries for the vector equilibrium problem. In addition, based on the meaning of the Baire category, Jia et al. obtained the generic convergence theorem for the solution of the strictly quasimonotone scalar equilibrium problems and applied a series of results to Nash equilibrium problems with multiobjective games, vector optimization problems, and vector variational inequality problems.

At the moment, the model for the IQVI problem has not been considered from the perspective of bounded rationality ideas. Therefore, this is another motivation why we study the approximation theorem for the IQVI problem.

Motivated by the aforementioned works, we consider the approximation theorem and the generic convergence of the IQVI problem. In this paper, we first define the concept of the approximate solution of IQVI problems under bounded rationality theory [10]. Then, the approximation theorem of the IQVI problem is proved, which reflects the approximation of bounded rationality to full rationality. Moreover, we establish the function space and discuss the generic convergence of IQVI problems. Finally, based on the meaning of the Baire category, we prove the results that the IQVI problem has generic convergence in the case of perturbation of the objective function. These new results generalize those of some previous literature.

This paper consists of four parts. First, in Section 2, we introduce some indispensable lemmas and definitions for later use. Next, we define the concept of approximate solutions of IQVI problems and propose approximation theorems of solutions for IQVI problems in Section 3. Then, we construct the function space and discuss the generic convergence for IQVI problems in Section 4. Section 5 summarizes this paper.

2. Preliminaries

Definition 1 (see [17, 18]). Let and be two metric spaces, the set-valued mapping be expressed as , where is the nonempty set, is said to be (1)lower semicontinuous (l.s.c.) at if for any open set in , , there exists of , where is an open neighbourhood such that (2)upper semi-continuous (u.s.c.) at if for any open set in , , there exists of , where is an open neighbourhood such that (3)a usco mapping on if for any , is u.s.c and nonempty compact values(4)continuous at if is both u.s.c and l.s.c. at

Lemma 2 (see [19]). Let be a sequence of nonempty bounded subset of and represents a nonempty bounded subset of , where for each . If and , then there exists a subsequence of such that .

Lemma 3 (see [19]). Let denote a sequence of nonempty bounded subset of and represents a nonempty bounded subset of , where for each . Denote by an open set of , where . If , thus there exists a positive integer such that for each , .

Definition 4 (see [1]). Let represent a mapping, then is monotonic on if for any , one has

Definition 5 (see [1]). Assume that two mappings are denoted as , respectively. If there exists a constant such that for every , one has then is said to be -strongly monotone couple on .

Lemma 6 (see [17]). Let and be two topological spaces, where be a compact space. If the set-valued mapping is closed, then is a usco mapping on .

Definition 7 (see [20]). Let and denotes a nonempty convex set of , then (1) is quasiconvex on , if for all and , we have(2) is quasiconcave on , if for all and , we have

Lemma 8 (see [21], Fort Lemma). Let be a Hausdorff topological space and be a metric space. The set-valued mapping is a usco mapping. Then, there exists a residual subset of such that is l.s.c. on .

3. Approximation Theorem of Inverse Quasivariational Inequality

First, we introduce the concept of an approximate solution of IQVI problems.

Definition 9. Let be a nonempty compact subset of and denote by a set of all nonempty compact subsets in . Let represent two continuous mappings. A set-valued mapping represents as such that for any , is a nonempty convex compact set. For a real number , finding a vector such that , we have Then, is said to be an -approximate solution of IQVI problems.

Theorem 10. Let be a nonempty compact subset of and satisfy the following assumptions: (i)For every , the two function sequences and a set-valued mapping sequence are satisfied by where denotes the Hausdorff distance defined on . and are continuous and for all is a nonempty convex compact set (ii)For every , is a nonempty subset sequence of and where is a nonempty compact set of (iii)For every , is satisfied with , ,we have where and (1)Then there exists a convergent subsequence of which converges to some (2)For all , we have (3)If the solution of IQVI problems is a singleton set, there must be

Proof. (1)Since , we can see that there exists such that . Because and is a compact set by Lemma 2, for any sequence in , there must be a subsequence such that . Therefore, there exists a subsequence of the sequence such that (2)According to conclusion (1), it may be assumed that . By contradiction, we assume that conclusion (2) does not hold. Thus, there exists such thatFirst, since are continuous at and is continuous at variables and , there exists a real number and two open neighbourhoods and of and such that for any , , we have According to the Cauchy-Swartz inequality, we can obtain that is, This means that Again, because , there exists a positive integer such that for every , we have Finally, since , and , according to Lemma 3, we know that there exists a positive integer . It may be assumed that such that for all , we have and . Let , thus, we have Then, Then, this conflicts with condition (iii). Hence, for all , we have . (3)First, using the contradiction method, we assume that conclusion (3) does not hold. In other words, if the solution set of IQVI problems is a singleton set, then. Hence, there exists and a subsequence of such that . Next, by conclusion (1), it can be seen that sequence must have subsequences. Let , that is, , according to conclusion (2), we can obtain . Finally, because the solution of IQVI problems is singleton set, therefore , that is, , which conflicts with . Therefore, we can obtain . The proof is completed.

Remark 11. According to Theorem 10, although the objective function, feasible solution set are all approximated (that is, , , , and ), we can obtain an approximation sequence , which must have convergent subsequences , that is, and must be the solution of IQVI problems. If the -approximate solution of IQVI problems is regarded as a “satisfactory solution” under bounded rationality, and the solution of IQVI problems is regarded as an“exact solution” under full rationality. Theorem 10 implies the approximate of bounded rationality to full rationality, that is, full rationality can be approximated by a series of approximate solutions of bounded rationality, which verifies Simon’s bounded rationality theory from a certain perspective.

Remark 12. Theorem 10 shows that the limit points and convergent subsequences existing for sequence are equivalent. It can be seen from the above proof that the sequence must have limit points. Each limit point belongs to compact set and is the solution of IQVI problems. If the solution of IQVI problems is a singleton set, there is a stronger convergence result as follows: .

In Theorem 10, if , then the result of Theorem 10 still holds, that is, we can obtain the Corollary 13 as follows.

Corollary 13. Let be a nonempty compact subset of and satisfy the following assumptions: (i)For every , the two function sequences and a set-valued mapping sequence are satisfied bywhere and are continuous. And for all , is a nonempty convex compact set.
(ii)For every , is a nonempty subset of and where is a nonempty compact set of (iii)For every , is satisfied with , we havewhere and

Then, (1)There exists a convergent subsequence of which converges to some (2)For all , we have (3)If the solution of IQVI problems is a singleton set, there is

In Theorem 10, if , then the result of Theorem 10 still holds, that is, we can obtain the Corollary 14 as follows.

Corollary 14. Let be a nonempty compact subset of and all the following assumptions be satisfied: (i)For every , the two function sequences and a set-valued mapping sequence are satisfied bywhere and are continuous. And for all , is a nonempty convex compact set.
(ii) is a nonempty compact set of :(iii)For every , is an -approximate solution of function sequences for the IQVI problem, which satisfies , we have where and

Then, (1)There exists a convergent subsequence of which converges to some (2)For all , we have (3)If the solution of IQVI problems is singleton set, there is

4. Generic Convergence of Inverse Quasivariational Inequality

Let be a nonempty compact subset of , be a set of all nonempty compact subsets in . The function space of the IQVI problem is as follows:

For any , , we define the distance on

Theorem 15. is a complete metric space.

Proof. Clearly, is a metric space. Therefore, we only need to prove that is complete.
First, we assume that any Cauchy sequence in is . This means that for every , there exists an integer such that any , we have Then, there exist and such that , , we can obtain Because , and , it is clear that and are continuous at and is a convex set. Since is the Cauchy sequence in . According to a theorem in [20], is the complete metric space. Therefore, there exists such that , for any is a nonempty compact set, then is a nonempty convex compact set and is continuous at .
Next, we verify that (or ) is monotone quasiconcave (or monotone quasiconvex) and is a strongly monotone couple. Since , are monotone, is a strongly monotone couple, and are continuous at , that is, , then , are monotone and () is a strongly monotone couple. Again, because (or ) is quasiconcave (or quasiconvex), for every , , we have Since are continuous at , then there exists a real parameter such that Therefore, we can obtain By the arbitrariness of , then is quasiconcave and is quasiconvex at .
Again, according to , there exists such that , we have Since is a compact set, there exists such that is the convergence point of {}. It may be assumed that and , is continuous at , then Because is continuous at , , we define the distance on is , then Since is a nonempty convex compact set, we can obtain the result that .
Finally, by the contradiction method, we suppose that there exists satisfying ; thus, there exists a small enough such that Because and , there exists such that . Since are continuous at and , there exists such that for any , we can obtain Then, this conflicts with (33). So . Therefore, the metric space is complete. The proof is completed.

For all , we define

Then, we can see that by the definition of . A set-valued mapping is defined by .

Lemma 16. is a usco mapping on .

Proof. Since be a nonempty compact subset, according to Lemma 6, we only need to prove that is closed, which means that we have to prove that is closed, that is, for each , and for each and , then .
For all , since , we have Since are continuous at and , therefore, Then, we can obtain , . Since we have Thus, . The proof is completed.

Theorem 17. There exists a dense residual subset of such that for all , is a singleton set.

Proof. First, because is a complete metric space and is a usco mapping on , by the Fort lemma, there exists a dense residual subset of such that for all , the set-valued mapping is l.s.c. at .
For all , we suppose that is not a singleton set; thus, there exist , where . Simultaneously, there exist two open neighbourhoods of and of , respectively, such that and . Let , then for all , we can obtain For each , , we define Therefore, it is easy to see that is continuous.
Next, because , are monotone and for all , by Definition 4, we can obtain Similarly, thus, and are monotone.
Since is a strongly monotone couple on , by Definition 5, there exists a constant and for each , we have Then, Therefore, is a strongly monotone couple on .
Again, because is quasiconcave and is quasiconvex, so for all , according to Definition 7 we can obtain Similarly, Then, is quasiconcave and is quasiconvex.
Finally, for all , we have Therefore, , . Obviously, .
Note that , then . Since the set-valued mapping is l.s.c. at , there exists a sufficiently large integer such that . Take , we have Let , then that is, Because , for every , we have Let , then that is, This contradicts with the fact that is a strongly monotone couple on . Therefore, for any , is a singleton set. This completes the proof.

Theorem 18. There exists a dense residual subset of such that for all , there must be , , and for each .

Proof. According to Theorem 17, there exists a dense residual subset of such that for any , the solution of the IQVI problem is a singleton set. By conclusion (3) of Theorem 10, there must be . This completes the proof.

Remark 19. Theorem 18 shows that the solution set of the IQVI problem has generic convergence in the case of perturbation with the objective function on .

5. Conclusion

In this paper, we mainly obtained two new results for IQVI problems: one is the approximation theorem, and the other is the generic convergence theorem. According to the approximation theorem, we can see that the approximate solution representing bounded rationality can converge to the exact solution representing complete rationality from the perspective of Simon’s bounded rationality. Especially, notice Theorem 10, these conditions are general, that is, the objective functions , , and are not necessarily continuous and the set of feasible solutions is not necessarily compact. Hence, Theorem 10 provides a unified theoretical framework for the convergence of the approximate solution on IQVI problems. Moreover, the generic convergence of the IQVI problems implies that a certain sequence is found to converge to an exact solution, rather than a subsequence converging to an exact solution. And on the meaning of the Baire category, we obtained that most of IQVI problems have generic convergence under the perturbation with the objective function. The results obtained in this paper are new and different from the literatures [2, 3].

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare no conflict of interest.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant Nos. 12061020, 71961003), the Science and Technology Foundation of Guizhou Province (Grant Nos. 20201Y284, 20205016, 2021088, [2021]5640), and the Foundation of Guizhou University (Grant Nos. 201405, 201811). The authors acknowledge these supports.

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Copyright © 2022 Ting Liu and Wensheng Jia. 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.

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