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

Authors: Ting Liu; Wensheng Jia
Journal: Journal of Function Spaces
 (2022)

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## 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.


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## Body

## 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 vectors∈Rn such that ψs∈Hs, we have
(1)φs,h−ψs≥0,forallh∈HS,where φ,ψ:Rn⟶Rn represents two continuous mappings, respectively. H:Rn⟶2Rn denotes a set-valued mapping. For all s∈Rn, Hs is a nonempty closed convex set in Rn. In addition, ·,· represents the inner products and · denotes the norms in Rn. When the mapping ψ is the identity mapping or Hs represents a constant on Rn, 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, andD-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 generalizedf-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 generalizedf-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 ε-weakly efficient solution for a class of vector equilibrium problems under the Hausdorff topological vector space. Then, they proved the connectedness of the ε-efficient solution sets and ε-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 Section2, 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]).
LetW and D be two metric spaces, the set-valued mapping be expressed as ℵ:W⟶P0D, where P0D is the nonempty set, ℵ is said to be
(1)
lower semicontinuous (l.s.c.) atw∈W if for any open set ℧ in D, ℵw∩℧≠∅, there exists Ow of w, where Ow is an open neighbourhood such that ℵw′∩℧≠∅,foreveryw′∈Ow(2)
upper semi-continuous (u.s.c.) atw∈W if for any open set ℧ in D, ℵw⊂℧, there exists Ow of w, where Ow is an open neighbourhood such that ℵw′⊂℧,foreveryw′∈Ow(3)
a usco mapping onW if for any w∈W, ℵ:W⟶P0D is u.s.c and nonempty compact values(4)
continuous atw∈W if ℵ is both u.s.c and l.s.c. at wLemma 2 (see [19]).
LetAm be a sequence of nonempty bounded subset of Rn and A represents a nonempty bounded subset of Rn, where for each m=1,2,3,⋯. If Am⟶A and tm∈Am, then there exists a subsequence tmk of tm such that tmk⟶t∗∈Am⟶∞.Lemma 3 (see [19]).
LetAm denote a sequence of nonempty bounded subset of Rn and A represents a nonempty bounded subset of Rn, where for each m=1,2,3,⋯. Denote by G an open set of Rn, where G⋂A≠∅. If Am⟶A, thus there exists a positive integer N such that for each m≥N, G⋂Am≠∅m⟶∞.Definition 4 (see [1]).
Letξ:Rn⟶Rn represent a mapping, then ξ is monotonic on Rn if for any ς,σ∈Rn, one has
(2)ξς−ξσ,ς−σ≥0.Definition 5 (see [1]).
Assume that two mappings are denoted asξ,ϕ:Rn⟶Rn, respectively. If there exists a constant δ>0 such that for every ς,σ∈Rn, one has
(3)ξς−ξσ,ϕς−ϕσ≥δς−σ2,then ξ,ϕ is said to be δ-strongly monotone couple on Rn.Lemma 6 (see [17]).
LetM and F be two topological spaces, where F be a compact space. If the set-valued mapping ℵ:M⟶P0F is closed, then ℵ is a usco mapping on M.Definition 7 (see [20]).
Letξ:E⟶R and E denotes a nonempty convex set of Rn, then
(1)
ξ is quasiconvex on E, if for all e1,e2∈E and ρ∈0,1, we have(4)ξϱe1+1−ϱe2≤maxξe1,ξe2(2)
ξ is quasiconcave on E, if for all e1,e2∈E and ρ∈0,1, we have(5)ξϱe1+1−ϱe2≥minξe1,ξe2Lemma 8 (see [21], Fort Lemma).
LetM be a Hausdorff topological space and W be a metric space. The set-valued mapping ℵ:M⟶P0W is a usco mapping. Then, there exists a residual subset Q of M such that ℵ is l.s.c. on Q.


## 3. Approximation Theorem of Inverse Quasivariational Inequality

First, we introduce the concept of an approximate solution of IQVI problems.Definition 9.
LetS be a nonempty compact subset of Rn and denote by TS a set of all nonempty compact subsets in S. Let φ,ψ:S⟶S represent two continuous mappings. A set-valued mapping represents as H:S⟶TS such that for any s∈S, Hs is a nonempty convex compact set. For a real number ε>0, finding a vector s∗∈S such that ψs∗∈Hs∗, we have
(6)φs∗,h−ψs∗+ε≥0,forallh∈Hs∗.
Then,s∗ is said to be an ε-approximate solution of IQVI problems.Theorem 10.
LetS be a nonempty compact subset of Rn and satisfy the following assumptions:
(i)
For everym=1,2,⋯, the two function sequences φm,ψm:S⟶S and a set-valued mapping sequence Hm:S⟶TS are satisfied by
(7)sups∈Sφms−φs⟶0m⟶∞,sups∈Sψms−ψs⟶0m⟶∞,sups∈ShHms,Hs⟶0m⟶∞,where h denotes the Hausdorff distance defined on S. φ,ψ:S⟶S and H:S⟶TS are continuous and for all s∈S,Hs is a nonempty convex compact set
(ii)
For everym=1,2,⋯, Am is a nonempty subset sequence of S and
(8)hAm,A⟶0m⟶∞,where A is a nonempty compact set of S(iii)
For everym=1,2,⋯, sm∈S is satisfied with dsm,Am⟶0, ψmsm∈Hmsm,we have
(9)φmsm,h−ψmsm+εm≥0,forallh∈Hmsm,where εm≥0 and εm⟶0m⟶∞(1)
Then there exists a convergent subsequencesmk of sm which converges to some s∗∈Am⟶∞(2)
For allh∈Hs∗, we have
(10)φs∗,h−ψs∗≥0(3)
If the solution of IQVI problems is a singleton set, there must besm⟶s∗Proof.
(1)
Sincesm,Am⟶0m⟶∞, we can see that there exists sm′∈Am such that dsm,sm′⟶0. Because hAm,A⟶0 and A is a compact set by Lemma 2, for any sequence sm′ in S, there must be a subsequence sm′k such that sm′k⟶s∗∈A. Therefore, there exists a subsequence smk of the sequence sm such that smk⟶s∗∈A(2)
According to conclusion (1), it may be assumed thatsm⟶s∗∈A. By contradiction, we assume that conclusion (2) does not hold. Thus, there exists h∈Hs∗ such that(11)φs∗,h−ψs∗<0
First, sinceφ,ψ:S⟶S are continuous at s and φs,h−ψs is continuous at variables s and h, there exists a real number γ0>0 and two open neighbourhoods Vs∗ and Vh0 of s∗ and h0 such that for any s′∈Vs∗, h′∈Vh0, we have
(12)φs′,h′−ψs′<−γ0.
According to the Cauchy-Swartz inequality, we can obtain(13)φms−φs,h−ψms−ψs≤φms−φsh−ψms−ψs≤φms−φsh+ψms−ψs,that is,
(14)sups∈Sφms−φs⟶0m⟶∞,sups∈Sψms−ψs⟶0m⟶∞.
This means that(15)sups∈S,h∈Hsφms−φs,h−ψms−ψs⟶0m⟶∞.
Again, becauseεm⟶0m⟶∞, there exists a positive integer N1 such that for every m≥N1, we have
(16)sups∈S,h∈Hsφms−φs,h−ψms−ψs<γ02,εm<γ02.
Finally, sincesm⟶s∗∈A, hHm,H⟶0m⟶∞, and h0∈Hs, according to Lemma 3, we know that there exists a positive integer N2. It may be assumed that N2≥N1 such that for all m≥N2, we have sm∈Vs∗ and Vh0⋂Hmsm≠∅. Let hm∈Vh0⋂Hmsm, thus, we have
(17)φsm,hm−ψsm<−γ0,φmsm−φsm,hm−ψmsm−ψsm=φmsm,hm−ψmsm+φmsm,ψsm−φsm,hm−ψmsm−φsm,ψsm=φmsm,hm−ψmsm−φsm,hm−ψsm+φmsm−φsm,ψsm−φsm,ψsm−ψmsm<γ02.
Then,(18)φmsm,hm−ψmsm<φsm,hm−ψsm−φmsm−φsm,ψsm+φsm,ψsm−ψmsm+γ02≤φsm,hm−ψsm+φsm−φmsmψsm+φsmψsm−ψmsm+γ02≤φsm,hm−ψsm+γ02<−γ02<−εm.
Then, this conflicts with condition (iii). Hence, for allh∈Hs∗, we have φs∗,h−ψs∗≥0.
(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, thensm↛s∗. Hence, there exists γ>0 and a subsequence smk of sm such that s∗−smk≥γ. Next, by conclusion (1), it can be seen that sequence smk must have subsequences. Let smk⟶s¯∈A, that is, s¯−smk⟶0, according to conclusion (2), we can obtain φs¯,h−ψs¯≥0. Finally, because the solution of IQVI problems is singleton set, therefore s¯=s∗, that is, s¯−smk≥γ, which conflicts with s¯−smk⟶0. Therefore, we can obtain sm⟶s∗. The proof is completed.Remark 11.
According to Theorem10, although the objective function, feasible solution set are all approximated (that is, φm⟶φ, ψm⟶ψ, Hm⟶H, and Am⟶A), we can obtain an approximation sequence sm, which must have convergent subsequences smk, that is, smk⟶s∗∈A and s∗ must be the solution of IQVI problems. If the εm-approximate solution sm of IQVI problems is regarded as a “satisfactory solution” under bounded rationality, and the solution s∗ 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.
Theorem10 shows that the limit points and convergent subsequences existing for sequence sm are equivalent. It can be seen from the above proof that the sequence sm must have limit points. Each limit point belongs to compact set A 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: sm⟶s.In Theorem10, if sm∈Amm=1,2,⋯, then the result of Theorem 10 still holds, that is, we can obtain the Corollary 13 as follows.Corollary 13.
LetS be a nonempty compact subset of Rn and satisfy the following assumptions:
(i)
For everym=1,2,⋯, the two function sequences φm,ψm:S⟶S and a set-valued mapping sequence Hm:S⟶TS are satisfied by(19)sups∈Sφms−φs⟶0m⟶∞,sups∈Sψms−ψs⟶0m⟶∞,sups∈ShHms,Hs⟶0m⟶∞,where φ,ψ:S⟶S and H:S⟶TS are continuous. And for all s∈S, Hs is a nonempty convex compact set.
(ii)
For everym=1,2,⋯, Am is a nonempty subset of S and
(20)hAm,A⟶0m⟶∞,where A is a nonempty compact set of S(iii)
For everym=1,2,⋯, sm∈Am is satisfied with ψmsm∈Hmsm, we have(21)φmsm,h−ψmsm+εm≥0,forallh∈Hmsm,where εm≥0 and εm⟶0m⟶∞Then,(1)
There exists a convergent subsequencesmk of sm which converges to some s∗∈Am⟶∞(2)
For allh∈Hs∗, we have
(22)φs∗,h−ψs∗≥0(3)
If the solution of IQVI problems is a singleton set, there issm⟶s∗In Theorem10, if A=Amm=1,2,⋯, then the result of Theorem 10 still holds, that is, we can obtain the Corollary 14 as follows.Corollary 14.
LetS be a nonempty compact subset of Rn and all the following assumptions be satisfied:
(i)
For everym=1,2,⋯, the two function sequences φm,ψm:S⟶S and a set-valued mapping sequence Hm:S⟶TS are satisfied by(23)sups∈Sφms−φs⟶0m⟶∞,sups∈Sψms−ψs⟶0m⟶∞,sups∈ShHms,Hs⟶0m⟶∞,where φ,ψ:S⟶S and H:S⟶TS are continuous. And for all s∈S, Hs is a nonempty convex compact set.
(ii)
A is a nonempty compact set of S:(iii)
For everym=1,2,⋯, sm∈Am is an εm-approximate solution of function sequences for the IQVI problem, which satisfies ψmsm∈Hmsm, we have
(24)φmsm,h−ψmsm+εm≥0,forallh∈Hmsm,where εm≥0 and εm⟶0m⟶∞Then,(1)
There exists a convergent subsequencesmk of sm which converges to some s∗∈Am⟶∞(2)
For allh∈Hs∗, we have
(25)φs∗,h−ψs∗≥0(3)
If the solution of IQVI problems is singleton set, there issm⟶s∗


## 4. Generic Convergence of Inverse Quasivariational Inequality

LetS be a nonempty compact subset of Rn, TS be a set of all nonempty compact subsets in S. The function space U of the IQVI problem is as follows:
(26)U=φ:S⟶Sisacontinuousmappingwithmonotonequasiconcave;ψ:S⟶Sisacontinuousmappingwithmonotonequasiconvex;u=φ,ψ,H:φ,ψisastronglymonotonecouple;H:S⟶TSisacontinuousset‐valuedmapping,andforanys∈S,Hsisanonemptyconvexcompactset;thereiss∈Ssuchthatψs∈Hs,andforanyh∈Hs,wehaveφs,h−ψs≥0.For anyu1=φ1,ψ1,H1, u2=φ2,ψ2,H2∈U, we define the distance on U(27)ℓu1,u2=sups∈Sφ1s−φ2s+sups∈Sψ1s−ψ2s+sups∈ShH1s,H2s.Theorem 15.
U,ℓ is a complete metric space.Proof.
Clearly,U,ℓ is a metric space. Therefore, we only need to prove that U,ℓ is complete.
First, we assume that any Cauchy sequence inU is um=φm,ψm,Hmm=1∞. This means that for every θ>0, there exists an integer N1 such that any p,m≥N1, we have
(28)ℓum,up=sups∈Sφms−φps+sups∈Sψms−ψps+sups∈ShHms,Hps<θ.
Then, there existφ,ψ:S⟶S and H:S⟶TS such that limp⟶∞φps=φs,limp⟶∞ψps=ψs, limp⟶∞Hps=Hs, we can obtain
(29)sups∈Sφs−φms≤θ,sups∈Sψs−ψms≤θ,sups∈ShHs,Hms≤θ.
Becausesm⟶s, φm⟶φ, and ψm⟶ψ, it is clear that φ and ψ are continuous at s and Hs is a convex set. Since Hms is the Cauchy sequence in TS. According to a theorem in [20], TS,h is the complete metric space. Therefore, there exists H:S⟶TS such that Hms⟶Hs, for any s∈S,Hs is a nonempty compact set, then Hs is a nonempty convex compact set and H is continuous at s.
Next, we verify thatφ (or ψ) is monotone quasiconcave (or monotone quasiconvex) and φ,ψ is a strongly monotone couple. Since φm, ψm are monotone, φm,ψm is a strongly monotone couple, and φ,ψ are continuous at s, that is, φm⟶φ,ψm⟶ψ, then φ, ψ are monotone and (φ,ψ) is a strongly monotone couple. Again, because φm (or ψm) is quasiconcave (or quasiconvex), for every s1,s2∈S, ρ∈0,1, we have
(30)φmρs1+1−ρs2≥minφms1,φms2,ψmρs1+1−ρs2≤maxψms1,ψms2.
Sinceφ,ψ are continuous at s, then there exists a real parameter θ>0 such that
(31)φs−θ<φms<θ+φs,ψs−θ<ψms<θ+ψs.
Therefore, we can obtain(32)φρs1+1−ρs2+θ>φmρs1+1−ρs2≥minφms1,φms2>minφs1−θ,φs2−θ,ψρs1+1−ρs2−θ<ψmρs1+1−ρs2≤maxψms1,ψms2<maxψs1+θ,ψs2+θ.
By the arbitrariness ofθ, then φ is quasiconcave and ψ is quasiconvex at s.
Again, according toum=φm,ψm,Hm∈U, there exists sm∈S such that ψmsm∈Hmsm, we have
(33)φmsm,h−ψmsm≥0,foranyh∈Hmsm.
SinceS is a compact set, there exists s∈S such that s is the convergence point of {sm}. It may be assumed that sm⟶s and Hms⟶Hs, H is continuous at s, then
(34)hHmsm,Hs≤hHmsm,Hsm+hHsm,Hs⟶0.
Becauseψ is continuous at s, ψms⟶ψs, we define the distance on S is d, then
(35)dψs,Hs≤dψs,ψmsm+dψmsm,Hmsm+hHmsm,Hs⟶0.
SinceHs is a nonempty convex compact set, we can obtain the result that ψs∈Hs.
Finally, by the contradiction method, we suppose that there existsh0∈Hs satisfying φs,h0−ψs<0; thus, there exists a small enough β>0 such that
(36)φs,h0−ψs+β<0.
BecausehHmsm,Hs⟶0 and h0∈Hs, there exists hm∈Hmsm such that hm⟶h0. Since φ·,ψ· are continuous at s and φm⟶φ,ψm⟶ψ,sm⟶s,hm⟶h0, there exists m1≥0 such that for any m≥m1, we can obtain
(37)φmsm,hm−ψmsm<φs,h0−ψs+β<0.
Then, this conflicts with (33). So u=φ,ψ,H∈U. Therefore, the metric space U,ℓ is complete. The proof is completed.For allu=φ,ψ,H∈U, we define
(38)Ωu=s∈S:ψs∈Hs,forallh∈Hs,φs,h−ψs≥0.Then, we can see thatΩu≠∅ by the definition of U. A set-valued mapping Ω:U⟶P0S is defined by u⟶Ωu.Lemma 16.
Ω is a usco mapping on U.Proof.
SinceS be a nonempty compact subset, according to Lemma 6, we only need to prove that Ω is closed, which means that we have to prove that GraphΩ is closed, that is, for each um∈U, um⟶u and for each sm∈Ωum and sm⟶s, then s∈Ωu.
For allm=1,2,⋯, since sm∈Ωum, we have
(39)φmsm,h−ψmsm≥0,forallh∈Hmsm.
Sinceφ·,ψ· are continuous at s and φm⟶φ,ψm⟶ψ,sm⟶s, therefore,
(40)φmsm−φs≤φmsm−φsm+φsm−φs⟶0,ψmsm−ψs≤ψmsm−ψsm+ψsm−ψs⟶0.
Then, we can obtainφmsm⟶φs, ψmsm⟶ψs. Since h−ψmsm⟶h−ψs, we have
(41)φs,h−ψs=limm⟶∞φmsm,h−ψmsm≥0.
Thus,s∈Ωu. The proof is completed.Theorem 17.
There exists a dense residual subsetQ of U such that for all u=φ,ψ,H∈Q, Ωu is a singleton set.Proof.
First, becauseU is a complete metric space and Ω is a usco mapping on U, by the Fort lemma, there exists a dense residual subset Q of U such that for all u=φ,ψ,H∈Q, the set-valued mapping Ω is l.s.c. at u.
For allu=φ,ψ,H∈Q, we suppose that Ωu is not a singleton set; thus, there exist s1,s2∈Ωu, where s1≠s2. Simultaneously, there exist two open neighbourhoods K of s1 and L of s2, respectively, such that K⋂L≠∅ and K⋂Ωu≠∅. Let s1∈K⋂Ωu, then for all h∈Hs1, we can obtain
(42)φs1,h−ψs1≥0.
For eachs∈S, m=1,2,⋯, we define
(43)φms=φs+1mψs−ψs1,ψms=ψs+1mψs−ψs1.
Therefore, it is easy to see thatφm,ψm is continuous.
Next, becauseφ, ψ are monotone and for all s,l∈S, by Definition 4, we can obtain
(44)φms−φml,s−l=φs−φl+1mψs−ψl,s−l=φs−φl,s−l+1mψs−ψl,s−l≥0.
Similarly,(45)ψms−ψml,s−l≥0,thus, φm and ψm are monotone.
Sinceφ,ψ is a strongly monotone couple on S, by Definition 5, there exists a constant δ>0 and for each s,l∈S, we have
(46)φs−φl,ψs−ψl≥δs−l2.
Then,(47)φms−φml,ψms−ψml=φs−φl+1mψs−ψl,m+1mψs−ψl=m+1mφs−φl,ψs−ψl+m+1m2ψs−ψl2≥δs−l2≥0.
Therefore,φm,ψm is a strongly monotone couple on S.
Again, becauseφ is quasiconcave and ψ is quasiconvex, so for all ρ∈0,1, according to Definition 7 we can obtain
(48)φmρs1+1−ρs2=φρs1+1−ρs2+1mψρs1+1−ρs2−ψs1≥φρs1+1−ρs2≥minφs1,φs2.
Similarly,(49)ψmρs1+1−ρs2=ψρs1+1−ρs2+1mψρs1+1−ρs2−ψs1=m+1mψρs1+1−ρs2−1mψs1≤m+1mmaxψs1.ψs2.
Then,φm is quasiconcave and ψm is quasiconvex.
Finally, for allh∈Hms1, we have
(50)φms1,h−ψms1=φs1,h−ψs1≥0.
Therefore,s1∈Ωum, um∈U. Obviously, ℓum,u⟶0m⟶∞.
Note thats2∈L, then L⋂Ωu≠∅. Since the set-valued mapping Ω is l.s.c. at u, there exists a sufficiently large integer m0 such that L⋂Ωum0≠∅. Take sm0∈L⋂Ωum0, we have
(51)φm0sm0,h−ψm0sm0≥0,forallh∈Hm0sm0.
Leth=ψs1, then
(52)φm0sm0,ψs1−ψm0sm0=φsm0+1m0ψsm0−ψs1,ψs1−ψsm0−1m0ψsm0−ψs1=φsm0,m0+1m0ψs1−ψsm0−m0+1m02ψs1−ψsm02≥0,that is,
(53)φsm0,ψs1−ψsm0≥1m0ψs1−ψsm02>0.
Becauses1∈Ωu, for every h∈Hs1, we have
(54)φs1,h−ψs1≥0.
Leth=ψsm0, then
(55)φs1,ψsm0−ψs1≥0,that is,
(56)φs1−φsm0,ψs1−ψsm0=φs1,ψs1−ψsm0−φsm0,ψs1−ψsm0=−φs1,ψsm0−ψs1+φsm0,ψs1−ψsm0<0.
This contradicts with the fact thatφ,ψ is a strongly monotone couple on S. Therefore, for any u=φ,ψ,H∈Q, Ωu is a singleton set. This completes the proof.Theorem 18.
There exists a dense residual subsetQ of U such that for all u=φ,ψ,H∈Q, there must be sm⟶s, ψs∈Hs, and φs,h−ψs≥0 for each h∈Hs.Proof.
According to Theorem17, there exists a dense residual subset Q of U such that for any u=φ,ψ,H∈Q, the solution of the IQVI problem is a singleton set. By conclusion (3) of Theorem 10, there must be sm⟶s. This completes the proof.Remark 19.
Theorem18 shows that the solution set of the IQVI problem has generic convergence in the case of perturbation with the objective function on S.


## 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 Theorem10, these conditions are general, that is, the objective functions φm, ψm, and Hmm=1,2,3,⋯ are not necessarily continuous and the set of feasible solutions Am 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].



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