# α-Well-Posedness for Quasivariational Inequality Problems

Authors: Jian Wen Peng; Jing Tang
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We introduce and study the concepts ofα-well-posedness and L-α-well-posedness for quasivariational inequality problems having a unique solution and the concepts of α-well-posedness in the generalized sense and L-α-well-posedness in the generalized sense
for quasivariational inequality problems having more than one solution. We present some
necessary and/or sufficient conditions for the various kinds of well-posedness to occur. Our
results generalize and strengthen previously known results for quasivariational inequality
problems.


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## Body

## 1. Introduction

LetE be a reflexive real Banach space and let K be a nonempty closed convex subset of E. Let S be a set-valued mapping from K to K and let A be an operator from E to the dual space E*. Bensoussan and Lions [1], Baiocchi and Capelo [2], and Mosco [3] considered the following quasivariational inequality (in short, (QVIP)), which is to find a point u0∈K such that(1.1)u0∈S(u0),〈Au0,u0-v〉≤0,∀v∈S(u0).The interest in quasivariational inequality problems lies in the fact that many economic or engineering problems are modeled through them, as explained in [4, 5] where a wide bibliography on variational inequalities, quasivariational inequality problems, and related problems is contained. Moreover, under suitable assumptions, a quasivariational inequality is equivalent to a generalized Nash equilibrium problem [3].On the other hand, well-posedness plays a crucial role in the stability theory for optimization problems, which guarantees that, for an approximating solution sequence, there exists a subsequence which converges to a solution [6]. The study of well-posedness for scalar minimization problems started from Tikhonov [7] and Levitin and Polyak [8]. Since the convergence of numerical methods for quasivariational inequality Problems can be obtained also with the aid of well-posedness theory, Lignola [9] introduced and investigated the concepts of well-posedness and L-well-posedness for quasivariational inequalities having a unique solution and the concepts of well-posedness and L-well-posedness in the generalized sense for quasivariational inequality problems having more than one solution.In this paper, inspired by the above concepts of well-posedness for (QVIP), we introduce and study the concepts ofα-well-posedness and L-α-well-posedness for quasivariational inequality Problems having a unique solution and the concepts of α-well-posedness in the generalized sense and L-α-well-posedness in the generalized sense for quasivariational inequality Problems having more than one solution. The results in this paper generalize and improve the results in [9, 10].


## 2. Preliminaries

Denote byΓ the solution set of (QVIP). Let α>0. In order to investigate the α-well-posed for (QVIP), we need the following definitions.First we recall the notion of Mosco convergence [11]. A sequence (Hn)n of subsets of E Mosco converges to a set H if(2.1)H=lim⁡inf⁡nHn=w-limsup⁡nHn,
where limninfHn and w-limnsupHn are, respectively, the Painlevé-Kuratowski strong limit inferior and weak limit superior of a sequence (Hn)n, that is,(2.2)lim⁡inf⁡nHn={y∈E:∃yn∈Hn,n∈N,withyn⟶y},w-limsup⁡nHn={y∈E:∃nk↑+∞,nk∈N,∃ynk∈Hnk,k∈N,withynk⇀y},
where “⇀” means weak convergence, “→” means strong convergence.IfH=limninfHn, we call the sequence (Hn)n of subsets of E Lower Semi-Mosco which converges to a set H.It is easy to see that a sequence(Hn)n of subsets of E Mosco converges to a set H which implies that the sequence (Hn)n, also Lower Semi-Mosco, converges to the set H, but the converse is not true in general.We will use the usual abbreviations usc and lsc for “upper semicontinuous” and “lower semicontinuous,” respectively. LetE be a reflexive real Banach space with dual E*. An operator A:E→E* will be called hemicontinuous if it is continuous from every segment of E to E* endowed with the weak topology. A:E→E* will be called monotone if 〈Au-Av,u-v〉≥0 for every u, and v∈E. A:E→E* will be called pseudomonotone if 〈Au,u-v〉≤0⇒〈Av,u-v〉≤0 for every u and v∈E.Definition 2.1.
A sequence(un)n is an α-approximating sequence for (QVIP) if
(i)
(un)∈K, for all n∈N;(ii)
there exists a sequence(ɛn)n,ɛn>0, decreasing to 0 such that(2.3)d(un,S(un))≤ɛn,thatis,un∈B(S(un),ɛn),∀n∈N,〈Aun,un-v〉-α2‖un-v‖2≤ɛn,∀v∈S(un),∀n∈N.Definition 2.2.
A quasivariational inequality (QVIP) is said to beα-well-posed (resp., α-well-posed in the generalized sense) if it has a unique solution u0 and every α-approximating sequence (un)n strongly converges to u0 (resp., if the solution set Γ of (QVIP) is nonempty and for every α-approximating sequence (un)n has a subsequence which strongly converges to a point of Γ).Definition 2.3.
A sequence(un)n is an L-α-approximating sequence for (QVIP) if:
(i)
(un)∈K, for all n∈N;(ii)
there exists a sequence(ɛn)n,ɛn>0, decreasing to 0 such that d(un,S(un))≤ɛn, and(2.4)〈Av,un-v〉-α2‖un-v‖2≤ɛn,∀v∈S(un),∀n∈N.Definition 2.4.
A quasivariational inequality (QVIP) is said to be L-α-well-posed (resp., L-α-well-posed in the generalized sense) if it has a unique solution u0 and every L-α-approximating sequence (un)n strongly converges to u0 (resp., if the solution set Γ of (QVIP) is nonempty and for every L-α-approximating sequence (un)n has a subsequence which strongly converges to a point of Γ).
It is worth noting that ifα=0, then the definitions of α-well-posedness, α-well-posedness in the generalized sense, L-α-well-posedness, and L-α-well-posedness in the generalized sense for (QVIP), respectively, reduce to those of the well-posedness, well-posedness in the generalized sense, L-well-posedness, and L-well-posedness in the generalized sense for (QVIP) in [9]. We also note that Definition 2.2 generalizes and extends α-well-posedness and α-well-posedness in the generalized sense of variational inequalities in [10] which are related to the continuously differentiable gap function of variational inequality Problems introduced by Fukushima [12].We recall some lemmas which will be needed in the rest of this paper.Lemma 2.5 (see [13]).
Let(Hn)n be a sequence of nonempty subsets of the space E such that
(i)
Hn is convex for every n∈N;(ii)
H0⊆limninfHn;(iii)
there existsm∈N such that int⋂n≥mHn≠∅.
Then, for everyu0∈intH0, there exists a positive real number δ such that B(u0,δ)⊆Hn, for all n≥m.
IfE is a finite dimensional space, then assumption (iii) can be replaced by
(iii)’
intH0≠∅.The following Lemmas2.6 and 2.7 play important roles in this paper. Now we present a Minty type lemma for quasivariational inequalities as follows.Lemma 2.6.
Suppose that set-valued mappingS is nonempty convex-valued, the operator A is hemicontinuous and monotone, u0∈S(u0). Then the following conditions are equivalent:
(i)
〈Au0,u0-v〉-(α/2)∥u0-v∥2≤0, for all v∈S(u0),(ii)
〈Av,u0-v〉-(α/2)∥u0-v∥2≤0, for all v∈S(u0).Proof.
We first prove that (ii) implies (i). Letv be a arbitrary point of S(u0). For every number t∈[0,1], since the set-valued mapping S is convex-valued and u0∈S(u0), the point vt=tv+(1-t)u0 belongs to S(u0). It follows from (ii) that
(2.5)〈Avt,u0-vt〉-α2‖u0-vt‖2≤0.
From the definition of vt, one has
(2.6)lim⁡t→0(〈Avt,u0-v〉-α2t‖u0-v‖2)≤0,and it follows from the hemicontinuity of A that
(2.7)〈Au0,u0-v〉≤0,
then
(2.8)〈Au0,u0-v〉-α2‖u0-v‖2≤0,∀v∈S(u0).
The converse is an easy consequence of monotonicity of A.Lemma 2.7.
Assume that set-valued mappingS is nonempty convex-valued, then u0∈Γ if and only if the following conditions hold:(2.9)u0∈S(u0),〈Au0,u0-v〉-α2‖u0-v‖2≤0,∀v∈S(u0).Proof.
The necessity is clearly held. Now we prove the sufficiency. Let for allv∈S(u0), for all t∈[0,1], vt=tv+(1-t)u0. Since S is convex-valued, vt∈S(u0), one has
(2.10)〈Au0,u0-vt〉-α2‖u0-vt‖2≤0,∀t∈(0,1],
which implies that
(2.11)〈Au0,u0-v〉-tα2‖u0-v‖2≤0,∀t∈(0,1],∀v∈S(u0).
The above inequality implies, for t converging to zero, that u0 is a solution of (QVIP). This completes the proof.


## 3. Case of a Unique Solution

In this section, we investigate some metric characterizations ofα-well-posedness and L-α-well-posedness for (QVIP).For anyɛ>0, we consider the set(3.1)Qɛ={u∈K:u∈B(S(u),ɛ),〈Au,u-v〉-α2‖u-v‖2≤ɛ,∀v∈S(u)}Lɛ={u∈K:u∈B(S(u),ɛ),〈Av,u-v〉-α2‖u-v‖2≤ɛ,∀v∈S(u)}.Theorem 3.1.
Let the same assumptions be as in Lemma2.7. Then, one has(a)
(QVIP) isα-well-posed if and only if the solution set Γ of (QVIP) is nonempty and limɛ→0diamQɛ=0;(b)
moreover, ifA:E→E* is pseudomonotone, then (QVIP) is L-α-well-posed if and only if the solution set Γ of (QVIP) is nonempty and limɛ→0diamLɛ=0.Proof.
We only prove (a). The proof of (b) is similar and is omitted here. Suppose that (QVIP) isα-well-posed, then Γ≠∅. It follows from Lemma 2.7 that Qɛ≠∅. Suppose by contradiction that there exists a real number β, such that limɛ→0diamQɛ>β>0, then there exists ɛn>0, with ɛn→0, and (wn)n,(zn)n∈Qɛn, such that ∥wn-zn∥>β, for all n∈N. Since the sequences (wn)n,(zn)n are both α-approximating sequences for (QVIP), (wn)n and (zn)n strongly converge to the unique solution u0, and this gives a contradiction. Therefore, limɛ→0diamQɛ=0.
Conversely, let(un)n,un∈K, be an α-approximating sequence for (QVIP). Then there exists a sequence ɛn>0, with ɛn→0, such that
(3.2)d(un,S(un))≤ɛn,∀n∈N,〈Aun,un-v〉-α2‖un-v‖2≤ɛn,∀v∈S(un),∀n∈N.
that is, un⊂Qɛn, for all n∈N. It is easy to see limɛ→0diamQɛ=0 and Γ≠∅ implying that Γ is a singleton point set. Indeed, if there exist two different solutions z1,z2, then from Lemma 2.7, we know that z1,z2∈Qɛ, for all ɛ>0. Thus, limɛ→0diamQɛ≥∥z1-z2∥≠0, a contraction. Let u0 be the unique solution of (QVIP). It follows from Lemma 2.7 that u0∈Qɛn. Thus, limn→0∥un-u0∥≤limn→0diamQɛn=0. So (un)n strongly converge to u0. Therefore, (QVIP) is α-well-posed.Theorem 3.2.
Letα>0 and the following assumptions hold:
(i)
the set-valued mappingS is nonempty convex-valued, and, for each sequence (un)n in K converges to u0, the sequence (S(un))n Lower Semi-Mosco converging to S(u0);(ii)
for every converging sequence(hn)n, there exists m∈N, such that(3.3)int⁡⋂n≥mS(hn)≠∅;(iii)
the operatorA is hemicontinuous and monotone on K.Then, (QVIP) is α-well-posed if and only if
(3.4)Qɛ≠∅,∀ɛ>0,lim⁡ɛ→0diam⁡Qɛ=0.Proof.
The necessity has been proved in Theorem3.1(a).
Conversely, assume that (3.4) holds. It is easy to see that (3.4) implies that the solution set Γ of (QVIP) is a singleton point set. Let (un)n be an α-approximating sequence for (QVIP), that is, there exists a sequence ɛn>0, with ɛn→0, such that
(3.5)d(un,S(un))≤ɛn,∀n∈N,〈Aun,un-v〉-α2‖un-v‖2≤ɛn,∀v∈S(un),∀n∈N.
Therefore, un⊂Qɛn, for all n∈N. In light of (3.4), (un)n is a Cauchy sequence and strongly converges to a point u0∈K. In order to obtain that u0 solves (QVIP), we start to prove that u0∈S(u0). For each n∈N, choose un′∈S(un), such that ∥un-un′∥<d(un,S(un))+ɛn≤2ɛn. It follows from un→u0 and ɛn→0 that un′→u0. It follows from the assumption (i) that limninfS(un)=S(u0). Thus, u0∈S(u0).
To complete the proof, consider an arbitrary pointv∈S(u0). By Lower Semi-Mosco convergence again, we have S(u0)⊆limninfS(un). Also observe that assumption (ii) applied to the constant sequence hn=u0, for all n∈N, implies that intS(u0)≠∅. From Lemma 2.5, it follows that if v∈intS(u0), then there exist m∈N and δ>0 such that intB(v,δ)⊆S(un), for all n>m. Thus, v∈S(un) for n sufficiently large. Notice the A is monotone and the sequence (un)n is an α-approximating sequence for (QVIP), then we have
(3.6)〈Av,u0-v〉=lim⁡n〈Av,un-v〉≤lim⁡inf⁡n〈Aun,un-v〉≤lim⁡n(ɛn+α2‖un-v‖2)=α2‖u0-v‖2.
If v∈S(u0)∖intS(u0), let (vn)n be a sequence converging to v, whose point belongs to a segment contained in intS(u0). Since vn∈intS(u0), for all n∈N, one has
(3.7)〈Avn,u0-vn〉≤α2‖u0-vn‖2.
Since the hemicontinuity of A,
(3.8)〈Av,u0-v〉≤α2‖u0-v‖2,∀v∈S(u0).
It follows from Lemma 2.6 that
(3.9)〈Au0,u0-v〉-α2‖u0-v‖2≤0,∀v∈S(u0),
then, by Lemma 2.7, we obtain that u0 solves (QVIP). This completes the proof.Now, we present a result in which assumption (ii) of above theorem is dropped, while the continuity assumption on the operatorA is strengthened.Theorem 3.3.
Let the following assumptions hold:(i)
the set-valued mappingS is nonempty convex-valued, and, for each sequence (un)n in K converging to u0, the sequence (S(un))n Lower Semi-Mosco converges to S(u0);(ii)
the operatorA is (s,w)-continuous on K.
Then, (QVIP) isα-well-posed if and only if (3.4) holds.Proof.
The necessity follows from Theorem3.1 and Lemma 2.7.
Conversely, let(un)n be an α-approximating sequence for (QVIP) and (3.4) holds. From (3.4) and the proof of Theorem 3.2, we can obtain that (un)n strongly converges to u0, with u0∈S(u0). Since Lower Semi-Mosco convergence implies for every v∈S(u0), there exists sequence (vn)n strongly converging to v such that vn∈S(un). Since the operator A is (s,w)-continuous and (un)n is an α-approximating sequence for (QVIP), we have
(3.10)〈Au0,u0-v〉=lim⁡n〈Aun,un-vn〉≤lim⁡n(ɛn+α2‖un-vn‖2)=α2‖u0-v‖2.
By Lemma 2.7, we obtain that u0 solves (QVIP). This completes the proof.Theorem 3.4.
Let the following assumptions hold:
(i)
the set-valued mappingS is nonempty convex-valued, and, for each sequence (un)n in K converges to u0, the sequence (S(un))n Lower Semi-Mosco converging to S(u0);(ii)
for every converging sequence(hn)n, there exists m∈N, such that(3.11)int⁡⋂n≥mS(hn)≠∅;(iii)
the operatorA is hemicontinuous and monotone on K.
Then, (QVIP) is L-α-well-posed if and only if
(3.12)Lɛ≠∅,∀ɛ>0,lim⁡ɛ→0diam⁡Lɛ=0.Proof.
Assume that (QVIP) is L-α-well-posed, then it follows from the monotonicity of A that ∅≠Γ≠Lɛ, for all ɛ>0. It follows from Theorem 3.1(b) that the necessity can be completed.
Assume that (3.12) holds. Let (un)n be an L-α-approximating sequence for (QVIP), then there exists a sequence ɛn>0, with ɛn→0, such that un∈Lɛn, for all n∈N. Following the same argument as the proof of Theorem 3.1, it is easy to see limɛ→0diamLɛ=0 and Γ≠∅ imply that Γ is a singleton point set. In light of the assumption, (un)n is a Cauchy sequence and strongly converges to a point u0∈K and u0∈S(u0).  Let v∈intS(u0) and using Lemma 2.5, one has v∈S(un), for n sufficiently large. Then, we get
(3.13)〈Av,u0-v〉-α2‖u0-v‖2=lim⁡n[〈Av,un-v〉-α2‖un-v‖2]≤lim⁡nɛn=0.
If  v∈S(u0)∖intS(u0),  let a sequence vn converges to v, whose points belong to a segment contained in  intS(u0).   Since
(3.14)〈Avn,u0-vn〉-α2‖u0-vn‖2≤0
and the operator A is hemicontinuous, one gets
(3.15)〈Av,u0-v〉-α2‖u0-v‖2≤0.
According to Lemmas 2.6 and 2.7, u0 is the solution of (QVIP).Theorem 3.5.
Let the following assumptions hold:(i)
the set-valued mappingS is nonempty convex-valued, and, for each sequence (un)n in K converging to u0, the sequence (S(un))n Lower Semi-Mosco converges to S(u0);(ii)
the operatorA is (s,w)-continuous and monotone on K.
Then, (QVIP) is L-α-well-posed if and only if (3.12) holds.Proof.
Assume (3.12) holds. Let (un)n be an L-α-approximating sequence for (QVIP), then there exists a sequence ɛn>0, with ɛn→0, such that (un)n⊂Lɛn, for all n∈N. Since limɛ→0diamLɛ=0, (un)n is a Cauchy sequence and converges to u0. As the similar proof of Theorem 3.2, u0∈S(u0). Let v∈S(u0).   Since Lower Semi-Mosco convergence implies for every v∈S(u0), there exists a sequence (vn)n converging to v, such that vn∈S(un).   Since A is (s,w)-continuous and (un)n is an L-α-approximating sequence for (QVIP), one has
(3.16)〈Av,u0-v〉-α2‖u0-v‖2=lim⁡n[〈Avn,un-vn〉-α2‖un-vn‖2]≤lim⁡nɛn=0.
Applying Lemmas 2.6 and 2.7, we have that (QVIP) is L-α-well-posed.
The necessity can be completed as Theorem3.3.


## 4.α-Well-Posedness in the Generalized Sense

In this section, we introduce and investigate some metric characterizations ofα-well-posedness in the generalized sense and L-α-well-posedness in the generalized sense for (QVI).Definition 4.1 (see [11]).
Let(X,d) be a metric space and let A,B be nonempty subsets of X. The Hausdorff distance H(·,·) between A and B is defined by
(4.1)H(A,B)=max⁡{e(A,B),e(B,A)},
where e(A,B)=supa∈Ad(a,B) with d(a,B)=infb∈B∥a-b∥.Definition 4.2 (see [11]).
LetA be a nonempty subset of X. The measure of non compactness μ of the set A is defined by
(4.2)μ(A)=inf⁡{ɛ>0:A⊆⋃i=1nAi,diam⁡Ai<ɛ,i=1,2,…,n},
where diam means the diameter of a set.Theorem 4.3.
Let the same assumptions be as in Lemma2.7. Then, one has the following.(a)
(QVIP) isα-well-posed in the generalized sense if and only if the solution set Γ of (QVIP) is nonempty compact and e(Qɛ,Γ)→0,asɛ→0.(b)
Moreover, ifA is pseudomonotone, then (QVIP) is L-α-well-posed in the generalized sense if and only if the solution set Γ of (QVIP) is nonempty compact and e(Lɛ,Γ)→0,asɛ→0.Proof.
We only prove (a), the proof of (b) is similar and is omitted here. Assume that (QVIP) isα-well-posed in the generalized sense, then the Γ is nonempty and compact. It follows from Lemma 2.7 that Qɛ≠∅. Now we prove e(Qɛ,Γ)→0,asɛ→0. Suppose by contradiction that there exists β>0,ɛn→0, and wn∈Qɛn, such that d(wn,Γ)≥β. It follows from wn∈Qɛn that (wn)n is an α-approximating sequence for (QVIP). (QVIP) is α-well-posedness in the generalized sense, then there exists a subsequence (wnk)k of (wn)n strongly converging to a point of Γ. This contradicts d(wn,Γ)≥β. Thus, e(Qɛ,Γ)→0,asɛ→0.
For the converse, let(un)n be an α-approximating sequence for (QVIP), then un∈Qɛn. It follows from e(Qɛn,Γ)→0 that there exists a sequence zn⊂Γ, such that d(un,zn)→0. Since Γ is compact, there exists a subsequence (znk)k of (zn)n strongly converging to u0∈Γ. Thus there exists the corresponding subsequence (unk)k of (un)n strongly converging to u0. Therefore, (QVIP) is α-well-posed in the generalized sense.Theorem 4.4.
(a) If (QVIP) isα-well-posed in the generalized sense, then
(4.3)Qɛ≠∅,∀ɛ>0,lim⁡ɛ→0μ(Qɛ)=0.
(b) If (4.3) and the following assumptions hold:
(i)
the set-valued mappingS is nonempty convex-valued, and, for each sequence (un)n in K converges to u0, the sequence (S(un))n Lower Semi-Mosco converging to S(u0);(ii)
the operatorA is (s,w)-continuous on K,then, (QVIP) is α-well-posed in the generalized sense.Proof.
(a) Suppose that (QVIP) isα-well-posed in the generalized sense. So Qɛ≠∅, for all ɛ>0. By Theorem 4.3(a), Γ is nonempty compact and e(Qɛ,Γ)→0,asɛ→0. For any ɛ>0, we have
(4.4)H(Qɛ,Γ)=max⁡{e(Qɛ,Γ),e(Γ,Qɛ)}=e(Qɛ,Γ),and since Γ is compact, μ(Γ)=0. For every n∈N, the following relation holds [14]:
(4.5)μ(Qɛ)≤2H(Qɛ,Γ)+μ(Γ)=2H(Qɛ,Γ)=2e(Qɛ,Γ).
It follows from e(Qɛ,Γ)→0,asɛ→0, that limɛ→0μ(Qɛ)=0.
(b) Assume that (4.3) holds. Then, for any ɛ>0, cl(Qɛ) is nonempty closed and increasing with ɛ>0. By (4.3), limɛ→0μ(cl(Qɛ))=limɛ→0μ(Qɛ)=0, where cl(Qɛ) is the closure of Qɛ. By the generalized Cantor theorem  [11, page 412], we know that
(4.6)lim⁡ɛ→0H(cl⁡(Qɛ),Δ)=0,asɛ⟶0,
where Δ=⋂ɛ>0cl(Qɛ) is nonempty compact.
Now we show that(4.7)Γ=Δ.
It follows from Lemma 2.7 that Γ⊆Δ. So we need to prove that Δ⊆Γ. Indeed, let u0∈Δ. Then, d(u0,Qɛ)=0 for every ɛ>0. Given ɛn>0, ɛn→0, for every n, there exists un∈Qɛn such that d(u0,un)<ɛn. Hence, un→u0 and
(4.8)d(un,S(un))≤ɛn,(4.9)〈Aun,un-v〉≤ɛn+α2‖un-v‖2,∀v∈S(un).
It follows from (4.8), un→u0, and the proof of Theorem 3.2 that u0∈S(u0).
Since Lower Semi-Mosco convergence implies that, for everyv∈S(u0), there exists a sequence vn∈S(un), for all n∈N, such that limnvn=v in the strongly topology.
Since the operatorA is (s,w)-continuous on K, hence
(4.10)〈Au0,u0-v〉-α2‖u0-v‖2=lim⁡n[〈Aun,un-vn〉-α2‖un-vn‖2]≤lim⁡nɛn=0.
By Lemma 2.7, we know u0∈Γ. Thus, Δ⊆Γ. It follows from (4.6) and (4.7) that limɛ→0e(Qɛ,Γ)=0. It follows from the compactness of Γ and Theorem 4.3(a) that (QVIP) is α-well-posed in the generalized sense. The proof is completed.Theorem 4.5.
LetK be a nonempty, compact, and convex subset of E, let the set-valued mapping S be nonempty convex-valued, and, for each sequence (un)n in K converging to u0, the sequence (S(un))n Lower Semi-Mosco converges to S(u0), and the operator A is (s,w)-continuous on K. Then, (QVIP) is α-well-posed in the generalized sense.Proof.
Let(un)n be an α-approximating sequence for (QVIP). Since the set K is compact, there exists subsequence (unk)k of (un)n strongly converging to a point u0∈K. Reasoning as in Theorem 3.3, we get u0∈S(u0) and u0 solves (QVIP). Therefore, (QVIP) is α-well-posed in the generalized sense.Theorem 4.6.
Let the following assumptions hold:
(i)
the set-valued mappingS is nonempty convex-valued, and, for each sequence (un)n in K converging to u0, the sequence (S(un))n Lower Semi-Mosco converges to S(u0);(ii)
the operatorA is (s,w)-continuous and monotone on K.
Then, (QVIP) is L-α-well-posed in the generalized sense if and only if
(4.11)Lɛ≠∅,∀ɛ>0,lim⁡ɛ→0μ(Lɛ)=0.Proof.
Assume that (QVIP) is L-α-well-posed in the generalized sense. It follows from Lemma 2.7 and the monotonicity of A that Γ⊂Lɛ, for all ɛ>0. And so Lɛ≠∅, for each ɛ>0. By Theorem 4.3(b), we can get e(Lɛ,Γ)→0 as ɛ→0. From the proof of Theorem 4.4, we also obtain
(4.12)μ(Lɛ)≤2H(Lɛ,Γ)+μ(Γ)=2H(Lɛ,Γ)=2e(Lɛ,Γ).
Thus, limɛ→0μ(Lɛ)=0.
Conversely, assume (4.11) holds. Then, for any ɛ>0, cl(Lɛ) is nonempty closed and increasing with ɛ>0. By (4.11), limɛ→0μ(cl(Lɛ))=limɛ→0μ(Lɛ)=0, where cl(Lɛ) is the closure of Lɛ. By the generalized Cantor theorem [11, page 412], we know that
(4.13)lim⁡ɛ→0H(cl⁡(Lɛ),Δ)=0,asɛ⟶0,
where Δ=⋂ɛ>0cl(Lɛ) is nonempty compact.
Now we show that(4.14)Γ=Δ.
It follow from Lemma 2.7 and the monotonicity of A that Γ⊆Δ. So we need to prove that Δ⊆Γ. Indeed, let u0∈Δ. Then d(u0,Lɛ)=0 for every ɛ>0. Given ɛn>0, ɛn→0, for every n, there exists un∈Lɛn such that d(u0,un)<ɛn. Hence, un→u0 and
(4.15)d(un,S(un))≤ɛn,(4.16)〈Av,un-v〉≤ɛn+α2‖un-v‖2,∀v∈S(un).
It follows from (4.15), xn→x0, and the proof of Theorem 3.2 that u0∈S(u0).
SinceS(un) Lower Semi-Mosco converges to S(u0), for every v∈S(u0), there exists a sequence vn∈S(un), for all n∈N, such that limnvn=v in the strong topology.
Since the operatorA is (s,w)-continuous on K, hence
(4.17)〈Av,u0-v〉-α2‖u0-v‖2=lim⁡n[〈Avn,un-vn〉-α2‖un-vn‖2]≤lim⁡nɛn=0.
By Lemma 2.6 we know that u0∈S(u0), such that
(4.18)〈Au0,u0-v〉-α2‖u0-v‖2≤0,∀v∈S(u0).
It follow from Lemma 2.7 that u0∈Γ. Thus, Δ⊆Γ. It follows from (4.13) and (4.14) that limɛ→0e(Lɛ,Γ)=0. It follows from the compactness of Γ and Theorem 4.3(b) that (QVIP) is L-α-well-posed in the generalized sense. The problem is completed.Remark 4.7.
It is easy to see that ifα=0, then by the main results in our paper, we can recover the corresponding results in [9] with the weaker condition that S(xn) Lower Semi-Mosco converges to  S(x0) instead of the condition that  S  is (s,w)-closed and (s,w)-subcontinuous, and (s,s)-lower semicontinuous.



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*Source: 157532-2012-05-20.xml*