# α-Well-Posedness for  Mixed Quasi Variational-Like Inequality Problems

Authors: Jian-Wen Peng; Jing Tang
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

The concepts ofα-well-posedness, α-well-posedness in the
generalized sense, L-α-well-posedness and L-α-well-posedness in the generalized sense for
mixed quasi variational-like inequality problems are investigated. We present some metric
characterizations for these well-posednesses.


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## Body

## 1. Introduction

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. The study of well-posedness for scalar minimization problems started from Tykhonov [1] and Levitin and Polyak [2]. Since then, various notions of well-posedness for scalar minimization problems have been defined and studied in [3–8] and the references therein. It is worth noting that the recent study for various types of well-posedness has been generalized to variational inequality problems [9–13], generalized variational inequality problems [14, 15], quasi variational inequality problems [16], generalized quasi variational inequality problems [17], generalized vector variational inequality problems [18], vector quasi variational inequality problems [19], mixed quasi variational-like inequality problems [20], and many other problems.In this paper, we are interested in investigating four classes of well-posednesses for a mixed quasi variational-like inequality problem. The paper is organized as follows. In Section2, we introduce 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 a mixed quasi variational-like inequality problem. In Section 3, some characterizations of α-well-posedness, and L-α-well-posedness for a mixed quasi variational-like inequality problem are obtained. In Section 4, some characterizations of α-well-posedness in the generalized sense and L-α-well-posedness in the generalized sense for a mixed quasi variational-like inequality problem are presented.


## 2. Preliminaries

Throughout this paper, without other specification, letE be a real Banach space with the dual E*, let K be a nonempty closed convex subset of E, and let S:K→2K be a set-valued map. Let F:K→2E* be a set-valued map with nonempty values, let η:K×K→E be a single-valued map, and let f:K→R be a real-valued function. Ceng et al. [20] introduced the following mixed quasi variational-like inequality problem, which is to find a point x0∈K such that, for some u0∈F(x0), (2.1)(MQVLI)x0∈S(x0),〈u0,η(x0,y)〉+f(x0)-f(y)≤0,∀y∈S(x0).Denote byΓ the solution set of (MQVLI). Let α>0; we introduce the notions of several classes of α-well-posednesses for (MQVLI).Definition 2.1.
A sequence(xn)n in K is an α-approximating sequence for (MQVLI) if(i)
there exists a sequence(un)n in E*, with un∈F(xn),for alln∈N;(ii)
there exists a sequence(εn)n,εn>0, εn→0 such that(2.2)d(xn,S(xn))≤εn,∀n∈N,〈un,η(xn,y)〉+f(xn)-f(y)-α2‖xn-y‖2≤εn,∀y∈S(xn),∀n∈N.Definition 2.2.
(MQVLI) is said to beα-well-posed (resp., α-well-posed in the generalized sense) if it has a unique solution x0 and every α-approximating sequence (xn)n strongly converges to x0 (resp., if the solution set Γ of (MQVLI) is nonempty and for every α-approximating sequence (xn)n has a subsequence which strongly converges to a point of Γ).Definition 2.3.
A sequence(xn)n is an L-α-approximating sequence for (MQVLI) if there exists a real number sequence (εn)n,εn>0, εn→0 such that
(2.3)d(xn,S(xn))≤εn,∀n∈N,〈v,η(xn,y)〉+f(xn)-f(y)-α2‖xn-y‖2≤εn,∀y∈S(xn),v∈F(y),n∈N.Definition 2.4.
(MQVLI) is said to be L-α-well-posed (resp., L-α-well-posed in the generalized sense) if it has a unique solution x0 and every L-α-approximating sequence (xn)n strongly converges to x0 (resp., if the solution set Γ of (MQVLI) is nonempty and for every L-α-approximating sequence (xn)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 (MQVLI), 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 (MQVLI) in [20]. 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 inequalities introduced by Fukushima [21].In order to investigate theα-well-posedness for (MQVLI), we need the following definitions.We recall the notion of Mosco convergence [22]. A sequence (Hn)n of subsets of E Mosco converges to a set H if (2.4)H=lim⁡inf⁡n⁡Hn=w-lim⁡⁡sup⁡n⁡Hn,
where liminfnHn and w-limsupnHn are, respectively, the Painlevé-Kuratowski strong limit inferior and weak limit superior of a sequence (Hn)n, that is, (2.5)lim⁡inf⁡n⁡Hn={y∈E:∃yn∈Hn,n∈N,withyn→y},w-limsup⁡n⁡Hn={y∈E:∃nk↑+∞,nk∈N,∃ynk∈Hnk,k∈N,withynk⇀y},
where “⇀” means weak convergence, and “→” means strong convergence.IfH=liminfnHn, we call the sequence (Hn)n of subsets of E Lower Semi-Mosco convergent to a set H.It is easy to see that a sequence(Hn)n of subsets of E Mosco converges to a set H 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. For anyx,y∈E, [x,y] will denote the line segment {tx+(1-t)y:t∈[0,1]}, while [x,y) and (x,y) are defined analogously. We will frequently use s,w, and w* to denote, respectively, the norm topology on E, the weak topology on E, and the weak* topology on E. Given a convex set K, a multivalued map F:K→2E* will be called upper hemicontinuous, if its restriction on any line segment [x,y]⊆K is usc with respect to the w* topology on E*. F:K→2E* will be called η-monotone if, for any x,y∈K, for all u∈F(x),v∈F(y), 〈u-v,η(x,y)〉≥0. We refer the reader to [23, 24] for basic facts about multivalued maps.Lemma 2.5 (see [25]).
Let(Hn)n be a sequence of nonempty subsets of a Banach space E such that(i)
Hn is convex for every n∈N;(ii)
H0⊆liminfnHn;(iii)
there existsm∈N such that int⋂n≥mHn≠∅.Then, for every u0∈intH0, there exists a positive real number δ such that
(2.6)int⁡B(u0,δ)⊆Hn,∀n≥m,
where B(u0,δ) is a closed ball with a center u0 and radius δ. If E is a finite dimensional space, then assumption (iii) can be replaced by(iii)′
intH0≠∅.The following lemmas play important role in this paper.Lemma 2.6.
LetE be a real separable Banach space with the dual E*, let S0 be a nonempty convex subset of E, and let F:S0→2E* be a set-valued map with nonempty, weakly* compact convex valued, η-monotone, and upper hemicontinuous. Let η:S0×S0→E be a single-valued map with η(x,x)=0,for allx∈S0, and let f:S0→R be a convex lsc function.  Assume that the map y↦〈u,η(x,y)〉 is concave for each (u,x)∈F(S0)×S0 and usc. If S1 is a convex subset of S0 with the property that, for each x∈S0 and each y∈S1,(x,y]⊆S1, then for each x0∈S0, the following conditions are equivalent:(i)
There existsu0∈F(x0), such that for ally∈S0,〈u0,η(x0,y)〉+f(x0)-f(y)-(α/2)∥x0-y∥2≤0,(ii)
for ally∈S1,there existsv∈F(y), such that 〈v,η(x0,y)〉+f(x0)-f(y)-(α/2)∥x0-y∥2≤0.Proof.
According to theη-monotonicity of F, (i) ⇒ (ii) is obvious.
Next prove (ii)⇒ (i). Suppose that (ii) holds. Given any y∈S1, let yn=(1/n)y+(1-(1/n))x0, for n∈N. By the assumptions of S1,yn∈S1 for each n∈N. It follows from the condition (ii) that for each n∈N, there exists vn∈F(yn) such that
(2.7)〈vn,η(x0,yn)〉+f(x0)-f(yn)-α2‖x0-yn‖2≤0.
Then
(2.8)0≥〈vn,η(x0,yn)〉+f(x0)-f(yn)-α2‖x0-yn‖2=〈vn,η(x0,1ny+(1-1n)x0)〉+f(x0)-f(1ny+(1-1n)x0)-α2‖x0-1ny-(1-1n)x0‖2≥1n〈vn,η(x0,y)〉+(1-1n)〈vn,η(x0,x0)〉+f(x0)-1nf(y)-(1-1n)f(x0)-α21n2‖x0-y‖2=1n〈vn,η(x0,y)〉+1nf(x0)-1nf(y)-α2n2‖x0-y‖2,
which implies that
(2.9)〈vn,η(x0,y)〉+f(x0)-f(y)-α2n‖x0-y‖2≤0,∀n∈N.
It follows that for each n∈N,
(2.10)∃vn∈F(yn),〈vn,η(x0,y)〉+f(x0)-f(y)-α2‖x0-y‖2≤0.
Since F is a weak* compact valued and (s,w*)-usc on the line segment [x0,y], F is (s,w*)-closed, and (s,w*)-subcontinuous on [x0,y], it follows from limnyn=x0 and vn∈F(yn) that {vn} has a subsequence weak* converging to some v∈F(x0). By taking the limit of subsequence in (2.10) we get
(2.11)∀y∈S1,∃v∈F(x0),〈v,η(x0,y)〉+f(x0)-f(y)-α2‖x0-y‖2≤0.
Define the bifunction ϕ(v,y) on F(x0)×S0 by
(2.12)ϕ(v,y)=〈v,η(x0,y)〉+f(x0)-f(y)-α2‖x0-y‖2.
For each y∈S0,ϕ(·,y) is weakly* lsc and quasiconvex on the weakly* compact convex set F(x0) while for each v∈F(x0),ϕ(v,·) is usc and quasiconcave on the convex set S1. Hence, according to the Sion Minimax Theorem [26],
(2.13)sup⁡y∈S1⁡min⁡v∈F(x0)⁡ϕ(v,y)=min⁡v∈F(x0)⁡sup⁡y∈S1⁡ϕ(v,y).
By (2.11), we have supy∈S1minv∈F(x0)ϕ(v,y)≤0; hence, minv∈F(x0)supy∈S1ϕ(v,y)≤0, which implies that there exists v0∈F(x0), such that
(2.14)〈v0,η(x0,y)〉+f(x0)-f(y)-α2‖x0-y‖2≤0,∀y∈S1.
Finally, for eachy∈S0, choose z∈S1, and a sequence (yn)n in (y,z]⊆S1 converging to y. The function ϕ(v,·) is usc and concave on S0; hence its restriction on any line segment is continuous [27, Theorem  2.35]. Accordingly, (2.14) implies there existsv0∈F(x0),for ally∈S0,
(2.15)〈v0,η(x0,y)〉+f(x0)-f(y)-α2‖x0-y‖2=lim⁡n⁡[〈v0,η(x0,yn)〉+f(x0)-f(yn)-α2‖x0-yn‖2]≤0.
Hence, (i) holds.Lemma 2.7.
LetE be a real Banach space with the dual E*, let K be a nonempty convex subset of E, and let S be a convex-valued set-valued map from K to 2K. Let F:K→2E* be a set-valued map with nonempty values, let η:K×K→E be a single-valued map with η(x,x)=0,for allx∈K, and let f:K→R be a convex function. Assume that the function y↦〈u,η(x,y)〉 is concave, for each (u,x)∈F(K)×K. Then x0∈Γ if and only if the following condition holds:
(2.16)∃u0∈F(x0),x0∈S(x0),〈u0,η(x0,y)〉+f(x0)-f(y)-α2‖x0-y‖2≤0,∀y∈S(x0).Proof.
The necessity is easy to get; next we start to prove the sufficiency. Letfor ally∈S(x0),for allt∈(0,1),yt=ty+(1-t)x0. Since u0∈F(x0),x0∈S(x0), and S is convex-valued, yt∈S(x0), it follows that
(2.17)〈u0,η(x0,yt)〉+f(x0)-f(yt)-α2‖x0-yt‖2≤0,∀t∈(0,1).
Thus,
(2.18)0≥〈u0,η(x0,yt)〉+f(x0)-f(yt)-α2‖x0-yt‖2≥t〈u0,η(x0,y)〉+(1-t)〈u0,η(x0,x0)〉+f(x0)-tf(y)-(1-t)f(x0)-α2t2‖x0-y‖2=t[〈u0,η(x0,y)〉+f(x0)-f(y)-α2t‖x0-y‖2],
which implies that
(2.19)u0∈F(x0),x0∈S(x0),〈u0,η(x0,y)〉+f(x0)-f(y)-α2t‖x0-y‖2≤0∀y∈S(x0),∀t∈(0,1).
The above inequality implies, for t converging to zero, that x0 is a solution of (MQVLI). This completes the proof.


## 3. The Characterizations of Well-Posedness for (MQVLI)

In this section, we investigate some metric characterizations ofα-well-posedness and L-α-well-posedness for (MQVLI).For anyε>0, we consider the sets (3.1)Qε={x∈K:d(x,S(x))≤ε,∃u∈F(x):〈u,η(x,y)〉+f(x)-f(y)-α2‖x-y‖2≤ε,∀y∈S(x)},Lε={x∈K:d(x,S(x))≤ε,〈v,η(x,y)〉+f(x)-f(y)-α2‖x-y‖2≤ε,∀y∈S(x),∀v∈F(y)}.Theorem 3.1.
Let the same assumptions be as in Lemma2.7. Then, one has the following.(a)
(MQVLI) isα-well-posed if and only if the solution set Γ of (MQVLI) is nonempty and limε→0diamQε=0.(b)
Moreover, ifF is η-monotone, then (MQVLI) is L-α-well-posed if and only if the solution set Γ of (MQVLI) is nonempty and limε→0diamLε=0.Proof.
We only prove (a). The proof of (b) is similar and is omitted here. Suppose that (MQVLI) isα-well-posed; then Γ≠∅. It follows from Lemma 2.7 that Qε≠∅. Suppose by contradiction that exists a real number β, such that limε→0diamQε>β>0; then there exists εn>0, with εn↘0, and (wn)n,(zn)n∈Qεn, such that ∥wn-zn∥>β,for alln∈N. Since the sequences (wn)n, and (zn)n are both α-approximating sequences for (MQVLI), (wn)n and (zn)n strongly converge to the unique solution u0, and this gives a contradiction. Therefore, limε→0diamQε=0.
Conversely, let(xn)n⊂K be an α-approximating sequence for (MQVLI). Then there exists a sequence (un)n in E* with un∈F(xn) and a sequence (εn)n in R+ with εn→0, such that
(3.2)d(xn,S(xn))≤εn,〈un,η(xn,y)〉+f(xn)-f(y)-α2‖xn-y‖2≤εn,∀y∈S(xn),∀n∈N.
That is, xn∈Qεn,for alln∈N. It is easy to see limε→0diamQε=0 and Γ≠∅ imply 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∈Qε,for allε>0. Thus, limε→0diamQε≥∥z1-z2∥≠0, a contradiction. Let x0 be the unique solution of (MQVLI). It follows from Lemma 2.7 that x0∈Qεn. Thus, limn→∞∥xn-x0∥≤limn→∞diamQεn=0. So (xn)n strongly converges to x0. Therefore, (MQVLI) is α-well-posed.Theorem 3.2.
LetE be a real separable Banach space with the dual E*, let K be a nonempty closed convex subset of E, and let η:K×K→E be a single-valued map with η(x,x)=0,for allx∈K, which is (s,w)-continuous in each of its variables separately. And let f:K→R be a convex lsc function; let S:K→2K and F:K→2E* be two set-valued maps. Assume the following conditions hold:(i)
S is nonempty convex-valued and, for each sequence (xn)n in K converging to x0, the sequence (S(xn))n Lower Semi-Mosco converging to S(x0);(ii)
for every converging sequence(wn)n, there exists m∈N, such that int⋂n≥mS(wn)≠∅;(iii)
F:K→2E* is nonempty, weak* compact convex valued, η-monotone, and upper hemicontinuous;(iv)
the mapy↦〈u,η(x,y)〉 is concave for each (u,x)∈F(K)×K.Then, (MQVLI) is α-well-posed if and only if
(3.3)Qε≠∅,∀ε≥0,lim⁡ε→0⁡diam⁡Qε=0.The proof of the above theorem relies on the following lemma.Lemma 3.3.
Let the same assumptions be made as in Theorem3.2. Let (xn)n in K be an α-approximating sequence. If (xn)n converges to some x0∈K, then x0 is a solution of (MQVLI).Proof.
Since(xn)n is an α-approximating sequence for (MQVLI), there exists a sequence (un)n in E* with un∈F(xn) and a sequence (εn)n in R+ with εn→0, such that
(3.4)d(xn,S(xn))≤εn,〈un,η(xn,y)〉+f(xn)-f(y)-α2‖xn-y‖2≤εn,∀y∈S(xn),∀n∈N.
For eachn∈N, choose xn′∈S(xn), such that ∥xn-xn′∥<d(xn,S(xn))+εn≤2εn. It follows from xn→x0 and εn→0 that xn′→x0. It follows from the assumption (i) that liminfnS(xn)=S(x0). Thus, x0∈S(x0).
Assumption (ii) applied to the constant sequencewn=x0,for alln∈N, implies that intS(x0)≠∅. For every y∈intS(x0), it follows from assumptions (i) and (ii) and Lemma 2.5 that there exist m∈N and δ>0 such that intB(y,δ)⊆S(xn),for alln>m. Therefore, for n sufficiently large, we have y∈S(xn). Notice that η(·,y) is (s,w)-continuous, f is lsc, F is η-monotone, and (xn)n is an approximating sequence; we have, for every v∈F(y)(3.5)〈v,η(x0,y)〉+f(x0)-f(y)≤lim⁡⁡inf⁡n⁡{〈v,η(xn,y)〉+f(xn)-f(y)}≤lim⁡⁡inf⁡n⁡{〈un,η(xn,y)〉+f(xn)-f(y)}≤lim⁡⁡inf⁡n⁡[εn+α2‖xn-y‖2]=α2‖x0-y‖2.
Thus, for everyy∈intS(x0) and every v∈F(y), we get 〈v,η(x0,y)〉+f(x0)-f(y)-(α/2)∥x0-y∥2≤0. Let S0=S(x0) and S1=intS(x0); it follows from Lemma 2.6 that there exists u0∈F(x0) such that for all y∈S(x0),〈u0,η(x0,y)〉+f(x0)-f(y)-(α/2)∥x0-y∥2≤0. According to Lemma 2.7, x0 is a solution of (MQVLI).Proof of Theorem3.2.
The necessity follows from Theorem3.1 and Lemma 2.7. Now we prove the sufficiency. Suppose that (3.3) holds. Let us show that there exists at most one solution of (MQVLI). Indeed, if there exist two different solutions z1,z2, then from Lemma 2.7, we know that z1,z2∈Qε,for allε>0. Thus, limε→0diamQε≥∥z1-z2∥≠0, a contradiction. Note also that there exist α-approximate sequences for (MQVLI); indeed, for any sequence (εn)n in R+ with εn→0, and any choice of xn∈Qεn (which is nonempty by assumption), (xn)n is an α-approximate sequence.
Let(xn)n be an α-approximating sequence for (MQVLI); then xn∈Qεn,for alln∈N. In light of (3.3), (xn)n is a Cauchy sequence and strongly converging to a point x0∈K. Applying Lemma 3.3, we get that x0 is a solution of (MQVLI) and so (MQVLI) is α-well-posed.Now, we present a result in which assumption (ii) and the monotonicity ofF are dropped, while the continuity requirements are strengthened.Theorem 3.4.
LetE be a real separable Banach space with the dual E*, let K be a nonempty closed convex subset of E, and let η:K×K→E be a single-valued map with η(x,x)=0,for allx∈K, which is (s,s)-continuous. And let f:K→R be a convex and continuous function, let S:K→2K and F:K→2E* be two set-valued maps. Assume the following assumptions hold:(i)
the multifunctionS is nonempty convex-valued and for each sequence (xn)n in K converging to x0, the sequence (S(xn))n Lower Semi-Mosco converging to S(x0);(ii)
F:K→2E* is nonempty, weak* compact, and convex valued, (s,w*)-usc;(iii)
the mapy↦〈u,η(x,y)〉 is concave for each (u,x)∈F(K)×K.Then, (MQVLI) is α-well-posed if and only if (3.3) holds.The proof of the above theorem relies on the following lemma.Lemma 3.5.
Let the assumptions be as in Theorem3.4. Let (xn)n in K be an α-approximating sequence. If (xn)n converges to some x0∈K, then x0 is a solution of (MQVLI).Proof.
Since(xn)n is an α-approximating sequence for (MQVLI), there exist a sequence (un)n in E* with un∈F(xn) and a sequence (εn)n in R+,εn→0, such that
(3.6)d(xn,S(xn))≤εn,〈un,η(xn,y)〉+f(xn)-f(y)-α2‖xn-y‖2≤εn,∀y∈S(xn),∀n∈N.
As in Lemma3.3, we infer x0∈S(x0). Since S(xn) Lower Semi-Mosco converges to S(x0), for every y∈S(x0), there exists a sequence yn∈S(xn),for alln∈N, such that limnyn=y in the strongly topology. Since η is (s,s)-continuous, the sequence (η(xn,yn))n converges strongly to η(x0,y). It follows from (ii) and Proposition  2.19 in [24] that there exists a subsequence (unj)j of (un)n weak* converging to some u0∈E*. It follows from (ii) and Proposition  2.17 in [24] that F is (s,w*)-closed, and so u0∈F(x0). Thus, we have
(3.7)|〈unj,η(xnj,ynj)〉-〈u0,η(x0,y)〉|≤|〈unj,η(xnj,ynj)-η(x0,y)〉|+|〈u0-unj,η(x0,y)〉|≤‖unj‖‖η(xnj,ynj)-η(x0,y)‖-|〈u0-unj,η(x0,y)〉|⟶0.
Hence, 〈unj,η(xnj,ynj)〉→〈u0,η(x0,y)〉 and so
(3.8)〈u0,η(x0,y)〉+f(x0)-f(y)-α2‖x0-y‖2=lim⁡j⁡[〈unj,η(xnj,ynj)〉+f(xnj)-f(ynj)-α2‖xnj-ynj‖2]≤lim⁡j⁡εnj=0.
Applying Lemma 2.7, x0 is a solution of (MQVLI).Proof of Theorem3.4.
The necessity follows from Theorem3.1 and Lemma 2.7. Now we prove the sufficiency. Suppose that (3.3) holds. It follows from the proof of Theorem 3.2 that there exists at most one solution of (MQVLI) and there exist α-approximate sequences for (MQVLI). Let (xn)n be an α-approximating sequence for (MQVLI); then xn∈Qεn,for alln∈N. In light of (3.3), (xn)n is a Cauchy sequence and strongly converging to a point x0∈K. Applying Lemma 3.5, we get that x0 is a solution of (MQVLI) and so (MQVLI) is α-well-posed.We have analogous results for L-α-well-posedness.Theorem 3.6.
LetE be a real separable Banach space with the dual E*, let K be a nonempty closed convex subset of E, and let η:K×K→E be a single-valued map with η(x,x)=0,for allx∈K, which is (s,w)-continuous in each of its variables separately. And let f:K→R be a convex lsc function; let S:K→2K and F:K→2E* be two set-valued maps. Assume the following assumptions hold:(i)
the multifunctionS is nonempty convex-valued and for each sequence (xn)n in K converging to x0, the sequence (S(xn))n Lower Semi-Mosco converging to S(x0);(ii)
for every converging sequence(wn)n, there exists m∈N, such that int⋂n≥mS(wn)≠∅;(iii)
F:K→2E* is a set-valued map with nonempty, weak* compact convex valued, η-monotone and upper hemicontinuous;(iv)
the mapy↦〈u,η(x,y)〉 is concave for each (u,x)∈F(K)×K.Then (MQVLI) is L-α-well-posed if and only if
(3.9)Lε≠∅,∀ε≥0,lim⁡ε→0⁡diam⁡Lε=0.Lemma 3.7.
Let the same assumptions be as in Theorem3.6. Let (xn)n in K be an L-α-approximating sequence. If (xn)n converges to some x0∈K, then x0 is a solution of (MQVLI).Proof.
Since(xn)n is an L-α-approximating sequence for (MQVLI), there exists a sequence (εn)n in R+, εn→0, such that d(xn,S(xn))≤εn, and
(3.10)〈v,η(xn,y)〉+f(xn)-f(y)-α2‖xn-y‖2≤εn,∀y∈S(xn),v∈F(y),n∈N.
From the proof of Lemma 3.3, (i) and (ii), we can obtain x0∈S(x0), intS(x0)≠∅, and for each y∈intS(x0), one has y∈S(xn) for n sufficiently large. It follows from (iii) that for every y∈intS(x0) and every v∈F(y), we have
(3.11)〈v,η(x0,y)〉+f(x0)-f(y)-α2‖x0-y‖2≤lim⁡⁡inf⁡n⁡[〈v,η(xn,y)〉+f(xn)-f(y)-α2‖xn-y‖2]≤lim⁡⁡inf⁡n⁡εn=0.
LetS0=S(x0) and S1=intS(x0); it follows from Lemma 2.6 that there exists u0∈F(x0) such that for all y∈S(x0),〈u0,η(x0,y)〉+f(x0)-f(y)-(α/2)∥x0-y∥2≤0. According to Lemma 2.7, x0 is a solution of (MQVLI).Proof of Theorem3.6.
Assume that (3.9) holds. Let (xn)n in K be an L-α-approximating sequence for (MQVLI); then there exists a sequence (εn)n in R+, such that xn∈Lεn. It is easy to see that limε→0diamLε=0 and Γ≠∅ imply that Γ is a singleton point set. Indeed, if there exist two different solutions z1,z2, then from Lemma 2.7 and the η-monotonicity of F, we know that z1,z2∈Lε,for allε>0. Thus, limε→0diamLε≥∥z1-z2∥≠0, a contradiction. Let x0 be the unique solution of (MQVLI). It follows from Lemma 2.7 and the η-monotonicity of F that x0∈Lεn. Thus, limn→∞∥xn-x0∥≤limn→∞diamLεn=0. So (xn)n strongly converge to x0. It follows from Lemma 3.7 that x0∈Γ. Therefore, (MQVLI) is L-α-well-posed.
Conversely, assume that the problem is L-α-well-posed, It follows from the η-monotonicity of F that ∅≠Γ⊂Lε,∀ε>0. Suppose by contradiction that a real number β exists, such that limε→0diamLε>β>0; then there exists εn>0, with εn→0, and (wn)n,(zn)n∈Lεn, such that ∥wn-zn∥>β,∀n∈N. Since the sequences (wn)n and (zn)n are both L-α-approximating sequences for (MQVLI), (wn)n and (zn)n strongly converge to the unique solution u0, and this gives a contradiction. Therefore, limε→0diamLε=0.Theorem 3.8.
LetE be a real separable Banach space, let K be a nonempty closed convex subset of E, and let η:K×K→E be a single-valued map with η(x,x)=0,for allx∈K, which is (s,s)-continuous. And let f:K→R be a convex continuous function; S be a set-valued map from K to 2K. Assume the following assumptions hold:(i)
the multifunctionS is nonempty convex-valued and for each sequence (xn)n in K converging to x0, the sequence (S(xn))n Lower Semi-Mosco converging to S(x0);(ii)
F:K→2E* is a set-valued map with nonempty, weak* compact convex-valued, (s,w*)-usc, and η-monotone;(iii)
the mapy↦〈u,η(x,y)〉 is concave for each (u,x)∈F(K)×K.
Then (MQVLI) is L-α-well-posed if and only if (3.9) holds.Lemma 3.9.
Let the same assumptions be as in Theorem3.8. Let (xn)n in K be an L-α-approximating sequence. If (xn)n converges to some x0∈K, then x0 is a solution of (MQVLI).Proof.
Since(xn)n is an L-α-approximating sequence for (MQVLI), there exists a sequence (εn)n in R+, εn→0, such that d(xn,S(xn))≤εn, and
(3.12)〈v,η(xn,y)〉+f(xn)-f(y)-α2‖xn-y‖2≤εn,∀y∈S(xn),∀v∈F(y),n∈N.
It follows from the Lower Semi-Mosco convergence ofS and the proof of Lemma 3.3 that x0∈S(x0). Since S(xn) Lower Semi-Mosco converges to S(x0), for every y∈S(x0), there exists a sequence yn∈S(xn),for alln∈N, strongly converging to y. For each n∈N select vn∈F(yn). It follows from (ii) and Proposition  2.19 in [24] that there exists a subsequence (vnj)j of (vn)n weak* converging to some v∈E*. It follows from (ii) and Proposition  2.17 in [24] that F is (s,w*)-closed, and so v∈F(y). By the continuity of η and similar argument with the proof of Lemma 3.5, we know that
(3.13)〈vnj,η(xnj,ynj)〉⟶〈v,η(x0,y)〉.
It follows from (3.12) that
(3.14)〈vnj,η(xnj,ynj)〉+f(xnj)-f(ynj)-α2‖xnj-ynj‖2≤εnj.
We deduce from the above inequality that
(3.15)∀y∈S(x0),∃v∈F(y),〈v,η(x0,y)〉+f(x0)-f(y)-α2‖x0-y‖2≤0.
Let S0=S1=S(x0); by Lemma 2.6 we know that there exists u0∈F(x0), such that
(3.16)∀y∈S(x0),〈u0,η(x0,y)〉+f(x0)-f(y)-α2‖x0-y‖2≤0.
Then using Lemma 2.7, x0 is a solution of (MQVLI).Proof of Theorem3.8.
Assume that (3.9) holds. If (xn)n in K is an L-α-approximating sequence, then from the proof of Theorem 3.6, we know that (xn)n converges to some x0∈K. By Lemma 3.9, x0 is a solution of (MQVLI) and so (MQVLI) is L-α-well-posed. The converse is exactly same as that in the proof of Theorem 3.6.


## 4. The Characterizations ofα-Well-Posed in the Generalized Sense for (MQVLI)

In this section, we investigate some metric characterizations ofα-well-posedness in the generalized sense for (MQVLI).Definition 4.1 (see [8]).
LetA be a nonempty subset of X. The measure of noncompactness μ of the set A is defined by
(4.1)μ(A)=inf⁡{ε>0:A⊆⋃i=1nAi,diam⁡Ai<ε,i=1,2,…,n}.Definition 4.2 (see [8]).
Let(X,d) be a metric space and let A and B be nonempty subsets of X. The Hausdorff distance H(·,·) between A and B is defined by
(4.2)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∥.Theorem 4.3.
Let the same assumptions be as in Lemma2.7. Then, one has the following.(a)
(MQVLI) isα-well-posed in the generalized sense if and only if the solution set Γ of (MQVLI) is nonempty compact and e(Qε,Γ)→0,asε→0.(b)
Moreover, ifF is η-monotone, then (MQVLI) is L-α-well-posed in the generalized sense if and only if the solution set Γ of (MQVLI) is nonempty compact and e(Lε,Γ)→0,asε→0.Proof.
We only prove (a). The proof of (b) is similar and is omitted here. Assume that (MQVLI) isα-well-posed in the generalized sense; then Γ is nonempty and compact. It follows from Lemma 2.7 that Qε≠∅. Now we show that e(Qε,Γ)→0,asε→0. Suppose by contradiction that there exists β>0,εn→0 and wn∈Qεn, such that d(wn,Γ)>β. It follows from wn∈Qεn that (wn)n is an α-approximating sequence for (MQVLI). Since (MQVLI) is α-well-posedness in the generalized sense, there exists a subsequence (wnk)k of (wn)n strongly converging to a point of Γ. This contradicts d(wn,Γ)>β. Thus e(Qε,Γ)→0,asε→0.
For the converse, let(xn)n be an α-approximating sequence for (MQVLI); then xn∈Qεn. It follows from e(Qεn,Γ)→0 that there exists a sequence zn⊂Γ, such that d(xn,zn)→0. Since Γ is compact, there exists a subsequence (znk)k of (zn)n strongly converging to x0∈Γ. Thus the corresponding subsequence (xnk)k of (xn)n is strongly converging to x0. Therefore, (MQVLI) is α-well-posed in the generalized sense.Theorem 4.4.
Let the same assumptions be as in Theorem3.4. Then (MQVLI) is α-well-posed in the generalized sense if and only if
(4.3)Qε≠∅,∀ε>0,lim⁡ε→∞⁡μ(Qε)=0.Proof.
Assume that (MQVLI) isα-well-posed in the generalized sense; so Qε≠∅,for allε>0. By Theorem 4.3(a), Γ is nonempty compact and limε→0e(Qε,Γ)→0. For any ε>0, we have
(4.4)H(Qε,Γ)=max⁡{e(Qε,Γ),e(Γ,Qε)}=e(Qε,Γ),
since Γ is compact, μ(Γ)=0. For every ε>0, the following relation holds (see, e.g., [13])
(4.5)μ(Qε)≤2H(Qε,Γ)+μ(Γ)=2H(Qε,Γ)=2e(Qε,Γ).
It follows from limε→0e(Qε,Γ)→0 that limε→0μ(Qε)=0.
Conversely, assume that (4.3) holds. Then, for any ε>0, cl(Qε) is nonempty closed and increasing with ε>0. By (4.3), limε→0μ(cl(Qε))=limε→0μ(Qε)=0, where cl(Qε) is the closure of Qε. By the generalized Cantor theorem [23, page 412], we know that
(4.6)lim⁡ε→0⁡H(cl⁡(Qε),Δ)=0,asε⟶0,
where Δ=⋂ε>0cl(Qε) 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 x0∈Δ. Then d(x0,Qε)=0 for every ε>0. Given εn>0, εn→0, for every n there exists xn∈Qεn such that d(x0,xn)<εn. Hence, xn→x0 and
(4.8)d(xn,S(xn))≤εn,(4.9)∃un∈F(xn),〈un,η(xn,y)〉+f(xn)-f(y)≤εn+α2‖xn-y‖2,∀y∈S(xn).
It follows from (4.8), xn→x0, and the proof of Lemma 3.3 that x0∈S(x0).
SinceS(xn) Lower Semi-Mosco converges to S(x0), for every y∈S(x0), there exists a sequence yn∈S(xn),for alln∈N, such that limnyn=y in the strong topology.
Sinceη is (s,s)-continuous, the sequence (η(xn,yn))n converges strongly to η(x0,y). It follows from (ii) and Proposition  2.19 in [24] that there exists a subsequence (unj)j of (un)n weak* converging to some u0∈E*. It follows from (ii) and Proposition  2.17 in [24] that F is (s,w*)-closed, and so u0∈F(x0). It follows from the proof of Lemma 3.5 that
(4.10)〈unj,η(xnj,ynj)〉⟶〈u0,η(x0,y)〉.
Hence,
(4.11)〈u0,η(x0,y)〉+f(x0)-f(y)-α2‖x0-y‖2=lim⁡j⁡[〈unj,η(xnj,ynj)〉+f(xnj)-f(ynj)-α2‖xnj-ynj‖2]≤lim⁡j⁡εnj=0,
that is,
(4.12)∃u0∈F(x0),〈u0,η(x0,y)〉+f(x0)-f(y)-α2‖x0-y‖2≤0,∀y∈S(x0).
By Lemma 2.7, we know that x0∈Γ. Thus, Δ⊆Γ. It follows from (4.6) and (4.7) that limε→0e(Qε,Γ)=0. It follows from the compactness of Γ and Theorem 4.3(a) that (MQVLI) is α-well-posed in the generalized sense. The proof is completed.Theorem 4.5.
Let the same assumptions be as in Theorem3.8. Then (MQVLI) is L-α-well-posed in the generalized sense if and only if
(4.13)Lε≠∅,∀ε>0,lim⁡ε→0⁡μ(Lε)=0.Proof.
Assume that (MQVLI) is L-α-well-posed in the generalized sense. It follows from Lemma 2.7 and the η-monotonicity of F that Γ⊂Lε,for allε>0. And so Lε≠∅, for each ε>0. By similar argument with that in the proof of Theorem 4.3(a), we can get e(Lε,Γ)→0 as ε→0. From the proof of Theorem 4.4, we also obtain
(4.14)μ(Lε)≤2H(Lε,Γ)+μ(Γ)=2H(Lε,Γ)=2e(Lε,Γ).
Thus, limε→0μ(Lε)=0.
Conversely, assume that (4.13) holds. Then, for any ε>0, cl(Lε) is nonempty closed and increasing with ε>0. By (4.13), limε→0μ(cl(Lε))=limε→0μ(Lε)=0, where cl(Lε) is the closure of Lε. By the generalized Cantor theorem [23, Page 412], we know that
(4.15)lim⁡ε→0⁡H(cl⁡(Lε),Δ)=0,asε⟶0,
where Δ=⋂ε>0cl(Lε) is nonempty compact.
Now we show that(4.16)Γ=Δ.
It follows from Lemma 2.7 and the monotonicity of F that Γ⊆Δ. So we need to prove that Δ⊆Γ. Indeed, let x0∈Δ. Then d(x0,Lε)=0 for every ε>0. Given εn>0, εn→0, for every n there exists xn∈Lεn such that d(x0,xn)<εn. Hence, xn→x0 and
(4.17)d(xn,S(xn))≤εn,(4.18)〈v,η(xn,y)〉+f(xn)-f(y)≤εn+α2‖xn-y‖2,∀y∈S(xn),∀v∈F(y).
It follows from (4.17), xn→x0, and the proof of Lemma 3.3 that x0∈S(x0).
SinceS(xn) Lower Semi-Mosco converges to S(x0), for every y∈S(x0), there exists a sequence yn∈S(xn),for alln∈N, such that limnyn=y in the strong topology.
For eachn∈N select vn∈F(yn). Since F is (s,w*)-usc with weak* compact convex values, we can find a subsequence (vnj)j of (vn)n weak* converging to some v∈F(y). By the continuity of η and similar argument with the proof of Lemma 3.5, we know that
(4.19)〈vnj,η(xnj,ynj)〉⟶〈v,η(x0,y)〉.
Hence,
(4.20)〈vnj,η(xnj,ynj)〉+f(xnj)-f(ynj)-α2‖xnj-ynj‖2≤εnj.
We deduce from the above inequality that
(4.21)∀y∈S(x0),∃v∈F(y),〈v,η(x0,y)〉+f(x0)-f(y)-α2‖x0-y‖2≤0.
By Lemma 2.6 we know that there exist u0∈F(x0), such that
(4.22)∀y∈S(x0),〈u0,η(x0,y)〉+f(x0)-f(y)-α2‖x0-y‖2≤0.
It follows from Lemma 2.7 that x0∈Γ. Thus, Δ⊆Γ. It follows from (4.15) and (4.16) that limε→0e(Lε,Γ)=0. It follows from the compactness of Γ and Theorem 4.3(b) that (MQVLI) is L-α-well-posed in the generalized sense. The problem is completed.Remark 4.6.
(i) It is easy to see that ifα=0, then by the main results in our paper, we can recover the corresponding results in [20] with the weaker condition S(xn) Lower Semi-Mosco converging to S(x0) instead of the condition S(xn) Mosco converging to S(x0).
(ii) The proof methods of Theorems4.4 and 4.5 are different from those ofTheorems  4.1 and  4.2in [20].



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