# Well-Posedness by Perturbations of Generalized Mixed Variational Inequalities in Banach Spaces

Authors: Lu-Chuan Ceng; Ching-Feng Wen
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

We consider an extension of the notion of well-posedness by perturbations, introduced by Zolezzi (1995, 1996) for a minimization problem, to a class of generalized mixed variational inequalities in Banach spaces, which includes as a special case the class of mixed variational inequalities. We establish some metric characterizations of the well-posedness by perturbations. On the other hand, it is also proven that, under suitable conditions, the well-posedness by perturbations of a generalized mixed variational inequality is equivalent to the well-posedness by perturbations of the corresponding inclusion problem and corresponding fixed point problem. Furthermore, we derive some conditions under which the well-posedness by perturbations of a generalized mixed variational inequality is equivalent to the existence and uniqueness of its solution.


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## Body

## 1. Introduction

LetX be a real Banach space and f:X→R∪{+∞} a real-valued functional on X. In 1966, Tikhonov [1] first introduced the classical notion of well-posedness for a minimization problem minx∈Xf(x), which has been known as the Tikhonov well-posedness. A minimization problem is said to be Tikhonov well-posed if it has a unique solution toward which every minimizing sequence of the problem converges. It is obvious that the notion of Tikhonov well-posedness is inspired by the numerical methods producing optimizing sequences for optimization problems and plays a crucial role in the optimization theory. The notion of generalized Tikhonov well-posedness is also introduced for a minimization problem having more than one solution, which requires the existence of solutions and the convergence of some subsequence of every minimizing sequence toward some solution. Another important notion of well-posedness for a minimization problem is the well-posedness by perturbations or extended well-posedness due to Zolezzi [2, 3]. The notion of well-posedness by perturbations establishes a form of continuous dependence of the solutions upon a parameter. There are many other notions of well-posedness in optimization problems. For more details, we refer the readers to [1–7] and the references therein.On the other hand, the concept of well-posedness has been generalized to other variational problems, such as variational inequalities [4, 8–14], saddle point problems [15], Nash equilibrium problems [14, 16–18], equilibrium problems [19], inclusion problems [20, 21], and fixed point problems [20–22]. An initial notion of well-posedness for a variational inequality is due to Lucchetti and Patrone [4]. They introduced the notion of well-posedness for variational inequalities and proved some related results by means of Ekeland’s variational principle. Since then, many papers have been devoted to the extensions of well-posedness of minimization problems to various variational inequalities. Lignola and Morgan [12] generalized the notion of well-posedness by perturbations to a variational inequality and established the equivalence between the well-posedness by perturbations of a variational inequality and the well-posedness by perturbations of the corresponding minimization problem. Lignola and Morgan [14] introduced the concepts of α-well-posedness for variational inequalities. Del Prete et al. [13] further proved that the α-well-posedness of variational inequalities is closely related to the well-posedness of minimization problems. Recently, Fang et al. [9] generalized the notions of well-posedness and α-well-posedness to a mixed variational inequality. In the setting of Hilbert spaces, Fang et al. [9] proved that under suitable conditions the well-posedness of a mixed variational inequality is equivalent to the existence and uniqueness of its solution. They also showed that the well-posedness of a mixed variational inequality has close links with the well-posedness of the corresponding inclusion problem and corresponding fixed point problem in the setting of Hilbert spaces. Subsequently, the notions of well-posedness and α-well-posedness for a mixed variational inequality in [9] are extended by Ceng and Yao [11] to a generalized mixed variational inequality in the setting of Hilbert spaces. Very recently, Fang et al. [10] generalized the notion of well-posedness by perturbations to a mixed variational inequality in Banach spaces. In the setting of Banach spaces, they established some metric characterizations and showed that the well-posedness by perturbations of a mixed variational inequality is closely related to the well-posedness by perturbations of the corresponding inclusion problem and corresponding fixed point problem. They also derived some conditions under which the well-posedness by perturbations of the mixed variational inequality is equivalent to the existence and uniqueness of its solution.In this paper, we further extend the notion of well-posedness by perturbations to a class of generalized mixed variational inequalities in Banach spaces, which includes as a special case the class of mixed variational inequalities in [10]. Under very mild conditions, we establish some metric characterizations for the well-posed generalized mixed variational inequality and show that the well-posedness by perturbations of a generalized mixed variational inequality is closely related to the well-posedness by perturbations of the corresponding inclusion problem and corresponding fixed point problem. We also derive some conditions under which the well-posedness by perturbations of the generalized mixed variational inequality is equivalent to the existence and uniqueness of its solution.


## 2. Preliminaries

Throughout this paper, unless stated otherwise, we always suppose thatX is a real reflexive Banach space with its dual X* and the duality pairing 〈·,·〉 between X and X*. For convenience, we denote strong (resp., weak) convergence by → (resp., ⇀). Let F:X→2X be a nonempty-valued multifunction, A:X→X* a single-valued mapping, and f:X→R∪{+∞} a proper, convex, and lower semicontinuous functional. Denote by domf the domain of f, that is,(2.1)dom⁡f:={x∈X:f(x)<+∞}.The generalized mixed variational inequality associated with(A,F,f) is formulated as follows:(2.2)GMVI(A,F,f):findx∈Xsuchthat,forsomeu∈F(x),〈Au,x-y〉+f(x)-f(y)≤0,∀y∈X,
which has been studied intensively (see, e.g., [11, 23–25]).In the following, we give some special cases ofGMVI(A,F,f).(i)
WheneverF=I, the identity mapping of X, GMVI(A,F,f) reduces to the following mixed variational inequality associated with (A,f):
(2.3)MVI(A,f):findx∈Xsuchthat〈Ax,x-y〉+f(x)-f(y)≤0,∀y∈X,
which has been considered in [8–11, 26].(ii)
Wheneverf=δK, MVI(A,f) reduces to the following classical variational inequality:
(2.4)VI⁡(A,K):findx∈Ksuchthat〈Ax,x-y〉≤0,∀y∈K,
where δK denotes the indicator functional of a convex subset K of X.(iii)
WheneverA=0, MVI(A,f) reduces to the global minimization problem:
(2.5)MP(f,X):min⁡x∈Xf(x).Suppose thatL is a parametric normed space, P⊂L is a closed ball with positive radius, and p*∈P is a fixed point. The perturbed problem of GMVI(A,F,f) is always given by(2.6)GMVIp(A,F,f):findx∈Xsuchthatforsomeu∈F(x),〈Ã(p,u),x-y〉+f̃(p,x)-f̃(p,y)≤0,∀y∈X,
where Ã:P×X→X* is such that Ã(p*,·)=A and f̃:P×X→R∪{+∞} is such that f̃(p*,·)=f.Now we recall some concepts and results.Definition 2.1 (see [26]).
A mappingM:X→2X* is said to be(i)
monotone if(2.7)〈g-h,x-y〉≥0,∀x,y∈Dom⁡M,g∈M(x),h∈M(y);(ii)
maximal monotone ifM is monotone and
(2.8)〈g-w,x-z〉≥0,∀x∈Dom⁡M,g∈M(x)⟹w∈M(z),
where 2X* denotes the family of all subsets of X* and DomM={x∈X:M(x)≠∅}.Definition 2.2 (see [11]).
A nonempty-valued multifunctionF:X→2X is said to be monotone with respect to a single-valued mapping A:X→X* if, for all x,y∈X,
(2.9)〈Au-Av,x-y〉≥0,∀u∈F(x),v∈F(y).Proposition 2.3 (Nadler’s Theorem [27]).
Let(X,∥·∥) be a normed vector space and ℋ(·,·) the Hausdorff metric on the collection CB(X) of all nonempty, closed, and bounded subsets of X, induced by a metric d in terms of d(x,y)=∥x-y∥, which is defined by ℋ(U,V)=max{e(U,V),e(V,U)} for U and V in CB(X), where e(U,V)=supx∈Ud(x,V) with d(x,V)=infy∈V∥x-y∥. If U and V lie in CB(X), then, for any ɛ>0 and any u∈U, there exists v∈V such that ∥u-v∥≤(1+ε)ℋ(U,V). In particular, whenever U and V are compact subsets in X, one has ∥u-v∥≤ℋ(U,V).Definition 2.4.
Let{Un} be a sequence of nonempty subsets of X. One says that Un converges to U in the sense of Hausdorff metric if ℋ(Un,U)→0. It is easy to see that e(Un,U)→0 if and only if d(xn,U)→0 for all selection xn∈Un. For more details on this topic, the reader is referred to [28].Definition 2.5 (see [29]).
A mappingA:X→X* is said to be(i)
coercive if(2.10)lim⁡‖x‖→∞〈Ax,x〉‖x‖=+∞;(ii)
bounded ifA(B) is bounded for every bounded subset B of X;(iii)
hemicontinuous if, for anyx,y∈X, the function t↦〈A(x+t(y-x)),y-x〉 from [0,1] into R is continuous at 0+;(iv)
uniformly continuous if, for any neighborhoodV of 0 in X*, there exists a neighborhood U of 0 in X such that Ax-Ay∈V for all x,y∈U.Clearly, the uniform continuity implies the continuity, and the continuity implies the hemicontinuity, but the converse is not true in general.Definition 2.6.
(i) A nonempty weakly compact-valued multifunctionF:X→2X is said to be ℋ-hemicontinuous if, for any x,y∈X, the function t↦ℋ(F(x+t(y-x)),F(x)) from [0,1] into R+=[0,∞) is continuous at 0+, where ℋ is the Hausdorff metric defined on CB(X).
(ii) A nonempty weakly compact-valued multifunctionF:X→2X is said to be ℋ-continuous at a point x∈X if, for any ε>0, there exists δ>0 such that, for all y∈X with ∥x-y∥<δ, one has ℋ(F(x),F(y))<ε, where ℋ is the Hausdorff metric defined on CB(X). If this multifunction F:X→2X is ℋ-continuous at each x∈X, then one says that F:X→2X is ℋ-continuous.
(iii) A nonempty weakly compact-valued multifunctionF:X→2X is said to be ℋ-uniformly continuous if, for any ε>0, there exists δ>0 such that, for all x,y∈X with ∥x-y∥<δ, one has ℋ(F(x),F(y))<ε, where ℋ is the Hausdorff metric defined on CB(X).Remark 2.7.
IfX=H a real Hilbert space, then Definition 2.6(i)–(iii) reduce to Definition  2.3 (ii)–(iv) in [11], respectively.Lemma 2.8.
LetA:X→X* be weakly continuous (i.e., continuous from the weak topology of X to the weak topology of X*), let F:X→2X be a nonempty weakly compact-valued multifunction which is ℋ-hemicontinuous and monotone with respect to A, and let f:X→R∪{+∞} be proper and convex. Then, for a given x∈X, the following statements are equivalent:(i)
there existsu∈F(x) such that 〈Au,x-y〉+f(x)-f(y)≤0, for all y∈X;(ii)
〈Av,x-y〉+f(x)-f(y)≤0, for all y∈X,v∈F(y).Proof.
Suppose that, for someu∈F(x),
(2.11)〈Au,x-y〉+f(x)-f(y)≤0,∀y∈X.
Since F is monotone with respect to A, one has
(2.12)〈Av,x-y〉+f(x)-f(y)≤〈Au,x-y〉+f(x)-f(y)≤0,∀y∈X,v∈F(y).
Consequently,
(2.13)〈Av,x-y〉+f(x)-f(y)≤0,∀y∈X,v∈F(y).
Conversely, suppose that the last inequality is valid. Given anyy∈X, we define yt=x+t(y-x) for all t∈(0,1). Replacing y by yt in the left-hand side of the last inequality, one derives, for each vt∈F(yt),
(2.14)0≥〈Avt,x-yt〉+f(x)-f(yt)≥t[〈Avt,x-y〉+f(x)-f(y)],
which hence implies that
(2.15)〈Avt,x-y〉+f(x)-f(y)≤0,∀vt∈F(yt),t∈(0,1).
Since F:X→2X is a nonempty weakly compact-valued multifunction, both F(yt) and F(x) are nonempty weakly compact and hence are nonempty, weakly closed, and weakly bounded. Note that the weak closedness of sets in X implies the strong closedness and that the weak boundedness of sets in X is equivalent to the strong boundedness. Thus, it is known that both F(yt) and F(x) lie in CB(X). From Proposition 2.3, it follows that, for each t∈(0,1) and each fixed vt∈F(yt), there exists a ut∈F(x) such that
(2.16)‖vt-ut‖≤(1+t)H(F(yt),F(x)).
Since F(x) is weakly compact, it follows from the net {ut:t∈(0,1)}⊂F(x) that there exists some subnet which converges weakly to a point of F(x). Without loss of generality, we may assume that ut⇀u∈F(x) as t→0+. Since F is ℋ-hemicontinuous, one deduces that as t→0+(2.17)‖vt-ut‖≤(1+t)H(F(yt),F(x))=(1+t)H(F(x+t(y-x)),F(x))⟶0.
Observe that, for each φ∈X*,
(2.18)|〈φ,vt-u〉|=|〈φ,vt〉-〈φ,ut〉+〈φ,ut〉-〈φ,u〉|≤‖φ‖‖vt-ut‖+|〈φ,ut-u〉|⟶0ast⟶0+,
that is, vt⇀u as t→0+. Since A is weakly continuous, Avt⇀Au and hence, for y∈X,
(2.19)〈Avt,x-y〉⟶〈Au,x-y〉ast⟶0+.
Thus, letting t→0+ in the left-hand side of (2.15), we obtain that
(2.20)〈Au,x-y〉+f(x)-f(y)≤0,∀y∈X.
Finally let us show that the vector u in the last inequality is not dependent on y, that is,
(2.21)〈Au,x-z〉+f(x)-f(z)≤0,∀z∈X.
Indeed, take a fixed z∈X arbitrarily, and define zt=x+t(z-x) for all t∈(0,1). Utilizing Proposition 2.3, for each t∈(0,1) and ut∈F(x), there exists wt∈F(zt) such that
(2.22)‖ut-wt‖≤(1+t)H(F(x),F(zt)).
Since F is ℋ-hemicontinuous, we deduce that as t→0+(2.23)‖ut-wt‖≤(1+t)H(F(x),F(zt))=(1+t)H(F(x),F(x+t(z-x)))⟶0.
Thus, one has, for each φ∈X*,
(2.24)|〈φ,wt-u〉|=|〈φ,wt〉-〈φ,ut〉+〈φ,ut〉-〈φ,u〉|≤‖φ‖‖ut-wt‖+|〈φ,ut-u〉|⟶0ast⟶0+.
This shows that wt⇀u as t→0+. Since A is weakly continuous, Awt⇀Au and hence, for z∈X,
(2.25)〈Awt,x-z〉⟶〈Au,x-z〉ast⟶0+.
Replacing y, yt, and vt in (2.15) by z, zt, and wt, respectively, one concludes that
(2.26)〈Awt,x-z〉+f(x)-f(z)≤0,∀t∈(0,1).
This immediately implies that inequality (2.21) is valid. This completes the proof.Corollary 2.9 (see [11, Lemma  2.2]).
LetH be a real Hilbert space. Let A:H→H be weakly continuous (i.e., continuous from the weak topology of H to the weak topology of H), let F:H→2H be a nonempty weakly compact-valued multifunction which is ℋ-hemicontinuous and monotone with respect to A, and let f:H→R∪{+∞} be proper and convex. Then, for a given x∈H, the following statements are equivalent:(i)
there existsu∈F(x) such that 〈Au,x-y〉+f(x)-f(y)≤0, for all y∈H;(ii)
〈Av,x-y〉+f(x)-f(y)≤0, for all y∈H,v∈F(y).Definition 2.10 (see [30]).
LetK be a nonempty, closed, and convex subset of X. One says that K is well-positioned if there exist x0∈X and g∈X* such that
(2.27)〈g,x-x0〉≥‖x-x0‖,∀x∈K.Remark 2.11 (see [10, Remark  2.1]).
(i) IfK is well-positioned, then K+x* is well-positioned for all x*∈X.
(ii) As pointed out in [30, Remark  2.2], every nonempty compact convex set of a finite-dimensional space is well-positioned. Some useful properties and interesting applications have been discussed in [30, 31]. The following result is exacted from Proposition  2.1 of [30]. Also see [31, Proposition  2.1].Lemma 2.12.
LetK be a nonempty, closed, and convex subset of a reflexive Banach space X. If K is well-positioned, then there is no sequence {xn}⊂K with ∥xn∥→+∞ such that origin is a weak limit of {xn/∥xn∥}.Definition 2.13 (see [28]).
LetB be a nonempty subset of X. The measure of noncompactness μ of the set B is defined by
(2.28)μ(B)=inf⁡{ε>0:A⊂⋃i=1nBi,diam⁡Bi<ɛ,i=1,2,…,n,forsomeintegern≥1},
where diam means the diameter of a set.Lemma 2.14 (see [10, Lemma  2.3]).
LetAn, A be nonempty, closed, and convex subsets of a real reflexive Banach space X, and let A be well-positioned. Suppose that e(An,A)→0 as n→∞ and x∈X. Then, there is no sequence an∈An with ∥an∥→+∞ such that origin is a weak limit of {(an-x)/∥an-x∥}.


## 3. Well-Posedness by Perturbations and Metric Characterizations

In this section, we generalize the concepts of well-posedness by perturbations to the generalized mixed variational inequality and establish their metric characterizations. In the sequel we always denote by→ and ⇀ the strong convergence and weak convergence, respectively. Let α≥0 be a fixed number.Definition 3.1.
Let{pn}⊂P be with pn→p*. A sequence {xn}⊂X is called an α-approximating sequence corresponding to {pn} for GMVI(A,F,f) if there exists a sequence {un}⊂X with un∈F(xn) (for all n≥1) and a sequence {ɛn} of nonnegative numbers with εn→0 such that
(3.1)xn∈dom⁡f̃(pn,⋅),〈Ã(pn,un),xn-y〉+f̃(pn,xn)-f̃(pn,y)≤α2‖xn-y‖2+εn,∀y∈X,n≥1.
Whenever α=0, we say that {xn} is an approximating sequence corresponding to {pn} for GMVI(A,F,f). Clearly, every α2-approximating sequence corresponding to {pn} is α1-approximating corresponding to {pn} provided α1>α2≥0.Definition 3.2.
One says thatGMVI(A,F,f) is strongly (resp., weakly) α-well-posed by perturbations if GMVI(A,F,f) has a unique solution and, for any {pn}⊂P with pn→p*, every α-approximating sequence corresponding to {pn} converges strongly (resp., weakly) to the unique solution. In the sequel, strong (resp., weak) 0-well-posedness by perturbations is always called strong (resp., weak) well-posedness by perturbations. If α1>α2≥0, then strong (resp., weak) α1-well-posedness by perturbations implies strong (resp., weak) α2-well-posedness by perturbations.Remark 3.3.
(i) WhenX is a Hilbert space and pn=p* (for all n≥1), Definitions  3.1 and 3.2 coincide with Definitions  3.1 and 3.2 of [11], respectively. (ii) When f=δK and F=I the identity mapping of X, Definitions 3.1 and 3.2 reduce to the definitions of approximating sequences of the classical variational inequality (see [12, 13]).Definition 3.4.
One says thatGMVI(A,F,f) is strongly (resp., weakly) generalized α-well-posed by perturbations if GMVI(A,F,f) has a nonempty solution set S and, for any {pn}⊂P with pn→p*, every α-approximating sequence corresponding to {pn} has some subsequence which converges strongly (resp., weakly) to some point of S. Strong (resp., weak) generalized 0-well-posedness by perturbations is always called strong (resp., weak) generalized well-posedness by perturbations. Clearly, if α1>α2≥0, then strong (resp., weak) generalized α1-well-posedness by perturbations implies strong (resp., weak) generalized α2-well-posedness by perturbations.Remark 3.5.
(i) WhenX is a Hilbert space and pn=p* (for all n≥1), Definition 3.4 coincides with Definition  3.3 of [11]. (ii) When f=δK and F=I the identity mapping of X, Definition 3.4 reduces to the definition of strong (resp., weak) parametric α-well-posedness in the generalized sense for the classical variational inequality (see [12–14]). (iii) When A=0 and α=0, Definition 3.4 coincides with the definition of well-posedness by perturbations introduced for a minimization problem [2, 3].To derive the metric characterizations ofα-well-posedness by perturbations, we consider the following approximating solution set of GMVI(A,F,f):(3.2)Ωα(ε)=⋃p∈B(p*,ε){α2‖x-y‖2x∈dom⁡f̃(p,⋅):thereexistsu∈F(x)suchthat〈Ã(p,u),x-y〉+f̃(p,x)-f̃(p,y)≤α2‖x-y‖2+ε,∀y∈X},∀ε≥0,
where B(p*,ε) denotes the closed ball centered at p* with radius ɛ. In this section, we always suppose that x* is a fixed solution of GMVI(A,F,f). Define(3.3)θ(ε)=sup⁡{‖x-x*‖:x∈Ωα(ε)},∀ɛ≥0.
It is easy to see that θ(ε) is the radius of the smallest closed ball centered at x* containing Ωα(ε). Now, we give a metric characterization of strong α-well-posedness by perturbations by considering the behavior of θ(ε) when ε→0.Theorem 3.6.
GMVI(A,F,f) is strongly α-well-posed by perturbations if and only if θ(ε)→0 as ε→0.Proof.
Repeating almost the same argument as in the proof of [10, Theorem  3.1], we can easily obtain the desired result.Remark 3.7.
Theorem3.6 improves Proposition  2.2 of [13], Theorem  3.1 of [9], and Theorem  3.1 of [10].Now, we give an example to illustrate Theorem3.6.Example 3.8.
LetX=R, P=[-1,1], p*=0, α=2, Ã(p,x)=x(p2+1), F(x)={x,0}, and f̃(p,x)=x2 for all x∈X, p∈P. Clearly, x*=0 is a solution of GMVI(A,F,f). For any ε>0, it follows that
(3.4)Ωαp(ε)={x∈R:u(p2+1)(x-y)+x2-y2≤(x-y)2+ε,∀y∈R,forsomeu∈F(x)}=Δ∪Λ,
where
(3.5)Δ={x∈R:x(p2+1)(x-y)+x2-y2≤(x-y)2+ε,∀y∈R},Λ={x∈R:0(p2+1)(x-y)+x2-y2≤(x-y)2+ε,∀y∈R}.
Observe that
(3.6)Δ={x∈R:-2(y+p2-14x)2+(p2+3)28x2≤ε,∀y∈R}=[-22ε(p2+3),22ε(p2+3)],Λ={x∈R:(y-x2)2-x24+ε2≥0,∀y∈R}=[-2ε,2ε].
Thus, we obtain
(3.7)Ωαp(ε)=Δ∪Λ=[-22ε(p2+3),22ε(p2+3)]∪[-2ε,2ε]=[-2ε,2ε].
Therefore,
(3.8)Ωα(ε)=⋃p∈B(0,ε)Ωαp(ε)=[-2ε,2ε]
for sufficiently small ε>0. By trivial computation, we have
(3.9)θ(ε)=sup⁡{‖x-x*‖:x∈Ωα(ε)}=2ε⟶0asε⟶0.
By Theorem 3.6, GMVI(A,F,f) is 2-well-posed by perturbations.To derive a characterization of strong generalizedα-well-posedness by perturbations, we need another function q which is defined by(3.10)q(ε)=e(Ωα(ε),S),∀ε≥0,
where S is the solution set of GMVI(A,F,f) and e is defined as in Proposition 2.3.Theorem 3.9.
GMVI(A,F,f) is strongly generalized α-well-posed by perturbations if and only if S is nonempty compact and q(ε)→0 as ε→0.Proof.
Repeating almost the same argument as in the proof of [10, Theorem  3.2], we can readily derive the desired result.Example 3.10.
LetX=R, P=[-1,1], p*=0, α=2, Ã(p,x)=x(p2+1), F(x)={x,0}, and f̃(p,x)=x2 for all x∈X, p∈P. Clearly, x*=0 is a solution of GMVI(A,F,f). Repeating the same argument as in Example 3.8, we obtain that, for any ε>0,
(3.11)Ωα(ε)=⋃p∈B(0,ε)Ωαp(ε)=[-2ε,2ε]
for sufficiently small ε>0. By trivial computation, we have
(3.12)q(ε)=e(Ωα(ε),S)=sup⁡x(ε)∈Ωα(ε)d(x(ε),S)⟶0asε⟶0.
By Theorem 3.9, GMVI(A,F,f) is generalized α-well-posed by perturbations.The strong generalizedα-well-posedness by perturbations can be also characterized by the behavior of the noncompactness measure μ(Ωα(ε)).Theorem 3.11.
LetL be finite dimensional, Ã:P×X→X* weakly continuous (i.e., continuous from the product of the norm topology of P and weak topology of X to the weak topology of X*), F:X→2X a nonempty weakly compact-valued multifunction which is ℋ-continuous, and f̃:P×X→R∪{+∞} a continuous functional such that f̃(p,·) is proper and convex. Then, GMVI(A,F,f) is strongly generalized α-well-posed by perturbations if and only if Ωα(ε)≠∅, for all ε>0 and μ(Ωα(ε))→0 as ε→0.Proof.
First, we will prove thatΩα(ε) is closed for all ε≥0. Let {xn}⊂Ωα(ε) with xn→x¯. Then, there exist {pn}⊂B(p*,ε) and {un}⊂X with un∈F(xn) (for all n≥1) such that
(3.13)〈Ã(pn,un),xn-y〉+f̃(pn,xn)-f̃(pn,y)≤α2‖xn-y‖2+ε,∀y∈X,n≥1.
Without loss of generality, we may assume pn→p¯∈B(p*,ε) since L is finite dimensional. Since F:X→2X is a nonempty weakly compact-valued multifunction, F(xn) and F(x¯) are nonempty weakly compact and hence are nonempty, weakly closed, and weakly bounded. Note that the weak closedness of sets in X implies the strong closedness and that the weak boundedness of sets in X is equivalent to the strong boundedness. Thus, it is known that F(xn) and F(x¯) lie in CB(X). According to Proposition 2.3, for each n≥1 and un∈F(xn), there exists vn∈F(x¯) such that
(3.14)‖un-vn‖≤(1+1n)H(F(xn),F(x¯)).
Since F is ℋ-continuous, one deduces that
(3.15)‖un-vn‖≤(1+1n)H(F(xn),F(x¯))⟶0asn⟶∞.
Also, since F(x¯) is weakly compact, it follows from {vn}⊂F(x¯) that there exists some subsequence of {vn} which converges weakly to a point of F(x¯). Without loss of generality, we may assume that
(3.16)vn⇀u¯∈F(x¯)asn⟶∞.
Consequently, one has, for each φ∈X*,
(3.17)|〈φ,un-u¯〉|≤|〈φ,un-vn〉|+|〈φ,vn-u¯〉|≤‖φ‖‖un-vn‖+|〈φ,vn-u¯〉|⟶0asn⟶∞.
This implies that un⇀u¯ as n→∞. Taking into account the weak continuity of Ã, we immediately obtain that
(3.18)Ã(pn,un)⇀Ã(p¯,u¯)asn⟶∞,
and hence, for each y∈X,
(3.19)|〈Ã(pn,un),xn-y〉-〈Ã(p¯,u¯),x¯-y〉|≤|〈Ã(pn,un),xn-y〉-〈Ã(pn,un),x¯-y〉|+|〈Ã(pn,un),x¯-y〉-〈Ã(p¯,u¯),x¯-y〉|≤‖Ã(pn,un)‖‖xn-x¯‖+|〈Ã(pn,un)-Ã(p¯,u¯),x¯-y〉|⟶0asn⟶∞,
that is,
(3.20)〈Ã(pn,un),xn-y〉⟶〈Ã(p¯,u¯),x¯-y〉asn⟶∞.
Therefore, it follows from (3.13) and the continuity of f̃ that
(3.21)〈Ã(p¯,u¯),x¯-y〉+f̃(p¯,x¯)-f̃(p¯,y)≤α2‖x¯-y‖2+ε,∀y∈X.
This shows that x¯∈Ωα(ε) and so Ωα(ε) is closed.
Second, we show that(3.22)S=⋂ε>0Ωα(ε).
It is obvious that S⊂⋂ε>0Ωα(ε). Let x*∈⋂ε>0Ωα(ε). Let {εn} be a sequence of positive numbers such that εn→0. Then, x*∈Ωα(εn) and so there exist pn∈B(p*,εn) and un*∈F(x*) such that
(3.23)〈Ã(pn,un*),x*-y〉+f̃(pn,x*)-f̃(pn,y)≤α2‖x*-y‖2+εn,∀y∈X,n≥1.
It is clear that pn→p* as n→∞. Since F(x*) is weakly compact, it follows from {un*}⊂F(x*) that there exists some subsequence of {un*} which converges weakly to a point of F(x*). Without loss of generality, we may assume that
(3.24)un*⇀u*∈F(x*)asn⟶∞.
Note that Ã is weakly continuous. Thus,
(3.25)Ã(pn,un*)⇀Ã(p*,u*),
and hence, letting n→∞ in the last inequality, we get
(3.26)〈A(u*),x*-y〉+f(x*)-f(y)=〈Ã(p*,u*),x*-y〉+f̃(p*,x*)-f̃(p*,y)≤α2‖x*-y‖2,∀y∈X.
For any z∈X and t∈(0,1), putting y=x*+t(z-x*) in (3.26), we have
(3.27)t{〈A(u*),x*-z〉+f(x*)-f(z)}≤t〈A(u*),x*-z〉+f(x*)-f(x*+t(z-x*))≤αt22‖x*-z‖2.
This implies that
(3.28)〈A(u*),x*-z〉+f(x*)-f(z)≤αt2‖x*-z‖2,∀z∈X.
Letting t→0 in the last inequality, we get
(3.29)〈A(u*),x*-z〉+f(x*)-f(z)≤0,∀z∈X.
Consequently, x*∈S and so (3.22) is proved.
Now, we suppose thatGMVI(A,F,f) is strongly generalized α-well-posed by perturbations. By Theorem 3.9, S is nonempty compact and q(ε)→0. Then, Ωα(ε)≠∅ since S⊂Ωα(ε) for all ε>0. Observe that, for all ε>0,
(3.30)H(Ωα(ε),S)=max⁡{e(Ωα(ε),S),e(S,Ωα(ε))}=e(Ωα(ε),S).
Taking into account the compactness of S, we get
(3.31)μ(Ωα(ε))≤2H(Ωα(ε),S)=2e(Ωα(ε),S)=2q(ε)⟶0.
Conversely, we suppose thatΩα(ε)≠∅, for all ε>0, and μ(Ωα(ε))→0 as ε→0. Since Ωα(ε) is increasing with respect to ε>0, by the Kuratowski theorem [28, page 318], we have from (3.22)
(3.32)q(ε)=H(Ωα(ε),S)⟶0asε⟶0
and S is nonempty compact. By Theorem 3.9, GMVI(A,F,f) is strongly generalized α-well-posed by perturbations.Remark 3.12.
Theorem  3.3 of [10] generalizes Theorem  3.2 of [9]. Theorem 3.11 generalizes Theorem  3.2 of [11] from the case of strong generalized α-well-posedness in the setting of Hilbert spaces to the case of strong generalized α-well-posedness by perturbations in the setting of Banach spaces. Furthermore, Theorem 3.11 improves, extends, and develops [10, Theorem  3.3] in the following aspects.(i)
The mixed variational inequality problem (MVI) in [10, Theorem  3.3] is extended to develop the more general problem, that is, the generalized mixed variational inequality problem (GMVI) with a nonempty weakly compact-valued multifunction in the setting of Banach spaces. Moreover, the concept of strong generalized α-well-posedness by perturbations for MVI in [10, Theorem  3.3] is extended to develop the concept of strong generalized α-well-posedness by perturbations for GMVI.(ii)
Since the generalized mixed variational inequality problem (GMVI) is more general and more complicated than the mixed variational inequality problem (MVI), the assumptions in Theorem3.11 are very different from the ones in [10, Theorem  3.3]; for instance, in Theorem 3.11, let L be finite dimensional, Ã:P×X→X* weakly continuous, and F:X→2X a nonempty weakly compact-valued multifunction which is ℋ-continuous, but, in [10, Theorem  3.3], let L be finite dimensional, Ã:P×X→X* a continuous mapping.(iii)
The technique of proving strong generalizedα-well-posedness by perturbations for GMVI in Theorem 3.11 is very different from the one for MVI in [10, Theorem  3.3] because our technique depends on the well-known Nadler’s Theorem [27], the ℋ-continuity of nonempty weakly compact-valued multifunction F and the property of the Hausdorff metric ℋ.Remark 3.13.
Clearly, any solution ofGMVI(A,F,f) is a solution of the α problem: find x∈X such that, for some u∈F(x),
(3.33)〈A(u),x-y〉+f(x)-f(y)≤α2‖x-y‖2,∀y∈X,
but the converse is not true in general. To show this, let X=R, A(x)=x, F(x)={x,0}, and f(x)=-x2 for all x∈X. It is easy to verify that the solution set of GMVI(A,F,f) is empty and 0 is the unique solution of the corresponding α problem with α=2. If f is proper and convex, then GMVI(A,F,f) and α problem have the same solution (this fact has been shown in the proof of Theorem 3.11).


## 4. Links with the Well-Posedness by Perturbations of Inclusion Problems

Lemaire et al. [20] introduced the concept of well-posedness by perturbations for an inclusion problem. In this section, we will show that the well-posedness by perturbations of a generalized mixed variational inequality is closely related to the well-posedness by perturbations of the corresponding inclusion problem. Let us recall some concepts. Let M:X→2X*. The inclusion problem associated with M is defined by(4.1)IP(M):findx∈Xsuchthat0∈M(x).
The perturbed problem of IP(M) is given by(4.2)IPp(M):findx∈Xsuchthat0∈M̃(p,x),
where M̃:P×X→2X* is such that M̃(p*,·)=M.Definition 4.1 (see [20]).
Let{pn}⊂P be with pn→p*. A sequence {xn}⊂X is called an approximating sequence corresponding to {pn} for IP(M) if xn∈DomM̃(pn,·) for all n≥1 and d(0,M̃(pn,xn))→0, or, equivalently, there exists yn∈M̃(pn,xn) such that ∥yn∥→0 as n→∞.Definition 4.2 (see [20]).
One says thatIP(M) is strongly (resp., weakly) well-posed by perturbations if it has a unique solution and, for any {pn}⊂P with pn→p*, every approximating sequence corresponding to {pn} converges strongly (resp., weakly) to the unique solution of IP(M). IP(M) is said to be strongly (resp., weakly) generalized well posed by perturbations if the solution set S of IP(M) is nonempty and, for any {pn}⊂P with pn→p*, every approximating sequence corresponding to {pn} has a subsequence which converges strongly (resp., weakly) to a point of S.Letf:X→R∪{+∞} be a proper, convex, and lower semicontinuous functional. Denote by ∂f and ∂ɛf the subdifferential and ε-subdifferential of f, respectively, that is,(4.3)∂f(x)={x*∈X*:f(y)-f(x)≥〈x*,y-x〉,∀y∈X},∀x∈dom⁡f,∂εf(x)={x*∈X*:f(y)-f(x)≥〈x*,y-x〉-ε,∀y∈X},∀x∈dom⁡f.
It is known that ∂f is maximal monotone and ∂εf(x)⊃∂f(x)≠∅ for all x∈domf and for all ε>0. In terms of ∂f, GMVI(A,F,f) is equivalent to the following inclusion problem:(4.4)IP(AF+∂f):findx∈Xsuchthat0∈AF(x)+∂f(x).In other words, we have the following lemma.Lemma 4.3.
Letx∈X be a fixed point, and let f:X→R∪{+∞} be a proper, convex, and lower semicontinuous functional. Then, the following statements are equivalent:(i)
〈Au,x-y〉+f(x)-f(y)≤0  for all y∈X and some u∈F(x);(ii)
0∈AF(x)+∂f(x).Proof.
Observe that(4.5)〈Au,x-y〉+f(x)-f(y)≤0⟺f(y)-f(x)≥〈-Au,y-x〉⟺-Au∈∂f(x)⟺0∈Au+∂f(x)⟺0∈AF(x)+∂f(x),
for all y∈X and some u∈F(x). The desired result follows immediately from the above relations.Naturally, we consider the perturbed problem ofIP(AF+∂f) as follows:(4.6)IPp(AF+∂f):findx∈Xsuchthat0∈Ã(p,F(x))+∂f̃(p,⋅)(x),
where f̃:P×X→R∪{+∞} is such that f̃(p,·) is proper, convex, and lower semicontinuous for all p∈P, and f̃(p*,·)=f.The following theorems establish the relations between the strong (resp., weak) well-posedness by perturbations of generalized mixed variational inequalities and the strong (resp., weak) well-posedness by perturbations of inclusion problems.Theorem 4.4.
LetÃ(·,y):P→X* be continuous for each y∈X, let A(=A(p*,·)):X→X* be weakly continuous, let F:X→2X be a nonempty weakly compact-valued multifunction which is ℋ-hemicontinuous and monotone with respect to Ã(p,·) for each p∈P, and let f̃:P×X→R∪{+∞} be a continuous functional with respect to the product of the norm topology of P and weak topology of X such that the following conditions hold:(i)
f̃(p,·) is proper and convex for all p∈P;(ii)
domf is well-positioned and domf⊂⋂p∈Pdomf̃(p,·);(iii)
e(domf̃(pn,·),domf)→0  whenever pn→p*, where e is defined as in Proposition 2.3.
Then,IP(AF+∂f) is weakly well-posed by perturbations whenever GMVI(A,F,f) has a unique solution.Proof.
Suppose thatGMVI(A,F,f) has a unique solution x*. Let {pn}⊂P be with pn→p*, and let {xn} be an approximating sequence corresponding to {pn} for IP(AF+∂f). Then, there exists wn∈Ã(pn,F(xn))+∂f̃(pn,·)(xn) such that ∥wn∥→0. Further, there exists un∈F(xn) such that wn∈Ã(pn,un)+∂f̃(pn,·)(xn) with ∥wn∥→0. It follows that
(4.7)f̃(pn,y)-f̃(pn,xn)≥〈wn-Ã(pn,un),y-xn〉,∀y∈X,n≥1.
We claim that {xn} is bounded. Indeed, if {xn} is unbounded, without loss of generality, we may assume that ∥xn∥→+∞. Let
(4.8)tn=1‖xn-x*‖,zn=x*+tn(xn-x*).
By conditions (i)-(ii), we get zn∈domf̃(pn,·). Note that
(4.9)‖zn‖≤‖x*‖+tn‖xn-x*‖=‖x*‖+1,∀n≥1.
So, {zn} is bounded. Since X is reflexive, it follows from the boundedness of {zn} that there exists some subsequence of {zn} which converges weakly to a point of X. Hence, without loss of generality, we may assume that tn∈(0,1) and zn⇀z. It follows from Lemma 2.14 and conditions (ii)-(iii) that z≠x*. For any y∈X, observe that, for all v∈F(y),
(4.10)〈Ã(pn,v),z-y〉=〈Ã(pn,v),z-zn〉+〈Ã(pn,v),zn-x*〉+〈Ã(pn,v),x*-y〉=〈Ã(pn,v),z-zn〉+tn〈Ã(pn,v),xn-x*〉+〈Ã(pn,v),x*-y〉=〈Ã(pn,v),z-zn〉+tn〈Ã(pn,v),xn-y〉+(1-tn)〈Ã(pn,v),x*-y〉.
Since x* is the unique solution of GMVI(A,F,f), there exists some u*∈F(x*) such that
(4.11)〈Au*,x*-y〉+f(x*)-f(y)≤0,∀y∈X.
Also, since F is monotone with respect to Ã(pn,·), we deduce that, for u*∈F(x*), v∈F(y), and un∈F(xn),
(4.12)〈Ã(pn,v),x*-y〉≤〈Ã(pn,u*),x*-y〉,〈Ã(pn,v),xn-y〉≤〈Ã(pn,un),xn-y〉.
In addition, we have
(4.13)f̃(pn,zn)≤tnf̃(pn,xn)+(1-tn)f̃(pn,x*)
by virtue of the convexity of f̃(pn,·). It follows from (4.7)–(4.13) that
(4.14)〈Ã(pn,v),z-y〉≤〈Ã(pn,v),z-zn〉+tn〈Ã(pn,un),xn-y〉+(1-tn)〈Ã(pn,u*),x*-y〉≤〈Ã(pn,v),z-zn〉+(1-tn)〈Ã(pn,u*),x*-y〉+tn(f̃(pn,y)-f̃(pn,xn))+tn〈wn,xn-y〉≤〈Ã(pn,v),z-zn〉+(1-tn)〈Ã(pn,u*),x*-y〉+tn〈wn,xn-y〉+tnf̃(pn,y)+(1-tn)f̃(pn,x*)-f̃(pn,zn).
Moreover, it is easy to see from wn→0 that  tn〈wn,xn-y〉→0. Further, since Ã(·,y):P→X* is continuous for each y∈X and zn⇀z, it is known that Ã(pn,v)→Ã(p*,v) and {zn} is bounded. Consequently,
(4.15)|〈Ã(pn,v),z-zn〉|=|〈Ã(pn,v),z-zn〉-〈Ã(p*,v),z-zn〉+〈Ã(p*,v),z-zn〉|≤|〈Ã(pn,v)-Ã(p*,v),z-zn〉|+|〈Ã(p*,v),z-zn〉|≤‖Ã(pn,v)-Ã(p*,v)‖‖zn-z‖+|〈Ã(p*,v),zn-z〉|⟶0asn⟶∞.
In the meantime, since f̃:P×X→R∪{+∞}is a continuous functional with respect to the product of the norm topology of P and weak topology of X, we conclude from pn→p* and zn⇀z that f̃(pn,zn)→f̃(p*,z) and f̃(pn,x*)→f̃(p*,x*) as n→∞. Now, letting n→∞ in (4.14) we get
(4.16)〈Av,z-y〉=〈Ã(p*,v),z-y〉≤〈Ã(p*,u*),x*-y〉+f̃(p*,x*)-f̃(p*,z)=〈Au*,x*-y〉+f(x*)-f(z),∀y∈X.
Since x* is the unique solution of GMVI(A,F,f), from (4.16) we get
(4.17)〈Av,z-y〉≤〈Au*,x*-y〉+f(x*)-f(z)=〈Au*,x*-y〉+f(x*)-f(y)+f(y)-f(z)≤f(y)-f(z),∀y∈X,
which implies that
(4.18)〈Av,z-y〉+f(z)-f(y)≤0,∀y∈X,v∈F(y).
Note that A is weakly continuous, that f is proper and convex, and that F is a nonempty weakly compact-valued multifunction which is ℋ-hemicontinuous and monotone with respect to A. Hence, all conditions of Lemma 2.8 are satisfied. Thus, it follows from Lemma 2.8 that there exists w∈F(z) such that
(4.19)〈Aw,z-y〉+f(z)-f(y)≤0,∀y∈X.
Therefore, z is a solution of GMVI(A,F,f), a contradiction. This shows that {xn} is bounded.
Let{xnk} be any subsequence of {xn} such that xnk⇀x¯ as k→∞. It follows from (4.7) that
(4.20)〈Ã(pnk,unk),xnk-y〉+f̃(pnk,xnk)-f̃(pnk,y)≤〈wnk,xnk-y〉,∀y∈X,k≥1.
Since Ã(·,y):P→X*  is continuous for each y∈X and xnk⇀x¯, it is known that Ã(pnk,v)→Ã(p*,v) and {xnk} is bounded. Consequently,
(4.21)|〈Ã(pnk,v),xnk-y〉-〈Ã(p*,v),x¯-y〉|=|〈Ã(pnk,v),xnk-y〉-〈Ã(p*,v),xnk-y〉+〈Ã(p*,v),xnk-y〉-〈Ã(p*,v),x¯-y〉|≤|〈Ã(pnk,v)-Ã(p*,v),xnk-y〉|+|〈Ã(p*,v),xnk-x¯〉|≤‖Ã(pnk,v)-Ã(p*,v)‖‖xnk-y‖+|〈Ã(p*,v),xnk-x¯〉|⟶0asn⟶∞.
Moreover, since f̃:P×X→R∪{+∞}is a continuous functional with respect to the product of the norm topology of P and weak topology of X, we conclude from pnk→p* and xnk⇀x¯ that f̃(pnk,xnk)→f̃(p*,x¯) and f̃(pnk,y)→f̃(p*,y) as k→∞. Note that F is monotone with respect to Ã(pnk,·). Hence, it follows that for, unk∈F(xnk) and v∈F(y),
(4.22)〈Av,x¯-y〉+f(x¯)-f(y)=〈Ã(p*,v),x¯-y〉+f̃(p*,x¯)-f̃(p*,y)=lim⁡k→∞{〈Ã(pnk,v),xnk-y〉+f̃(pnk,xnk)-f̃(pnk,y)}≤lim⁡k→∞{〈Ã(pnk,unk),xnk-y〉+f̃(pnk,xnk)-f̃(pnk,y)}≤lim⁡k→∞〈wnk,xnk-y〉=0,∀y∈X.
This together with Lemma 2.8 yields that there exists u¯∈F(x¯) such that
(4.23)〈Au¯,x¯-y〉+f(x¯)-f(y)≤0,∀y∈X.
Consequently, x¯ solves GMVI(A,F,f). We must have x¯=x* since GMVI(A,F,f) has a unique solution x*. Therefore, {xn} converges weakly to x* and so IP(AF+∂f) is weakly well-posed by perturbations.Remark 4.5.
Theorem4.4 improves, extends, and develops [10, Theorem  4.1] in the following aspects. (i)
The mixed variational inequality problem (MVI) in [10, Theorem  4.1] is extended to develop the more general problem, that is, the generalized mixed variational inequality problem (GMVI) with a nonempty weakly compact-valued multifunction in the setting of Banach spaces. Moreover, the inclusion problem corresponding to MVI in [10, Theorem  4.1] is extended to develop the more general problem, that is, the inclusion problem corresponding to GMVI.(ii)
Since the generalized mixed variational inequality problem(GMVI) is more general and more complicated than the mixed variational inequality problem (MVI), the assumptions in Theorem 4.4 are very different from the ones in [10, Theorem  4.1], for instance, in Theorem 4.4, let Ã(·,y):P→X* be continuous for each y∈X, let A(=A(p*,·)):X→X* be weakly continuous, and let F:X→2X be a nonempty weakly compact-valued multifunction which is ℋ-hemicontinuous and monotone with respect to Ã(p,·) for each p∈P, but, in [10, Theorem  4.1], let Ã:P×X→X* be a continuous mapping such that Ã(p,·):X→X* is monotone for all p∈P.(iii)
The technique of proving weak well-posedness by perturbations for inclusion problemIP(AF+∂f) in Theorem 4.4 is very different from the one for inclusion problem IP(A+∂f) in [10, Theorem  4.1] because our technique depends on Lemma 2.8. Note that A is weakly continuous, that f is proper and convex, and that F is a nonempty weakly compact-valued multifunction which is ℋ-hemicontinuous and monotone with respect to A. Hence, all the conditions of Lemma 2.8 are satisfied. Recall that the proof of Lemma 2.8 depends on the well-known Nadler’s Theorem [27]. Thus, our technique depends essentially on the well-known Nadler’s Theorem [27], the ℋ-hemicontinuity of nonempty weakly compact-valued multifunction F and the monotonicity of F with respect to Ã(p,·) for each p∈P.Theorem 4.6.
LetÃ(·,y):P→X* be continuous for each y∈X, let A:X→X* be weakly continuous, let F:X→2X be a nonempty weakly compact-valued multifunction which is ℋ-hemicontinuous and monotone with respect to Ã(p,·) for each p∈P, and let f̃:P×X→R∪{+∞} be a continuous functional with respect to the product of the norm topology of P and weak topology of X such that the following conditions hold:(i)
f̃(p,·) is proper and convex for all p∈P;(ii)
domf is well-positioned and domf⊂⋂p∈Pdomf̃(p,·);(iii)
e(domf̃(pn,·),domf)→0 whenever pn→p*.
Then,GMVI(A,F,f) is weakly well-posed by perturbations whenever IP(AF+∂f) has a unique solution.Proof.
LetIP(AF+∂f) have a unique solution x*. By Lemma 4.3, x* is also the unique solution of GMVI(A,F,f). Let {pn}⊂P be with pn→p*, and let {xn} be an approximating sequence corresponding to {pn} for GMVI(A,F,f). Then, there exist {un}⊂X with un∈F(xn) (for all n≥1) and 0<εn→0 such that
(4.24)〈Ã(pn,un),xn-y〉+f̃(pn,xn)-f̃(pn,y)≤εn,∀y∈X,n≥1.
We claim that {xn} is bounded. Indeed, if {xn} is unbounded, without loss of generality, we may assume that ∥xn∥→+∞. Let
(4.25)tn=1‖xn-x*‖,zn=x*+tn(xn-x*).
By conditions (i)-(ii), we get zn∈domf̃(pn,·). Without loss of generality we may assume that tn∈(0,1) and zn⇀z. From Lemma 2.14 and conditions (ii)-(iii) we obtain that z≠x*. For any y∈X, observe that, for all v∈F(y),
(4.26)〈Ã(pn,v),z-y〉=〈Ã(pn,v),z-zn〉+〈Ã(pn,v),zn-x*〉+〈Ã(pn,v),x*-y〉=〈Ã(pn,v),z-zn〉+tn〈Ã(pn,v),xn-y〉+(1-tn)〈Ã(pn,v),x*-y〉.
Since x* is the unique solution of GMVI(A,F,f), there exists some u*∈F(x*) such that
(4.27)〈Ãu*,x*-y〉+f(x*)-f(y)≤0,∀y∈X.
Also, since F is monotone with respect to Ã(pn,·), we deduce that, for u*∈F(x*), v∈F(y), and un∈F(xn),
(4.28)〈Ã(pn,v),x*-y〉≤〈Ã(pn,u*),x*-y〉,〈Ã(pn,v),xn-y〉≤〈Ã(pn,un),xn-y〉.
In addition, since f̃(pn,·) is convex, we get
(4.29)f̃(pn,zn)≤tnf̃(pn,xn)+(1-tn)f̃(pn,x*).
It follows from (4.24)–(4.29) that
(4.30)〈Ã(pn,v),z-y〉≤〈Ã(pn,v),z-zn〉+tn〈Ã(pn,un),xn-y〉+(1-tn)〈Ã(pn,u*),x*-y〉≤〈Ã(pn,v),z-zn〉+(1-tn)〈Ã(pn,u*),x*-y〉+tn(f̃(pn,y)-f̃(pn,xn))+tnεn≤〈Ã(pn,v),z-zn〉+(1-tn)〈Ã(pn,u*),x*-y〉+tnεn+tnf̃(pn,y)+(1-tn)f̃(pn,x*)-f̃(pn,zn).
Letting n→∞ in the last inequality we get
(4.31)〈Av,z-y〉=〈Ã(p*,v),z-y〉≤〈Ã(p*,u*),x*-y〉+f̃(p*,x*)-f̃(p*,z)=〈Au*,x*-y〉+f(x*)-f(z),∀y∈X.
By using (4.31) and the same argument as in the proof of Theorem 4.4, we can prove that z is a solution of GMVI(A,F,f), a contradiction. Thus, {xn} is bounded.
The rest follows from the similar argument to that in the proof of Theorem4.4 and so is omitted.By Lemma4.3 and Theorems 4.4 and 4.6, we have the following result.Theorem 4.7.
LetÃ(·,y):P→X* be continuous for each y∈X, let A:X→X* be weakly continuous, let F:X→2X be a nonempty weakly compact-valued multifunction which is ℋ-hemicontinuous and monotone with respect to Ã(p,·) for each p∈P, and let f̃:P×X→R∪{+∞} be a continuous functional with respect to the product of the norm topology of P and weak topology of X such that the following conditions hold:(i)
f̃(p,·) is proper and convex for all p∈P;(ii)
domf is well-positioned and domf⊂⋂p∈Pdomf̃(p,·);(iii)
e(domf̃(pn,·),domf)→0 whenever pn→p*.Then, the following statements are equivalent:(i)
GMVI(A,F,f) is weakly well posed by perturbations;(ii)
IP(AF+∂f) is weakly well posed by perturbations;(iii)
GMVI(A,F,f) has a unique solution;(iv)
IP(AF+∂f) has a unique solution.Remark 4.8.
Theorem4.7 improves Theorems  4.1, 4.2, and 6.1 of [9], Theorems 4.1–4.3 of [10], and Theorems 4.1, 4.2, and 6.1 of [11].Now we give the following example as an application of Theorem4.7.Example 4.9.
LetX=R, P=[-1,1], and p*=0. Let Ã(p,x)=x+p, F(x)=x3 for all x∈X, p∈P, and
(4.32)f̃(p,x)={(p2+1)x2,x∈[-1,1],+∞,otherwise,
for all p∈P. Clearly, DomÃ(p,·)=R and Domf=Domf̃(p,·)=[-1,1] for all p∈P. It is easy to see that Ã and f̃ are continuous, f̃(p,·) is proper and convex, and F is ℋ-hemicontinuous and monotone with respect to Ã(p,·) for each p∈P. By (ii) of Remark 2.7, Domf is well-positioned. Hence, all the assumptions of Theorem 4.7 are satisfied. Let S be the solution set of GMVI(A,F,f). It follows that
(4.33)S={x∈[-1,1]:x3(x-y)+x2-y2≤0,∀y∈[-1,1]}={x∈[-1,1]:-(y+x32)2+x64+x4+x2≤0,∀y∈[-1,1]}={0}.
So x*=0 is the unique solution of GMVI(A,F,f). By Theorem 4.7, GMVI(A,F,f) is well-posed by perturbations.Next, we discuss the relationships between the generalized well-posedness by perturbations ofGMVI(A,F,f) and the generalized well-posedness by perturbations of IP(AF+∂f).Theorem 4.10.
LetÃ:P×X→X* be a uniformly continuous mapping, let F:X→2X be a nonempty weakly compact-valued multifunction which is ℋ-uniformly continuous, and let f̃:P×X→R∪{+∞} be a functional such that f̃(p,·) is proper, convex, and lower semicontinuous for each p∈P. Then, GMVI(A,F,f) is strongly (resp., weakly) generalized well-posed by perturbations whenever IP(AF+∂f) is strongly (resp., weakly) generalized well-posed by perturbations.Proof.
Let{pn}⊂P be with pn→p*, and let {xn} be an approximating sequence corresponding to {pn} for GMVI(A,F,f). Then, there exist {un}⊂X with un∈F(xn) (for all n≥1) and 0<ɛn→0 such that
(4.34)〈Ã(pn,un),xn-y〉+f̃(pn,xn)-f̃(pn,y)≤εn,∀y∈X,n≥1.
Define φn:X→R∪{+∞} as follows:
(4.35)φn(y)=f̃(pn,y)+〈Ã(pn,un),y-xn〉,∀y∈X,n≥1.
Clearly, φn is proper, convex, and lower semicontinuous and 0∈∂ɛnφn(xn) for all n≥1. By the Brondsted-Rockafellar theorem [32], there exist x¯n∈X and
(4.36)wn∈∂φn(x¯n)=Ã(pn,un)+∂f̃(pn,⋅)(x¯n)
such that
(4.37)‖xn-x¯n‖≤ɛn,‖wn‖≤ɛn.
Since F:X→2X is a nonempty weakly compact-valued multifunction, both F(xn) and F(x¯n) are nonempty weakly compact and hence are nonempty, weakly closed, and weakly bounded. Note that the weak closedness of sets in X implies the strong closedness and that the weak boundedness of sets in X is equivalent to the strong boundedness. Thus, it is known that both F(xn) and F(x¯n) lie in CB(X) for each n≥1. By Proposition 2.3, for each n≥1 and un∈F(xn), there exists u¯n∈F(x¯n) such that
(4.38)‖un-u¯n‖≤(1+1n)H(F(xn),F(x¯n)).
Since F is ℋ-uniformly continuous, we have from (4.37)
(4.39)‖un-u¯n‖≤(1+1n)H(F(xn),F(x¯n))⟶0asn⟶∞.
It follows from (4.36) that
(4.40)wn+Ã(pn,u¯n)-Ã(pn,un)∈Ã(pn,u¯n)+∂f̃(pn,⋅)(x¯n).
Since Ã is a uniformly continuous, from (4.37) we get
(4.41)‖wn+Ã(pn,u¯n)-Ã(pn,un)‖≤‖wn‖+‖Ã(pn,u¯n)-Ã(pn,un)‖⟶0.
So {x¯n} is an approximating sequence corresponding to {pn} for IP(AF+∂f).
By the strong (resp., weak) generalized well-posedness by perturbations ofIP(AF+∂f), there exists some subsequence {x¯nk} of {x¯n} such that x¯nk→x* (resp., x¯nk⇀x*), where x* is some solution of IP(AF+∂f). By Lemma 4.3, x* is also a solution of GMVI(A,F,f).
Case i.
IP(AF+∂f) is strongly generalized well-posed by perturbations. It follows from (4.37) that
(4.42)‖xnk-x*‖≤‖xnk-x¯nk‖+‖x¯nk-x*‖⟶0
and so GMVI(A,F,f) is strongly generalized well-posed by perturbations.Case ii.
IP(AF+∂f) is weakly generalized well-posed by perturbations. For any φ∈X*, from (4.37) we have
(4.43)|〈φ,xnk-x*〉|≤|〈φ,xnk-x¯nk〉|+|〈φ,x¯nk-x*〉|≤‖φ‖ɛnk+|〈φ,x¯nk-x*〉|⟶0.
Thus, GMVI(A,F,f) is weakly generalized well-posed by perturbations.Theorem 4.11.
LetÃ:P×X→X* and F:X→2X, and let f̃:P×X→R∪{+∞} be a functional such that f̃(p,·) is proper, convex, and lower semicontinuous for each p∈P. Then, IP(AF+∂f) is strongly (resp., weakly) generalized well-posed by perturbations whenever GMVI(A,F,f) is strongly (resp., weakly) generalized 1-well-posed by perturbations.Proof.
Let{pn}⊂P be with pn→p*, and let {xn} be an approximating sequence corresponding to {pn} for IP(AF+∂f). Then, there exists wn∈Ã(pn,F(xn))+∂f̃(pn,·)(xn) such that ∥wn∥→0. It follows that there exists a sequence {un}⊂X with un∈F(xn) (for all n≥1) such that
(4.44)wn∈Ã(pn,un)+∂f̃(pn,⋅)(xn),∀n≥1,
and hence
(4.45)〈Ã(pn,un),xn-y〉+f̃(pn,xn)-f̃(pn,y)≤〈wn,xn-y〉≤12‖xn-y‖2+12‖wn‖2,∀y∈X,n≥1.
This together with ∥wn∥→0 implies that {xn} is a 1-approximating sequence corresponding to {pn} for GMVI(A,F,f). Since GMVI(A,F,f) is strongly (resp., weakly) generalized 1-well-posed by perturbations, {xn} converges strongly (resp., weakly) to some solution x* of GMVI(A,F,f). By Lemma 4.3, x* is also a solution of IP(AF+∂f). So IP(AF+∂f) is strongly (resp., weakly) generalized well-posed by perturbations.Remark 4.12.
Whenpn=p* (for all n≥1) and F=I the identity mapping of X, Theorems 4.10 and 4.11 coincide with Theorems  4.3 and 4.4 of [9], respectively. Also, when F=I, Theorems 4.10 and 4.11 coincide with Theorems  4.4 and 4.5 of [10], respectively. Furthermore, it can be found that Theorems 4.10 and 4.11 also improve and extend Theorems  4.3 and 4.4 of [11], respectively. In the meantime, Theorems 4.10 and 4.11 partially generalize Theorem 2.1 of Lemaire et al. [20].


## 5. Links with the Well-Posedness by Perturbations of Fixed Point Problems

Lemaire et al. [20] also considered the concepts of well-posedness by perturbations for a (single-valued) fixed point problem. In this section, we consider the concepts of well-posedness by perturbations for a (set-valued) fixed point problem. Let T:X→2X be a set-valued mapping. The fixed point problem associated with T is defined by(5.1)FP(T):findx∈Xsuchthatx∈T(x).The perturbed problem ofFP(T) is given by(5.2)FPp(T):findx∈Xsuchthatx∈T̃(p,x),
where T̃:P×X→2X is such that T̃(p*,·)=T.Definition 5.1.
Let{pn}⊂P be with pn→p*. A sequence {xn}⊂X is called an approximating sequence corresponding to {pn} for FP(T) if there exists a sequence {yn}⊂X with yn∈T(xn) (for all n≥1) such that ∥xn-yn∥→0 as n→∞.Definition 5.2.
One says thatFP(T) is strongly (resp., weakly) well-posed by perturbations if FP(T) has a unique solution and, for any {pn}⊂P with pn→p*, every approximating sequence corresponding to {pn} for FP(T) converges strongly (resp., weakly) to the unique solution. FP(T) is said to be strongly (resp., weakly) generalized well-posed by perturbations if FP(T) has a nonempty solution set S and, for any {pn}⊂P with pn→p*, every approximating sequence corresponding to {pn} for FP(T) has a subsequence which converges strongly (resp., weakly) to some point of S.In particular, wheneverT is a single-valued mapping, we can readily see that Definitions  5.1 and 5.2 reduce to the corresponding definitions in [20]. It is known that in the setting of Hilbert spaces a generalized mixed variational inequality can be transformed into a fixed point problem (see [11, Proposition  2.1]). Utilizing this result, Ceng and Yao [11] proved that in the setting of Hilbert spaces the well-posedness of a generalized mixed variational inequality is equivalent to the well-posedness of the corresponding fixed point problem. In this section, we will further show that the well-posedness by perturbations of a generalized mixed variational inequality is closely related to the well-posedness by perturbations of the corresponding fixed point problem in the setting of Banach spaces. Let us first recall some concepts.LetU={x∈X:∥x∥=1} be the unit sphere. A Banach space X is said to be(a)
strictly convex if, for anyx,y∈U,(5.3)x≠y⟹‖x+y2‖<1,(b)
smooth if the limit(5.4)lim⁡t→0‖x+ty‖-‖x‖t
exists for all x,y∈U.The modulus of convexity ofX is defined by(5.5)δX(ε)=inf⁡{1-‖x+y2‖:x,y∈U,‖x+y‖≥ε},
and the modulus of smoothness of X is defined by(5.6)ρX(τ)=sup⁡{12(‖x+y‖+‖x-y‖)-1:x∈U,‖y‖≤τ}.
In the sequel we always suppose that q>1 and s>1 are fixed numbers. A Banach space X is said to be(c)
uniformly convex ifδX(ε)>0 for all ε∈(0,2),(d)
q-uniformly convex if there exists a constant c>0 such that δX(ε)≥cεq for all ε∈[0,2],(e)
uniformly smooth if(5.7)ρX(τ)τ⟶0asτ⟶0,(f)
q-uniformly smooth if there exists a constant k>0 such that(5.8)ρX(τ)≤kτq.It is well known that the LebesgueLq(q≥2) spaces are q-uniformly convex and 2-uniformly smooth and Lq(1<q<2) is 2-uniformly convex and q-uniformly smooth.The generalized duality mappingJq:X→2X* is defined by(5.9)Jq(x)={jq(x)∈X*:〈jq(x),x〉=‖x‖q,‖jq(x)‖=‖x‖q-1}.
In particular, J=J2 is called the normalized duality mapping. Jq has the following properties:(i)
Jq is bounded;(ii)
ifX is smooth, then Jq is single-valued;(iii)
ifX is strictly convex, then Jq is one-to-one and strictly monotone.For more details, we refer the readers to [29, 33] and the references therein.Lemma 5.3 (see [34]).
LetX be a q-uniformly smooth Banach space. Then, there exists a constant Lq>0 such that
(5.10)‖Jq(x)-Jq(y)‖≤Lq‖x-y‖q-1,∀x,y∈X.Lemma 5.4 (see [34]).
LetX be a q-uniformly convex Banach space. Then, there exists a constant kq>0 such that
(5.11)〈Jq(x)-Jq(y),x-y〉≤kq‖x-y‖q,∀x,y∈X.Lemma 5.5 (see [10, Lemma  5.3]).
LetX be a q-uniformly convex Banach space and M:X→2X* a maximal monotone operator. Then, for every λ>0 and s>1, (M+λJs)-1 is well-defined and single-valued.The following result indicates that, under suitable conditions, the mappingΠλf:X*→X is Lipschitz continuous, where Πλf=(Jq+λ∂f)-1.Lemma 5.6 (see [10, Lemma  5.4]).
LetX be a q-uniformly convex Banach space and M:X→2X* a maximal monotone operator. Then, for every λ>0,
(5.12)‖Πλf(w1)-Πλf(w2)‖≤(1λkq)1/(q-1)‖w1-w2‖1/(q-1),∀w1,w2∈X*.By means of Lemma5.5, we can transform GMVI(A,F,f) into a (set-valued) fixed point problem.Lemma 5.7.
LetX be a q-uniformly convex Banach space, and let A:X→X* and F:X→2X. Let f:X→R∪{+∞} be a proper, convex, and lower semicontinuous functional. Then, x∈X is a solution of GMVI(A,F,f) if and only if it is a solution of the following fixed point problem:
(5.13)FP(Πλf(Jq-λAF)):findx∈Xsuchthatx∈Πλf(Jq(x)-λAF(x)).Proof.
The conclusion follows directly from the definitions of∂f and Πλf and Lemma 5.5.Naturally, the perturbed problem ofFP(Πλf(Jq-λAF)) is given by(5.14)FPp(Πλf(Jq-λAF)):findx∈Xsuchthatx∈Πλf̃(p,⋅)(Jq(x)-λÃ(p,F(x))).Theorem 5.8.
LetX be an s-uniformly convex and q-uniformly smooth Banach space. Let Ã:P×X→X* be uniformly continuous, and let F:X→2X be a nonempty weakly compact-valued multifunction which is ℋ-uniformly continuous and monotone with respect to Ã(p,·) for each p∈P. Let f̃:P×X→R∪{+∞} be a continuous functional with respect to the product of the norm topology of P and weak topology of X such that the following conditions hold:(i)
f̃(p,·) is proper and convex for all p∈P;(ii)
domf is well-positioned and domf⊂⋂p∈Pdomf̃(p,·);(iii)
e(domf̃(pn,·),domf)→0 whenever pn→p*.
Then,FP(Πλf(Jq-λAF)) is weakly well-posed by perturbations whenever GMVI(A,F,f) has a unique solution.Proof.
By Lemma5.5, Πλf̃(pn,·)=(Jq+λ∂f̃(pn,·))-1 is well-defined and single-valued. Suppose that GMVI(A,F,f) has a unique solution x*. Then, by Lemma 5.7,  x* is also the unique solution of FP(Πλf(Jq-λAF)). Let {pn}⊂P be with pn→p*, and let {xn} be an approximating sequence corresponding to {pn} for FP(Πλf(Jq-λAF)). Then, there exists a sequence {yn}⊂X with yn∈Πλf̃(pn,·)(Jq(xn)-λÃ(pn,F(xn))) (for all n≥1) such that ∥xn-yn∥→0. Further, it is known that there exists a sequence {un}⊂X with un∈F(xn) (for all n≥1) such that
(5.15)yn=Πλf̃(pn,⋅)(Jq(xn)-λÃ(pn,un)),∀n≥1.
By the definition of Πλf̃(pn,·),
(5.16)Jq(xn)-Jq(yn)λ-Ã(pn,un)∈∂f̃(pn,⋅)(yn).
It follows that
(5.17)f̃(pn,y)-f̃(pn,yn)≥〈Jq(xn)-Jq(yn)λ-Ã(pn,un),y-yn〉,∀y∈X,n≥1.
From (5.17) we get yn∈domf̃(pn,·). We claim that {yn} is bounded. Indeed, if {yn} is unbounded, without loss of generality, we may assume that ∥yn∥→+∞. Let
(5.18)tn=1‖yn-x*‖,zn=x*+tn(yn-x*).
From conditions (i)-(ii), we have zn∈domf̃(pn,·). Note that
(5.19)‖zn‖≤‖x*‖+tn‖yn-x*‖=‖x*‖+1,∀n≥1.
So, {zn} is bounded. Since X is reflexive, it follows from the boundedness of {zn} that there exists some subsequence of {zn} which converges weakly to a point of X. Hence, without loss of generality, we may assume that tn∈(0,1) and zn⇀z. By Lemma 2.14 and conditions (ii)-(iii), we get z≠x*. For any y∈X, observe that, for all v∈F(y),
(5.20)〈Ã(pn,v),z-y〉=〈Ã(pn,v),z-zn〉+〈Ã(pn,v),zn-x*〉+〈Ã(pn,v),x*-y〉=〈Ã(pn,v),z-zn〉+tn〈Ã(pn,v),yn-y〉+(1-tn)〈Ã(pn,v),x*-y〉.
Since x* is the unique solution of GMVI(A,F,f), there exists some u*∈F(x*) such that
(5.21)〈Au*,x*-y〉+f(x*)-f(y)≤0,∀y∈X.
Also, since F:X→2X is a nonempty weakly compact-valued multifunction, both F(xn) and F(yn) are nonempty weakly compact and hence are nonempty, weakly closed, and weakly bounded. Note that the weak closedness of sets in X implies the strong closedness and that the weak boundedness of sets in X is equivalent to the strong boundedness. So, it is known that both F(xn) and F(yn) lie in CB(X). According to Proposition 2.3, for each n≥1 and un∈F(xn), there exists vn∈F(yn) such that
(5.22)‖un-vn‖≤(1+1n)H(F(xn),F(yn)).
Note that F is ℋ-uniformly continuous. Thus, it follows that
(5.23)‖un-vn‖≤(1+1n)H(F(xn),F(yn))⟶0asn⟶∞.
Furthermore, since F is monotone with respect to Ã(pn,·), one concludes that, for u*∈F(x*), v∈F(y), and vn∈F(yn),
(5.24)〈Ã(pn,v),x*-y〉≤〈Ã(pn,u*),x*-y〉,〈Ã(pn,v),yn-y〉≤〈Ã(pn,vn),yn-y〉.
It follows from (5.20) and (5.24) that
(5.25)〈Ã(pn,v),z-y〉≤〈Ã(pn,v),z-zn〉+tn〈Ã(pn,vn),yn-y〉+(1-tn)〈Ã(pn,u*),x*-y〉=〈Ã(pn,v),z-zn〉+tn〈Ã(pn,vn)-Ã(pn,un),yn-y〉+tn〈Ã(pn,un),yn-y〉+(1-tn)〈Ã(pn,u*),x*-y〉.
In addition, we have
(5.26)f̃(pn,zn)≤tnf̃(pn,yn)+(1-tn)f̃(pn,x*)
since f̃(pn,·) is convex. It follows from (5.17), (5.25), and (5.26) that
(5.27)〈Ã(pn,v),z-y〉≤〈Ã(pn,v),z-zn〉+tn〈Ã(pn,vn)-Ã(pn,un),yn-y〉+tnλ〈Jq(xn)-Jq(yn),yn-y〉+tnf̃(pn,y)-tnf̃(pn,yn)+(1-tn)〈Ã(pn,u*),x*-y〉=〈Ã(pn,v),z-zn〉+tn〈Ã(pn,vn)-Ã(pn,un),yn-y〉+tnλ〈Jq(xn)-Jq(yn),yn-y〉+(1-tn)〈Ã(pn,u*),x*-y〉+tnf̃(pn,y)+(1-tn)f̃(pn,x*)-{tnf̃(pn,yn)+(1-tn)f̃(pn,x*)}≤〈Ã(pn,v),z-zn〉+tn〈Ã(pn,vn)-Ã(pn,un),yn-y〉+tnλ〈Jq(xn)-Jq(yn),yn-y〉+(1-tn)〈Ã(pn,u*),x*-y〉+tnf̃(pn,y)+(1-tn)f̃(pn,x*)-f̃(pn,zn).
Note that ∥yn∥→+∞ and
(5.28)‖tn(yn-y)‖≤‖yn-x*‖+‖x*-y‖‖yn-x*‖=1+‖x*-y‖‖yn-x*‖.
It is easy to see that {tn(yn-y)} is bounded. Since ∥xn-yn∥→0, by Lemma 5.3,
(5.29)‖Jq(xn)-Jq(yn)‖≤Lq‖xn-yn‖q-1⟶0.
In the meantime, on account of ∥un-vn∥→0, we have
(5.30)‖Ã(pn,vn)-Ã(pn,un)‖⟶0
by means of the uniformly continuity of Ã. Consequently, letting n→∞ we obtain that
(5.31)|tn〈Ã(pn,vn)-Ã(pn,un),yn-y〉|≤‖A(pn,vn)-Ã(pn,un)‖‖tn(yn-y)‖⟶0,|tnλ〈Jq(xn)-Jq(yn),yn-y〉|≤‖Jq(xn)-Jq(yn)‖λ‖tn(yn-y)‖⟶0,(1-tn)〈Ã(pn,u*),x*-y〉⟶〈Ã(p*,u*),x*-y〉.
Moreover, also observe that
(5.32)|〈Ã(pn,v),z-zn〉|=|〈Ã(pn,v)-Ã(p*,v),z-zn〉+〈Ã(p*,v),z-zn〉|≤‖Ã(pn,v)-Ã(p*,v)‖‖zn-z‖+|〈Ã(p*,v),zn-z〉|⟶0asn⟶∞.
Further, since f̃:P×X→R∪{+∞} is a continuous functional with respect to the product of the norm topology of P and weak topology of X, we get that, as n→∞,
(5.33)tnf̃(pn,y)⟶0,(1-tn)f̃(pn,x*)⟶f̃(p*,x*),f̃(pn,zn)⟶f̃(p*,z).
Therefore, letting n→∞ in (5.27) we conclude that
(5.34)〈Av,z-y〉=〈Ã(p*,v),z-y〉≤〈Ã(p*,u*),x*-y〉+f̃(p*,x*)-f̃(p*,z)=〈Au*,x*-y〉+f(x*)-f(z),∀y∈X,v∈F(y).
By using (5.34) and the same argument as in the proof of Theorem 4.4, we can prove that z is a solution of GMVI(A,F,f), a contradiction. Thus, {yn} is bounded and so is {xn}.
By using (5.17) and the similar argument to that in the proof of Theorem 4.4, we can prove that {xn} converges weakly to x*. Since x* is the unique solution of FP(Πλf(Jq-λAF)), FP(Πλf(Jq-λAF)) is weakly well-posed by perturbations.Remark 5.9.
Theorem5.8 generalizes Theorem  5.1 of [9] and Theorem  5.1 of [11] since every Hilbert space is 2-uniformly convex and 2-uniformly smooth. Theorem 5.8 improves, extends, and develops [10, Theorem  5.1] in the following aspects.(i)
The mixed variational inequality problem (MVI) in [10, Theorem  5.1] is extended to develop the more general problem, that is, the generalized mixed variational inequality problem (GMVI) with a nonempty weakly compact-valued multifunction in the setting of Banach spaces. In the meantime, the (single-valued) fixed point problem corresponding to MVI in [10, Theorem  5.1] is extended to develop the more general problem, that is, the (set-valued) fixed point problem corresponding to GMVI. Furthermore, the concept of weak well-posedness by perturbations for the (single-valued) fixed point problem corresponding to MVI is extended to develop the concept of weak well-posedness by perturbations for the (set-valued) fixed point problem corresponding to GMVI.(ii)
Since the generalized mixed variational inequality problem (GMVI) is more general and more complicated than the mixed variational inequality problem (MVI), the assumptions in Theorem5.8 are very different from the ones in [10, Theorem  5.1]; for instance, in Theorem 5.8, let Ã:P×X→X* be uniformly continuous, and let F:X→2X be a nonempty weakly compact-valued multifunction which is ℋ-uniformly continuous and monotone with respect to Ã(p,·) for each p∈P, but, in [10, Theorem  5.1], let Ã:P×X→X* be a uniformly continuous mapping such that Ã(p,·):X→X* is monotone for all p∈P.(iii)
The technique of proving weak well-posedness by perturbations for (set-valued) fixed point problemFP(Πλf(Jq-λAF)) in Theorem 5.8 is very different from the one for (single-valued) fixed point problem FP(Πλf(Jq-λA)) in [10, Theorem  5.1] because our technique depends on the well-known Nadler’s Theorem [27], the ℋ-uniformly continuity of nonempty weakly compact-valued multifunction F and the monotonicity of F with respect to Ã(p,·) for each p∈P.Now we give the following example as an application of Theorem5.8.Example 5.10.
LetX=R, P=[-1,1], and p*=0. Let Ã(p,x)=x+p, F(x)=x-sinx for all x∈X, p∈P, and let
(5.35)f̃(p,x)={0,x∈K(p),+∞,otherwise,
for all p∈P, where K(p)=[-1,1]+{p}. Then, DomÃ(p,·)=R, Domf=[-1,1], Domf̃(p,·)=K(p) for all p∈P. Clearly, if pn→p*, then e(Domf̃(pn,·),Domf)→0. By the above definitions, it is easy to see that Ã is uniformly continuous, F is ℋ-uniformly continuous and monotone with respect to Ã(p,·) for each p∈P, and f̃(p,·) is proper and convex. By (ii) of Remark 2.7, Domf is well-positioned. It is known that R is 2-uniformly convex and 2-uniformly smooth. Hence, all the assumptions of Theorem 5.8 are satisfied. Let S be the solution set of GMVI(A,F,f). It follows that
(5.36)S={x∈[-1,1]:(x-sin⁡x)(x-y)≤0,∀y∈[-1,1]}={0}.
So x*=0 is the unique solution of GMVI(A,F,f). By Theorem 5.8, FP(Πλf(Jq-λAF)) is well-posed by perturbations.Based on Theorems4.7 and 5.8 and Lemma 5.7, we have the following result.Theorem 5.11.
LetX be an s-uniformly convex and q-uniformly smooth Banach space. Let Ã:P×X→X* be uniformly continuous, let A:X→X* be weakly continuous, and let F:X→2X be a nonempty weakly compact-valued multifunction which is ℋ-uniformly continuous and monotone with respect to Ã(p,·) for each p∈P. Let f̃:P×X→R∪{+∞} be a continuous functional with respect to the product of the norm topology of P and weak topology of X such that the following conditions hold:(i)
f̃(p,·) is proper and convex for all p∈P;(ii)
domf is well-positioned and domf⊂⋂p∈Pdomf̃(p,·);(iii)
e(domf̃(pn,·),domf)→0 whenever pn→p*. Then, the following statements are equivalent:(i)
GMVI(A,F,f) is weakly well-posed by perturbations;(ii)
IP(AF+∂f) is weakly well-posed by perturbations;(iii)
FP(Πλf(Jq-λAF)) is weakly well-posed by perturbations;(iv)
GMVI(A,F,f) has a unique solution;(v)
IP(AF+∂f) has a unique solution;(vi)
FP(Πλf(Jq-λAF)) has a unique solution.Next we consider the case of generalized well-posedness by perturbations.Theorem 5.12.
LetX be an s-uniformly convex and q-uniformly smooth Banach space. Let Ã:P×X→X* be uniformly continuous, and let F:X→2X be a nonempty weakly compact-valued multifunction which is ℋ-uniformly continuous and monotone with respect to Ã(p,·) for each p∈P. Let f̃:P×X→R∪{+∞} be a continuous functional with respect to the product of the norm topology of P and weak topology of X such that f̃(p,·) is proper and convex. Then, FP(Πλf(Jq-λAF)) is strongly (resp., weakly) generalized well-posed by perturbations whenever GMVI(A,F,f) is strongly (resp., weakly) generalized (1+1/λ)-well-posed by perturbations.Proof.
Suppose thatGMVI(A,F,f) is strongly (resp., weakly) generalized (1+1/λ)-well-posed by perturbations. Let {pn}⊂P be with pn→p*, and let {xn} be an approximating sequence corresponding to {pn} for FP(Πλf(Jq-λAF)). Then, there exists a sequence {yn}⊂X with yn∈Πλf̃(pn,·)(Jq(xn)-λÃ(pn,F(xn))) (for all n≥1) such that ∥xn-yn∥→0. Further, it is known that there exists a sequence {un}⊂X with un∈F(xn) (for all n≥1) such that
(5.37)yn=Πλf̃(pn,⋅)(Jq(xn)-λÃ(pn,un)),∀n≥1.
By the definition of Πλf̃(pn,·),
(5.38)Jq(xn)-Jq(yn)λ-Ã(pn,un)∈∂f̃(pn,⋅)(yn).
It follows that
(5.39)f̃(pn,y)-f̃(pn,yn)≥〈Jq(xn)-Jq(yn)λ-Ã(pn,un),y-yn〉,∀y∈X,n≥1.
Furthermore, since F:X→2X is a nonempty weakly compact-valued multifunction, both F(xn) and F(yn) are nonempty weakly compact and hence are nonempty, weakly closed, and weakly bounded. Note that the weak closedness of sets in X implies the strong closedness and that the weak boundedness of sets in X is equivalent to the strong boundedness. So, it is known that both F(xn) and F(yn) lie in CB(X). According to Proposition 2.3, for each n≥1 and un∈F(xn) there exists vn∈F(yn) such that
(5.40)‖un-vn‖≤(1+1n)H(F(xn),F(yn)).
Note that F is ℋ-uniformly continuous. Thus, one deduces that
(5.41)‖un-vn‖≤(1+1n)H(F(xn),F(yn))⟶0asn⟶∞.
Now utilizing (5.39) we have
(5.42)〈Ã(pn,vn),yn-y〉+f̃(pn,yn)-f̃(pn,y)≤〈Jq(xn)-Jq(yn)λ,yn-y〉+〈Ã(pn,vn)-Ã(pn,un),yn-y〉≤12(1+1λ)‖yn-y‖2+12εn,∀y∈X,n≥1,
where
(5.43)εn=1λ‖Jq(xn)-Jq(yn)‖2+‖Ã(pn,vn)-Ã(pn,un)‖2≤Lqλ‖xn-yn‖2(q-1)+‖Ã(pn,vn)-Ã(pn,un)‖2⟶0.
Therefore, {yn} is a (1+1/λ)-approximating sequence corresponding to {pn} for GMVI(A,F,f). By the strong (resp., weak) generalized (1+1/λ)-well-posedness by perturbations, {yn} has some subsequence which converges strongly (resp., weakly) to a solution x* of GMVI(A,F,f). By Lemma 5.7, x* is also a solution of FP(Πλf(Jq-λAF)). Consequently, FP(Πλf(Jq-λAF)) is strongly (resp., weakly) generalized well-posed by perturbations.Remark 5.13.
Theorem5.12 generalizes Theorem  5.3 of [9] and Theorem  5.3 of [11]. In addition, whenever F=I the identity mapping of X, Theorem 5.12 reduces to Theorem  5.3 of [10].Theorem 5.14.
LetX be a q-uniformly convex Banach space. Let Ã:P×X→X* be uniformly continuous, and let F:X→2X be a nonempty weakly compact-valued multifunction which is ℋ-uniformly continuous. Let f̃:P×X→R∪{+∞} be a functional such that f̃(p,·) is proper, convex, and lower semicontinuous for each p∈P. Then, GMVI(A,F,f) is strongly (resp., weakly) generalized well-posed by perturbations whenever FP(Πλf(Jq-λAF)) is strongly (resp., weakly) generalized well-posed by perturbations.Proof.
Let{pn}⊂P be with pn→p*, and let {xn} be an approximating sequence corresponding to {pn} for GMVI(A,F,f). Then, there exist {un}⊂X with un∈F(xn) (for all n≥1) and 0<εn→0 such that
(5.44)〈Ã(pn,un),xn-y〉+f̃(pn,xn)-f̃(pn,y)≤εn,∀y∈X,n≥1.
Define φn:X→R∪{+∞} as follows:
(5.45)φn(y)=f̃(pn,y)+〈Ã(pn,un),y-xn〉,∀y∈X,n≥1.
Clearly, φn is proper, convex, and lower semicontinuous and 0∈∂εnφn(xn) for all n≥1. By the Brondsted-Rockafellar theorem [32], there exist x¯n∈X and
(5.46)wn∈∂φn(x¯n)=Ã(pn,un)+∂f̃(pn,⋅)(x¯n)
such that
(5.47)‖xn-x¯n‖≤εn,‖wn‖≤εn.
Since F is a nonempty weakly compact-valued multifunction, both F(xn) and F(x¯n) lie in CB(X) for each n≥1. By Proposition 2.3, for each n≥1 and un∈F(xn) there exists u¯n∈F(x¯n) such that
(5.48)‖un-u¯n‖≤(1+1n)H(F(xn),F(x¯n)).
Since F is ℋ-uniformly continuous, we obtain from (5.47) that
(5.49)‖un-u¯n‖≤(1+1n)H(F(xn),F(x¯n))⟶0asn⟶∞.
Utilizing (5.46) we have
(5.50)x¯n=Πλf̃(pn,⋅)[λwn+Jq(x¯n)-λÃ(pn,un)].
Since Ã is uniformly continuous, it follows from (5.47) and (5.50) that
(5.51)‖[λwn+Jq(x¯n)-λÃ(pn,un)]-[Jq(x¯n)-λÃ(pn,u¯n)]‖≤λ‖wn‖+λ‖Ã(pn,u¯n)-Ã(pn,un)‖⟶0.
By Lemma 5.6 and (5.50) and (5.51),
(5.52)‖x¯n-Πλf̃(pn,⋅)[Jq(x¯n)-λÃ(pn,u¯n)]‖⟶0.
Thus, {x¯n} is an approximating sequence corresponding to {pn} for FP(Πλf(Jq-λAF)). Repeating the same argument as in the proof of Theorem 4.10, we can deduce that {x¯n} has some subsequence {x¯nk} which converges strongly (resp., weakly) to some solution x* of FP(Πλf(Jq-λAF)). By Lemma 5.7, x* is also a solution of GMVI(A,F,f). Thus, GMVI(A,F,f) is strongly (resp., weakly) generalized well-posed by perturbations.Remark 5.15.
Theorem5.14 generalizes Theorem  5.4 of [9] and Theorem  5.4 of [11]. In addition, whenever F=I the identity mapping of X, Theorem 5.14 reduces to Theorem  5.4 of [10].



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*Source: 194509-2011-12-27.xml*