# Well-Posedness by Perturbations for Variational-Hemivariational Inequalities

Authors: Shu Lv; Yi-bin Xiao; Zhi-bin Liu; Xue-song Li
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

We generalize the concept of well-posedness by perturbations for optimization problem to a class of variational-hemivariational inequalities. We establish some metric characterizations of the well-posedness by perturbations for the variational-hemivariational inequality and prove their equivalence between the well-posedness by perturbations for the variational-hemivariational inequality and the well-posedness by perturbations for the corresponding inclusion problem.


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## Body

## 1.   Introduction and Preliminaries

The concept well-posedness is important in both theory and methodology for optimization problems. An initial, already classical concept of well-posedness for unconstrained optimization problem is due to Tykhonov in [1]. Let f:V→ℝ∪{+∞} be a real-valued functional on Banach space V. The problem of minimizing f on V is said to be well-posed if there exists a unique minimizer, and every minimizing sequence converges to the unique minimizer. Soon after, Levitin and Polyak [2] generalized the Tykhonov well-posedness to the constrained optimization problem, which has been known as the Levitin-Polyak well-posedness. It is clear that the concept of well-posedness is motivated by the numerical methods producing optimizing sequences for optimization problems. Unfortunately, these concepts generally cannot establish appropriate continuous dependence of the solution on the data. In turn, they are not suitable for the numerical methods when the objective functional f is approximated by a family or a sequence of functionals. For this reason, another important concept of well-posedness for optimization problem, which is called the well-posedness by perturbations or extended well-posedness, has been introduced and studied by [3–6]. Also, many other notions of well-posedness have been introduced and studied for optimization problem. For details, we refer to [7] and the reference therein.The concept well-posedness also has been generalized to other related problems, especially to the variational inequality problem. Lucchetti and Patrone [8] first introduced the well-posedness for a variational inequality, which can be regarded as an extension of the Tykhonov well-posedness of optimization problem. Since then, many authors were devoted to generalizing the concept of well-posedness for the optimization problem to various variational inequalities. In [9], Huang et al. introduced several types of (generalized) Levitin-Polyak well-posednesses for a variational inequality problem with abstract and functional constraint and gave some criteria, characterizations, and their relations for these types of well-posednesses. Recently, Fang et al. [10] generalized the concept of well-posedness by perturbations, introduced by Zelezzi for a minimization problem, to a generalized mixed variational inequality problem in Banach space. They established some metric characterizations of well-posedness by perturbations and discussed its links with well-posedness by perturbations of corresponding inclusion problem and the well-posedness by perturbations of corresponding fixed point problem. Also they 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. For further more results on the well-posedness of variational inequalities, we refer to [8–15] and the references therein.When the corresponding energy functions are not convex, the mathematical model describing many important phenomena arising in mechanics and engineering is no longer variational inequality but a new type of inequality problem that is called hemivariational inequality, which was first introduced by Panagiotopoulos [16] as a generalization of variational inequality. A more generalized variational formulation which is called variational-hemivariational inequality is presented to model the problems subject to constraints because the setting of hemivariational inequalities cannot incorporate the indicator function of a convex closed subset. Due to the fact that the potential is neither convex nor smooth generally, the hemivariational inequalities have been proved very efficient to describe a variety of mechanical problems using the generalized gradient of Clarke for nonconvex and nondifferentiable functions [17], such as unilateral contact problems in nonlinear elasticity, obstacles problems, and adhesive grasping in robotics (see, e.g., [18–20]). So, in recent years all kinds of hemivariational inequalities have been studied [21–30] and the study of hemivariational inequalities has emerged as a new and interesting branch of applied mathematics. However, there are very few researchers extending the well-posedness to hemivariational inequality. In 1995, Goeleven and Mentagui [23] first defined the well-posedness for hemivariational inequalities. Recently, Xiao et al. [31] generalized the concept of well-posedness to hemivariational inequalities. They established some metric characterizations of the well-posed hemivariational inequality, derived some conditions under which the hemivariational inequality is strongly well-posed in the generalized sense, and proved the equivalence between the well-posedness of hemivariational inequality and the well-posedness of a corresponding inclusion problem. Moreover, Xiao and Huang [32] studied the well-posedness of variational-hemivariational inequalities and generalized some related results.In the present paper, we generalize the well-posedness by perturbations for optimization problem to a class of variational-hemivariational inequality. We establish some metric characterizations of the well-posedness by perturbations for variational-hemivariational inequality and prove the equivalence between the well-posedness by perturbations for the variational-hemivariational inequality and the well-posedness by perturbations for the corresponding inclusion problem.We suppose in what follows thatV is a real reflexive Banach space with its dual V*, and 〈·,·〉 is the duality between V and V*. We denote the norms of Banach space V and V* by ∥·∥V and ∥·∥V*, respectively. Let A:V→V* be a mapping, let J:V→ℝ be a locally Lipschitz functional, let G:V→ℝ∪{+∞} be a proper, convex, and lower semicontinuous functional, and let f∈V* be some given element. Denote by domG the domain of functional G, that is,
(1.1)dom⁡G={u∈V:G(u)<+∞}.
The functional G is called proper if its domain is nonempty. The variational-hemivariational inequality associated with (A,f,J,G) is specified as follows: (1.2)VHVI(A,f,J,G):findu∈dom⁡Gsuch  that〈A(u),v-u〉+J∘(u,v-u)+G(v)-G(u)≥〈f,v-u〉,∀v∈V,
where J∘(u,v) denotes the generalized directional derivative in the sense of Clarke of a locally Lipschitz functional J at u in the direction v (see [17]) given by
(1.3)J∘(u,v)=lim⁡sup⁡w→uλ↓0J(w+λv)-J(w)λ.
The variational-hemivariational inequality which includes many problems as special cases has been studied intensively. Some special cases of VHVI(A,f,J,G) are as follows:  (i)
ifG=0, then VHVI(A,f,J,G) reduces to hemivariational inequality:
(1.4)HVI(A,f,J):findu∈Vsuch  that〈A(u),v-u〉+J∘(u,v-u)≥〈f,v-u〉,∀v∈V,(ii)
ifJ=0, then VHVI(A,f,J,G) is equivalent to the following mixed variational inequality:
(1.5)MVI(A,f,G):findu∈dom⁡Gsuch  that〈A(u),v-u〉+G(v)-G(u)≥〈f,v-u〉,∀v∈V,(iii)
ifA=0,  J=0 and f=0, then VHVI(A,f,J,G) reduces to the global minimization problem:
(1.6)MP(G):min⁡u∈VG(u).Let∂G(u):V→2V*∖{∅} and ∂¯J(u):V→2V*∖{∅} denote the subgradient of convex functional G in the sense of convex analysis (see [33]) and the Clarke's generalized gradient of locally Lipschitz functional J (see [17]), respectively, that is,
(1.7)∂G(u)={u*∈V*:G(v)-G(u)≥〈u*,v-u〉,∀v∈V},∂¯J(u)={ω∈V*:J∘(u,v)≥〈ω,v〉,∀v∈V}.
About the subgradient in the sense of convex analysis, the Clarke's generalized directional derivative and the Clarke's generalized gradient, we have the following basic properties (see, e.g., [17, 19, 33, 34]).Proposition 1.1.
LetV be a Banach space and G:V→ℝ∪{+∞} a convex and proper functional. Then one has the following properties of ∂G:  (i)
∂G(u) is convex and weak*-closed;(ii)
ifG is continuous at u∈domG, then ∂G(u) is nonempty, convex, bounded, and weak*-compact;(iii)
ifG is Gâteaux differentiable at u∈domG, then ∂G(u)={DG(u)}, where DG(u) is the Gâteaux derivative of G at u.Proposition 1.2.
LetV be a Banach space, and let G1, G2:V→ℝ∪{+∞} be two convex functionals. If there is a point u0∈domG1∩domG2 at which G1 is continuous, then the following equation holds:
(1.8)∂(G1+G2)(u)=∂G1(u)+∂G2(u),∀u∈V.Proposition 1.3.
LetV be a Banach space, u, v∈V, and let J be a locally Lipschitz functional defined on V. Then  (1)
the functionv↦J∘(u,v) is finite, positively homogeneous, subadditive, and then convex on V,(2)
J∘(u,v) is upper semicontinuous as a function of (u,v), as a function of v alone, is Lipschitz continuous on V,(3)
J∘(u,-v)=(-J)∘(u,v),(4)
∂¯J(u) is a nonempty, convex, bounded, and weak*-compact subset of V*,(5)
for everyv∈V, one has
(1.9)J∘(u,v)=max⁡{〈ξ,v〉:ξ∈∂¯J(u)}.Suppose thatL is a parametric normed space with norm ∥·∥L, P⊂L is a closed ball with positive radius, and p*∈P is a given point. We denote the perturbed mappings of A, J, G as A^:P×V→V* and J^, G^:P×V→ℝ, respectively, which have the property that for any p∈P, J^(p,·) is a locally Lipschitz functional in V, G^(p,·) is proper, convex, and lower semicontinuous in V, and
(1.10)A^(p*,⋅)=A(⋅),J^(p*,⋅)=J(⋅),G^(p*,⋅)=G(⋅).
Then the perturbed Clarke's generalized directional derivative J^2∘(p,·):V×V→ℝ and the perturbed Clarke's generalized gradient ∂¯2J^(p,·):V→2V* corresponding to the perturbed locally Lipschitz functional J^ are, respectively, specified as
(1.11)J^2∘(p,⋅)(u,v)=lim⁡sup⁡w→uλ↓0J^(p,w+λv)-J^(p,w)λ,∂¯2J^(p,u)={ω∈V*:J^2∘(p,⋅)(u,v)≥〈ω,v〉,∀v∈V}.
The perturbed subgradient ∂2G^(p,·):domG→2V* corresponding to the perturbed convex functional G^ is
(1.12)∂2G^(p,u)={u*∈V*:G^(p,v)-G^(p,u)≥〈u*,v-u〉,∀v∈V}.
Based on the above-perturbed mappings, the perturbed problem of VHVI(A,f,J,G) is given by
(1.13)VHVIp(A,f,J,G):findu∈dom⁡G^(p,⋅)such  that〈A^(p,u)-f,v-u〉+J^2∘(p,⋅)(u,v-u)+G^(p,v)-G^(p,u)≥0,∀v∈V.In the sequel, we recall some important definitions and useful results.Definition 1.4 (see [35]).
LetS be a nonempty subsets of V. The measure of noncompactness μ of the set S is defined by
(1.14)μ(S)=inf⁡{ϵ>0:S⊂⋃i=1nSi,diam⁡(Si)<ϵ,i=1,2,…,n},
where diam(Si) means the diameter of set Si.Definition 1.5 (see [35]).
LetA and B be two given subsets of V. The excess of A over B is defined by
(1.15)e(A,B)=sup⁡a∈Ad(a,B),
where d(·,B) is the distance function generated by B, that is,
(1.16)d(x,B)=inf⁡b∈B‖a-b‖V,x∈V.
The Hausdorff metric ℋ(·,·) between A and B is defined by
(1.17)H(A,B)=max⁡{e(A,B),e(B,A)}.
Let {An} be a sequence of nonempty subset of V. One says that An converges to A in the sense of Hausdorff metric if ℋ(An,A)→0. It is easy to see that e(An,A)→0 if and only if d(an,A)→0 for all selection an∈An. For more details on this topic, the reader should refer to [35]. The following theorem is crucial to our main results.Theorem 1.6 (see [36]).
LetC⊂V be nonempty, closed, and convex, let C*⊂V* be nonempty, closed, convex, and bounded, let φ:V→ℝ be proper, convex, and lower semicontinuous, and let y∈C be arbitrary. Assume that, for each x∈C, there exists x*(x)∈C* such that
(1.18)〈x*(x),x-y〉≥φ(y)-φ(x).
Then, there exists y*∈C* such that
(1.19)〈y*,x-y〉≥φ(y)-φ(x),∀x∈C.


## 2. Well-Posedness by Perturbations ofVHVI(A,f,J,G) with Metric Characterizations

In this section, we generalize the concept of well-posedness by perturbations to the variational-hemivariational inequalityVHVI(A,f,J,G) and establish its metric characterizations.Definition 2.1.
Let{pn}⊂P  with  pn→p*. A sequence {un}⊂V is said to be an approximating sequence corresponding to {pn} for VHVI(A,f,J,G) if there exists a nonnegative sequence {ϵn} with ϵn→0 as n→∞ such that un∈domG^(pn,·) and
(2.1)〈A^(pn,un)-f,v-un〉+J^2∘(pn,⋅)(un,v-un)+G^(pn,v)-G^(pn,un)≥-ϵn‖v-un‖V,∀v∈V.Definition 2.2.
VHVI
(
A
,
f
,
J
,
G
) is said to be strongly (resp., weakly) well-posed by perturbations if VHVI(A,f,J,G) has a unique solution in V, and for any {pn}⊂P with pn→p*, every approximating sequence corresponding to {pn} converges strongly (resp., weakly) to the unique solution.Remark 2.3.
Strong well-posedness by perturbations implies weak well-posedness by perturbations, but the converse is not true in general.Definition 2.4.
VHVI
(
A
,
f
,
J
,
G
) is said to be strongly (resp., weakly) well-posed by perturbations in the generalized sense if VHVI(A,f,J,G) has a nonempty solution set S in V, and for any {pn}⊂P  with  pn→p*, every approximating sequence corresponding to {pn} has some subsequence which converges strongly (resp., weakly) to some point of solution set S.Remark 2.5.
Strong well-posedness by perturbations in the generalized sense implies weak well-posedness by perturbations in the generalized sense, but the converse is not true in general.
To derive the metric characterizations of well-posedness by perturbations forVHVI(A,f,J,G), we define the following approximating solution set of VHVI(A,f,J,G): For any ϵ>0,
(2.2)Ω(ϵ)=⋃p∈B(p*,ϵ){u∈dom⁡G^(p,⋅):〈A^(p,u)-f,v-u〉+J^2∘(p,⋅)(u,v-u)∪p∈B(p*,ϵ)+G^(p,v)-G^(p,u)≥-ϵ‖v-u‖V,∀v∈V},
where B(p*,ϵ) denotes the closed ball centered at p* with radius ϵ. For any ϵ>0,u∈Ω(ϵ) and any set K⊂Ω(ϵ), we define the following two functions which are specified as follows:
(2.3)p(ϵ,u)=sup⁡{‖v-u‖V:v∈Ω(ϵ)},q(ϵ,K)=e(Ω(ϵ),K).
It is easy to see that p(ϵ,u) is the smallest radius of the closed ball centered at u containing Ω(ϵ), and q(ϵ,K) is the excess of approximating solution set Ω(ϵ) over K.
Based on the two functionsp(ϵ,u) and q(ϵ,K), we now give some metric characterizations of well-posedness by perturbations for the VHVI(A,f,J,G).Theorem 2.6.
VHVI
(
A
,
f
,
J
,
G
) is strongly well-posed by perturbations if and only if there exists a solution u* for VHVI(A,f,J,G) and p(ϵ,u*)→0 as ϵ→0.Proof.
“Necessity”: suppose thatVHVI(A,f,J,G) is strongly well-posed by perturbations. Then Ω(ϵ)≠∅ for all ϵ>0 since there is a unique solution u* belonging to Ω(ϵ) by the strong well-posedness by perturbations for VHVI(A,f,J,G). We now need to prove p(ϵ,u*)→0 as ϵ→0. Assume by contradiction that p(ϵ,u*) does not converge to 0 as ϵ→0, then there exist a constant l>0 and a nonnegative sequence {ϵn} with ϵn→0 such that
(2.4)p(ϵn,u*)>l>0,∀n∈N.
By the definition of function p(ϵ,u), there exists un∈Ω(ϵn) such that
(2.5)‖un-u*‖V>l,∀n∈N.
Since un∈Ω(ϵn), there exists some pn∈B(p*,ϵn) such that
(2.6)〈A^(pn,un)-f,v-un〉+J^2∘(pn,⋅)(un,v-un)+G^(pn,v)-G^(pn,un)≥-ϵn‖v-un‖V,∀v∈V,n∈N.
It is obvious that pn→p* as n→∞ and so {un} is an approximating sequence corresponding to {pn} for VHVI(A,f,J,G). Therefore, by the strong well-posedness by perturbations for VHVI(A,f,J,G), we can get un→u* which is a contradiction to (2.4).
“Sufficiency”: suppose thatVHVI(A,f,J,G) has a solution u* and p(ϵ,u*)→0 as ϵ→0. First, we claim that u* is a unique solution for VHVI(A,f,J,G). In fact, if VHVI(A,f,J,G) has another solution u^ with u*≠u^, it follows from the definition of Ω(ϵ) that u* and u^ belong to Ω(ϵ) for all ϵ>0, which together with the definition of p(ϵ,u) implies that
(2.7)p(ϵ,u*)≥‖u^-u*‖V>0,∀ϵ>0,
which is a contradiction to the assumption p(ϵ,u*)→0 as ϵ→0. Now, let {pn}⊂P with pn→p* and {un} be an approximating sequence corresponding to {pn} for VHVI(A,f,J,G). Then there exists a nonnegative sequence {ϵn} with ϵn→0 such that
(2.8)〈A^(pn,un)-f,v-un〉+J^2∘(pn,⋅)(un,v-un)+G^(pn,v)-G^(pn,un)≥-ϵn‖v-un‖V,∀v∈V,n∈N.
Taking δn=∥pn-p*∥L and ϵn′=max{δn,ϵn}, it easy to see that ϵn′→0 as n→∞ and un∈Ω(ϵn′). Since u* is the unique solution for VHVI(A,f,J,G), u* also belongs to Ω(ϵn′). And so, it follows from the definition of p(ϵ,u) that
(2.9)‖un-u*‖V≤p(ϵn′,u*)⟶0,asn⟶∞,
which implies that VHVI(A,f,J,G) is strongly well-posed by perturbations. This completes the proof of Theorem 2.6.Theorem 2.7.
VHVI
(
A
,
f
,
J
,
G
) is strongly well-posed by perturbations in the generalized sense if and only if the solution set S of VHVI(A,f,J,G) is nonempty and compact, and q(ϵ,S)→0 as ϵ→0.Proof.
“Necessity”: suppose thatVHVI(A,f,J,G) is strongly well-posed by perturbations in the generalized sense. Then VHVI(A,f,J,G) has nonempty solution set S by the definition of strong well-posedness by perturbations in the generalized sense of VHVI(A,f,J,G). Let {un} be any sequence in S. It is obvious that {un} is an approximating sequence corresponding to constant sequence {p*} for VHVI(A,f,J,G). Again by the strong well-posedness by perturbations in the generalized sense of VHVI(A,f,J,G), {un} has a subsequence which converges strongly to some point of S, which implies that the solution set S of VHVI(A,f,J,G) is compact. Now we show that q(ϵ,S)→0 as ϵ→0. Assume by contradiction that q(ϵ,S)↛0 as ϵ→0, then there exist a constant l>0 and a nonnegative sequence {ϵn} with ϵn→0 and xn∈Ω(ϵn) such that
(2.10)xn∉S+B(0,l),∀n∈N.
Since xn∈Ω(ϵn), there exists pn∈B(p*,ϵn) such that
(2.11)〈A^(pn,xn)-f,v-xn〉+J^2∘(pn,⋅)(xn,v-xn)+G^(pn,v)-G^(pn,xn)≥-ϵn‖v-xn‖V,∀v∈V.
Clearly, pn→p* as n→∞. This together with the above inequality implies that {xn} is an approximating consequence corresponding to {pn} for VHVI(A,f,J,G). It follows from the strongly well-posedness by perturbations in the generalized sense for VHVI(A,f,J,G) that there is a subsequence {unk} of {un} which converges to some point of S. This is contradiction to (2.10) and so q(ϵ,S)→0 as ϵ→0.
“Sufficiency”: we suppose that the solution setS of VHVI(A,f,J,G) is nonempty compact and q(ϵ,S)→0 as ϵ→0. Let {pn}⊂P be any sequence with pn→p* and {un} an approximating sequence corresponding to pn for VHVI(A,f,J,G), which implies that
(2.12)〈A^(pn,un)-f,v-un〉+J^2∘(pn,⋅)(un,v-un)+G^(pn,v)-G^(pn,un)≥-ϵn‖v-un‖V,∀v∈V.
Taking ϵn′=max{ϵn,∥pn-p*∥L}, it is easy to see that ϵn′→0 and un∈Ω(ϵn′). It follows that
(2.13)d(un,S)≤e(Ω(ϵn′,S))=q(ϵn′,S)⟶0.
Since the solution set S of VHVI(A,f,J,G) is compact, there exists u¯n∈S such that
(2.14)‖un-u¯n‖V=d(un,S)⟶0.
Again from the compactness of solution set S, u¯n has a subsequence {u¯nk} converging strongly to some point u¯∈S. It follows from (2.14) that
(2.15)‖unk-u¯‖V≤‖unk-u¯nk‖V+‖u¯nk-u¯‖V⟶0,
which implies that {unk} converges strongly to u¯. Thus, VHVI(A,f,J,G) is strongly well-posed by perturbations in the generalized sense. This completes the proof of Theorem 2.7.The strong well-posedness by perturbations in the generalized sense forVHVI(A,f,J,G) can also be characterized by the behavior of noncompactness measure of its approximating solution set.Theorem 2.8.
LetL be a finite-dimensional space. Suppose that  (i)
A^(·,·):P×V→V*, the perturbed mapping of A, is continuous with respect to (p,v),(ii)
G^:P×V→ℝ∪{+∞}, the perturbed functional of G, is lower semicontinuous with respect to (p,v) and continuous with respect to p for any given v∈V,(iii)
J^:P×V→ℝ, the perturbed functional of J, is locally Lipschitz with respect to v for any p∈P, and its Clarke's generalized directional derivative J^2∘(p,·):P→ℒ(V×V,ℝ) is continuous with respect to p.
Then,VHVI(A,f,J,G) is strongly well-posed by perturbations in the generalized sense if and only if
(2.16)Ω(ϵ)≠∅,∀ϵ>0,μ(Ω(ϵ))⟶0asϵ⟶0.Proof.
From the metric characterization of strongly well-posedness by perturbations in the generalized sense forVHVI(A,f,J,G) in Theorem 2.7, we can easily prove the necessity. In fact, since VHVI(A,f,J,G) is strongly well-posed by perturbations in the generalized sense, it follows from Theorem 2.7 that the solution set S of VHVI(A,f,J,G) is nonempty compact and q(ϵ,S)→0 as ϵ→0. Then, we can easily get from the compactness of S and the fact S⊂Ω(ϵ) for all ϵ>0 that Ω(ϵ)≠∅ for all ϵ>0 and
(2.17)μ(Ω(ϵ))≤2H(Ω(ϵ),S)+μ(S)=2e(Ω(ϵ),S)=2q(ϵ,S)⟶0.
Now we prove the sufficiency. First, we claim that Ω(ϵ) is closed for all ϵ>0. In fact, let {un}⊂Ω(ϵ) and un→u. Then there exists pn∈B(p*,ϵ) such that
(2.18)〈A^(pn,un)-f,v-un〉+J^2∘(pn,⋅)(un,v-un)+G^(pn,v)-G^(pn,un)≥-ϵ‖v-un‖V,∀v∈V.
Without loss of generality, we can suppose that pn→p¯∈B(p*,ϵ) since L is finite dimensional. By taking limsup at both sides of above inequality, it follows from the assumptions (i)–(iii) and the upper semicontinuity of J^2∘(p,·)(u,v) with respect to (u,v) that
(2.19)〈A^(p¯,u)-f,v-u〉+J^2∘(p¯,⋅)(u,v-u)+G^(p¯,v)-G^(p¯,u)≥limsup⁡{〈A^(pn,un)-f,v-un〉+J^2∘(pn,⋅)(un,v-un)+G^(pn,v)-G^(pn,un)}≥limsup⁡{-ϵ‖v-un‖V}=-ϵ‖v-u‖V.
Thus, u∈Ω(ϵ) and so Ω(ϵ) is closed.
Second, we prove that(2.20)S=⋂ϵ>0Ω(ϵ).
It is obvious that S⊂∩ϵ>0Ω(ϵ) since the solution set S⊂Ω(ϵ) for all ϵ>0. Conversely, let u∈∩ϵ>0Ω(ϵ), and let {ϵn} be a nonnegative sequence with ϵn→0 as n→+∞. Then for any n∈ℕ, u∈Ω(ϵn), and so there exists pn∈B(p*,ϵn) such that
(2.21)〈A^(pn,u)-f,v-u〉+J^2∘(pn,⋅)(u,v-u)+G^(pn,v)-G^(pn,u)≥-ϵn‖v-u‖V,∀v∈V.
Since pn∈B(p*,ϵn) and ϵn→0, it is clear that pn→p*. By letting n→+∞ in the above inequality, we get from the continuity of A^, J^2∘(p,·), and G^ in assumptions that
(2.22)〈A(u)-f,v-u〉+J∘(u,v-u)+G(v)-G(u)=〈A^(p*,u)-f,v-u〉+J^2∘(p*,⋅)(u,v-u)+G^(p*,v)-G^(p*,u)≥lim⁡{-ϵn‖v-u‖V}=0.
Thus, u∈S and so ∩ϵ>0Ω(ϵ)⊂S.
Now, we suppose that(2.23)Ω(ϵ)≠∅,∀ϵ>0,μ(Ω(ϵ))⟶0asϵ⟶0.
From the definition of approximating solution set Ω(ϵ), Ω(ϵ) is increasing with respect to ϵ. Then by applying the Kuratowski theorem on page 318 in [35], we have from (2.20) that S is nonempty compact and
(2.24)q(ϵ,S)=e(Ω(ϵ),S)=H(Ω(ϵ),S)⟶0asϵ⟶0.
Therefore, by Theorem 2.7, VHVI(A,f,J,G) is strongly well-posed by perturbations in the generalized sense.Example 2.9.
LetL be a finite-dimensional space with norm ∥·∥L, let P⊂L be a closed ball in L, and let p* be a given point in P. We supposed that the perturbed mappings A^:P×V→V*, G^, J^:P×V→ℝ of the mapping A, G, J are, respectively, specified as follows: (2.25)A^(p,v)=exp⁡α‖p-p*‖LA(v),G^(p,v)=G(v)+β‖p-p*‖L,J^(p,v)=J(v)+γ‖p-p*‖L,
where α, β, γ are three positive numbers. It is obvious that A^ is continuous with respect to (p,v) due to the continuity of the mapping A:V→V*, and G^ is lower semicontinuous with respect to (p,v) and continuous with respect to p for any given v∈V because the functional G:V→ℝ is proper convex and lower semicontinuous. Also, the perturbed functional J^ is locally Lipschitz with respect to v since J:V→ℝ is locally Lipschitz. Furthermore, it is easy to check that the perturbed Clarke's generalized directional derivative corresponding to the perturbed function J^ can be specified as
(2.26)J^2∘(p,⋅)(u,v)=J∘(u,v)=lim⁡sup⁡w→uλ↓0J(w+λv)-J(w)λ,
which implies that J^2∘(p,·) is continuous with respect to p. Thus, the assumptions in Theorem 2.8 are satisfied, and so the VHVI(A,f,J,G) is strongly well-posed by perturbations in the generalized sense if and only if (2.16) holds.


## 3. Links with Well-Posedness by Perturbations for Corresponding Inclusion Problem

In this section, we recall some concepts of well-posedness by perturbations for inclusion problems, which are introduced by Lemaire et al. [4], and investigate the relations between the well-posedness by perturbations for VHVI(A,f,J,G) and the well-posedness by perturbations for the corresponding inclusion problem.In what follows, we always letF be a set-valued mapping from real reflexive Banach space V to its dual space V*. The inclusion problem associated with mapping F is defined by
(3.1)IP(F):findx∈Vsuch  that0∈F(x),
whose corresponding perturbed problem is specified as
(3.2)IPp(F):findx∈Vsuch  that0∈F^(p,x),
where F^:P×V→2V* is the perturbed set-valued mapping such that F^(p*,·)=F.Definition 3.1 (see [4]).
Let{pn}⊂P be a sequence in P with pn→p*. A sequence {un}⊂V is said to be an approximating sequence corresponding to {pn} for inclusion problem IP(F) if un∈domF^(pn,·) for all n∈ℕ and d(0,F^(pn,un))→0, or equivalently, there exists a sequence wn∈F^(pn,un) such that ∥wn∥V*→0 as n→∞.Definition 3.2 (see [4]).
One says that inclusion problemIP(F) 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(F).Definition 3.3 (see [4]).
One says that inclusion problemIP(F) is strongly (resp., weakly) well-posed by perturbations in the generalized sense if the solution set S of IP(F) is nonempty, and for any {pn}⊂P with pn→p*, every approximating sequence corresponding to {pn} has a subsequence converging strongly (resp., weakly) to some point of solution set S for IP(F).In order to obtain the relations between the strong (resp., weak) well-posedness by perturbations for variational-hemivariational inequalityVHVI(A,f,J,G) and the strong (resp., weak) well-posedness by perturbations for the corresponding inclusion problem, we first give the following important lemma which establishes the equivalence between the variational-hemivariational inequality VHVI(A,f,J,G) and the corresponding inclusion problem. Although the lemma is a corollary of Lemma 4.1 in [32] with T=0, we also give proof here for its importance and the completeness of our paper.Lemma 3.4.
LetA be a mapping from Banach space V to its dual V*, let J:V→ℝ be a locally Lipschitz functional, let G:V→ℝ∪{+∞} be a proper, convex, and lower semicontinuous functional, and let f be a given element in dual space V*. Then u∈domG is a solution of VHVI(A,f,J,G) if and only if u is a solution of the following inclusion problem:
(3.3)IP(A-f+∂¯J+∂G):findu∈dom⁡Gsuch  thatAu-f+∂¯J(u)+∂G(u)∋0.Proof.
“Sufficiency”: assume thatu∈domG is a solution of inclusion problem IP(A-f+∂¯J+∂G). Then there exist ω1∈∂¯J(u) and ω2∈∂G(u) such that
(3.4)Au-f+ω1+ω2=0.
By multiplying v-u at both sides of above equation (3.4), we obtain from the definitions of the Clarke's generalized gradient for locally Lipschitz functional and the subgradient for convex functional that
(3.5)0=〈Au-f+ω1+ω2,v-u〉≤〈Au-f,v-u〉+J∘(u,v-u)+G(v)-G(u),∀v∈V,
which implies that u∈domG is a solution of VHVI(A,f,J,G).
“Necessity”: conversely, suppose thatu∈domG is a solution of VHVI(A,f,J,G). Then,
(3.6)〈Au-f,v-u〉+J∘(u,v-u)+G(v)-G(u)≥0,∀v∈V.
From the fact that
(3.7)J∘(u,v-u)=max⁡{〈ω,v-u〉:ω∈∂¯J(u)},
we get that there exists a ω(u,v)∈∂¯J(u) such that
(3.8)〈Au-f,v-u〉+〈ω(u,v),v-u〉+G(v)-G(u)≥0,∀v∈V.
By virtue of Proposition 1.3, ∂¯J(u) is a nonempty, convex, and bounded subset in V* which implies that {Au-f+ω:ω∈∂¯J(u)} is nonempty, convex, and bounded in V*. Since G:V→ℝ∪{+∞} is a proper, convex, and lower semicontinuous functional, it follows from (3.8) and Theorem 1.6 with φ(u)=G(u) that there exists ω(u)∈∂¯J(u), which is independent on v, such that
(3.9)〈Au-f,v-u〉+〈ω(u),v-u〉+G(v)-G(u)≥0,∀v∈V.
For the sake of simplicity in writing, we denote ω=ω(u). Then by (3.9), we have
(3.10)G(v)-G(u)≥〈-Au+f-ω,v-u〉,∀v∈V,
that is, -Au+f-ω∈∂G(u). Thus, it follows from ω∈∂¯J(u) that
(3.11)Au-f+∂¯J(u)+∂G(u)∋0,
which implies that u∈domG is a solution of the inclusion problem IP(A-f+∂¯J+∂G). This completes the proof of Lemma 3.4.Remark 3.5.
The corresponding perturbed problem of inclusion problemIP(A-f+∂¯J+∂G) is specified as
(3.12)IPp(Au-f+∂¯J(u)+∂G(u)):findu∈dom⁡G^(p,u)suchthatA^(p,u)-f+∂¯2J^(p,u)+∂2G^(p,u)∋0.
Now we prove the following two theorems which establish the relations between the strong (resp., weak) well-posedness by perturbations for variational-hemivariational inequalityVHVI(A,f,J,G) and the strong (resp., weak) well-posedness by perturbations for the corresponding inclusion problem IP(A-f+∂¯J+∂G).Theorem 3.6.
LetA be a mapping from Banach space V to its dual V*, let J:V→ℝ be a locally Lipschitz functional, let G:V→ℝ∪{+∞} be a proper, convex, and lower semicontinuous functional, and let f be a given element in dual space V*. The variational-hemivariational inequality VHVI(A,f,J,G) is strongly (resp., weakly) well-posed by perturbations if and only if the corresponding inclusion problem IP(A-f+∂¯J+∂G) is strongly (resp., weakly) well-posed by perturbations.Proof.
“Necessity”: assume thatVHVI(A,f,J,G) is strongly (resp., weakly) well-posed by perturbations, which implies that there is a unique solution u* of VHVI(A,f,J,G). Clearly, the existence and uniqueness of solution for inclusion problem IP(A-f+∂¯J+∂G) is obtained easily by Lemma 3.4. Let {pn}⊂P be a sequence with pn→p* and {un} an approximating sequence corresponding to {pn} for IP(A-f+∂¯J+∂G). Then there exists a sequence ωn∈A^(pn,un)-f+∂¯2J^(pn,un)+∂2G^(pn,un) such that ∥ωn∥V*→0 as n→∞. And so, there exist ξn∈∂¯2J^(pn,un) and ηn∈∂2G^(pn,un) such that
(3.13)ωn=A^(pn,un)-f+ξn+ηn.
From the definition of the perturbed Clarke's generalized gradient ∂¯2J^(p,·) corresponding to the perturbed locally Lipschitz functional J^ and the definition of the perturbed subgradient ∂2G^(p,·) corresponding to the perturbed convex functional G^, we obtain by multiplying v-un at both sides of above equation (3.13) that
(3.14)〈A^(pn,un)-f,v-un〉+J^2∘(pn,⋅)(un,v-un)+G^(pn,v)-G^(pn,un)≥〈A^(pn,un)-f,v-un〉+〈ξn,v-un〉+〈ηn,v-un〉=〈ωn,v-un〉≥-‖ωn‖V*‖v-un‖V,∀v∈V.
Letting ϵn=∥ωn∥V*, we obtain from (3.14) and the fact ∥ωn∥V*→0 as n→∞ that {un} is an approximating sequence corresponding to {pn} for VHVI(A,f,J,G). Therefore, it follows from the strong (resp., weak) well-posedness by perturbations for VHVI(A,f,J,G) that un converges strongly (resp., weakly) to the unique solution u*. Thus, the inclusion problem IP(A-f+∂¯J+∂G) is strongly (resp., weakly) well-posed.
“Sufficiency”: conversely, suppose that inclusion problemIP(A-f+∂¯J+∂G) is strongly (resp., weakly) well-posed by perturbations. Then IP(A-f+∂¯J+∂G) has a unique solution u*, which implies that u* is the unique solution of VHVI(A,f,J,G) by Lemma 3.4. Let {pn}⊂P be a sequence with pn→p* and {un} an approximating sequence corresponding to {pn} for VHVI(A,f,J,G). Then there exists a nonnegative sequence {ϵn} with ϵn→0 as n→0 such that
(3.15)〈A^(pn,un)-f,v-un〉+J^2∘(pn,⋅)(un,v-un)+G^(pn,v)-G^(pn,un)≥-ϵn‖v-un‖V,∀v∈V.
By the same arguments in proof of Lemma 3.4, there exists a ω(pn,un,v)∈∂¯2J^(pn,un) such that
(3.16)〈A^(pn,un)-f,v-un〉+〈ω(pn,un,v),v-un〉+G^(pn,v)-G^(pn,u)≥-ϵn‖v-un‖V,∀v∈V,
and the set {A^(pn,un)-f+ω:ω∈∂¯2J^(pn,un)} is nonempty, convex, and bounded in V*. Then, it follows from (3.16) and Theorem 1.6 with φ(u)=G^(pn,u)+ϵn∥u-un∥, which is proper convex and lower semicontinuous, that there exists ω(pn,un)∈∂¯2J^(pn,un) such that
(3.17)〈A^(pn,un)-f,v-un〉+〈ω(pn,un),v-un〉+G^(pn,v)-G^(pn,un)≥-ϵn‖v-un‖V,∀v∈V.
For the sake of simplicity in writing, we denote ωn=ω(pn,un). Then it follows from (3.17) that
(3.18)G^(pn,un)≤G^(pn,v)+〈A^(pn,un)-f+ωn,v-un〉+ϵn‖v-un‖V,∀v∈V.
Define functional Tn:V→ℝ∪{+∞} as follows:
(3.19)Tn(v)=G^(pn,v)+Rn(v)+ϵnQn(v),
where Rn(v),Qn(v) are two functional on V defined by
(3.20)Rn(v)=〈A^(pn,un)-f+ωn,v-un〉,Qn(v)=‖v-un‖V.
Clearly, the functionals Rn and Qn are convex and continuous on V, and so Tn is proper, convex, and lower semicontinuous because G^(pn,v) is proper, convex, and lower semicontinuous with respect to v. Furthermore, it follows from (3.18) that un is a global minimizer of Tn on V. Thus, the zero element in V*, we also denote to be 0, belongs to the subgradient ∂Tn(un) which is specified as follows due to Proposition 1.2:
(3.21)∂Tn(v)=∂2G^(pn,v)+A^(pn,un)-f+ωn+ϵn∂Qn(v).
It is easy to calculate that
(3.22)∂Qn(v)={v*∈V*:‖v*‖V*=1,〈v*,v-un〉=‖v-un‖V},
and so there exists a ξn∈∂Qn(un) with ∥ξn∥V*=1 such that
(3.23)0∈∂2G^(pn,v)+A^(pn,un)-f+ωn+ϵnξn.
Let un*=-ϵnξn, then ∥un*∥V*→0 due to ϵn→0 as n→∞. This together with (3.23) and ωn∈∂¯2J^(pn,un) implies that
(3.24)un*∈A^(pn,un)-f+∂¯2J^(pn,un)+∂2G^(pn,v).
Therefore, {un} is an approximating sequence corresponding to {pn} for IP(A-f+∂¯J+∂G). Since inclusion problem IP(A-f+∂¯J+∂G) is strongly (resp., weakly) well-posed by perturbations, un converges strongly (resp., weakly) to the unique solution u*. Therefore, variational-hemivariational inequality VHVI(A,f,J,G) is strongly (resp., weakly) well-posed. This completes the proof of Theorem 3.6.Theorem 3.7.
LetA be a mapping from Banach space V to its dual V*, let J:V→ℝ be a locally Lipschitz functional, let G:V→ℝ∪{+∞} be a proper, convex, and lower semicontinuous functional, and let f be a given element in dual space V*. The variational-hemivariational inequality VHVI(A,f,J,G) is strongly (resp., weakly) well-posed by perturbations in the generalized sense if and only if the corresponding inclusion problem IP(A-f+∂¯J+∂G) is strongly (resp., weakly) well-posed by perturbations in the generalized sense.Proof.
The proof of Theorem3.7 is similar to Theorem 3.6, and so we omit it here.



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*Source: 804032-2012-08-10.xml*