Well-Posedness for a Class of

Journal of Applied Mathematics Volume 2012, Article ID 712306,21pages doi:10.1155/2012/712306

Research Article

Well-Posedness for a Class of

Strongly Mixed Variational-Hemivariational Inequalities with Perturbations

Lu-Chuan Ceng,

Ngai-Ching Wong,

and Jen-Chih Yao

1Department of Mathematics, Shanghai Normal University, Shanghai 200234, China

2Scientific Computing Key Laboratory, Shanghai Universities, Shanghai 200234, China

3Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung 80424, Taiwan

4Center for General Education, Kaohsiung Medical University, Kaohsiung 80708, Taiwan

Correspondence should be addressed to Ngai-Ching Wong,[email protected] Received 1 August 2011; Accepted 19 November 2011

Academic Editor: Ya Ping Fang

Copyrightq2012 Lu-Chuan Ceng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The concept of well-posedness for a minimization problem is extended to develop the concept of well-posedness for a class of strongly mixed variational-hemivariational inequalities with perturbations which includes as a special case the class of variational-hemivariational inequalities with perturbations. We establish some metric characterizations for the well-posed strongly mixed variational-hemivariational inequality and give some conditions under which the strongly mixed variational-hemivariational inequality is strongly well-posed in the generalized sense. On the other hand, it is also proven that under some mild conditions there holds the equivalence between the well posedness for a strongly mixed variational-hemivariational inequality and the well-posedness for the corresponding inclusion problem.

1. Introduction

It is well known that the classical notion of well-posedness for the minimization problem MPis due to Tykhonov1, which has been known as the Tykhonov well-posedness. Let V be a Banach space andf :V → R∪ {∞}be a real-valued functional onV. The problem MP, that is, minx∈Vfx, is said to be well posed if there exists a unique minimizer and every minimizing sequence converges to the unique minimizer. Furthermore, the notion of generalized Tykhonov well-posedness is also introduced for the problemMP, which means the existence of minimizers and the convergence of some subsequence of every minimizing sequence toward a minimizer. Clearly, the concept of well-posedness is inspired by numerical methods producing optimizing sequences for optimization problems and plays a crucial

role in the optimization theory. Therefore, various concepts of well-posedness have been introduced and studied for optimization problems. For more details, we refer to2–8and the references therein.

On the other hand, the concept of well-posedness has been extended to other related problems, such as variational inequalities 5, 9–14, saddle-point problem 15, inclusion problems10,11, and fixed-point problems10,11. An initial notion of well-posedness for variational inequalities is due to Lucchetti and Patrone5. They introduced the notion of well-posedness for variational inequalities and proved some related results by means of Eke- land’s variational principle. Since then, many authors have been devoted to generating the concept of well-posedness from the minimization problem to various variational inequalities.

In2, Crespi et al. gave the notions of well-posedness for a vector optimization problem and a vector variational inequality of the differential type, explored their basic properties, and investigated their links. Lignola13 introduced two concepts of well-posedness and L-well-posedness for quasivariational inequalities and investigated some equivalent char- acterizations of these two concepts. Recently, Fang et al. 11generalized the concepts of well-posedness and α-well-posedness to a generalized mixed variational inequality which includes as a special case the classical variational inequality and discussed its links with the well-posedness of corresponding inclusion problem and the well-posedness of corresponding fixed-point problem. They also derived some conditions under which the mixed variational inequality is well posed. For further results on the well-posedness for variational inequalities and equilibrium problems, we refer to5,8,11,13,16–18and the references therein.

In 1983, in order to formulate variational principles involving energy functions with no convexity and no smoothness, Panagiotopoulos19first introduced the hemivariational inequality which is an important and useful generalization of variational inequality and investigated it by using the mathematical notion of the generalized gradient of Clarke for nonconvex and nondifferentiable functions20. The hemivariational inequalities have been proved very efficient to describe a variety of mechanical problems, for instance, uni- lateral contact problems in nonlinear elasticity, problems describing the adhesive and fric- tional effects, and nonconvex semipermeability problems see, for instance, 19, 21, 22.

Therefore, in recent years all kinds of hemivariational inequalities have been studied by many authors14,21,23–29, 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 inequalities. In 1995, Goeleven and Mentagui14first introduced the notion of well-posedness for hemivariational inequalities and established some basic results concerning the well-posed hemivariational inequality.

Very recently, Xiao and Huang30generalized the well-posedness of minimization problems to a class of variational-hemivariational inequalities with perturbations, which in- cludes as special cases the classical hemivariational inequalities and variational inequalities.

Under appropriate conditions, they derived some metric characterizations for the well- posed variational-hemivariational inequality and presented some conditions under which the variational-hemivariational inequality is strongly well posed in the generalized sense.

Meantime, they also proved that the well-posedness for a variational-hemivariational ine- quality is equivalent to the well-posedness for the corresponding inclusion problem.

In this paper, we extend the notion of well-posedness for minimization problems to a class of strongly mixed variational-hemivariational inequalities with perturbations, which includes as a special case the class of variational-hemivariational inequalities with perturba- tions in30. Under very mild conditions, we establish some metric characterizations for the well-posed strongly mixed variational-hemivariational inequality and give some conditions

under which the strongly mixed variational-hemivariational inequality is strongly well- posed in the generalized sense. On the other hand, it is also proven that the well-posed- ness for a strongly mixed variational-hemivariational inequality is equivalent to the well-posed- ness for the corresponding inclusion problem.

2. Preliminaries

Throughout this paper, unless stated otherwise, we always suppose thatV is a real reflexive Banach space, where its dual space is denoted by V^∗ and the generalized duality pairing betweenV and V^∗ is denoted by·,·. We denote the norms of Banach spacesV and V^∗ by · _V and · _V^∗, respectively. In what follows, letN : V^∗×V^∗ → V^∗,A, T : V → V^∗ and g : V → V be four mappings, G : V → R∪ {∞}be a proper, convex, and lower semicontinuous functional, andf∈V^∗be some given element. Denote by domGthe efficient domain of functional, that is,

domG: {u∈V :Gu<∞}. 2.1

Consider the following strongly mixed variational-hemivariational inequality: findu∈ V such that

SMVHVI : N

Agu, Tu

−f, v−gu J^◦

u, v−gu

Gv−G gu

≥0, ∀v∈V, 2.2 whereJ^◦u, vdenotes the generalized directional derivative in the sense of Clarke of a locally Lipschitz functionalJ:V → R atuin the directionvsee20given by

J^◦u, v: lim sup

w→u λ↓0

Jwλv−Jw

λ . 2.3

In particular, ifNu^∗, v^∗ u^∗v^∗, for allu^∗, v^∗∈V^∗andg Ithe identity mapping ofV, then the problem2.2reduces to the following variational-hemivariational inequality of findingu∈Vsuch that

VHVI : AuTu, v−uJ^◦u, v−u Gv−Gu≥

f, v−u

, ∀v∈V, 2.4

whereTis perturbation, which was first introduced and studied by Xiao and Huang30.

LetΩ be an open bounded subset of R³ which is occupied by a linear elastic body and Γ the boundary of the Ω which is assumed to be appropriately regular C^0,1, i.e., a Lipschitzian boundary, is sufficient. We denote byS {Si}the stress vector onΓ, which can be decomposed into a normal componentS_Nand a tangential componentS_Twith respect to Γ, that is,

S_N σ_ijn_jn_i, S_Ti σ_ijn_j− σ_ijn_in_j

n_i, 2.5

where σ {σij} is an appropriately defined stress tensor and n {ni} is the outward unit normal vector on Γ. Analogously, u_N and u_T denote the normal and the tangential

components of the displacement vector uwith respect to Γ. As pointed out in30, the re- action-displacement law presents in compression ideal locking effectthe infinite branchEF, that is, alwaysu_N≤a, whereasu_N> ais impossible. Specifically,

ifuN < athen−SN∈βu N, ifuN athen− ∞<−SN≤βa, ifu_N > athen S_N ∅,

2.6

whereβis a multivalued function defined as follows. Suppose thatβ :R → R is a function such thatβ∈ L^∞_locR, that is, a function essentially bounded on any bounded interval ofR.

For anyρ >0 andξ ∈R, we defineβ_ρξ ess inf_{|ξ1−ξ|≤ρ}βξ1andβ_ρξ ess sup_{|ξ1−ξ|≤ρ}βξ1. By the monotonicity of the functionsβ_ρandβ_ρwith respect toρ, we infer that the limits as ρ → 0exist, that is,

βξ lim

ρ→0β_ρξ, βξ lim

ρ→0β_ρξ. 2.7

Then,

βξ

βξ, βξ

. 2.8

Furthermore, a locally Lipschitz functionj_Ncan be determined up to an additive constant by

jNξ ^ξ

0

βξ1dξ1 2.9

such that∂j_Nξ βξ for eachξ ∈R when the limitsβξ_±exist, where∂j_Nis the Clarke’s generalized gradient of locally Lipschitz functionjNwhich will be specified in what follows.

Now, letK {uN |uN ≤a},NKthe normal cone toKatuN, andIKthe indicator of the setK. Then2.6can be written as

−SN∈βu N N_KuN ∂j_NuN ∂I_KuN, 2.10

where ∂I_K is the subgradient of the convex functional I_K in the sense of convex analysis, which will also be specified in what follows. By the definitions of the Clarke’s generalized gradient of locally Lipschitz function and the subgradient of the convex functional, 2.10 gives rise to the following variational-hemivariational inequality

u_N∈R : SN, v−u_Nj_N^◦uN, v−u_N I_Kv−I_KuN≥0, ∀v∈R, 2.11 which is a special case of the variational-hemivariational inequality VHVI. Beyond question, the problem2.11is a special case of the strongly mixed variational-hemivariational inequal- ity SMVHVI as well. More special cases of the SMVHVI are stated as follows.

iIfG δKandJu

Ωjx, udΩ, whereδKdenotes the indicator functional of a nonempty, convex subsetKof a function spaceV defined onΩandj:Ω×R → R is a locally Lipschitz continuous function, then the SMVHVI reduces to the following strongly mixed variational-hemivariational inequality:

SMVHVI :

N

Agu, Tu

−f, v−gu J^◦

u, v−gu

≥0, ∀v∈K. 2.12

Remark that the SMVHVI 2.12with NAgu, Tu Agu Tu and g I is equivalent to the VHVI which was considered by Goeleven and Mentagui in14.

iiIfG 0, then the SMVHVI2.2withNAgu, Tu Agu Tureduces to the strongly mixed hemivariational inequality of findingu∈V such that

SMHVI :

Agu Tu−f, v−gu J^◦

u, v−gu

≥0, ∀v∈V. 2.13

Remark that the SMHVI2.13withT 0 andg Iis equivalent to the hemivar- iational inequalityHVIstudied intensively by many authorssee, e.g.,21,22.

iiiIfJ 0, then the SMVHVI2.2withNAgu, Tu Agu Tu reduces to the strongly mixed variational inequality of findingu∈Vsuch that

SMVI :

Agu Tu−f, v−gu

Gv−G gu

≥0, ∀v∈V. 2.14

Remark that the SMVI2.14withT 0 andg I is equivalent to the mixed var- iational inequalitysee, e.g.,11,31and the references therein.

ivIfT 0, J 0, g I andG δK, then the SMVHVI2.2withNAgu, Tu Agu Tureduces to the classical variational inequality:

VI :

Au−f, v−u

≥0, ∀v∈K. 2.15

vIfN 0, J 0, g I, andf 0, then the SMVHVI2.2 reduces to the global minimization problem:

MP : min

u∈V Gu. 2.16

Let ∂Gu : V → 2^V∗\ {∅}and ∂Ju : V → 2^V∗ \ {∅}denote the subgradient of convex functionalGin the sense of convex analysissee32and the Clarke’s generalized gradient of locally Lipschitz functionalJsee20, respectively. That is,

∂Gu {u^∗∈V^∗:Gv−Gu≥ u^∗, v−u,∀v∈V},

∂Ju {ω∈V^∗:J^◦u, v≥ ω, v,∀v∈V}. 2.17

Remark 2.1see33. The Clarke’s generalized gradient of a locally Lipschitz functionalJ : V → R at a pointuis given by

∂Ju ∂J^◦u,·0. 2.18

About the subgradient in the sense of convex analysis, the Clarke’s generalized di- rectional derivative, and the Clarke’s generalized gradient, we have the following basic propertiessee, e.g.,20,30,32,33.

Proposition 2.2. LetVbe a Banach space andG:V → R∪{∞}be a convex and proper functional.

Then we have the following properties of∂G:

i∂Guis convex and weak^∗-closed;

iiifGis continuous atu∈domG, then∂Guis nonempty, convex, bounded, and weak^∗- compact;

iiiifGis Gateaux differentiable atu∈domG, then∂Gu {DGu}, whereDGuis the Gateaux derivative ofGatu.

Proposition 2.3. LetV be a Banach space andG1, G2:V → R∪ {∞}be two convex functionals.

If there is a pointu₀ ∈ domG₁∩domG₂ at whichG₁ is continuous, then the following equation holds:

∂G1G2u ∂G1u ∂G2u, ∀u∈V. 2.19

Proposition 2.4. LetV be a Banach space,u, v∈V, andJa locally Lipschitz functional defined on V. Then

ithe functionv→J^◦u, vis finite, positively homogeneous, subadditive, and then convex onV;

iiJ^◦u, v is upper semicontinuous as a function of u, v, as a function of v alone, is Lipschitz continuous onV;

iiiJ^◦u−v −J^◦u, v;

iv∂Juis a nonempty, convex, bounded, weak^∗-compact subset ofV^∗; vfor everyv∈V, one has

J^◦u, v max

ξ, v:ξ∈∂Ju

. 2.20

Now we recall some important definitions and useful results.

Definition 2.5see34. LetV be a real Banach space with its dualV^∗andT be an operator fromVto its dual spaceV^∗.Tis said to be monotone if

Tu−Tv, u−v ≥0, ∀u, v∈V. 2.21

Definition 2.6see34. A mapping T : V → V^∗ is said to be hemicontinuous if for any u, v∈V, the functiont→ Tutv−u, v−ufrom0,1intoR is continuous at 0.

It is clear that the continuity implies the hemicontinuity, but the converse is not true in general.

Theorem 2.7see35. LetC ⊂ V be nonempty, closed, and convex,C^∗ ⊂V^∗ nonempty, closed, convex, and bounded,ϕ:V → R∪ {∞}proper, convex, and lower semicontinuous, andy∈Cbe arbitrary. Assume that, for eachx∈C, there existsx^∗x∈C^∗such that

x^∗x, x−y

≥ϕ y

−ϕx. 2.22

Then, there existsy^∗∈C^∗such that y^∗, x−y

≥ϕ y

−ϕx, ∀x∈C. 2.23

Definition 2.8see36. LetSbe a nonempty subset ofV. The measure, sayμ, of noncom- pactness for the setSis defined by

μS: inf

>0 :S⊂ⁿ

i 1

Si, diamSi< , i 1,2, . . . , n

, 2.24

where diamSimeans the diameter of setSi.

Definition 2.9 see36. LetA, B be nonempty subsets of V. The Hausdorffmetric H·,· betweenAandBis defined by

HA, B: max{eA, B, eB, A}, 2.25

whereeA, B: sup_a∈Ada, Bwithda, B: inf_b∈B a−b _V.

Let{An}be a sequence of nonempty subsets ofV. We say thatA_n converges toAin the sense of Hausdorffmetric ifHAn, A → 0. It is easy to see thateAn, A → 0 if and only ifdan, A → 0 for all sectionan ∈An. For more details on this topic, we refer the reader to 36.

3. Well-Posedness of the SMVHVI with Metric Characterizations

In this section, we generalize the concept of well-posedness to the strongly mixed varia- tional-hemivariational inequality SMVHVI with perturbations, establish its metric charac- terizations, and derive some conditions under which the strongly mixed variational-hemi- variational inequality is strongly well-posed in the generalized sense in Euclidean spaceRⁿ.

Definition 3.1. A sequence{un} ⊂Vis said to be an approximating sequence for the SMVHVI if there exists a nonnegative sequence{n}with_n → 0 asn → ∞such that

N

Agun, Tun

−f, v−gun J^◦

un, v−gun

Gv−G gun

≥ −nv−gun

V, ∀v∈V. 3.1

Definition 3.2. The SMVHVI is said to be stronglyresp., weaklywell posed if the SMVHVI has a unique solution in V and every approximating sequence converges stronglyresp., weaklyto the unique solution.

Remark 3.3. Strong well-posedness implies weak well-posedness, but the converse is not true in general.

Definition 3.4. The SMVHVI is said to be stronglyresp., weaklywell posed in the general- ized sense if the SMVHVI has a nonempty solution set S in V and every approximating sequence has a subsequence which converges stronglyresp., weaklyto some point of the solution setS.

Remark 3.5. Strong well-posedness in the generalized sense implies weak well-posedness in the generalized sense, but the converse is not true in general.

Definition 3.6. LetN:V^∗×V^∗ → V^∗andA:V → V^∗be two mappings. Then

iAis said to be monotone with respect to the first argument ofNif there holds NAu, w^∗−NAv, w^∗, u−v ≥0, ∀u, v∈V, w^∗∈V^∗; 3.2

iiAis said to be continuous with respect to the first argument ofNif for eachw^∗∈V^∗ the mappingv→NAv, w^∗fromVintoV^∗is continuous;

iiiA is said to be hemicontinuous with respect to the first argument ofN if for all u, v∈V andw^∗∈V^∗, the functiont → NAutv−u, w^∗, v−ufrom0,1 intoR is continuous at 0.

For any >0, we define the following two sets:

Ω

u∈V : N

Agu, Tu

−f, v−gu J^◦

u, v−gu

Gv−G gu

≥ −v−gu

V, ∀v∈V , Ψ

u∈V :

NAv, Tu−f, v−gu J^◦

u, v−gu

Gv−G gu

≥ −v−gu

V, ∀v∈V .

3.3

Lemma 3.7. Suppose thatA: V → V^∗ is both monotone and hemicontinuous with respect to the first argument ofN, G:V → R∪ {∞}is a proper, convex, and lower semicontinuous functional.

ThenΩ Ψfor all >0.

Proof. Letu ∈ Ω. Then, by the monotonicity of the mappingAwith respect to the first argument ofN, we have for allv∈V

0≤ N

Agu, Tu

−f, v−gu J^◦

u, v−gu

Gv−G gu

v−gu

V

≤

NAv, Tu−f, v−gu J^◦

u, v−gu

Gv−G gu

v−gu

V.

3.4 This implies thatu∈Ψ. Thus, we get the inclusionΩ⊂Ψ.

Next let us show thatΨ⊂Ω. Indeed, for anyu∈Ψ, we have NAv, Tu−f, v−gu

J^◦

u, v−gu

Gv−G gu

≥ −v−gu

V, ∀v∈V.

3.5 For anyw ∈V andt∈0,1, puttingv tw 1−tgu gu tw−guin3.5, we obtain

−t

w−gu

V ≤ N

A

tw 1−tgu , Tu

−f, t

w−gu J^◦

u, t

w−gu G

tw 1−tgu

−G gu

. 3.6

Since the Clarke’s generalized directional derivativeJ^◦u, vis positively homogeneous with respect tovandGis convex, it follows that

N A

tw 1−tgu , Tu

−f, w−gu J^◦

u, w−gu

Gw−G gu

≥ −w−gu

V. 3.7

Taking the limit for3.7ast → 0, we obtain from the hemicontinuity of the mappingA with respect to the first argument ofNthat

N

Agu, Tu

−f, w−gu J^◦

u, w−gu

Gw−G gu

≥ −w−gu

V. 3.8

By the arbitrariness ofw ∈ V, we conclude thatu∈ Ω, which implies thatΨ ⊂ Ω.

This completes the proof.

Lemma 3.8. Suppose thatT : V → V^∗ is continuous with respect to the second argument ofN, g :V → V is continuous, andG:V → R∪ {∞}is a proper, convex, and lower semicontinuous functional. ThenΨis closed inV for all >0.

Proof. Let{un} ⊂Ψbe a sequence such thatu_n → uinV. Then NAv, Tun−f, v−gun

J^◦

u_n, v−gun

Gv−G gun

≥ −v−gun

V,

∀v∈V.

3.9

SinceT : V → V^∗ is continuous with respect to the second argument of N,g : V → V is continuous,G : V → R∪ {∞} is lower semicontinuous, and the Clarke’s generalized directional derivativeJ^◦u, vis upper semicontinuous with respect tou, v, we deduce that gun → gu, NAv, Tun → NAv, Tu, and

nlim→ ∞

NAv, Tun, v−gun

NAv, Tu, v−gu ,

lim sup

n→ ∞ J^◦

u_n, v−gun

≤J^◦

u, v−gu ,

lim sup

n→ ∞ −G gun

≤ −G gu

.

3.10

Taking the lim sup for3.9asn → ∞, we obtain from3.10that NAv, Tu−f, v−gu

J^◦

u, v−gu

Gv−G gu

≥ −v−gu

V, ∀v∈V, 3.11

which implies thatu∈Ψ. Therefore,Ψis closed inV. This completes the proof.

Corollary 3.9. Suppose thatA:V → V^∗is both monotone and hemicontinuous with respect to the first argument ofNandT :V → V^∗is continuous with respect to the second argument ofN. Let g :V → V be continuous andG:V → R∪ {∞}be a proper, convex, and lower semicontinuous functional. Then, for all >0,Ω Ψis closed inV.

Theorem 3.10. Suppose thatA:V → V^∗is both monotone and hemicontinuous with respect to the first argument ofNandT :V → V^∗is continuous with respect to the second argument ofN. Let g : V → V be continuous andG : V → R∪ {∞}a proper, convex, and lower semicontinuous functional. Then, the SMVHVI is strongly well posed if and only if

Ω/∅, ∀ >0, diamΩ−→0 as−→0. 3.12

Proof. “Necessity”. Suppose that the SMVHVI is strongly well posed. Then the SMVHVI has a unique solution which lies inΩand soΩ/∅for all >0. If diamΩ0 as → 0, then there exist a constantl >0, a nonnegative sequence{n}withn → 0 andun, vn∈Ωn such that

un−v_{n V} > l, ∀n≥1. 3.13

Sinceu_n, v_n ∈Ωn, it is known that{un}and{vn}are both approximating sequences for the SMVHVI. From the strong well-posedness of the SMVHVI, it follows that both{un}and{vn} converge strongly to the unique solution of the SMVHVI, which is a contradiction to3.13.

“Sufficiency”. Let {un} ⊂ V be an approximating sequence for the SMVHVI. Then there exists a nonnegative sequence{n}with_n → 0 such that

N

Agun, Tun

−f, v−gun J^◦

u_n, v−gun

Gv−G gun

≥ −nv−gun

V,

∀v∈V, 3.14 which implies that u_n ∈ Ωn. By condition 3.12, {un} is a Cauchy sequence and so {un} converges strongly to some point u ∈ V. Since the mapping A is monotone with respect to the first argument ofN, the mappingT is continuous with respect to the second argument ofN,g is continuous, the Clarke’s generalized directional derivative J^◦u, vis upper semicontinuous with respect tou, v, andGis lower semicontinuous, it follows from 3.14that

NAv, Tu−f, v−gu J^◦

u, v−gu

Gv−G gu

≥lim sup

n→ ∞

NAv, Tun−f, v−gun J^◦

un, v−gun

Gv−G

gun

≥lim sup

n→ ∞

N

Agun, Tun

−f, v−gun J^◦

u_n, v−gun

Gv−G

gun

≥lim sup

n→ ∞

−nv−gun

V

0, ∀v∈V.

3.15 Furthermore, sinceAis also hemicontinuous with respect to the first argument ofNandGis convex, by the argument similar to that inLemma 3.7we can readily prove that

N

Agu, Tu

−f, v−gu J^◦

u, v−gu

Gv−G gu

≥0, ∀v∈V, 3.16 which implies thatusolves the SMVHVI.

To complete the proof ofTheorem 3.10, we need only to prove that the SMVHVI has a unique solution. Assume by contradiction that the SMVHVI has two distinct solutionsu₁ andu₂. Then it is easy to see thatu₁, u₂∈Ωfor all >0 and

0< u1−u2 V ≤diamΩ−→0, 3.17

which is a contradiction. Therefore, the SMVHVI has a unique solution. This completes the proof.

Corollary 3.11 see30, Theorem 3.1. Suppose thatA : V → V^∗ is a monotone and hemi- continuous mapping,T :V → V^∗is a continuous mapping, andG :V → R∪ {∞}is a proper, convex, and lower semicontinuous functional. Then, the VHVI is strongly well posed if and only if

Ω/∅, ∀ >0, diamΩ−→0 as−→0. 3.18

Proof. InTheorem 3.10, putNu^∗, v^∗ u^∗ v^∗, for allu^∗, v^∗ ∈ V^∗ and g I the identity mapping ofV. Then from the monotonicity and hemicontinuity ofAit follows thatA:V → V^∗is both monotone and hemicontinuous with respect to the first argument ofN. Moreover, from the continuity ofTit follows thatT :V → V^∗is continuous with respect to the second argument ofN. Thus, utilizingTheorem 3.10, we obtain the desired result.

Theorem 3.12. Suppose thatA:V → V^∗is both monotone and hemicontinuous with respect to the first argument ofNandT :V → V^∗is continuous with respect to the second argument ofN. Let g :V → V be continuous andG:V → R∪ {∞}be a proper, convex, and lower semicontinuous functional. Then, the SMVHVI is strongly well posed in the generalized sense if and only if

Ω/∅, ∀ >0, μΩ−→0 as−→0. 3.19

Proof. “Necessity”. Suppose that the SMVHVI is strongly well posed in the generalized sense.

Then the solution set of the SMVHVI is nonempty andS⊂Ωfor any >0. Furthermore, the solution set of the SMVHVI also is compact. In fact, for any sequence{un} ⊂S, it follows fromS ⊂ Ωfor any > 0 that{un} ⊂ Sis an approximating sequence for the SMVHVI.

Since the SMVHVI is strongly well posed in the generalized sense,{un}has a subsequence which converges strongly to some point of the solution setS. Thus, the solution setSof the SMVHVI is compact. Now let us show thatμΩ → 0 as → 0. FromS ⊂Ωfor any >0, we get

HΩ, S max{eΩ, S, eS,Ω} eΩ, S. 3.20

Taking into account the compactness of the solution setS, we obtain from3.20that

μΩ≤2HΩ, S 2eΩ, S. 3.21

In order to prove thatμΩ → 0 as → 0, it is sufficient to show thateΩ, S → 0 as → 0. Assume by contradiction thateΩ, S 0 as → 0. Then there exist a constant l >0, a sequence{n} ⊂Rwithn → 0 andun∈Ωnsuch that

u_n∈/SB0, l, 3.22

whereB0, lis the closed ball centered at 0 with radiusl. Since{un}is an approximating sequence for the SMVHVI and the SMVHVI is strongly well posed in the generalized sense, there exists a subsequence{unk}which converges strongly to some pointu ∈ Swhich is a contradiction to3.22. ThenμΩ → 0 as → 0.

“Sufficiency”. Assume that condition3.19holds. ByCorollary 3.9, we conclude that Ωis nonempty and closed for all >0. Observe that

S

>0

Ω. 3.23

SinceμΩ → 0 as → 0, by applying the theorem36, page 412, it can be easily found thatSis nonempty and compact with

eΩ, S HΩ, S−→0 as−→0. 3.24

Let{un} ⊂Vbe an approximating sequence for the SMVHVI. Then there exists a nonnegative sequence{n}with_n → 0 such that

N

Agun, Tun

−f, v−gun J^◦

u_n, v−gun

Gv−G gun

≥ −nv−gun

V,

∀v∈V, 3.25

and sou_n∈Ωnby the definition ofΩn. It follows from3.24that

dun, S≤eΩ, S−→0. 3.26

Since the solution setSis compact, there existsu_n∈Ssuch that

un−u_{n V} dun, S−→0. 3.27 Again from the compactness of the solution setS,{un}has a subsequence{unk}converging strongly to someu∈S. It follows from3.27that

unk−u _V ≤ unk −unk V unk−u _V −→0, 3.28 which implies that{unk}converges strongly to u. Therefore, the SMVHVI is strongly well- posed in the generalized sense. This completes the proof.

Corollary 3.13 see 30, Theorem 3.2. Suppose that A : V → V^∗ is a monotone and hemicontinuous mapping,T : V → V^∗ is a continuous mapping, andG : V → R∪ {∞}is a proper, convex, and lower semicontinuous functional. Then, the VHVI is strongly well posed in the generalized sense if and only if

Ω/∅, ∀ >0, μΩ−→0 as−→0. 3.29 The following theorem gives some conditions under which the strongly mixed variational-hemi- variational inequality is strongly well posed in the generalized sense in Euclidean spaceRⁿ.

Theorem 3.14. Suppose thatA: Rⁿ → Rⁿis both monotone and hemicontinuous with respect to the first argument ofN and T : Rⁿ → Rⁿ is continuous with respect to the second argument of N. Letg : Rⁿ → Rⁿ be continuous and G : Rⁿ → R∪ {∞} be a proper, convex, and lower semicontinuous functional. If there exists some > 0 such thatΩ is nonempty and bounded.

Then the strongly mixed variational-hemivariational inequality SMVHVI is strongly well posed in the generalized sense.

Proof. Suppose that{un}is an approximating sequence for the SMVHVI. Then there exists a nonnegative sequence{n}with_n → 0 asn → ∞such that

N

Agun, Tun

−f, v−gun J^◦

un, v−gun

Gv−G gun

≥ −nv−gun_Rn,

∀v∈Rⁿ. 3.30 Let0 >0 be such thatΩ0is nonempty and bounded. Then there existsn0such thatun ∈ Ω0for alln > n0. This implies that{un}is bounded by the boundedness ofΩ0. Thus, there exists a subsequence{unk} such that u_nk → u ask → ∞. Since the mappingA is monotone with respect to the first argument ofN, the mappingTis continuous with respect to the second argument ofN,gis continuous, the Clarke’s generalized directional derivative J^◦u, vis upper semicontinuous with respect to u, v, and G is lower semicontinuous, it follows from3.30that

NAv, Tu−f, v−gu J^◦

u, v−gu

Gv−G gu

≥lim sup

k→ ∞

NAv, Tunk−f, v−gunk J^◦

u_nk, v−gunk

Gv−G

gunk

≥lim sup

k→ ∞

N

Agunk, Tunk

−f, v−gunk J^◦

unk, v−gunk

Gv−G

gunk

≥lim sup

k→ ∞

−nkv−gunk_Rn 0, ∀v∈Rⁿ.

3.31 Meantime, sinceAis also hemicontinuous with respect to the first argument ofNandGis convex, by the argument similar to that inLemma 3.7we can readily prove that

N

Agu, Tu

−f, v−gu J^◦

u, v−gu

Gv−G gu

≥0, ∀v∈Rⁿ, 3.32 which implies thatusolves the SMVHVI. Therefore, the SMVHVI is strongly well-posed in the generalized sense. This completes the proof.

Corollary 3.15see30, Theorem 3.3. Suppose thatA : Rⁿ → Rⁿ is a monotone and hemi- continuous mapping, T : Rⁿ → Rⁿ is a continuous mapping, and G : Rⁿ → R∪ {∞}is a proper, convex, and lower semicontinuous functional. If there exists some > 0 such thatΩis nonempty and bounded. Then the variational-hemivariational inequality VHVI is strongly well posed in the generalized sense.

4. Well-Posedness of Inclusion Problem

In this section, we first recall the concept of well-posedness for inclusion problems and then investigate the relations between the well-posedness for the strongly mixed variational- hemivariational inequality and the well-posedness for the corresponding inclusion problem.

In what follows we always assume thatFis a set-valued mapping from real reflexive Banach spaceV to its dual spaceV^∗. The inclusion problem associated with mappingFis defined by

IPF: findx∈V such that 0∈Fx. 4.1

Definition 4.1see18,37. A sequence{un} ⊂ V is called an approximating sequence for the inclusion problem IPFifd0, Fun → 0 or, equivalently, there exists a sequencew_n∈ Funsuch that wn V^∗ → 0 asn → ∞.

Definition 4.2 see 18, 37. We say that the inclusion problem IPF is strongly resp., weaklywell posed if it has a unique solution and every approximating sequence converges stronglyresp., weaklyto the unique solution of IPF.

Definition 4.3 see 18, 37. We say that the inclusion problem IPF is strongly resp., weaklywell posed in the generalized sense if the solution setSof the IPFis nonempty and every approximating sequence has a subsequence which converges stronglyresp., weakly to some point of the solution setSfor the IPF.

The following two theorems establish the relations between the strongresp., weak well-posedness for the strongly mixed variational-hemivariational inequality and the strong resp., weakwell-posedness for the corresponding inclusion problem.

Theorem 4.4. LetN : V^∗ ×V^∗ → V^∗,A, T : V → V^∗, andg : V → V be four mappings, J : V → R a locally Lipschitz functional, and G : V → R∪ {∞} a proper, convex, and lower semicontinuous functional. Then the strongly mixed variational-hemivariational inequality SMVHVI is strongly (resp., weakly) well posed if and only if the corresponding inclusion problem IPNAg, T−f∂J∂Ggis strongly (resp., weakly) well posed.

Theorem 4.5. LetN : V^∗ ×V^∗ → V^∗,A, T : V → V^∗, andg : V → V be four mappings, J : V → R a locally Lipschitz functional, andG : V → R∪ {∞} a proper, convex, and lower semicontinuous functional. Then the strongly mixed variational-hemivariational inequality SMVHVI is strongly (resp., weakly) well posed in the generalized sense if and only if the corresponding inclusion problem IPNAg, T−f∂J∂Ggis strongly (resp., weakly) well posed in the generalized sense.

Lemma 4.6. Let N : V^∗×V^∗ → V^∗, A, T : V → V^∗, and g : V → V be four mappings, J : V → R a locally Lipschitz functional, andG : V → R∪ {∞} a proper, convex, and lower semicontinuous functional. Thenu∈V is a solution of the SMVHVI if and only ifuis a solution of the corresponding inclusion problem IPNAg, T−f∂J∂Ggof findingu∈V such that

0∈N A

gu , Tu

−f∂Ju ∂G gu

. 4.2

Proof. “Sufficiency”. Assume thatuis a solution of the inclusion problem IPNAg, T− f∂J∂Gg. Then there existw1∈∂Juandw2∈∂Ggusuch that

N A

gu , Tu

−fw₁w₂ 0. 4.3

By multiplyingv−guat both sides of the above equation4.3, we obtain from the de- finitions of the Clarke’s generalized gradient for locally Lipschitz functional and the sub- gradient for convex functional that

0

N A

gu , Tu

−fw₁w₂, v−gu

≤ N

A gu

, Tu

−f, v−gu J^◦

u, v−gu

Gv−G gu

, ∀v∈V, 4.4

which implies thatuis a solution of the SMVHVI.

“Necessity”. Suppose thatuis a solution of the SMVHVI. Then, N

A gu

, Tu

−f, v−gu J^◦

u, v−gu

Gv−G gu

≥0, ∀v∈V. 4.5

From the fact that J^◦

u, v−gu

max

w, v−gu

:w∈∂Ju

, 4.6

we deduce that there exists awu, v∈∂Jusuch that N

A gu

, Tu

−f, v−gu

w

gu, v

, v−gu

Gv−G gu

≥0, ∀v∈V.

4.7 In terms of Proposition 2.4 iv, ∂Ju is a nonempty, convex, bounded, weak^∗-compact subset of V^∗. Note that V is a real reflexive Banach space. Hence, ∂Ju is a nonempty, convex, bounded, weak-compact subset inV^∗. Thus∂Juis a nonempty, closed, convex, and bounded subset inV^∗which implies that{NAgu, Tu−fw:w∈∂Ju}is nonempty, closed, convex, and bounded inV^∗. SinceG:V → R∪ {∞}is a proper, convex, and lower semicontinuous functional, it follows fromTheorem 2.7withϕu Guand4.7that there existswu∈∂Jusuch that

N A

gu , Tu

−f, v−gu

wu, v−gu

Gv−G gu

≥0, ∀v∈V. 4.8

For the sake of simplicity we writew wu, and hence from4.8we have Gv−G

gu

≥

−N A

gu , Tu

f−w, v−gu

, ∀v∈V, 4.9

which implies that−NAgu, Tu f−w ∈∂Ggu. Consequently, it follows fromw ∈

∂Juthat

0∈N A

gu , Tu

−f∂Ju ∂G gu

, 4.10

which implies thatuis a solution of the inclusion problem IPNAg, T−f∂J∂Gg.

This completes the proof.

Proof ofTheorem 4.4. “Necessity”. Assume that the SMVHVI is stronglyresp., weaklywell posed. Then there is a unique solution u^∗ for the SMVHVI. ByLemma 4.6, u^∗ also is the unique solution for the inclusion problem IPNAg, T−f∂J ∂Gg. Let{un}be an approximating sequence for the IPNAg, T−f∂J∂Gg. Then there exists a sequence wn ∈NAgun, Tun−f∂Jun ∂Ggunsuch that wn V^∗ → 0 asn → ∞. And so there existξn∈∂Junandηn∈∂Ggunsuch that

wn N A

gun , Tun

−fξnηn. 4.11

From the definitions of the Clarke’s generalized gradient for locally Lipschitz functional and the subgradient for convex functional, we obtain by multiplyingv−gunat both sides of the above equation4.11that

N A

gun , Tun

−f, v−gun J^◦

un, v−gun

Gv−G gun

≥ N

A gun

, Tu_n

−f, v−gun

ξ_n, v−gun

η_n, v−gun wn, v−gun

≥ − w_{n V}∗v−gun

V, ∀v∈V.

4.12

Lettingn wn V^∗, we obtain that{un}is an approximating sequence for the SMVHVI from 4.12with wn V^∗ → 0 asn → ∞. Therefore, it follows from the strongresp., weakwell- posedness of the SMVHVI that{un}converges stronglyresp., weaklyto the unique solution u^∗. Thus, the inclusion problem IPNAg, T−f∂J∂Ggis stronglyresp., weakly well posed.

“Sufficiency”. Suppose that the inclusion problem IPNAg, T−f∂J∂Ggis stronglyresp., weaklywell posed. Then the IPNAg, T−f∂J∂Gghas a unique solutionu^∗, which implies thatu^∗is the unique solution for the SMVHVI byLemma 4.6. Let {un}be an approximating sequence for the SMVHVI. Then there exists a sequence{n}with _n → 0 asn → ∞such that

N A

gun , Tun

−f, v−gun J^◦

un, v−gun

Gv−G gun

≥ −nv−gun

V,

∀v∈V.

4.13

By the same argument as in the proof ofLemma 4.6, there exists awun, v∈∂Junsuch that N

A gun

, Tu_n

−f, v−gun

w

gun, v

, v−gun

Gv−G gun

≥ −nv−gun

V, ∀v∈V, 4.14

and{NAgun, Tun−fw :w∈∂Jun}is nonempty, closed, convex, and bounded in V^∗. Then, it follows from4.14andTheorem 2.7withϕu Gu n u−gun V, which is proper, convex, and lower semicontinuous, that there existswun∈∂Junsuch that

N A

gun , Tun

−f, v−gunwun, v−gunGv−G gun

≥ −nv−gun

V, ∀v∈V. 4.15

For the sake of simplicity we writewn wun, and hence from4.15we have G

gun

≤Gv N

A gun

, Tu_n

−fw_n, v−gun

_nv−gun

V, ∀v∈V.

4.16

Define functionalGn:V → R∪ {∞}as follows:

G_nv Gv P_nv _nQ_nv, 4.17

whereP_nvandQ_nvare two functionals onV defined by Pnv

N A

gun , Tun

−fwn, v−gun

, Qnv v−gun

V. 4.18

Clearly,G_nis proper, convex, and lower semicontinuous andv gunis a global minimizer ofG_nonV. Thus, 0∈∂G_ngun. Since the functionalsP_nandQ_nare continuous onV and Gis proper, convex, and lower semicontinuous, it follows fromProposition 2.3that

∂G_nv ∂Gv N A

gun , Tu_n

−fw_nn∂Q_nv. 4.19

It is easy to calculate that

∂Q_nv

v^∗∈V^∗: v^∗ _V∗ 1,

v^∗, v−gun v−gun

V

, 4.20

and so there exists aξ_n∈∂Q_ngunwith ξn V^∗ 1 such that 0∈∂G

gun N

A gun

, Tun

−fwnnξn. 4.21

Lettingu^∗_n −nξ_n, we have u^∗_{n V}∗ → 0 as_n → 0. Moreover, sincew_n ∈∂Jun, it follows from4.21that

u^∗_n∈N A

gun , Tu_n

−f∂Jun ∂G gun

, 4.22

which implies that{un}is an approximating sequence for the IPNAg, T−f∂J∂Gg.

Since the inclusion problem IPNAg, T−f ∂J ∂Gg is strongly resp., weakly

well posed,{un}converges stronglyresp., weaklyto the unique solutionu^∗. Therefore, the strongly mixed variational-hemivariational inequality SMVHVI is strongly resp., weakly well posed. This completes the proof.

Proof ofTheorem 4.5. The proof is similar to that inTheorem 4.4and so we omit it here.

Corollary 4.7see30, Theorem 4.1. LetAandT be two mappings from Banach spaceV to its dualV^∗,J:V → R be a locally Lipschitz functional, andG:V → R∪{∞}be a proper, convex, and lower semicontinuous functional. Then the variational-hemivariational inequality VHVI is strongly (resp., weakly) well posed if and only if the corresponding inclusion problem IPAT−f∂J∂G is strongly (resp., weakly) well posed.

Proof. InTheorem 4.4, putg Ithe identity mapping ofV andNu^∗, v^∗ u^∗v^∗, for all u^∗, v^∗∈V^∗. Then, in terms ofTheorem 4.4we derive the desired result.

Corollary 4.830, Theorem 4.2. LetAandT be two mappings from Banach spaceV to its dual V^∗,J :V → R be a locally Lipschitz functional, andG:V → R∪ {∞}be a proper, convex, and lower semicontinuous functional. Then the variational-hemivariational inequality VHVI is strongly (resp., weakly) well posed in the generalized sense if and only if the corresponding inclusion problem IPAT−f∂J∂Gis strongly (resp., weakly) well posed in the generalized sense.

5. Concluding Remarks

In this paper, we introduce some concepts of well-posedness for a class of strongly mixed variational-hemivariational inequalities with perturbations, which includes as a special case the class of variational-hemivariational inequalities in30. We establish some metric char- acterizations for the well-posed strongly mixed variational-hemivariational inequality and give some conditions under which the strongly mixed variational-hemivariational inequality is strongly well posed in the generalized sense inRⁿ. On the other hand, we first recall the concept of well-posedness for inclusion problems and then investigate the relations between the strongresp., weakwell-posedness for a strongly mixed variational-hemivariational ine- quality and the strongresp., weakwell-posedness for the corresponding inclusion problem.

It is well known that there are many other concepts of well-posedness for optimization problems, variational inequalities, and Nash equilibrium problems, such asα-well-posedness 17, well-posedness by perturbations12, and Levitin-Polyak well-posedness38. How- ever, we wonder whether the concepts mentioned as above can be extended to the strong- ly mixed variational-hemivariational inequality. Beyond question, this is an interesting prob- lem.

Acknowledgments

This paper was partially supported by the National Science Foundation of China11071169, Innovation Program of Shanghai Municipal Education Commission09ZZ133, and Leading Academic Discipline Project of Shanghai Normal UniversityDZL707 to L.-C. Ceng. This paper was partially supported by the Taiwan NSC Grant 99-2115-M-110-007-MY3to N.-C.

Wang. This paper was partially supported by the Taiwan NSC Grant 99-2221-E-037-007-MY3 to J.-C. Yao.