volume 3, issue 2, article 28, 2002.

*Received 5 March, 2001;*

*accepted 30 January, 2002.*

*Communicated by:**Z. Nashed*

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**Journal of** **Inequalities in** **Pure and** **Applied** **Mathematics**

**TWO REMARKS ON THE STABILITY OF GENERALIZED**
**HEMIVARIATIONAL INEQUALITIES**

MOHAMED AIT MANSOUR

Cadi Ayyad Univesity Semlalia Faculty of Sciences Department of Mathematics , B.P. 2390, 40 000-Marrakesh, Morocco.

*EMail:*mansour@ucam.ac.ma

*URL:*http://www.angelfire.com/nb/mansour/

c

2000Victoria University ISSN (electronic): 1443-5756 020-01

**Two Remarks on the Stability of**
**Generalized Hemivariational**

**Inequalities**
Mohamed Ait Mansour

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**Abstract**

The present paper is devoted to the stability analysis of a general class of hemi- variational inequalities. Essentially, we present two approaches for this class of problems. First, using a general version of Minty’s Lemma and the convergence result of generalized gradients due to T. Zolezzi [23], we prove a stability result in the spirit of Mosco’s results on the variational inequalities [14]. Second, we provide a quite different stability result with an estimate for the rate of conver- gence of solutions when the given perturbed data are converging with respect to an appropriate distance. Illustration is given with respect to a hemivariational inequality modelling the buckling of adhesively connected von kármán plates.

*2000 Mathematics Subject Classification:*49J40, 40J45, 49J52

*Key words: Generalized hemivariational inequalities, Clarke’s gradient, Perturbation,*
Epi-convergence, Stability, Rate of convergence, Equilibrium problems,
von kármán plates.

The author is deeply grateful to the anonymous referee for his suggestions, useful comments and pertinent remarks.

**Two Remarks on the Stability of**
**Generalized Hemivariational**

**Inequalities**
Mohamed Ait Mansour

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**Contents**

1 Introduction. . . 4

2 Mechanical Example. . . 6

3 Main Convergence Results. . . 13

3.1 Epi-convergence approach . . . 13

3.2 Distances approach . . . 21

4 Application . . . 27

4.1 Equilibrium of the von kármán plates . . . 27

5 Comments . . . 32 References

**Two Remarks on the Stability of**
**Generalized Hemivariational**

**Inequalities**
Mohamed Ait Mansour

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**1.** **Introduction**

The theory of inequalities has received remarkable developments in both pure and applied mathematics as well as in mechanics, engineering sciences and eco- nomics. This theory has been a key feature in the understanding and solution of many practical problems such as market price equilibria, traffic assignments, monetary policy setting and so on. In this context, variational inequalities have been the appropriate framework for studying some of these problems during the last forty years. More recently, new and efficient mathematical inequali- ties, called hemivariational inequalities, have facilitated the solution to many challenging open questions in mechanics and engineering. This class of prob- lems has been pioneered by the work of Panagiotopoulos [18] who introduced a variational formulation involving nonconvex and nonsmooth energy functions.

Subsequently, it has been developed from the point of view of existence results by many authors, we refer to [5,6], [10], [17], [16], [20] and references therein.

In this paper, we attempt to investigate stability results for the following
generalized hemivariational inequalities: for anyn ∈N,findu_{n} ∈X such that
for allv ∈X

(GHI_{n}) Φ_{n}(u_{n}, v) + Ψ_{n}(u_{n}, v) +J_{n}^{0}(u_{n};v−u_{n}) +ϕ_{n}(v)−ϕ_{n}(u_{n})≥0.

holds.

Here, X is a Banach space, (Φ_{n})n≥0,(Ψ_{n})n≥0 are sequences of real valued
bifunctions defined onX×X,(ϕn)n≥0a sequence of extended real valued func-
tions and(J_{n})_{n}a sequence of real locally Lipschitz functions;J_{n}^{0}is the Clarke’s
derivative ofJ_{n}.The main question is then the following : under what condi-
tions do the solutionsunto (GHIn) converge to a solution of the initial problem

**Two Remarks on the Stability of**
**Generalized Hemivariational**

**Inequalities**
Mohamed Ait Mansour

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(GHI_{0})?

The remainder of the paper is organized as follows. In Section2, we discuss a
concrete mechanical example that has motivated our study. Section3is devoted
to our main stability results. We present two approaches. Namely, we first pro-
pose a general version of Minty’s Lemma and proceed by the epi-convergence
method, Theorem3.1. Further, we define a “distance” between two bifunctions
and present a stability result with an estimate for the rate of convergence of
solutions in terms of the given data rate of convergence: Theorem 3.10is first
stated in equilibrium problems formulation and Corollary 3.11 is then derived
for(GHI_{0}). In Section4, we illustrate the abstract results by an application to
a hemivariational inequality that models the buckling of adhesively connected
von kármán plates allowing for delamination. Finally, we conclude with some
comments.

**Two Remarks on the Stability of**
**Generalized Hemivariational**

**Inequalities**
Mohamed Ait Mansour

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**2.** **Mechanical Example**

To illustrate the idea of hemivariational inequalities and explain how impor-
tant this class of inequalities is, we suggest the following model^{1} ^{1}summarized
from [15], further details and similar models can be found in [16,17, 18, 20].

The model is concerned with the buckling of adhesively connected von kármán
plates allowing for delamination. Roughly speaking, it consists of character-
izing the position on equilibrium of the plates and lead to research of solution
to special problem formulated as a hemivariational inequality. Let us now for-
mulate the problem. Consider a plate Ωand the binding material onΩ^{0}.In the
undeformed state, the middle of the plate occupies an open, bounded and con-
nected subset Ω of R^{2}, referred to a fixed right-handed Cartesian coordinate
system Ox_{1}x_{2}x_{3}. Let Γ be the boundary of the plate: Γis assumed to be ap-
propriately regular. Let also the binding material occupy a subsetΩ^{0} such that
Ω^{0} ⊂ΩandΩ¯^{0}T

Γ = ∅.We denote byζ(x)the vertical deflection of the point
x ∈ Ωof the plate, and byf = (0,0, f_{3}(x))the distributed vertical load. Fur-
ther, letu={u_{1}, u_{2}}be the in-plane displacement of the plate. We assume that
the plate has constant thickness h. Moreover, we assume that the plate obeys
the Von kármán theory, i.e. it is a thin plate having large deflections. The von
kármán plates verify the following system of differential equations:

K44ζ−h(σ_{αβ}ζ_{,β})_{,α}=f inΩ_{j},
(2.1)

σ_{αβ}_{,}_{β} = 0 inΩ_{j},
(2.2)

11We have recalled in details this model as it was stated in [15] in concern with existence of solutions, here we deal with stability issue under data perturbation for hemivariational inequal- ities modelling such problems.

**Two Remarks on the Stability of**
**Generalized Hemivariational**

**Inequalities**
Mohamed Ait Mansour

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σαβ =Cαβγδ(γδ(u) + 1

2ζ,γζ,δ) inΩj. (2.3)

Here the subscripts, α, β, γ, δ = 1,2 correspond to the coordinate directions:

{σ_{αβ}},{ε_{αβ}} andC_{αβγδ} denotes the stress, strain and elasticity tensors in the
plane of the plate. The components of C are elements ofL^{∞}(Ω) and have the
usual symmetry and ellipticity properties (further explanations and figures can
be found in [15]). Moreover,K = _{12(1−ν}^{Eh}^{3}2) is the bending rigidity of the plate
with E the modulus of elasticity and ν the Poisson ratio. For the sake of sim-
plicity, we consider here isotropic homogeneous plates of constant thickness.

In laminated and layered plates, the interlaminar normal stress σ_{33} is one of
the main cause for delamination effects. Note that this is a simplification of the
problem. In order to model the action ofσ_{33}, f is split into a vectorf ,¯ which de-
scribes the action of the adhesive andf ∈L^{2}(Ω),which represents the external
loading applied on the plate:

f =f +f inΩ.

We introduce now a phenomenological law connectingf¯with the correspond- ing deflection of the plate describing the action of adhesive material. We assume that:

(2.4) −f ∈β(ζ) inΩ^{0},

whereβ is a multivalued function defined as in [19] (by filling in the jumps in
the graph of a function β ∈ L^{∞}_{loc}(R)). We note here that cracking as well as
crushing effects of either a brittle or semi-brittle nature can be accounted for

**Two Remarks on the Stability of**
**Generalized Hemivariational**

**Inequalities**
Mohamed Ait Mansour

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by means of this law. The following relation completes in a natural way the definition off:

f = 0 in Ω−Ω^{0}.

In order to obtain a variational formulation of the problem, we express relation (2.4) in a superpotential form. Ifβ(ξ±0)exists for everyξ ∈Rthen, from [7]

and [19] a locally Lipschitz (nonconvex) functionJ :R→Rcan be determined up to an additive constant such that

β(ξ) = ∂J(ξ),

where ∂ is the generalized gradient of Clarke^{12}. Moreover, we suppose the
following boundary condition on the plate boundary:

ζ = 0 onΓ.

Now, let us denote by n the outward normal unit vector to Γ and by g_{α}
(α = 1,2) the self-equilibrating forces and assume for the in-plane action the
boundary conditions

(2.5) σ_{αβ}n_{β} =g_{α} onΓα = 1,2.

Notice that in [15], (2.5) involves an eigenvalueλsuch thatσαβnβ =λgα.Here
we take λ = 1. For the moment we assume that g_{α} = 0 α = 1,2. We can
now derive the variational formulation of the problem. From (2.1), by assuming

21For the convenience of the reader we recall (see [8]) that∂jis defined by∂j(x) = {ζ∈
Z^{∗}:hζ, vi ≤j^{0}(x;v) for allv inZ}andj^{0}(x;v) := lim sup

y→x t&0

1

t(j(y+tv)−j(y)).

**Two Remarks on the Stability of**
**Generalized Hemivariational**

**Inequalities**
Mohamed Ait Mansour

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sufficiently regular functions, multiplying byz^{(j)}−ζ^{(j)},integrating and applying
the Green-Gauss theorem, we obtain the expression(E):

α(ζ, z−ζ) + Z

Ω

hσ_{αβ}ζ_{,}α(z−ζ)_{,β}dΩ

= Z

Γ

hσ_{αβ}ζ_{,β}n_{α}(z−ζ)dΓ +
Z

Ω

f(z−ζ)dΩ +

Z

Γ

K_{n}(ζ)(z−ζ)dΓ−
Z

Γ

M_{n}(ζ)∂(z−ζ)

∂n dΓ.

Here,α, β = 1, ndenotes the outward normal unit vector toΓ, (2.6) α(ζ, z) = K

Z

Ω

[(1−ν)ζ_{,αβ}z_{,αβ}+ν4ζ4z]dΩ, 0< ν <0.5,

(2.7) M_{n}(ζ) = −K

ν4ζ+ (1−ν) 2n_{1}n_{2}ζ_{,12}+n^{2}_{1}ζ_{,11}+n^{2}_{2}ζ_{,22}
and

(2.8) K_{n}(ζ)

=−K ∂4ζ

∂n + (1−ν) ∂

∂τ

n_{1}n_{2}(ζ_{,22}−ζ_{,11}) + (n^{2}_{1}−n^{2}_{2})ζ_{,12}

,
where τ is the unit vector tangential to Γ such that ν.τ and the Ox_{3}-form a
right-handed system. A similar argument applied to (2.2) leads to the following
expression

(2.9) Z

Ω

σ_{αβ}ε_{αβ}(v−u)dΩ =
Z

Γ

σ_{αβ}n_{β}(v_{α}−u_{α})dΓ. α, β = 1,2.

**Two Remarks on the Stability of**
**Generalized Hemivariational**

**Inequalities**
Mohamed Ait Mansour

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Further, the following notations are introduced:

(2.10) R(m, k) = Z

Ω

CαβγδmαβkαβdΩ. α, β, γ, δ= 1,2.

and

(2.11) P(ζ, z) ={ζ_{,α}z_{,β}}, P(ζ, ζ) =P(ζ),
wherem={m_{αβ}}andk ={k_{αβ}}, α, β = 1,2are2×2tensors.

Let us also introduce the functional framework. We assume that u.v ∈
[H^{1}(Ω)]^{2}and thatζ, z ∈Z,where

Z ={z|z ∈H^{2}(Ω), z = 0onΓ}.

Taking into account expression(E),(2.9), the boundary conditions and the in- equalities defining the multivalued operator∂we obtain the following problem:

findu ∈ [H^{1}(Ω)]^{2} andζ ∈ Z such as to satisfy the hemivariational inequality
(HI):

α(ζ, z−ζ) +hR(ε(u) + 1

2P(ζ), P(ζ, z−ζ)) + Z

Ω^{0}

J^{0}(ζ, z−ζ)dΩ

≥ Z

Ω

f(z−ζ)dΩ. ∀z ∈Z and the variational equality(V E) :

R(ε(u) + 1

2P(ζ), ε(v−u)) = 0, ∀v ∈[H^{1}(Ω)]^{2}.

**Two Remarks on the Stability of**
**Generalized Hemivariational**

**Inequalities**
Mohamed Ait Mansour

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Further we shall eliminate the in-plane displacement of the plate. To this end
we note first that R(., .) as defined in (2.10) is a continuous symmetric, coer-
cive bilinear form on [L^{2}(Ω)]^{4} and that P : [H^{2}]^{2} → [L^{2}(Ω)]^{4} of (2.11) is a
completely continuous operator (see [20] p. 219). Thus the equality(V E)and
the Lax-Milgram theorem imply that to every deflection ζ ∈ Z, there corre-
sponds a plane displacementu(ζ)∈[H^{1}(Ω)]^{2}.Indeed, due to Korn’s inequality
R(ε(u), ε(v)) is a bilinear coercive form on the quotient space [H^{1}(Ω)]^{2}/R,¯
whereR¯is the space of in-plane rigid displacements defined by

(2.12) R¯ ={¯r/¯r∈[H^{1}(Ω)]^{2},r¯_{1} =α_{1}+bx_{2},r¯_{2} =α_{2}−bx_{2}, α_{1}, α_{2}, b∈R}.

From(V E)it results that

(2.13) ε(u(ζ)) :Z →[L^{2}(Ω)]^{4}

is uniquely determined and is completely quadratic function ofζ,sinceε(u(ζ))
is a linear continuous function of P(ζ).We also introduce the completely con-
tinuous quadratic functionG:Z →[L^{2}(Ω)]^{4} which is defined by

(2.14) ζ →G(ζ) = ε(u(ζ)) + 1

2P(ζ) and satisfies the equation

(2.15) R(G(ζ), ε(u(ζ))) = 0.

We now define the operator: A:Z →Z andC:Z →Zsuch that

(2.16) α(ζ, z) = (Aζ, z)

**Two Remarks on the Stability of**
**Generalized Hemivariational**

**Inequalities**
Mohamed Ait Mansour

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and

(2.17) hR(G(ζ), P(ζ.z)) = (C(ζ), z).

Ais a continuous linear operator,C a completely continuous operator and(·,·) denotes the scalar product inZ.Thus the following problem results:

findζ ∈Z,so as to satisfy the hemivariational inequality

(2.18) a(ζ, z−ζ)+(C(ζ), z−ζ)+

Z

Ω^{0}

j^{0}(ζ, z−ζ)dΩ≥
Z

Ω

f(z−ζ)dΩ∀z ∈Z.

The last hemivariational inequality characterizes the position of equilibrium of the studied problem. Note that the second member of (2.18) can be expressed by means of a linear, self-adjoint and compact operatorB. For the explicit form ofB, we refer to [20] (equation 7.2.13).

Therefore, this problem can be viewed as, and actually is, a particular case of(GHI0).

* Remark 2.1. Notice that if we take*J = 0,(GHI

_{0})

*covers the Generalized vari-*

*ational and quasi variational inequalities. Some other mathematical problems*

*contained in*(GHI

_{0})

*can be found in [4].*

**Two Remarks on the Stability of**
**Generalized Hemivariational**

**Inequalities**
Mohamed Ait Mansour

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**3.** **Main Convergence Results**

In this section, we present our stability results. By means of a general version of the celebrated Minty’s Lemma, we proceed first by the epi-convergence method.

In the sequel, unless another framework is specified, the space X is a Banach
space with dual X^{∗} equipped with the weak^{∗} topology denoted by w^{∗}. The
symbols →will stand for the strong convergence both in X and X^{∗}.We first
recall the following definitions:

* Definition 3.1. A sequence* f

_{n}: X → (−∞,+∞)

*is said to be equi-lower*

*semidifferentiable iff for every*x ∈ X

*there exists a ball*B

*around*x

*such that*

*for every*ε >0

*we can find*δ >0

*so as*

(3.1) f_{n}(z)≥f_{n}(y) +hu, z−yi −εkz−yk

*for every*y∈B,*every*n,*every*u∈∂^{−}f_{n}(y)*and every*z*such that*kz−yk ≤δ.

*Where*∂^{−}*denotes the lower semigradient given for some function*g*and*x∈X
*by:* u∈∂^{−}g(x)*iff*u∈X^{∗}*and*

lim inf

y→x (g(y)−g(x)− hu, y−xi)/ky−xk ≥0.

* Definition 3.2. A sequence* f

_{n}: X → (−∞,+∞)

*is called strongly epi-*

*convergent to*f :X →(−∞,+∞)

*iff*v

_{n}→ v

*implies*f(v) ≤lim inf

_{n}f

_{n}(v

_{n}),

*and for every*v ∈X

*there exists a sequence*v

_{n}→v

*such that:*lim sup

_{n}f

_{n}(v

_{n})≤ f(v).

**3.1.** **Epi-convergence approach**

Having our applications in mind, we make the following assumptions:

**Two Remarks on the Stability of**
**Generalized Hemivariational**

**Inequalities**
Mohamed Ait Mansour

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(H_{0}) Xis separable and has a equivalent norm that is Fréchet differentiable off
0;

(H_{1}) i) Φ_{0} is monotone, that is for eachu, v ∈K, Φ_{0}(u, v) + Φ_{0}(v, u)≤0;

ii) Φ_{0} is upper hemicontinuous i.e., for all u, v, w ∈ X, the map t ∈
[0,1]7→Φ_{0}(tu+ (1−t)v, w)is upper semicontinuous;

iii) Φ_{0} is convex on the second argument andΦ_{0}(u, u) = 0for allu∈X;

(H_{2}) Ψ_{0} is convex on the second argument andΨ_{0}(u, u) = 0for allu∈X;

(H_{3}) ϕ_{0} is proper and convex;

(H_{4}) Φ_{n}is monotone for eachnand(Φ_{n})lower-converges toΦ_{0} :∀u∈X, v ∈
X,∀u_{n}→uand∀(v_{n})_{n} →vit resultsΦ_{0}(u, v)≤lim inf_{n}Φ_{n}(u_{n}, v_{n}) ;
(H_{5}) (Ψ_{n})upper-converges toΨ_{0} :∀u∈X, v ∈X,∀u_{n}→uand∀v_{n}→vfor

a subsequencen_{k}one haslim sup_{k}Ψ_{n}_{k}(u_{n}_{k}, v_{n}_{k})≤Ψ_{0}(u, v) ;
(H6) the sequence(ϕn)nis strongly epi-convergent toϕ0;

(H_{7}) The sequenceJ_{n} is locally equi-Lipschitz, that is for every ball B in X
there existsM >0such that

|J_{n}(u)−J_{n}(v)| ≤Mku−vk
for allu, v ∈B and alln;

(H_{8}) (J_{n})_{n}is equi-lower semidifferentiable and strongly epi-convergent toJ_{0};

**Two Remarks on the Stability of**
**Generalized Hemivariational**

**Inequalities**
Mohamed Ait Mansour

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**Remark 3.1. We should notice that we do not need to make appeal to the as-***sumption*(H1)i)*since it is included in*(H4).*Indeed, for any*u, v ∈X *and for*
*some*u_{n}→u*and*v_{n} →v,*let us remark that*

Φ_{0}(u, v) + Φ_{0}(v, u) ≤ lim inf

n Φ_{n}(u_{n}, v_{n}) + lim inf

n Φ_{n}(v_{n}, n_{n})

≤ lim inf

n [Φ_{n}(u_{n}, v_{n}) + Φ(v_{n}, u_{n})]

≤ 0.

In the following theorem we denote byS_{n}the set of solutions to (GHI_{n}).

* Theorem 3.1. Suppose that assumptions* (H

_{0})−(H

_{8})

*are verified. Then, we*

*have*

s−lim inf

n S_{n}⊂S_{0}.

* Remark 3.2. The result of Theorem3.1means that whenever a sequence*u

_{n}

*of*

*solutions to (GHI*n

*) is strongly converging to*u,u

*is a solution to*(GHI0).

To prove this theorem, we first collect some lemmas.

* Lemma 3.2. [8] Let*g

*be a real Lipschitz function of rank*k

*near*x. Then, the

*function*v →g

^{0}(x;v)

*is positively homogeneous and subadditive (thus convex),*

*continuous and Lipschitz of rank*k

*on*X.

* Lemma 3.3 (Minty’s). Let* f

*be an extended real-valued bifunction such that*f

*is convex in the second argument and*f(v, v) = 0

*for each*v ∈ X.

*Assume*

*moreover that*(H

_{1})

*hold, then the following statements are equivalent.*

a) *There exists*u∈X *such that for every*v ∈X,
Φ_{0}(u, v) +f(u, v)≥0 .

**Two Remarks on the Stability of**
**Generalized Hemivariational**

**Inequalities**
Mohamed Ait Mansour

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b) *There exists*u∈X *such that for every*v ∈X,
Φ_{0}(v, u)≤f(u, v).

*Proof.* a)⇒b) Letu∈Xsuch thata)is satisfied. Thus we have

−Φ_{0}(u, v)≤f(u, v).

since Φ_{0} is monotone, it follows that Φ_{0}(v, u) ≤ −Φ_{0}(u, v), ∀v ∈ X. There-
fore, for everyv ∈X we have

Φ_{0}(v, u)≤f(u, v),
which means thatb)is verified.

b) ⇒ a) Let u be a solution inb) and fixv ∈ X and t ∈]0,1[. Then, using
(H_{1})iii)and the convexity off(u, .), forw_{t}=tu+ (1−t)v, we have

0 = Φ_{0}(w_{t}, w_{t}) ≤ (1−t)Φ_{0}(w_{t}, v) +tΦ_{0}(w_{t}, u)

≤ (1−t)Φ_{0}(w_{t}, v) +tf(u, w_{t})

≤ (1−t)Φ0(wt, v) +t(1−t)[f(u, v)].

Becausef(u, u) = 0.Therefore,

−t[f(u, v)]≤Φ_{0}(w_{t}, v).

Hence, by upper hemicontinuity ofΦ_{0},we end at

−[f(u, v)]≤lim sup

t→1

Φ(w_{t}, v)≤Φ_{0}(u, v)

**Two Remarks on the Stability of**
**Generalized Hemivariational**

**Inequalities**
Mohamed Ait Mansour

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which leads to

0≤Φ0(u, v) +f(u, v).

Finally,vbeing arbitrary chosen inX, the last inequality means thata)is satis- fied

**Remark 3.3. Notice that a particular case of Lemma***3.3* *is the variational*
*Minty’s Lemma given in [13, p. 249] as follows:*

*find*u∈X *such that*: hl, v−ui ≤ hA(u), v−ui *for all*v ∈X
*is equivalent to*

*find*u∈X *such that*: hl, v−ui ≤ hA(v), v−ui *for all*v ∈X

*where*A*is an hemicontinuous and monotone operator from a Banach space*X
*into its topological dual*X^{∗}*, and*l ∈X^{∗}*.*

* Lemma 3.4. [8] Let*g

*be as stated in Lemma3.2. Then,*∂g(x)

*is a nonempty,*

*convex, week*

^{∗}−compact subset ofX

^{∗}

*and*kζk ≤k

*for each*ζ ∈∂g(x).

* Lemma 3.5. Assumption* (H

_{7})

*holds. Then, the sequence of set-valued map*(∂J

_{n})

_{n}

*is uniformly bounded.*

*Proof. Let* u_{n} be a bounded sequence. (u_{n})_{n} is then contained in a ball B =
B(0, r)wherer >0.Let alsoM be a positive constant such that

|J_{n}(u)−J_{n}(v)| ≤Mku−vk for all u, v ∈B and all n.

Therefore by Lemma3.4we deduce that:k∂J_{n}(u_{n})k ≤M,that is wheneverξ∈

∂Jn(un)we havekξk ≤M.This means that∂Jnis uniformly bounded.

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* Lemma 3.6. [8] Consider*g

*as stated in Lemma3.2. Then, For every*v

*in*X,

*one has*

g^{0}(x;v) = max{hζ, vi:ζ ∈∂g(x)}.

* Lemma 3.7. Under* (H0), (H7)

*and*(H8)

*, for any*u, v ∈ X

*and any*un → u, v

_{n}→v

*there exists a subsequence*(n

_{k})

_{k}

*such that*

lim sup

k

J_{n}^{0}

k(u_{n}_{k};v_{n}_{k})≤J_{0}^{0}(u;v).

**Remark 3.4. To simplify the notation we consider, without loss of generality,***the inequality of Lemma3.7as:*

lim sup

n

J_{n}^{0}(u_{n};v_{n})≤J_{0}^{0}(u;v).

*Proof. Let*u_{n} → u, ξ_{n} ∈ ∂J_{n}(u_{n})and letv_{n} → v.As, for eachn, ∂J_{n}(u_{n})is
weakly compact, there exists a mapξ_{n} :X →X^{∗} defined, for eachw∈ X,as
follows: ξ_{n}(w)∈∂J_{n}(u_{n})such that

(ξ_{n}(w), w) = max

ξ∈∂Jn(un)(ξ, w) =J_{n}^{0}(u_{n};w).

Sinceu_{n}is bounded, by Lemma3.5it results that(ξ_{n}(v_{n}))_{n}is bounded. There-
fore,(ξ_{n}(v_{n}))_{n}has a weakly converging subsequence also denoted by(ξ_{n}(v_{n}))_{n}.
Let ξ(v) ∈ X^{∗} be the weak^{∗} limit of ξ_{n}(v_{n}).On the other hand, (H_{7}) implies
that (J_{n})_{n} is locally equi-bounded. Then, by (H_{0}) and (H_{8}), we apply [23,
Theorem 1] and obtain

lim sup

n

gph ∂J_{n}⊂gph ∂J_{0} in (X, s)×(X^{∗}, w^{∗}),

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which implies thatξ(v) ∈ ∂J_{0}(u).Hence, taking Lemma 3.6into account, we
end at

lim sup

n

J_{n}^{0}(u_{n};v_{n}) = lim sup

n

ξ∈∂Jmaxn(un)hξ, v_{n}i

= lim sup

n

hξ_{n}(v_{n}), v_{n}i

=hξ(v), vi

≤ max

ξ∈∂J_{0}(u)hξ, vi=J_{0}^{0}(u;v).

*Proof of Theorem3.1. Let*u_{n} ∈ s−lim infS_{n}andube the strong limit ofu_{n}.
We wish to prove that u ∈ S_{0}. To this end, fix v ∈ X. By (H_{6}) there exists
a sequence (v_{n})_{n} such that v_{n} → v andlim sup_{n}ϕ_{n}(v_{n}) ≤ ϕ_{0}(v).As u_{n} is a
solution to(GHI_{n}),by monotonicity ofΦ_{n}it follows:

Φn(vn, un)≤Ψn(vn, vn) +J_{n}^{0}(un;vn−un) +ϕn(vn)−ϕn(un).

hence, taking into account(H_{4})−(H_{6})and Lemma3.7, there exists(n_{k})_{k}such
that

Φ_{0}(v, u) ≤ lim inf

k Φ_{n}_{k}(v_{n}_{k}, v_{n}_{k})

≤ lim sup

k

J_{n}^{0}

k(u_{n}_{k};v_{n}_{k} −u_{n}_{k})−lim inf

k ϕ_{n}_{k}(u_{n}_{k})
+ lim sup

k

ϕ_{n}_{k}(v_{n}_{k}) + lim sup

k

Ψ_{n}_{k}(v_{n}_{k}, v_{n}_{k})

≤ Ψ0(u, v) +J_{0}^{0}(u;v−u) +ϕ0(v)−ϕ0(u).

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therefore,

Φ_{0}(v, u)≤Ψ(u, v) +J_{0}^{0}(u;v−u) +ϕ_{0}(v)−ϕ_{0}(u).

Sinceϕ0is proper, it follows thatu∈dom(ϕ).Hence, asJ_{0}^{0}(u;.−u)is convex
(Lemma 3.2), we can take f = Ψ_{0}(u, v) +J_{0}^{0}(u;v −u) +ϕ_{0}(v)−ϕ_{0}(u)in
Lemma3.3and obtain

0≤Φ_{0}(u, v) + Ψ_{0}(u, v) +J_{0}^{0}(u;v −u) +ϕ_{0}(v)−ϕ_{0}(u).

Now,v being arbitrary chosen, we conclude thatuis a solution to(GHI0).The proof is therefore complete.

* Remark 3.5. Let us mention that, if we take* J

_{0}= 0

*in*(GHI

_{0}),

*this result is*

*not affected if the sequence of solutions is weakly converging. In this case we*

*shall obtain:*

w−lim infS_{n}⊂S_{0}.

*In fact, we have made recourse to strong convergence in* (H_{8})*because of the*
*presence of Clarke’s derivative in*(GHI_{0}).

From Theorem3.1we deduce the following variant of the stability results in [12,14].

* Corollary 3.8. Let* T

*and*T

_{n}

*, for each*n ≥ 1,

*be operators from*X

*to*X

^{∗}.

*Suppose that:*

*a)* T *is hemicontinuous on*X;

*b)* Tn*is monotone;*

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*c)* (T_{n})_{n} *converges to*T_{n}*in the sense that: for any*u ∈ X,*any sequence*u_{n}
*strongly converging to*u*we have*Tnun * T u;

*d)* (H_{6})*is satisfied.*

*Then, if a sequence*(u_{n})*of solutions to the variational inequality:*

(V I_{n}) *find*u∈X *such that*(T_{n}u, v−u) +ϕ_{n}(v)−ϕ_{n}(u)≥0 ∀v ∈X
*converging to a point*u, u*is a solution to the variational inequality:*

(V I) *find*u∈X *such that*(T u, v−u) +ϕ(v)−ϕ(u)≥0 ∀v ∈X.

*Proof. It suffices to take, for each* u, v ∈ X, Φ(u, v) = (T u, v − u) and
Φ_{n}(u, v) = (T_{n}u, v − u). The result is hence an easy consequence of Theo-
rem3.1.

The paragraph below presents a stability result without recourse to(H_{0})and
(H_{8}).

**3.2.** **Distances approach**

In this paragraph, we first present the stability result for the equilibrium prob-
lem. Further, we derive the result for (GHI_{0}).In this respect, we suppose that
X is a normed vector space with normk·kand assume that ϕ_{0} = 0. We shall
also consider a sequence of bifunctions F_{n} : X ×X → R and the following
equilibrium problems: for anyn ≥0findu_{n} ∈Xsuch that:

(EPn) Fn(un, v)≥0for all v ∈X

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To carry out our stability analysis, we need the following monotonicity as- sumption:

(A_{1}) F_{n}(u, v)+F_{n}(v, u)≤ −Mku−vk^{2}for allu, v ∈X, n≥1,whereM >0.

Fnwill be said−M-monotone.

* Remark 3.6. Let* C : X → X

^{∗}

*be a*r-Lipschitz operator, (wherer > 0) and B :X →X

^{∗}

*be a linear bounded operator. Let us define the bifunctions*h

*and*h

_{1}

*given by*

h(u, v) = hCu, v−ui*and*h_{1}(u, v) = hBu;v−ui.

*It is easily shown that*h*is*r-monotone andh_{1} *is*kBk-monotone.

Let us give now an-essential-example of bifunction satisfying a relaxed mono-
tonicity assumption of (A1). Let X = H be a Hilbert space, I the identity
mapping onHandJ_{0}a real locally Lipschitz function onX.

* Lemma 3.9. Suppose that, for some* α ∈ R, ∂J

_{0}+αI

*is monotone. Then, the*

*bifunction*g

*defined, for all*u, v ∈H,

*by*g(u, v) =J

_{0}

^{0}(u;v−u)

*is*α-monotone.

*Proof. Straightforward.*

**Remark 3.7. Let us remark that in lemma***3.9, we can easily check that*∂J_{0} *is*
*strongly monotone if* α < 0, monotone if α = 0 *and weakly nonmonotone if*
α > 0.*It is known from convex analysis that the monotonicity of* ∂J0 *leads to*
*convexity of* J_{0}.*Then, whenever*α ≤ 0*the problem*(GHI_{0})*comes back to the*
*generalized variational inequality, since in this case*J_{0} *is a convex , whereas if*
α >0*the function*J0 *is not necessarily convex.*

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* Remark 3.8. A special case of*J

_{0}

*is when it is defined as follows:*

J_{0}(u) =
Z

Ω

j(u(x))dx

*Here, the space*X*is supposed imbedded in*L^{p}(Ω)*with*Ω*an open bound subset*
*of*R^{n},*and*

j(t) = Z t

0

β(s)ds; β ∈L^{∞}_{loc}(R).

*Notice that in [1], the authors provided some condition on*β *so as to satisfy the*
*monotonicity condition of Lemma3.9. Precisely, they considered the following*
*property:*

t_{1} ≤t_{2} ⇒β^{+}(t_{1})< β−(t_{2}) +γ(t_{2}−t_{1})^{r},
(3.2)

*where*γ, r >0*and*β^{+}*and*β−*are given by*
β^{+}(t) = lim

δ→0 ess sup

|s−t|≤δ β(s), β−(t) = lim

δ→0 ess inf

|s−t|≤δ β(s)*for some* t∈R.
*Using this assumption, it is argued in [1] that*J_{0} *is*K-monotone for some con-
*stant*K >0.

**Remark 3.9. The condition of Lemma**3.9is nothing else than a relaxed form of*monotonicity for*∂J_{0} *but it keeps the nonconvex framework for the energy func-*
*tion*J_{0}.*After we have finished this work we have realized that a such condition*
*was used by Naniewicz and Panagiotopoulos in [16] ( Chapter 7) for existence*
*results.*

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Before stating the main result of this paragraph, we introduce the following

"distance" between two bifunctionsf andg as follows:

ρ_{τ}(f, g) := sup

u6=v,kuk≤τ

|(f −g)(u, v)|

ku−vk whereτ >0.

* Remark 3.10. Let*A

*and*B

*be two operators from*X

*toX*

^{∗}.

*We associate to*A

*and*B

*two bifunctions as follows:*

f_{A}(u, v) = (Au, v−u); f_{B}(u, v) = (Bu, v−u).

*It is readily shown that*

ρ_{τ}(A, B) := ρ_{τ}(f_{A}, f_{B})≤d_{τ}(A, B)
*where*dτ *is the classical ”distance” defined by*

d_{τ}(A, B) := max

kuk≤τkA(u)−B(u)k.

Assume that the set of solutions to(EP_{0}),also denoted byS_{0},is nonempty
and bounded and letτ >0such thatS_{0} ⊂B(0, τ).We claim the following:

* Theorem 3.10. Assume that assumption*(A

_{1})

*holds and the sequence*F

_{n}

*con-*

*verge, following*ρτ,

*to*F

*. Then, whenever the solution*un

*to*(EPn)

*exists it*

*must be unique and strongly convergent to the unique solution*u

_{0}

*to*(EP

_{0})

*and*

*we have*

ku_{n}−u_{0}k ≤ 1

Mρ_{τ}(F_{n}, F_{0}).

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*Proof. Let us first establish, for each*i≥1,the following estimation:

ku_{i}−u_{0}k ≤ 1

Mρ_{τ}(F_{i}, F_{0}).

Leti≥1.Since, forj = 0, i, u_{j} is a solutions to(EP_{j}), we have
F_{j}(u_{j}, v)≥0 for all v ∈X

thus, we make in(EP_{j}), v =u_{m} form = 0, iandm 6=j, and adding the two
relations we obtain:

F_{i}(u_{i}, u_{0}) +F_{0}(u_{0}, u_{i})≥0.

Therefore

F_{i}(u_{i}, u_{0})−F_{0}(u_{i}, u_{0}) +F_{0}(u_{i}, u_{0}) +F_{0}(u_{0}, u_{i})≥0.

Taking into account(A1), we deduce that

Mku_{i}−u_{0}k^{2} ≤ F_{0}(u_{0}, u_{i})−F_{i}(u_{0}, u_{i})

≤ ρτ(Fi, F0)kui−u0k which leads to

(3.3) Mku_{i}−u_{0}k ≤ρ_{τ}(F_{i}, F_{0}).

Now forn ≥ 1, the uniqueness of solutionu_{n} to(EP_{n})comes (3.3). Fur-
thermore, we have

ku_{n}−u_{0}k ≤ 1

Mρ_{τ}(F_{n}, F)

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therefore, we conclude that
e(u_{n}, S) := sup

w∈S

ku_{n}−wk →0asn goes to +∞.

Hence, u_{n} strongly converges to someuwhich must be the unique solution to
(EP_{0}).The proof is then finished.

We are now in a position to derive a result with estimation of solutions to
(GHI_{0}).We hence claim the following:

* Corollary 3.11. Assume that* X

*is Hilbert space, for each*n ≥ 1 Φ

_{n}

*is*−γ-

*monotone for some*γ > 0, ∂J

_{n}+αI

*is monotone for some*α > 0, (Ψ

_{n})

_{n}

*is*c-monotone andγ > α+c. Then, ifτ >0

*is such that*S

_{0}⊂B(0, τ),

*whenever*

*the solution*u

_{n}

*to*(GHI)

_{n}

*exists is unique and the following estimation holds:*

ku_{n}−u_{0}k ≤ 1

(γ−α−c)[ρ_{τ}(Φ_{n},Φ_{0}) +ρ_{τ}(Ψ_{n},Ψ_{0}) +ρ_{τ}(g_{n}, g_{0})].

*If moreover, the sequences*(Φ_{n}),(Ψ_{n})*and*(g_{n})*converge with respect to*ρ_{τ}*,*
*then*u_{n}*strongly converges to*u_{0}.

Here we have adopted the notation:

g_{n}(u, v) :=J_{n}^{0}(u;v−u).

*Proof. Let us take:*

F_{n}(u, v) = Φ_{n}(u, v) +J_{n}^{0}(u, v−u) + Ψ_{n}(u, v).

Using Lemma3.9, we see thatF_{n}is(γ−α−c)-monotone. The result is hence
direct from Theorem3.10.

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**4.** **Application**

**4.1.** **Equilibrium of the von kármán plates**

We treat here a mathematical problem which simply models the equilibrium
problem of the von kármán plates presented In Section2. In this way, letV be a
Hilbert space with scalar product(·,·)and the associated normk.k. SpaceV is
supposed densely and compactly imbedded intoL^{p}(Ω,R)for somep≥2.Here
Ωis a bounded domain inR^{N}.We shall consider a bilinear forma :V×V →R,
a nonlinear operator C : V → V, a function β ∈ L^{∞}_{loc}(R) and the locally
Lipschitz function j defined by: j(t) = Rt

0 β(s)ds, t ∈ R. The problem is formulated as a hemivariational inequality: findu∈V so as to satisfy:

(EV KP) a(u, v) + (Cu, v) + Z

Ω

j^{0}(u(x);v(x))dx≥0∀v ∈V
which is equivalently expressed as

((EV KP)_{equi}) a(u, v−u) + (Cu, v−u) +
Z

Ω

j^{0}(u(x);v(x)−u(x))dx

≥0∀v ∈V.

Indeed, suppose thatuis a solution to (EV KP) and letv ∈V.By makingv^{0} =
v−uin (EV KP) we see thatusolves(EV KP)_{equi}.Ifusolves(EV KP)_{equi},
for anyv ∈V we takev^{0} =v+uto see thatuis a solution to (EV KP).

**Remark 4.1. The solutions to this problem have been provided in [15] by use***of critical point theory and other results are also established for a similar form*
*of (EV KP) by means of Ky Fan’s minimax inequality in [1].*

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Let us remark that if we setJ :L^{p}(Ω) →Rdefined byJ(u) =R

Ωj(u(x))dx u∈ V, the problem (EV KP) can be regarded in the form of (GHI). Moreover, it is possible to prove that these two problems are equivalent under suitable as- sumptions. The following lemma argue the passage from (GHI) to (EV KP).

* Lemma 4.1. Assume that for some*α

_{1}∈R

*and*α

_{2}>0, we have (H) |β(s)| ≤α

_{1}+α

_{2}|s|

^{p−1},∀s∈R.

*Then every solution to*

(GHI) a(u, v) + (Cu, v) +J^{0}(u;v)≥0∀v ∈V
*is also a solution to (EV KP).*

*Proof. We should first mention that, in view of assumption (H),* J is well de-
fined and locally Lipschitz on L^{p}(Ω) (see [7]). Now let u be a solution to
(GHI). Let us remark that, following Example 1 in [7], the assumption (H)
ensures that

∀s ∈R, ∀ξ ∈∂j(t),|ξ| ≤α1+ 2^{p−1}α2|s|^{p−1}.

Hence, sinceV is dense inL^{p}(Ω)we can apply Theorem 2.7.5 of [8] and The-
orem 2.2 of [7] to conclude that:

∂J_{/V}(u)⊂
Z

Ω

∂j(u(x))dx.

On the other hand, sinceuis a solution to (GHI), it follows that

−α(u, v)−(Cu, v)≤J^{0}(u;v)∀v ∈V.

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Therefore, by definition Clarke’s gradient, it results that:

−α(u, .)−(Cu, .)∈∂J_{/V}(u)⊂
Z

Ω

∂j(u(x))dx.

Which is interpreted as:

−α(u, v)−(Cu, v) ≤ Z

Ω

max

z∈∂j(u_{(}x))z(v(x))dx

≤ Z

Ω

j^{0}(u(x), v(x))dx ∀v ∈V.

uis henceforth a solution to (EV KP).

Now, by varyinga, CandJwe consider the perturbed problem: findu_{n}∈V
so as to satisfy:

(EV KP)_{n} a_{n}(u_{n}, v) + (C_{n}u_{n}, v) +J_{n}^{0}(u_{n};v)dx≥0∀v ∈V.

Consequently from Theorem3.1we have the following stability result for (EV KP).

**Corollary 4.2. Assume that:**

*i)* a*is positive, that is*a(u, u)≥0∀u∈V *and continuous;*

*ii)* a_{n}*is positive for each*n*and for all*u, v ∈V,*all* u_{n}→u *and all* v_{n}→v
*it results*

a(u, v)≤lim inf

n an(un, vn);

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*iii)* (C_{n})_{n}*converges to*C,*that is for all*u, v ∈V,*all* u_{n}→u *and all* v_{n}→v
*it results*

lim sup

n

C_{n}(u_{n}, v_{n})≤(Cu, v);

*vi) Assume that*(H_{0})*holds and*(J_{n})_{n}*satisfies assumptions*(H_{7})*and*(H_{8})*of*
*Theorem3.1.*

*Then whenever the sequence* (u_{n})_{n} *of solutions to* (EV KP_{n})_{equi} *strongly*
*converges to*u, u*is a solution to*(EV KP)_{equi}.

We now apply the result of the second approach.

**Corollary 4.3. Assume that**

h1) *for each*n, an*is*γ-coercive, that isan(u, u)≥γkuk^{2} ∀u∈V;
h_{2}) *for each*n, C_{n}*is Lipschitz of rank*c >0;

h_{3}) ∂J_{n}+αI *is monotone, for each*n,*for some*α >0;

h_{4}) *the sequences*(a_{n})_{n},(C_{n})_{n} *and*(g_{n})_{n} ρ_{τ}*-converges to*a, C *and*g *respec-*
*tively where*τ *is such that the solutions to*(EV KP)_{equi}*are contained in*
B(0, τ).

*Then, if*γ > α+c, the solutionu_{n}*to*(EV KP_{n})_{equi}*is unique and strongly*
*converging to the unique solution*u *to*(EV KP)_{equi} *and the following estima-*
*tion holds:*

ku_{n}−uk ≤ 1

(γ−α−c)[ρ_{τ}(a_{n}, a) +ρ_{τ}(C_{n}, C) +ρ_{τ}(g_{n}, g)].

**Two Remarks on the Stability of**
**Generalized Hemivariational**

**Inequalities**
Mohamed Ait Mansour

Title Page Contents

### JJ II

### J I

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**J. Ineq. Pure and Appl. Math. 3(2) Art. 28, 2002**

http://jipam.vu.edu.au

*Here* ρ_{τ}(a_{n}, a) := ρ_{τ}(f_{a}_{n}, f_{a}) *with* f_{a}_{n}(u, v) = a_{n}(u, v −u) *and* f_{a}(u, v) =
a(u, v−u).

* Remark 4.2. Assume moreover, for each*n,

*that*a

_{n}

*is continuous. Then, thanks*

*to remark3.10the estimation of the last corollary leads to*

kun−uk ≤ 1

(γ−α−c)[kan−ak+dτ(Cn, C) +ρτ(gn, g)].