Convergence analysis of perturbed hemivariational inequalities

J. Appl. Math. & Computing Vol. 14(2004), No. 1 - 2, pp. 329 - 341

CONVERGENCE ANALYSIS OF PERTURBEDHEMIVARIATIONAL INEQUALITIES

MOHAMED AIT MANSOUR AND HASSAN RIAHI

Abstract. We consider the rate of convergence for a class of perturbedhemivariational inequalities in reflexive Banach Spaces. Our results can beviewed as an extension and refinement of some previous known results inthis area.

AMS Mathematics Subject Classification : 49J40, 40J45, 49J52Key words and phrases : Hemivariational inequalities, Clarke’s generalizedgradient, perturbation, stability, rate of convergence.

1. Introduction

The point of departure for the modern theory of hemivariational inequalities((HI) for short) are papers written about 20 years ago by P. D. Panagiotopoulos[18, 19], wherein the author introduced a variational formulation in mechanicsfor problems involving nonconvex and possibly nonsmouth energy functions.Given a Hilbert space V which is supposed imbedded in Lp(Ω), Ω being an opensubset of Rn and p ≥ 2, the basic case of (HI) is as follows : find u ∈ V suchthat

α(u, v − u) +∫

Ω

j0(u(x); v(x) − u(x))dx ≥ 0 ∀v ∈ V,

where α is real bilinear form defined on V ×V and j0 is the Clarke’s directionalderivative of a real locally Lipschitz function j. This problem can be consideredas a nonconvex generalization of the classical variational inequalities initiatedby G. Stampacchia. Indeed, whenever j is assumed to be smouth or convex,then the hemivariational expression becomes a standard variational formulation.Some typical models in mechanics and engineering can be found in the books ofNaniewicz and Panagiotopoulos [13] and Panagiotopoulos [20, 22]. During thelast decade, much attention has been focused to the existence theory of such

Received January 15, 2002. Revised June 12, 2003.

c© 2004 Korean Society for Computational & Applied Mathematics and Korean SIGCAM.

329

330 On perturbed hemivariational inequalities

inequalities using different approaches, see for instance [4, 5], [9]-[13], [20]-[23]and references therein. Whereas, to the best of our knowledge, the analysis ofperturbation for these inequalities have not been proposed sufficiently.

In [1], the author initiated some remarks on the generalized hemivariationalinequalities and presented epi-convergence result of Mosco type with restrictionon the space and provided an alternative based on the distances techniques out-side of Hilbert structure, there a control of Noor type was crucial to concludestability properties as in sensitivity analysis for many studied variational inequal-ities. In [2], the distance approach was efficient for special class of eigenvaluesproblem for hemivariational inequalities. Indeed, the control of Noor in [2] wasrelaxed to a choice of special eigenvalues out of the spectrum of the consideredbilinear form.

The present work aims to complete these previous studies by consideringhemivariational inequality where the energy function is not necessary defined asan integral: find u0 ∈ V so as to satisfy

(HI) (Au0, v − u0) + (Bu0, v − u0) + J0(u0; v − u0) ≥ (l, v − u0) for all v ∈ V,

where V is a reflexive Banach space and V ∗ it’s dual. V is assumed imbeddedin Lp(Ω), A : V → V ∗ a linear continuous operator, B : V → V ∗ a nonlinear op-erator, l ∈ V ∗ and J0 denotes the Clarke’s directional derivative of the functionof locally Lipshcitz function J. Under suitable assumptions, one can show thatthis problem remains the same if we take J 1 as follows J : Lp(Ω) → R defined,for each u ∈ Lp(Ω), by

J(u) =∫

Ω

j(u(x))dx with j(t) =∫ t

0

β(s)ds

for some β ∈ L∞loc(R).

Note that if J = 0, then hemivariational inequalities reduces to a class of varia-tional inequalities considered and studied by Noor [14].

We shall also consider, for n ≥ 1, a sequence (An)n of linear continuousoperators, a sequence (Bn)n of nonlinear operators, a sequence(Jn)n of locallyLipschitz real functions and a sequence (ln)n of linear continuous forms and, forany n ≥ 1, the following hemivariational inequality (HI)n : find un ∈ V suchthat

(Anun, v − un) + (Bnun, v − un) + J0n(un; v − un) ≥ (ln, v − un) for all v ∈ V.

Here our aim is to establish a stability result with an estimate for the rate ofconvergence of solutions in terms of the rate of convergence of An, Bn, Jn andln. As far as we know, the convergence analysis with estimate of the rate ofconvergence for variational inequalities was presented first in [8], wherein theauthors have dealt with the following two monotone variational inequalities in

1Following [6], under suitable assumption, one can prove that the function J is locallyLipschtiz and hence the Clarke’s derivative J0 is well defined.

Mohamed Ait Mansour and Hassan Riahi 331

a Hilbert space H : given two operators A1, A2 : H → H, two closed convexsubsets K1, K2 and f1, f2 ∈ H, these problems consist on seeking, for i = 1, 2,ui ∈ Ki so as to satisfy

(V Ii) 〈Aiui, v − ui〉 ≥ 〈fi, v − ui〉 for all v ∈ Ki.

The crucial step in their stability result was the following estimate between thesolutions u1 and u2:

‖u1 − u2‖ ≤ c[%(K1, K2) + ‖f1 − f2‖ + a(A1, A2)], (1.1)

for appropriate ”distances” %, a and a positive constant c. This connection be-tween solutions have been based on the obvious fact that solutions to (V In) arecharacterized by means of the projection PKn into Kn. In fact, it is a simplematter to see that u solves (V In) if and only if

u = PKn(u − γ(Anu − fn)),

γ being an arbitrary positive number.This technique is not applicable for our problem. Actually, we can not find a

Lipschitz operator B : V → V ∗, which satisfies

〈B(u), v〉 := J0(u; v) ∀u, v ∈ V. (1.2)

In the course of the proof of our main result, we explain that (1.2) is possible,whenever the Clarke’s derivative coincide with the Gateaux derivative. Unfor-tunately, with the Gateaux derivative, the considered hemivariational inequalityreduces to a classical variational inequality. Therefore, our result proposes analternative that avoids the projections method and covers both the smouth andnonsmouth situations of energy functional J.

The remainder of the paper is organized as follows. In section 2, we recallsome nonsmouth analysis materials that we will need. In Section 3, we state ourmain convergence result. Namely, we show under certain assumptions that thesequence un of solutions to (HIn) strongly converges to the solution u0 of theinitial problem (HI) with an estimate for the rate of convergence, Theorem 3.Section 4 is devoted to proof of Theorem 3. We proceed first by estimatingthe distance ‖ui − u0‖ between a solution to (HI)i (i ≥ 1) and u0 in terms of‖li − l‖, ‖Ai − A‖ and adequate distances between the operators Bi and B andbetween the functions Ji and J : Proposition 2 is stated when nonconvex energyfunctions are smouth and Proposition 3 recovers the nonsmouth case. Further,we conclude the proof of Theorem 3 when the sequences An, Bn, Jn and ln areconverging in adequate sense to A, B, J and l, respectively.

2. Nonsmouth analysis background

As the problem (HI) is expressed by means of the Clarke’s generalized deriva-tive, for the convenience of the reader we first recall some definitions and results


related to this notion. Let X be a Banach space with norm ‖.‖, X∗ denotes thedual space to X and 〈., .〉 the corresponding duality paring. Let f be Lipschitzof rank k near x.The Clarke’s generalized derivative of f is defined as follows

f0(x; v) := lim supy→xt0

1t

(f (y + tv) − f (y))

and the generalized gradient of f at x, denoted by ∂f(x), is the subset of X∗givenby :

∂f(x) = ζ ∈ X∗ : 〈ζ, v〉 ≤ f0(x; v) for all v in X.When we suppose that f is continuously differentiable, ∂f(x) is reduced to∇f(x).

Proposition 1[7]. The following statements hold.(a) ∂f(x) is a nonempty, convex, weak∗−compact subset of X∗ and ‖ζ‖ ≤ k

for each ζ ∈ ∂f(x).(b) For every v in X, one has f0(x; v) = max〈ζ, v〉 : ζ ∈ ∂f(x).

Let us consider now (T, B, µ) a positive complete measure space with µ(T ) <+∞, and Y a separable space. Let X be a closed subspace of Lp(T, Y ) (1 ≤p ≤ +∞), we define a functional f on X as follows :

f(x) =∫

T

ft(x(t))µ(dt),

where ft :→ R is a given family of functions. We suppose that, for each y in Y,the function → ft(y) is measurable, and that x is a point where f(x) is finite.The following theorem gives a characterization of ∂f(x).

Theorem 1[7]. Suppose that each function ft is locally Lipschitz on Y, and, forsome positive constant c, one has

ζ ∈ ∂ft(y) implies ‖ζ‖Y ∗ ≤ c1 + ‖y‖p−1Y .

Then, f is Lipschitz on bounded subsets of X and one has

∂f(x) ⊆∫

T

∂ft(x(t))µ(dt) for all x ∈ X, (2.1)

where the interpretation of (2.1) is as follows : to every ζ in ∂f(x), there corre-sponds a mapping → ζ(t) from T to Y ∗ satisfying the following assertions:

(i) ζ(t) ∈ ∂ft(x(t)) µ − a.e , (almost everywhere for the measure µ),(ii) For every v in X, the function → 〈ζ(t), v(t)〉 belongs to L1(X, R),(iii) For all v in X, one has 〈ζ, v〉 =

∫T 〈ζ(t), v(t)〉µ(dt).


Lemma 1[7]. (see also [6], Example 2.2.5.) Let Φ ∈ L∞loc(R). Then the function

defined by Ψ(t) =∫ t

0Φ(s)ds is locally Lipschitz, and for Φ+(t) = lim

δ→0essup|s−t|≤δ

Φ(s)

and Φ−(t) = limδ→0

essinf|s−t|≤δ

Φ(s), we have Ψ0(t; z) ≤ Φ+(t)z if z > 0 and

Ψ0(t; z) ≤ Φ−(t)z if z < 0. Moreover ∂Ψ(t) ⊂ [Φ−(t), Φ+(t)].

Theorem 2[6]. Let X, Y be two Banach spaces. Suppose that X is reflexive,dense in Y and the embedding mapping is continuous . Let f be a locally Lipschitzcontinuous function in Y, and let f = f/X , then we have

∂f(x) ⊂ ∂f(x) for each x ∈ X.

3. Convergence result

Throughout the remainder of the paper, the space V will be supposed denselyand continuously imbedded in Lp(Ω) (which is natural and suggested by theformulation of (HI), see for instance [10]), where Ω is an open bounded subsetof Rn and p > 1. If we denote by ‖.‖ the norm of V and by ‖.‖p the norm ofLp (Ω), then ∀u ∈ V , ‖u‖p ≤ cp ‖u‖ for some positive constant cp. We haveto precise that the sequence (Jn)n of locally Lipschitz real functions is defined,similarly to J, as follows

Jn(u) =∫

Ω

jn(u(x))dx, u ∈ Lp(Ω)

where jn is given by jn(t) =∫ t

0 βn(s)ds, βn ∈ L∞loc(R). We will restrict ourselves

to the case where β and βn are in L∞(R); the setting of L∞loc(R) will be ignored

only for the sake of simplicity. From now on, we assume that (HI) has at leastone solution u0 and set the following notations :

• A0 = A, B0 = B, β0 = β, J0 = J, l0 = l and u0 = u a solution to(HI0) = (HI);

• τ > 0 is radius of the ball centered in 0 so as u0 ∈ B(0, τ);• L := mes (Ω)1− 2

p cp(Ω)2;• K := cp

2(mes (Ω))(p−2)/2, σ = cp( mes (Ω))(p−1)/p andπ = maxσ, mes (Ω);

• δ stands for a real function such that for any n ≥ 1,

|δ(s)| ≥ max|β(s)|, |βn(s)|.In this respect, we make the following assumptions:(H1) (Anu, u) ≥ M‖u‖2 for all u ∈ V , n ≥ 1, where M > 0;(H2) ‖Bnu − Bnv‖ ≤ N‖u − v‖ for all u, v ∈ V, n ≥ 1, where N > 0;(H3) Suppose that either


(a) βn is Lipschitz of rank k > 0 for any n ≥ 1 and β is continuous or(b) β+

n (t1) ≤ β−n (t2) + γ(t2 − t1) for all t1 < t2, for any n ≥ 1 where

γ > 0;(H4) |δ(s)| ≤ α1 + α2|s|p−1 ∀ s ∈ R for some α1 ∈ R and α2 > 0;(H5) M > N + T where T = maxkL, γK;(H6) Convergence conditions :

(i) An converges to A relatively to the norm topology;(ii) Bn converges to B relatively to dτ ;(iii) Jn converges to J with respect to D;(iv) ln converges to l with respect to the norm of V ∗.

Remark 1. Note that• When βn is continuous, the condition (H3)(b) reduces to the usual Lip-

schitz continuity.• (H3) b) was introduced in [2] for the study of special eigenvalues problem

for hemivariational inequalities, it becomes known as γ-lower essentialr-Holder property for β.

• (H5) is in the spirit of Noor [14] and plays a crucial part in the con-vergence analysis in this paper as in sensitivity analysis of variationalinequalities.

Before stating our main result, we consider some distances that we need inour convergence analysis.

• D(W, Z) := ‖υ − ω‖∞ between two real locally Lipchitz functions Wand Z defined as W (s) =

∫ s

0υ(t)dt and Z(s) =

∫ s

0ω(t)dt wherein

υ, ω ∈ L∞(R).• dτ (E, Q) := max

‖u‖≤τ‖Eu − Qu‖, where τ > 0 between two nonlinear

operators E and Q.

Theorem 3. Assume that assumptions (H1) − (H6) hold, then, whenever thesolution un to (HIn) exists, it must be unique and strongly convergent to theunique solution u0 to (HI0) and we have

‖un −u0‖ ≤ 1M − N − T

[‖ln − l0‖V ∗ + τ‖An −A0‖+ πD(Jn, J0) + dτ (Bn, B0)].

Remark 2. Notice here that under (H3) (a), the energy functions J, Jn arecontinuously differentiable, while the case of (H3) (b) recovers the nonsmouthframework.

Remark 3. If a function θ belongs to L∞(R), then function I(s) =∫ s

0 θ(s)ds islocally Lipschitz of rank ‖θ‖∞.


4. Proof of Theorem 3

To prove this theorem, we shall pass by three steps.

Step 1 : The nonconvex smouth case. We suppose that (H3)(a) is satisfiedand state the following Proposition.

Proposition 2. Assume that conditions (Hi)i=1,2,4,5 and (H3)(a) hold. Then,for any i ≥ 1; we have the following estimation

‖ui−u0‖ ≤ 1M − L − N

[‖li−l0‖V ∗+τ‖Ai−A0‖+mes (Ω)D(Ji, J0)+dτ (Bi, B0)].

We first collect some lemmas that we need for the proof of this step.

Lemma 2. Suppose that (H4) is fulfilled. Then, for each w ∈ V, we have∂Jn(w) ⊂

∫Ω ∂jn(w(x))dx.

Proof. By assumption (H4), for all t ∈ R and all ξ ∈ ∂jn(t), we deduce that

|ξ| ≤ max(β−n (t), β+

n (t)) ≤ α1 + α2|t|p−1. (4.1)

The conclusion is then a direct application of Theorem 1.

Lemma 3. Assume that, for any n ≥ 0, βn is continuous. Then for each n ≥ 0we have

i) ∂jn(t) = ∇jn(t) = βn(t) ∀t ∈ Rii) ∂Jn/V (w) = ∇Jn(w) =

∫Ω

βn(w(x))dx for all w ∈ V.

Proof. We can prove i) on two different ways:1) From the continuity of βn it follows that jn is continuously differentiable.Then, we have ∂jn(t) = ∇jn(t) = βn(t) for each t ∈ R.2) By Lemma 1, one has ∂jn(t) ⊂ [β−

n (t), β+n (t)]. Since βn is continuous, we

see that β−n (t) = β+

n (t) = βn(t). Hence ∂jn(t) ⊂ βn(t). Then, as ∂jn(t) isnonempty, it holds

∂jn(t) = βn(t).For ii), as V is densely and continuously imbedded in Lp(Ω), it results fromTheorem 2 and Lemma 2, for each w ∈ V, that

∂Jn/V (w) ⊂ ∂Jn(w) ⊂∫

Ω

∂jn(w(x))dx.

Since ∂jn(w(x)) = βn(w(x)), it follows that ∂Jn/IV (u) =∫Ω

βn(w(x))dx. Theproof is therefore completed.


Lemma 4. For all u, v ∈ V , u.v ∈ L1(Ω) and one has

‖u.v‖L1(Ω) ≤ mes (Ω)1− 2p ‖u‖p‖v‖p.

Proof. By generalized Holder’s inequality, it follows that

‖u.v‖L

p2(Ω) ≤ ‖u‖p‖v‖p.

On the other hand, Lp2 (Ω) is continuously imbedded into L1(Ω) and one has

‖ϕ‖L1(Ω) ≤ mes (Ω)1− 2p ‖ϕ‖

Lp2 (Ω)

∀ϕ ∈ Lp2 (Ω).

Then‖u.v‖L1(Ω) ≤ mes (Ω)1− 2

p ‖u‖p‖v‖p.

This ends the proof.

Lemma 5. Under the assumption (H3)(a), for each i ≥ 1 we have

‖∇Ji/V (u) −∇Ji/V (u)‖V ∗ ≤ L‖u− u‖ ∀u, u ∈ V.

Proof. By lemma 3, For each u, u and v in V , one can write

〈∇Ji/V (u) −∇Ji/V (u), v〉 =∫

Ω

〈βi(u(x)) − βi(u(x)), v(x)〉dx.

Hence, by the Lipschitz property of βi we obtain

〈∇Ji/V (u) −∇Ji/V (u), v〉 ≤ k

∫

Ω

|u(x) − u(x)||v(x)|dx. (4.2)

Then if p = +∞, it hold

|〈∇Ji/V (u) −∇Ji/V (u), v〉| ≤ k‖u− u‖∞∫Ω |v(x)|dx

≤ k mes (Ω)‖u − u‖∞‖v‖∞≤ k mes (Ω)C∞(Ω)2‖u− u‖‖v‖.

Thus

|〈∇Ji/V (u) −∇Ji/V (u), v〉| ≤ k mes (Ω)C∞(Ω)2‖u − u‖‖v‖. (4.3)

In particular, if we take in (4.3) any v belonging to the unit ball of V , we deducethat

‖∇Ji/V (u) −∇Ji/V (u)‖V ∗ ≤ k mes (Ω)C∞(Ω)2‖u − u‖.If 2 ≤ p < +∞, from Lemma 4 and (4.2), it follows that

|〈∇Ji/V (u) −∇Ji/V (u), v〉| ≤ k mes (Ω)1− 2p ‖u− u‖p‖v‖p

≤ k mes (Ω)1− 2p [Cp(Ω)]2‖u− u‖‖v‖.

It may be concluded that

‖∇Ji/V (u) −∇Ji/V (u)‖V ∗ ≤ [ mes (Ω)]1−2p Cp(Ω)2‖u − u‖.


This ends the proof of the lemma.

Proof of Proposition 2. Since uj , for j = 0, i , is a solution to (HIj), we have,for all v ∈ V,

(Ajuj , v − uj) + (Bju, v − uj) + J0j (uj ; v − uj) ≥ (lj , v − uj),

for all v ∈ V, it follows

J0j (uj ; v − uj) ≥ (lj − Ajuj − Bjuj , v − uj).

Combining this with Lemma 3, we see at once that

lj − Ajuj − Bjuj ∈ ∂Jj/V (uj) =∫

Ω

βj(uj(x))dx.

Therefore,

li − l0 + A0u0 − Aiui + B0u0 − Biui =∫

Ω

[βi(ui(x)) − β0(u0(x))]dx.

Now taking into account (Hi)i=1,2,4,5, (H3)(a) and Lemma 5, we obtain

M‖ui − u0‖ ≤∫

Ω

[(βi − β0)(u0(x))]dx + ‖∇Ji/V (ui) −∇Ji/V (u0)‖

+‖li − l0‖V ∗ + ‖A0u0 − Aiu0‖‖u0‖ + N‖ui − u0‖+‖B0u0 − Biu0‖

≤ mes (Ω)‖βi − β0‖∞ + L‖ui − u0‖ + N‖ui − u0‖+ τ‖Ai − A0‖ + dτ (Bi, B0).

Further, we confirm that

‖ui−u0‖ ≤ 1M − L − N

[‖li−l0‖V ∗+τ‖Ai−A0‖+ mes (Ω)D(Ji, J0)+dτ (Bi, B0)].

Step 2 : The nonconvex nonsmouth case. Assume that (H3)(b) is satisfied.By relaxing the Lipschitz property on βi we can recover the nonsmouth case.We can again state a similar estimation to previous proposition.

Proposition 3. Suppose that assumptions (Hi)i=1,2,4,5 and (H3)(b) hold. Then

‖ui − u0‖ ≤ 1M − N − K

[‖li − l0‖V ∗ + τ‖Ai − A0‖+ σD(Ji, J0) + dτ (Bi, B0)].

Two fundamental lemmas are behind this result.

Lemma 6. Under assumption (H3)(b), for each n ≥ 1 and for every s, t ∈R ∀ξ ∈ ∂jn(s), ∀η ∈ ∂jn(t) we have

(ξ − η, s − t) ≥ −γ|s − t|2.


Proof. Let ξ ∈ ∂jn(s) and η ∈ ∂jn(t).Using (b) of Proposition 1, we see that

(ξ − η, s − t) ≥ −j0n(t, s − t) − j0

n(s, t − s).

On the other hand, from Lemma 1 if t ≤ s, we have,

j0n(t, s − t) + j0

n(s, t − s) ≤ (s − t)β+n (t) + (t − s)β−

n (s)≤ (s − t)(β+

n (t) − β−n (s))

≤ γ(s − t)2

and if t ≥ s we have

j0n(t, s − t) + j0

n(s, t − s) ≤ (t − s)(β+n (s) − β−

n (t)) ≤ γ(t − s)2.

Therefore, (ξ − η, s − t) ≥ −γ|s − t|2.

Lemma 7. Assume that (H4) is satisfied. Then, whenever uj (j = 0, 1...) is asolution to (HIj), there exists a mapping ξj : Ω → R such that

(1) For all y in R, the map x → (ξj(x), y) ∈ L1(Ω).(2) ξj(x) ∈ ∂jj(uj(x)) almost everywhere on Ω relatively to the Lebesgue’s

measure.(3) (lj , v) − (Ajuj , v) + (Bjuj , v) =

∫Ω(ξj(x), v(x))dx, ∀v ∈ V.

Proof. By assumption (H4), taking into account (4.1) of Lemma 2 and Remark2, the conclusion is immediate from Theorem 1.

Proof of Proposition 3. Since, for j = 0, i, uj is a solution to (HI)j , fromLemma 7 we deduce that

(lj − Ajuj − Bjuj , v − uj) =∫

Ω

(ξj(x), v(x) − uj(x)) ∀v ∈ V (4.4)

for some mapping ξj satisfying assertions of Lemma 7. By setting, for m = 0, i,v = um in (4.4) for m 6= j, and adding the two relations, we obtain

∫Ω(ξi − ξ0, u0 − ui)dx = (A0u0 − Aiui, u0 − ui)+

(B0u0 − Biui, u0 − ui) + (li − l0, u0 − ui).(4.5)

According to Lemma 7, ξj verifies

ξj(x) ∈ ∂jj(uj(x)) almost everywhere on Ω.

Hence, for any ξ ∈ ∂ji(u0), it results from Lemma 6 that

∫

Ω

(ξi − ξ0, u0 − ui)dx ≤∫

Ω

((ξ, u0 − ui) + γ|ui − u0|2)dx −∫

Ω

(ξ0, u0 − ui)dx

≤∫

Ω

(ξ − ξ0, u0 − ui)dx + K‖ui − u0‖2.


Since

ξ − ξ0 ∈ ∂ji(u0) − ∂j0(u0) ⊂ ∂(ji − j0)(u0),

it follows from (a) of Proposition 1 and Remark 2 that∫

Ω

(ξi − ξ0, u0 − ui)dx ≤ ‖βi − β0‖∞∫

Ω

|ui(x) − u0(x)|dx + K‖ui − u0‖2

≤ σ‖βi − β0‖∞‖ui − u0‖ + K‖ui − u0‖2.

Now, using (H1) and (H2), (4.5) leads to

(M−N−K)‖ui−u0‖ ≤ ‖A0u0−Aiu0‖+‖B0u0−Biu0‖+σ‖βi−β0‖∞+‖li−l0‖V ∗ .

Therefore

‖ui − u0‖ ≤ 1M − N − K

[‖li − l0‖V ∗ + τ‖Ai − A0‖+ σD(Ji, J0) + dτ (Bi, B0)],

which ends the proof.

Step 3: Let us now conclude the proof of our convergence result.For n ≥ 1, the uniqueness of solution un to (HIn) comes from Propositions 2and 3. Besides, we have

‖un −u0‖ ≤ 1M − N − T

[‖ln − l0‖V ∗ + τ‖An −A0‖+ πD(Jn, J0) + dτ (Bn, B0)],

therefore, if we set S the set of solutions to (HI0), according to (H6) we concludethat

e(un, S) := supw∈S

‖un − w‖ → 0 as n goes to + ∞.

Hence, un strongly converges to some u which must be the unique solution to(HI). The proof of the main theorem is henceforth finished.

Acknowledgement

The authors would like to thank Professor M. Aslam Noor and the anonymousreferee for their valuable comments.


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Hassan Riahi got his first doctor degree in Montpellier in 1989, in 1993 he defendeda doctorate of Sciences in mathematics in Rabat. Currently, he is first class professor atCadi Ayyad university in Marrakesh. His main research interests are maximal monotoneoperator theory and variational analysis.

Departement de Mathematiques, Faculte des Sciences Semlalia, Universite Cadi Ayyad,B.P. 2390, Marrakech-40000, Morocco.e-mail: [email protected]

Mohamed Ait Mansour received his Ph. D, in 2002, from Cadi Ayyad University undersupervion of H. Riahi on the topic of equilibrium problems. Presently, he is a resaerchfellow from regional councel of “Limousin France” and from “Agence Universitaire deFrancophonie”.

Departement de Mathematiques, Faculte des Sciences Semlalia, Universite Cadi Ayyad,B.P. 2390, Marrakech-40000, Morocco.e-mail: [email protected]

Convergence analysis of perturbed hemivariational inequalities

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