Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 3, Issue 2, Article 30, 2002
ON A NONCOERCIVE SYSTEM OF QUASI-VARIATIONAL INEQUALITIES
RELATED TO STOCHASTIC CONTROL PROBLEMS
M. BOULBRACHENE1 , M. HAIOUR2 , AND B. CHENTOUF1
1
S ULTAN Q ABOOS U NIVERSITY,
C OLLEGE OF S CIENCE ,
D EPARTMENT OF M ATHEMATICS & S TATISTICS ,
P.O. B OX 36 M USCAT 123,
S ULTANATE OF O MAN .
boulbrac@squ.edu.om
2
D EPARTEMENT DE M ATHEMATIQUES ,
FACULTE DES S CIENCES ,
U NIVERSITE DE A NNABA ,
B.P. 12 A NNABA 23000 A LGERIA .
haiourmed@caramail.com
chentouf@squ.edu.om
Received 29 October, 2001; accepted 11 February, 2002.
Communicated by R. Verma
A BSTRACT. This paper deals with a system of quasi-variational inequalities with noncoercive
operators. We prove the existence of a unique weak solution using a lower and upper solutions
approach. Furthermore, by means of a Banach’s fixed point approach, we also prove that the
standard finite element approximation applied to this system is quasi-optimally accurate in L∞ .
Key words and phrases: Quasi-variational inequalities, Contraction, Fixed point finite element, Error estimate.
2000 Mathematics Subject Classification. 49J40, 65N30, 65N15.
1. I NTRODUCTION
We are interested in the following system of quasi-variational inequalities (QVI’s): find a
vector U = (u1 , . . . , uM ) ∈ (H01 (Ω))M such that

i i
i
i
i
1

 a (u , v − u ) = (f , v − u ) ∀v ∈ H0 (Ω)



ui ≤ k + ui+1 , v ≤ k + ui+1
(1.1)




 uM +1 = u1 ,
ISSN (electronic): 1443-5756
c 2002 Victoria University. All rights reserved.
The support provided by Sultan Qaboos University (project Sci/Doms/01/22) is gratefully acknowledged.
075-01
2
M. B OULBRACHENE , M. H AIOUR ,
AND
B. C HENTOUF
where Ω is a smooth bounded domain of RN , N ≥ 1 with boundary Γ, ai (u , v) are M
variational forms, f i are regular functions and k is a positive number.
This system arises in stochastic control problems. It also plays a fundamental role in solving
the Hamilton-Jacobi-Bellman equation, [1], [2].
Its coercive version is well understood from both the mathematical and numerical analysis
viewpoints (cf. eg., [1], [2], [6]).
In this paper we shall be concerned with the noncoercive case, that is, where the bilinear
forms ai (u, v) do not satisfy the usual coercivity condition.
To handle this new situation, we transform (1.1) into the following auxiliary system: find
U = (u1 , . . . , uM ) ∈ (H01 (Ω))M such that:

bi (ui , v − ui ) = (f i + λui , v − ui ) ∀v ∈ H01 (Ω)





ui ≤ k + ui+1 , v ≤ k + ui+1
(1.2)




 uM +1 = u1 ,
where
bi (u, v) = ai (u, v) + λ(v, v)
(1.3)
and λ > 0 is large enough such that:
bi (v, v) ≥ γ kvk2H 1 (Ω) γ > 0; ∀v ∈ H01 (Ω).
(1.4)
Under this condition, using a monotone approach inspired from [5], we shall prove that both
the continuous and discrete problems admit a unique solution.
On the numerical analysis side, using piecewise linear finite elements, we shall establish a
quasi-optimal L∞ −convergence order. To that end, we propose a new approach which consists of characterizing both the continuous and the finite element solution as fixed points of
contractions in L∞ .
This new approach appears to be quite simple. It also offers the advantage of providing an
iterative scheme useful for the numerical computation of the solution.
The paper is organized as follows. In Section 2, we discuss existence and uniqueness of a
solution to problem (1.1). Section 3 deals with its discretization by the standard finite element
method where, under a discrete maximum principle assumption, analogous discrete qualitative
results are given as well. Finally, in Section 4 we respectively associate with both the continuous
and discrete systems appropriate contractions and give an L∞ −error estimate.
2. T HE C ONTINUOUS P ROBLEM
Let us begin with some necessary notations, assumptions and qualitative properties of elliptic
variational inequalities.
2.1. Notations, Assumptions and Preliminaries. We are given functions
(2.1)
aijk (x), bik (x), ai0 (x) ∈ C 2 (Ω), x ∈ Ω, 1 ≤ k, j ≤ N ; 1 ≤ i ≤ M
such that:
(2.2)
X
aijk (x)ξj ξk = α |ξ|2 ; (x ∈ Ω, ξ ∈ RN , α > 0)
1≤ j, k≤N
(2.3)
aijk = aikj ; ai0 (x) = β > 0; x ∈ Ω.
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We define the bilinear forms: for any u, v ∈ H 1 (Ω),
!
Z
N
X
X
∂u
∂v
∂u
bik (x)
(2.4)
ai (u, v) =
aijk (x)
+
v + ai0 (x).uv .
∂x
∂x
∂x
j
k
k
Ω
1≤ j, k≤N
k=1
We are also given regular functions f i such that
f i ∈ C 2 (Ω) and f i ≥ 0; ∀ i = 1, . . . , M .
Q
∞
and the following norm: ∀ W = (w1 , . . . , wM ) ∈ M
i=1 L (Ω)
(2.5)
(2.6)
kW k∞ = max
1≤i≤M
wi
L∞ (Ω)
,
where k·kL∞ (Ω) denotes the well-known L∞ −norm.
2.2. Elliptic Variational Inequalities. Let f in L∞ (Ω) and ψ in W 2,∞ (Ω) such that ψ ≥ 0 on
∂Ω. Let also b(·, ·) be a continuous and coercive bilinear form of the same form as those defined
in (1.2) and consider u = σ(f, ψ) a solution to the following elliptic variational inequality VI:
find u ∈ H01 (Ω) such that

 b(u, v − u) = (f, v − u) ∀v ∈ H01 (Ω)
(2.7)
 u ≤ ψ; v ≤ ψ.
Theorem 2.1. (cf. [3],[4]) Under the above assumptions, there exists a unique solution to the
variational inequality (VI) (2.7). Moreover, u ∈ W 2,p (Ω), 1 ≤ p < ∞ .
2.2.1. A Monotonicity property for VI (2.7). Let (f, ψ), f˜, ψ̃ be a pair of data and u =
σ(f, ψ), u
e = σ f˜, ψ̃ the respective solutions to (2.7).
Theorem 2.2. (cf. [4]) If f ≥ f˜ and ψ ≥ ψ̃ then σ(f, ψ) ≥ σ(f˜, ψ̃).
2.3. Existence and uniqueness. As mentioned earlier, we solve the noncoercive system of
QVI’s by considering the following auxiliary system: find a vector U = u1 , . . . , uM ∈
M
(H01 (Ω)) such that

bi (ui , v − ui ) = (f i + λui , v − ui ) ∀v ∈ H01 (Ω)





ui ≤ k + ui+1 , v ≤ k + ui+1
(2.8)




 uM +1 = u1 .
It can readily be noticed in the above system, that besides the obstacles “k + ui+1 ”, the right
hand sides depend upon the solution as well. Therefore, the increasing property of the solution
of VI with respect to the obstacle and the right hand side, reduces the problem (2.8) to finding
a fixed point of anQ
increasing mapping as in [5].
∞
∞
∞
∞
Let L (Ω) = M
i=1 L+ (Ω), where L+ (Ω) is the positive cone of L (Ω). We introduce the
following mapping
(2.9)
T : L∞ (Ω) −→ L∞ (Ω)
W −→ T W = ζ 1 , . . . , ζ M
J. Inequal. Pure and Appl. Math., 3(2) Art. 30, 2002
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M. B OULBRACHENE , M. H AIOUR ,
AND
B. C HENTOUF
where ∀ i = 1, . . . , M , ζ i = σ (f i + λwi ; k + wi+1 ) is solution to the following VI:

 bi (ζ i , v − ζ i ) = (f i + λwi , v − ζ i ) ∀v ∈ H01 (Ω)
(2.10)
 ζ i ≤ k + wi+1 , v ≤ k + wi+1 .
Problem (2.10) being a coercive variational inequality, thanks to [3], [4], has a unique solution.
2.3.1. Properties of The Mapping T. Let us first introduce the vector Û 0 = û1,0 , . . . , ûM,0 ,
where ∀i = 1, . . . , M, ûi,0 is solution to the equation
ai (ûi,0 , v) = (f i , v) ∀ v ∈ H01 (Ω).
(2.11)
Since f i ≥ 0, there exists a unique positive solution to problem (2.11). Moreover, ûi,0 ∈
W 2,,p (Ω), p < ∞ (cf. e.g., [5]).
Proposition 2.3. Under the preceding notations and assumptions, the mapping T is increasing,
concave and satisfies: T W ≤ Û 0 , ∀ W ∈ L∞ (Ω) such that W ≤ Û 0 .
Proof.
1. T is increasing .
Let V = (v 1 , . . . , v M ), W = (w1 , , . . . , wM ) in L∞ (Ω) such that v i ≤ wi , ∀i =
1, , . . . , M . Then, by Theorem 2.2, it follows that σ(f i + λwi ; k + wi+1 ) ≥ σ(f i +
λv i ; k + v i+1 ). Thus, T V ≤ T W.
2. T W ≤ Û 0 ∀W ≤ Û 0 .
Let us first recall that u+ = sup(u, 0) and u− = sup(−u, 0). The fact that both of the
solutions ζ i of (2.10) and ûi,0 of (2.11) belong to H01 (Ω), we clearly have:
+
ζ i − ζ i − ûi,0 ∈ H01 (Ω).
Moreover, as (ζ i − ûi,0 )+ ≥ 0, it follows that
+
ζ i − ζ i − ûi,0 ≤ ζ i ≤ k + wi+1 .
Therefore, we can take v = ζ i − (ζ i − ûi,0 )+ as a trial function in (2.10). This gives:
+ i
+ bi ζ i , − ζ i − ûi,0
= f + λwi , − ζ i − ûi,0
.
+
On the other hand, taking v = (ζ i − ûi,0 ) in equation (2.11) we get
i
i,0
i
i,0 +
i
i,0
i
i,0 +
= f + λû , ζ − û
b û , ζ − û
and, since W ≤ Û 0 , by addition, we obtain
+
+ −bi ζ i − ûi,0 , ζ i − ûi,0
=0
which, by (1.4), yields
ζ i − ûi,0
+
= 0.
Thus
ζ i ≤ ûi,0 ∀ i = 1, 2, . . . , M
i.e.,
T W ≤ Û 0 .
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3. T is concave.
Let us agree on the following notations:
i
= θ(k + wi ) + (1 − θ)(k + w̃i ); θ ∈ [0, 1].
wθi = θwi + (1 − θ)w̃i ; wθ,k
Then we have:
T (θW + (1 − θ)W̃ )
= σ f 1 + λwθ1 ; k + wθ2 , . . . , σ f i + λwθi ; k + wθi+1 , . . . , σ f M + λwθM ; k + wθ1
i+1
1
2
, . . . , σ f M + λwθM ; wθ,k
.
= σ f 1 + λwθ1 ; wθ,k
, . . . , σ f 1 + λwθi ; wθ,k
Now, denoting by:
ζ i = σ f i + λwi ; k + wi+1 ,
ζ̃ i = σ f i + λw̃i ; k + w̃i+1 ,
ζθi = θζ i + (1 − θ)ζ̃ i ,
Uθi = σ f i + λwθi ; wθi+1 .
It is clear that ζθi is admissible for the problem which has Uθi as a solution. So
−
Uθi + Uθi − ζθi
is admissible for this problem. Therefore,
i
i
i −
i
i
i −
(2.12)
b Uθ , Uθ − ζθ
≥ f + λwθ , Uθ − ζθ
.
−
Also, we can take ζ i − (Uθi − ζθi ) as a test function in the problem where ζ i is the
−
solution and ζ̃ i − (Uθi − ζθi ) can be taken as a test function in the problem whose
solution is ζ̃ i . From this we deduce that
− − (2.13)
−b ζ i , Uθi − ζθi
≥ − f + λwi , Uθi − ζθi
and
− − −b ζ̃ i , Uθi − ζθi
≥ − f + λw̃i , Uθi − ζθi
.
(2.14)
Now multiplying (2.13) by θ, and (2.14) by 1 − θ, addition yields
− − ≥ − f + λwθi , Uθi − ζθi
−b ζθi , Uθi − ζθi
which added to (2.12) gives
− ≥ 0.
b Uθi − ζθi , Uθi − ζθi
Thus
Uθi − ζθi
−
=0
which completes the proof i.e.,
T θW + (1 − θ)W̃ ≥ θT W + (1 − θ)T W̃ .
J. Inequal. Pure and Appl. Math., 3(2) Art. 30, 2002
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M. B OULBRACHENE , M. H AIOUR ,
AND
B. C HENTOUF
2.3.2. A Continuous Iterative Scheme of Bensoussan-Lions Type. Starting from Û 0 solution
of (2.11) and Ǔ 0 = 0, we define the iterations:
Û n+1 = T Û n ; n = 0, 1, . . .
(2.15)
and
Ǔ n+1 = T Ǔ n ; n = 0, 1, . . . ,
(2.16)
respectively.
The analysis of the convergence of these iterations requires to prove the following intermediate results.
Lemma 2.4. Assume f i ≥ f 0 > 0 ; 1 ≤ i ≤ M, where f 0 is a positive constant, and let



 k
0
f
.
0 < µ < inf
;
 Û 0
λ Û 0
+ f0 
∞
∞
Then we have
T (0) = µÛ 0 .
(2.17)
Proof. Indeed, from (2.16), T (0) = Ǔ 1 = (ǔ1,1 , . . . , ǔ1,M ), where ǔi,1 is the solution of the
following variational inequality:

 bi (ǔi,1 , v − ǔi,1 ) = (f i + λǔi,0 , v − ǔi,1 ) ∀v ∈ H01 (Ω)
(2.18)
 i,1
ǔ ≤ k; v ≤ k.
Then by the choice of µ it is clear that
v = ǔi,1 − µûi,0
−
+ ǔi,1
can be taken as a trial function in the VI (2.18) inequality. So taking
−
v = − ǔi,1 − µûi,0
as a trial function in (2.11) and using the fact that f i ≥ f 0 and ǔi,0 = 0, we get by addition:
− i
− bi ǔi,1 − µûi,0 , ǔi,1 − µûi,0
= f − µf i − µλûi,0 , ǔi,1 − µûi,0
0
i,0
i,1
i,0 −
= f (1 − µ) − µλû , ǔ − µû
.
But, again, by the choice of µ
f 0 (1 − µ) − µλûi,0 ≥ f 0 (1 − µ) − µλ Û 0
Thus, by (1.4)
ǔi,1 − µûi,0
−
∞
≥ 0.
=0
i.e.,
ǔi,1 = µûi,0 ∀ i = 1, 2, . . . , M.
Proposition 2.5. Let C = {W ∈ L∞ (Ω) such that 0 ≤ W ≤ Û 0 }. Let also γ ∈ ]0 ; 1],
W, W̃ ∈ C such that:
W − W̃ ≤ γW .
(2.19)
Then, the following holds
(2.20)
T W − T W̃ ≤ γ(1 − µ)T W.
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Proof. By (2.19), we have (1 − γ) W ≤ W̃ . Then, using the fact that T is increasing and
concave (see Proposition 2.3.), it follows that
(1 − γ)T W + γT (0) ≤ T [(1 − γ)W + γ.0]
≤ T W̃
Finally, using Lemma 2.4. we get (2.20).
Theorem 2.6. Under conditions of Propositions 2.3 – 2.5, the sequences Û n and Ǔ n are
monotone and well defined in C . Moreover, they converge respectively from above and below
to the unique solution U of system of QVI’s (1.1).
Proof. The proof will be carried out in five steps.
Step 1. The sequence (Û n ) stays in C and is decreasing.
From (2.15) it is easy to see that ∀i, ûi,n is solution to the following VI:

bi (ûi,n , v − ûi,n ) = (f i + λûi,n−1 , v − ûi,n ) ∀v ∈ H01 (Ω)





ûi,n ≤ k + ûi+1,n−1 ; v ≤ k + ûi+1,n−1
(2.21)




 ûM +1,n = û1,n .
Since f i ≥ 0 and ûi,0 ≥ 0, a simple induction combined with standard comparison
results in variational inequalities lead to ûi,n ≥ 0 i.e.,
Û n ≥ 0 ∀n ≥ 0.
(2.22)
Furthermore, by Proposition 2.3. and (2.15), we have:
Û 1 = T Û 0 ≤ Û 0 .
Thus, inductively
0 ≤ Û n+1 = T Û n ≤ Û n ≤ · · · ≤ Û 0 .
(2.23)
Step 2. (Û n ) converges to the solution of the system (1.1).
From (2.22) and (2.23) it is clear that ∀i = 1, 2, . . . , M
(2.24)
lim ûi,n (x) = ūi (x), x ∈ Ω and (ū1 , . . . , ūM ) ∈ C.
n→∞
Moreover, from (2.22) we have k + ûi+1,n−1 ≥ 0. Then we can take v = 0 as a trial
function in (2.21), which yields
γ ûi,n
2
H 1 (Ω)
≤ bi (ûi,n , ûi,n ) ≤ f i + λûi,n−1
L2 (Ω)
ûi,n
H 1 (Ω)
or more simply
ûi,n
H 1 (Ω)
≤ C,
where C is a constant independent of n. Hence ûi,n stays bounded in H 1 (Ω) and consequently we can complete (2.24) by
(2.25)
lim ûi,n = ūi weakly in H 1 (Ω).
n→∞
Step 3. U = (ū1 , . . . , ūM ) coincides with the solution of system (1.1).
Indeed, since
ûi,n (x) ≤ k + ûi+1,n−1 (x)
then (2.24) implies
ūi (x) ≤ k + ūi+1 (x).
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B. C HENTOUF
Now let v ≤ k + ūi+1 then v ≤ k + ûi+1,n−1 , ∀n ≥ 0. We can therefore take v as a trial
function for the VI (2.21). Consequently, combining (2.24) and (2.25) with the weak
lower semi continuity of bi (v, v) and passing to the limit in problem (2.21), we clearly
get
bi (ūi , v − ūi ) = (f i + λūi , v − ūi ) ∀v ∈ H01 (Ω), v ≤ k + ūi+1 .
Step 4. Uniqueness. Let U , Ũ be two solutions of the system (1.1). These are fixed points of T .
Therefore, since U − Ũ ≤ U , by taking W = U and W̃ = Ũ in (2.19) with γ = 1 − µ
we have
U − Ũ ≤ (1 − µ)U.
Doing this again with γ = 1 − µ, we obtain
U − Ũ ≤ (1 − µ)2 U
and inductively
U − Ũ ≤ (1 − µ)n U ≤ (1 − µ)n Û 0
∞
.
Thus, making n tend to ∞ yields U ≤ Ũ . Finally, interchanging the roles of U and Ũ ,
we obtain U = Ũ
Step 5. The monotone property of the sequence (Ǔ n ) can be shown in a similar way to that of
sequence (Û n ). Let us prove its convergence to the solution of system (1.1). Indeed,
apply (2.19),(2.20) with
W = Û 0 ; W̃ = Ǔ 0 ; γ = 1
then
T Û 0 − T Ǔ 0 ≤ (1 − µ)T Û 0 ,
so
0 ≤ Û 1 − Ǔ 1 ≤ (1 − µ)Û 1 .
Applying (2.20) again, yields
0 ≤ Û 2 − Ǔ 2 ≤ (1 − µ)2 Û 2
and quite generally
0 ≤ Û n − Ǔ n ≤ (1 − µ)n Û n ≤ (1 − µ)n Û 0 ≤ (1 − µ)n Û 0
∞
.
Thus
Û n − Ǔ n → 0 a.e
from which it follows that
Ǔ n → U = U
is the unique solution of system of QVI’s (1.1).
2.3.3. Regularity of the solution of system (1.1). The following is a theorem on the regularity
of (1.1).
Theorem 2.7. (cf. e.g,[1]). Let assumptions (2.1)-(2.5) hold. Then each component of the
solution of system (1.1) belongs to C(Ω̄)∩ W 2, p (Ω); ∀2 ≤ p < ∞.
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3. T HE D ISCRETE P ROBLEM
Let Ω be decomposed into triangles and let τh denote the set of all those elements; h > 0 is
the mesh size. We assume that the family τh is regular and quasi-uniform.
Let Vh denote the standard piecewise linear finite element space,
(3.1)
Vh = {v ∈ C(Ω) ∩ H01 (Ω) such that v/K ∈ P1 , ∀K ∈ τh }.
Let Bi be the matrices with generic coefficients
(3.2)
Bi ls = bi (ϕl , ϕs ) 1 ≤ i ≤ M ; 1 ≤ l, s ≤ m(h),
where, {ϕl }, l = 1, 2, . . . m(h) is the basis of Vh .
Let rh be the usual restriction operator defined by
m(h)
(3.3)
∀v ∈ C(Ω) ∩
H01 (Ω) ,
rh v =
X
vl ϕ l .
l=1
In the sequel of the paper we shall make use of the discrete maximum principle (d.m.p)
assumption. In other words, we shall assume that Bi , 1 ≤ i ≤ M are M-matrices (see [7]).
3.1. Discrete Variational Inequality. Let uh ∈ Vh be the solution of the following discrete
variational inequality

 b(uh , v − uh ) = (f, v − uh ) ∀v ∈ Vh
(3.4)

uh ≤ rh ψ; v ≤ rh ψ.
3.1.1. A Discrete Monotonicity Property for VI (3.4). Let (f , ψ), (f˜, ψ̃) be a pair of data
and u = σh (f, ψ), u
e = σh (f˜, ψ̃) the respective solutions of (3.4). Then we have the discrete
analogue of Theorem 2.2.
Theorem 3.1. Under the d.m.p, If f ≥ f˜ and ψ ≥ ψ̃ then σh (f, ψ) ≥ σh (f˜, ψ̃).
3.2. The Noncoercive Discrete System of QVI’s. Let Vh = (Vh )M . We define the noncoercive discrete system of QVI’s as follows: find Uh = (u1h , . . . , uM
h ) ∈ Vh solution of
 i i
a (uh , v − ui ) = (f i , v − uih ) ∀v ∈ Vh





i+1
uih ≤ k + ui+1
(3.5)
h , v ≤ k + uh




 M +1
uh
= u1h .
And, similarly to the continuous problem, we solve (3.5) via the following implicit coercive
system: find Uh = (u1h , . . . , uM
h ) ∈ Vh solution to
 i i
b (uh , v − ui ) = (f i + λuih , v − uih ) ∀v ∈ Vh ;





i+1
uih ≤ k + ui+1
(3.6)
h , v ≤ k + uh ;




 M +1
uh
= u1h .
Let us also note that all the properties established in the continuous case remain conserved in
the discrete case, provided the d.m.p is satisfied. The proofs of these will not be given as they
are respectively identical to their continuous analogue ones.
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M. B OULBRACHENE , M. H AIOUR , AND B. C HENTOUF
3.3. Existence and Uniqueness. Let us first define Ûh0 to be the piecewise linear approximation of Û 0 defined in (2.11):
(3.7)
i
ai (ûi,0
h , v) = (f , v) ∀v ∈ Vh ; 1 ≤ i ≤ M.
We consider the following mapping
(3.8)
Th : L∞ (Ω) −→ Vh ,
W −→ T W = (ζh1 , . . . , ζhM ),
where, ∀ i = 1, . . . , M , ζhi = σh (f i + λwi , k + wi+1 ) is the solution of the following discrete
VI:

 bi (ζhi , v − ζhi ) = (f i + λwi , v − ζhi ) ∀v ∈ Vh ,
(3.9)
 i
ζh ≤ rh (k + wi+1 ), v ≤ rh (k + wi+1 ).
Proposition 3.2. Under the d.m.p, Th is increasing, concave and satisfies Th W ≤ Ûh0 ∀W ∈
L∞ (Ω), W ≤ Ûh0 .
3.4. A Discrete Iterative Scheme of Bensoussan-Lions Type. We associate with the mapping
Th the following discrete iterative scheme: starting from Ûh0 defined in (3.7) and Ǔh0 = 0, we
define:
Ûhn+1 = Th Ûhn
(3.10)
and
Ǔhn+1 = Th Ǔhn
(3.11)
respectively.
Similarly to Theorem 2.6, the convergence of the above algorithm rests on the discrete analogues of Lemma 2.4. and Proposition 2.5, respectively.
Lemma 3.3. Assume f i ≥ f 0 > 0; 1 ≤ i ≤ M, where f 0 is a positive constant, and let


 k

0
f
0 < µ < inf
;
.
 Û 0
λ Û 0
+ f0 
h
∞
h
∞
Then we have
Th (0) = µÛh0
(3.12)
Proposition 3.4. Let Ch = {W ∈ L∞ (Ω) such that 0 ≤ W ≤ Ûh0 }. Let also γ ∈]0, 1],
W, W̃ ∈ Ch such that:
W − W̃ ≤ γW .
(3.13)
Then the following holds
(3.14)
Th W − Th W̃ ≤ γ(1 − µ)Th W.
Theorem 3.5. Under conditions of Proposition 3.2 – 3.4, the sequences (Ûhn ) and (Ǔhn ) are
monotone and well defined in Ch . Moreover, they converge respectively from above and below
to the unique solution Uh of system of QVI’s (3.5).
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4. T HE F INITE E LEMENT E RROR A NALYSIS
In what follows, we prove the convergence of the approximation and establish a uniform
error estimate. Our approach consists of characterizing both the solution of systems (1.1) and
(3.5) as the unique fixed points of appropriate contractions in L∞ (Ω). To that end we need first
to introduce a coercive system of quasi-variational inequalities and prove that its solution is
monotone with respect to the right hand side.
Let F = (F 1 , . . . , F M ) ∈ L∞ (Ω) . We denote by Z = (z 1 , . . . , z M ) the solution of the
coercive system of QVI’s:

bi (z i , v − z i ) = (F i , v − z i ) ∀v ∈ H01 (Ω)





z i ≤ k + z i+1
(4.1)




 z M +1 = z 1 .
Denoting by z i = σ(F i , k + z i+1 ), we introduce the sequences Z
Z n = (z 1,n , . . . , z M,n ) defined by
n
= (z̄ 1,n , . . . , z̄ M,n ) and
z̄ i,n+1 = σ(F i , k + z̄ i+1,n ),
and
z i,n+1 = σ(F i , k + z i+1,n ),
where z̄ i,0 is the unique solution of b(z̄ i,0 , v) = (F i , v) ∀v ∈ H01 (Ω) and z i,0 = 0.
n
Theorem 4.1. (cf. [6]) The sequence (Z ) and (Z n ) converge respectively from above and
below to the unique solution of system (4.1). Moreover z i ∈ W 2, p (Ω) 1 ≤ i < M ; 1 ≤ p < ∞.
1
1
M
M
Proposition 4.2. Let F , . . . , F ; F̃ , . . . , F̃
be two families of right hands side and
1
M
1
M
Z = z , . . . , z ; Z̃ = z̃ , . . . , z̃
be the respective solutions of system (4.1). Then the
following holds. If F ≥ F̃ , then Z ≥ Z̃.
0
1,0
,M,0
i,0
Proof. Let Z̄ 0 = z̄ 1,0 , . . . , z̄ M,0 and Z̃ = z̃ , . . . , z̃
such that z̄ i,0 and z̃ are so
i,0
lutions to equations b(z̄ i,0 , v) = (F i , v) and b z̃ , v = F̃ i , v , respectively. Then the
respective associated decreasing sequences
n
n
1,n
M,n
Z = (z̄ 1,n , . . . , z̄ M,n ) and Z̃ = z̃ , . . . , z̃
satisfy the following assertion.
If F i ≥ F̃ i then z̄ i,n ≥ z̃
i,n
∀i = 1, . . . , M.
Indeed, since
z̄ i,n+1 = σ F i , k + z̄ i+1,n ,
z̃
i,n+1
i+1,n
= σ F̃ i , k + z̃
,
i,0
i+1,0
F i ≥ F̃ i implies z̄ i,0 ≥ z̃ , ∀i = 1, 2, . . . , M. So, k + z̄ i+1,0 ≥ k + z̃
and thus, from
standard comparison results in coercive variational inequalities, it follows that
i,1
z̄ i,1 ≥ z̃ .
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M. B OULBRACHENE , M. H AIOUR , AND B. C HENTOUF
Now assume that z̄ i,,n−1 ≥ z̃
as before, we get:
i,n−1
. Then, as F i ≥ F̃ i , applying the same comparison argument
i,n
z̄ i,,n ≥ z̃ .
Finally, by Theorem 4.1, passing to the the limit as n → ∞, we get Z ≥ Z̃.
Remark 4.3. Proposition 4.2 remains true in the discrete case provided the d.m.p is satisfied.
4.1. A Contraction Associated with System of QVI’s (1.1). Consider the following mapping
S : L∞ (Ω) → L∞ (Ω)
(4.2)
W → SW = Z = z 1 , . . . , z M ,
where Z is solution to the coercive system of QVI’s below

bi (z i , v − z i ) = (f i + λwi , v − z i ) ∀v ∈ H01 (Ω)





z i ≤ k + z i+1 , v ≤ k + z i+1 ; i = 1, .., M
(4.3)




 z M +1 = z 1 .
By Theorem 4.1, problem (4.3) has one and only one solution.
Proposition 4.4. The mapping S is a contraction in L∞ (Ω). i.e.,
SW − SW̃
∞
≤
λ
W − W̃
λ+β
∞
.
Therefore, there exists a unique fixed point which coincides with the solution U of the system of
QVI’s (1.1).
Proof. Let W, W̃ ∈ L∞ (Ω). We consider Z = SW = (z 1 , . . . , z M ) and Z̃ = SW̃ =
(z̃ 1 , . . . , z̃ M ) solutions to system of QVI’s (4.3) with right hands side F = (F 1 , . . . , F M ) and
F̃ = (F̃ 1 , . . . , F̃ M ), where F i = f i + λwi and F̃ i = f i + λw̃i . Now setting
Φ=
1
F − F̃
λ+β
∞
; Φi =
1
F i − F̃ i
λ+β
∞
it follows that
F i ≤ F̃ i + F i − F̃ i
∞
and
a0 (x) + λ
F − F̃ i
λ+β
so, by Proposition 4.2, we obtain:
F̃ i +
∞
≤ F̃ i + (a0 (x) + λΦ) (because ai0 (x) = β > 0)
z i ≤ z̃ i + Φi .
Interchanging the roles of W and W̃ , we similarly get
z̃ i ≤ z i + Φi .
Thus
z i − z̃ i
which completes the proof.
L∞ (Ω)
≤ Φi
In a similar way to that of the continuous problem, we are also able to characterize the
solution of the system of QVI’s (3.5) as the unique fixed point of a contraction.
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4.2. A Contraction Associated with The Discrete System of QVI’s (3.5). We consider the
following mapping:
(4.4)
Sh : L∞ (Ω) → Vh
W → Sh W = Zh = zh1 , . . . , zhM ,
where zhi is solution to the discrete coercive system of QVI’s:

i
i
i
i

 b(zh , v − zh ) = (f + λw , v − zh ) ∀v ∈ Vh



zhi ≤ k + zhi+1 ; v ≤ k + zhi+1
(4.5)




 M +1
= zh1 .
zh
Thanks to [6], [8] system (4.5) has one and only one solution.
Next, making use of Proposition 4.2 and Remark 4.3 we have the contraction property of Sh .
Proposition 4.5. The mapping Sh is a contraction in L∞ (Ω). i.e.,
Sh W − Sh W̃
∞
≤
λ
W − W̃
λ+β
∞
.
Therefore, there exists a unique fixed point which coincides with the solution Uh of the system
of QVI (3.5)
Now, guided by Propositions 4.4 and 4.5, we are in a position to establish a uniform error
estimate for the noncoercive system of QVI’s (1.1). To this end, we need first to introduce the
following auxiliary discrete coercive system of QVI’s.
4.3. An Auxiliary Coercive System
of QVI’s. We consider the following coercive system of
1
M
QVI’s: find Z̄h = z̄h , . . . , z̄h solution to

b (z̄hi , v − z̄hi ) = (f + λui , v − z̄hi ) ∀v ∈ Vh





z̄hi ≤ k + z̄hi+1 ; v ≤ k + z̄hi+1 ; i = 1, . . . , M
(4.6)




 i,M +1
z̄h
= z̄h1 .
Clearly, (4.6) is a coercive system whose right hand side depends on U = u1 , . . . , uM the
continuous solution of system (1.1). So, in view of (4.4), we readily have:
Z̄h = Sh U.
(4.7)
Therefore, using the result of [6], we have the following error estimate.
Theorem 4.6. (cf. [6])
(4.8)
Z̄h − U
∞
≤ Ch2 |Logh|3 .
4.4. L∞ - Error Estimate For the Noncoercive System of QVI’s (1.1). Let U and Uh be the
solutions of system (1.1) and (3.5), respectively. Then we have:
Theorem 4.7.
kU − Uh k∞ ≤ Ch2 |Logh|3 .
Proof. In view of (4.8) and Propositions 4.4 and 4.5, we clearly have
U = SU ; Uh = Sh Uh ; Z̄h = Sh U.
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M. B OULBRACHENE , M. H AIOUR , AND B. C HENTOUF
Then , using estimation (4.8), we have
kSh U − SU k∞ = Z̄h − U
2
∞
3
≤ Ch |Logh| .
Therefore
kUh − U k∞ ≤ kUh − Sh U k∞ + kSh U − SU k∞
≤ kSh Uh − Sh U k∞ + kSh U − SU k∞
λ
≤
kU − Uh k∞ + Ch2 |Logh|3 .
λ+β
Thus
Ch2 |Logh|3
.
kU − Uh k∞ ≤ λ
1−
λ+β
This completes the proof.
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[3] D. KINDERLEHRER AND G. STAMPACCHIA, An Introduction to Variational Inequalities and
their Applications, Academic Press (1980).
[4] A. BENSOUSSAN AND J.L. LIONS, Applications des Inequations Variationnelles en Controle
Stochastique, Dunod, Paris, (1978).
[5] A. BENSOUSSAN AND J.L. LIONS, Impulse Control and Quasi-variational Inequalities, Gauthier
Villars, Paris, (1984).
[6] M. BOULBRACHENE AND M. HAIOUR, The finite element approximation of Hamilton Jacobi
Bellman equations, Comp. & Math. with Appl., 41 (2001), 993–1007.
[7] P.G. CIARLET AND P.A. RAVIART, Maximum principle and uniform convergence for the finite
element method, Comp. Meth. in Appl Mech and Eng., 2 (1973), 1–20.
[8] P. CORTEY-DUMONT, Sur l’ analyse numerique des equations de Hamilton-Jacobi-Bellman, Math.
Meth. in Appl. Sci., (1987).
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