Journal of Inequalities in Pure and
Applied Mathematics
ON A NONCOERCIVE SYSTEM OF QUASI-VARIATIONAL
INEQUALITIES RELATED TO STOCHASTIC CONTROL PROBLEMS
volume 3, issue 2, article 30,
2002.
M. BOULBRACHENE, M. HAIOUR AND B. CHENTOUF
Sultan Qaboos University,
College of Science,
Department of Mathematics & Statistics,
P.O. Box 36 Muscat 123,
Sultanate of Oman.
EMail: boulbrac@squ.edu.om
Departement de Mathematiques,
Faculte des Sciences,
Universite de Annaba,
B.P. 12 Annaba 23000 Algeria.
EMail: haiourmed@caramail.com
Sultan Qaboos University,
College of Science,
Department of Mathematics & Statistics,
P.O. Box 36 Muscat 123,
Sultanate of Oman.
EMail: chentouf@squ.edu.om
Received 29 October, 2001;
accepted 11 February, 2002.
Communicated by: R. Verma
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
075-01
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Abstract
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∞ .
2000 Mathematics Subject Classification: 49J40, 65N30, 65N15
Key words: Quasi-variational inequalities, Contraction, Fixed point finite element, Error estimate.
The support provided by Sultan Qaboos University (project Sci/Doms/01/22) is gratefully acknowledged.
On a Noncoercive System of
Quasi-Variational Inequalities
Related to Stochastic Control
Problems
M. Boulbrachene, M. Haiour and
B. Chentouf
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J. Ineq. Pure and Appl. Math. 3(2) Art. 30, 2002
Contents
1
2
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
The Continuous Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1
Notations, Assumptions and Preliminaries . . . . . . . . . 6
2.2
Elliptic Variational Inequalities . . . . . . . . . . . . . . . . . . 7
2.2.1
A Monotonicity property for VI (2.7) . . . . 7
2.3
Existence and uniqueness . . . . . . . . . . . . . . . . . . . . . . . 8
2.3.1
Properties of The Mapping T . . . . . . . . . . . 9
2.3.2
A Continuous Iterative Scheme of BensoussanLions Type . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3.3
Regularity of the solution of system (1.1) . 19
3
The Discrete Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.1
Discrete Variational Inequality . . . . . . . . . . . . . . . . . . . 20
3.1.1
A Discrete Monotonicity Property for VI
(3.4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2
The Noncoercive Discrete System of QVI’s . . . . . . . . 21
3.3
Existence and Uniqueness . . . . . . . . . . . . . . . . . . . . . . 22
3.4
A Discrete Iterative Scheme of Bensoussan-Lions
Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4
The Finite Element Error Analysis . . . . . . . . . . . . . . . . . . . . . . 25
4.1
A Contraction Associated with System of QVI’s (1.1) 27
4.2
A Contraction Associated with The Discrete System of QVI’s (3.5) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.3
An Auxiliary Coercive System of QVI’s . . . . . . . . . . . 30
4.4
L∞ - Error Estimate For the Noncoercive System of
QVI’s (1.1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
References
On a Noncoercive System of
Quasi-Variational Inequalities
Related to Stochastic Control
Problems
M. Boulbrachene, M. Haiour and
B. Chentouf
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1.
Introduction
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
a (u , v − ui ) = (f i , v − ui ) ∀v ∈ H01 (Ω)





ui ≤ k + ui+1 , v ≤ k + ui+1
(1.1)




 M +1
u
= u1 ,
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:
 i i
b (u , v − ui ) = (f i + λui , v − ui ) ∀v ∈ H01 (Ω)





ui ≤ k + ui+1 , v ≤ k + ui+1
(1.2)




 M +1
u
= u1 ,
On a Noncoercive System of
Quasi-Variational Inequalities
Related to Stochastic Control
Problems
M. Boulbrachene, M. Haiour and
B. Chentouf
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J. Ineq. Pure and Appl. Math. 3(2) Art. 30, 2002
where
(1.3)
bi (u, v) = ai (u, v) + λ(v, v)
and λ > 0 is large enough such that:
(1.4)
bi (v, v) ≥ γ kvk2H 1 (Ω) γ > 0; ∀v ∈ H01 (Ω).
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.
On a Noncoercive System of
Quasi-Variational Inequalities
Related to Stochastic Control
Problems
M. Boulbrachene, M. Haiour and
B. Chentouf
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2.
The Continuous Problem
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)
On a Noncoercive System of
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Related to Stochastic Control
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M. Boulbrachene, M. Haiour and
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1≤ j, k≤N
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aijk = aikj ; ai0 (x) = β > 0; x ∈ Ω.
(2.3)
1
We define the bilinear forms: for any u, v ∈ H (Ω),
(2.4) ai (u, v)
Z
=
Ω
X
1≤ j, k≤N
aijk (x)
!
N
X
∂u ∂v
∂u
i
i
+
b (x)
v + a0 (x).uv .
∂xj ∂xk k=1 k ∂xk
We are also given regular functions f i such that
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(2.5)
f i ∈ C 2 (Ω) and f i ≥ 0; ∀ i = 1, . . . , M .
J. Ineq. Pure and Appl. Math. 3(2) Art. 30, 2002
Q
∞
and the following norm: ∀ W = (w1 , . . . , wM ) ∈ M
i=1 L (Ω)
(2.6)
kW k∞ = max wi L∞ (Ω) ,
1≤i≤M
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.
On a Noncoercive System of
Quasi-Variational Inequalities
Related to Stochastic Control
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M. Boulbrachene, M. Haiour and
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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 < ∞.
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).
2.2.1.
Theorem 2.2. (cf. [4]) If f ≥ f˜ and ψ ≥ ψ̃ then σ(f, ψ) ≥ σ(f˜, ψ̃).
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2.3.
Existence and uniqueness
As mentioned earlier, we solve the noncoercive system of QVI’sby considering
M
the following auxiliary system: find a vector U = u1 , . . . , uM ∈ (H01 (Ω))
such that
 i i
b (u , v − ui ) = (f i + λui , v − ui ) ∀v ∈ H01 (Ω)





ui ≤ k + ui+1 , v ≤ k + ui+1
(2.8)




 M +1
u
= 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 an
increasing mapping
in [5].
Qas
M
∞
∞
∞
Let L (Ω) = 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
where ∀ i = 1, . . . , M , ζ i = σ (f i + λwi ; k + wi+1 ) is solution to the following VI:
 i i
 b (ζ , v − ζ i ) = (f i + λwi , v − ζ i ) ∀v ∈ H01 (Ω)
(2.10)
 i
ζ ≤ k + wi+1 , v ≤ k + wi+1 .
On a Noncoercive System of
Quasi-Variational Inequalities
Related to Stochastic Control
Problems
M. Boulbrachene, M. Haiour and
B. Chentouf
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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 .
1
M
On a Noncoercive System of
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Related to Stochastic Control
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M. Boulbrachene, M. Haiour and
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1
M
∞
i
i
Let V = (v , . . . , v ), W = (w , , . . . , w ) in L (Ω) such that v ≤ w ,
∀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 .
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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).
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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
i
i
i
i,0 +
i
i
i,0 +
b ζ , − ζ − û
= f + λw , − ζ − û
.
i
i,0 +
On the other hand, taking v = (ζ − û ) in equation (2.11) we get
+ i
+ bi ûi,0 , ζ i − ûi,0
= f + λûi,0 , ζ i − ûi,0
and, since W ≤ Û 0 , by addition, we obtain
+ +
=0
−bi ζ i − ûi,0 , ζ i − ûi,0
which, by (1.4), yields
i
ζ − û
i,0 +
= 0.
Thus
ζ i ≤ ûi,0 ∀ i = 1, 2, . . . , M
On a Noncoercive System of
Quasi-Variational Inequalities
Related to Stochastic Control
Problems
M. Boulbrachene, M. Haiour and
B. Chentouf
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T W ≤ Û 0 .
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J. Ineq. Pure and Appl. Math. 3(2) Art. 30, 2002
3. T is concave.
Let us agree on the following notations:
i
wθi = θwi + (1 − θ)w̃i ; wθ,k
= θ(k + wi ) + (1 − θ)(k + w̃i ); θ ∈ [0, 1].
Then we have:
T (θW + (1 − θ)W̃ )
= σ f 1 + λwθ1 ; k + wθ2 , . . . ,
σ
1
= σ f +
f + λwθi ; k + wθi+1 , . . . , σ f M + λwθM ; k
i+1
2
,...,σ
, . . . , σ f 1 + λwθi ; wθ,k
λwθ1 ; wθ,k
i
+ wθ1
M
f
1
+ λwθM ; wθ,k
.
On a Noncoercive System of
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Related to Stochastic Control
Problems
M. Boulbrachene, M. Haiour and
B. Chentouf
Now, denoting by:
ζ i = σ f i + λwi ; k + wi+1 ,
ζ̃ i = σ f i + λw̃i ; k + w̃i+1 ,
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ζθi = θζ i + (1 − θ)ζ̃ i ,
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Uθi
i
=σ f +
λwθi ; wθi+1
.
It is clear that ζθi is admissible for the problem which has Uθi as a solution.
So
−
Uθi + Uθi − ζθi
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is admissible for this problem. Therefore,
− − ≥ f + λwθi , Uθi − ζθi
.
(2.12)
b Uθi , Uθi − ζθi
−
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
− − ≥ − f + λwi , Uθi − ζθi
(2.13)
−b ζ i , Uθi − ζθi
and
(2.14)
i
−b ζ̃ ,
Uθi
−
−
ζθi
i
≥ − f + λw̃ ,
Uθi
−
−
ζθi
.
Now multiplying (2.13) by θ, and (2.14) by 1 − θ, addition yields
− − −b ζθi , Uθi − ζθi
≥ − f + λwθi , Uθi − ζθi
which added to (2.12) gives
− b Uθi − ζθi , Uθi − ζθi
≥ 0.
Thus
Uθi − ζθi
−
=0
which completes the proof i.e.,
T θW + (1 − θ)W̃ ≥ θT W + (1 − θ)T W̃ .
On a Noncoercive System of
Quasi-Variational Inequalities
Related to Stochastic Control
Problems
M. Boulbrachene, M. Haiour and
B. Chentouf
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2.3.2. A Continuous Iterative Scheme of Bensoussan-Lions Type
ing from Û 0 solution of (2.11) and Ǔ 0 = 0, we define the iterations:
(2.15)
Start-
Û n+1 = T Û n ; n = 0, 1, . . .
and
(2.16)
Ǔ n+1 = T Ǔ n ; n = 0, 1, . . . ,
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 + f 0 
∞
∞
Then we have
(2.17)
T (0) = µÛ 0 .
Proof. Indeed, from (2.16), T (0) = Ǔ 1 = (ǔ1,1 , . . . , ǔ1,M ), where ǔi,1 is the
solution of the following variational inequality:
 i i,1
 b (ǔ , v − ǔi,1 ) = (f i + λǔi,0 , v − ǔi,1 ) ∀v ∈ H01 (Ω)
(2.18)
 i,1
ǔ ≤ k; v ≤ k.
On a Noncoercive System of
Quasi-Variational Inequalities
Related to Stochastic Control
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M. Boulbrachene, M. Haiour and
B. Chentouf
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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
− = f 0 (1 − µ) − µλûi,0 , ǔi,1 − µûi,0
.
But, again, by the choice of µ
On a Noncoercive System of
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Related to Stochastic Control
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M. Boulbrachene, M. Haiour and
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f 0 (1 − µ) − µλûi,0
0
≥ f (1 − µ) − µλ Û 0 Thus, by (1.4)
ǔi,1 − µûi,0
∞
−
=0
≥ 0.
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i.e.,
ǔi,1 = µûi,0 ∀ i = 1, 2, . . . , M.
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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.
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]
On a Noncoercive System of
Quasi-Variational Inequalities
Related to Stochastic Control
Problems
M. Boulbrachene, M. Haiour and
B. Chentouf
≤ T W̃
Finally, using Lemma 2.4. we get (2.20).
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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).
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Proof. The proof will be carried out in five steps.
Step 1. The sequence (Û n ) stays in C and is decreasing.
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From (2.15) it is easy to see that ∀i, ûi,n is solution to the following VI:
 i i,n
b (û , 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)




 M +1,n
û
= 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:
On a Noncoercive System of
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Related to Stochastic Control
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M. Boulbrachene, M. Haiour and
B. Chentouf
Û 1 = T Û 0 ≤ Û 0 .
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Thus, inductively
(2.23)
0 ≤ Û n+1 = T Û n ≤ Û n ≤ · · · ≤ Û 0 .
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
2
γ ûi,n H 1 (Ω) ≤ bi (ûi,n , ûi,n ) ≤ f i + λûi,n−1 L2 (Ω) ûi,n H 1 (Ω)
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or more simply
i,n û 1 ≤ C,
H (Ω)
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).
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 .
On a Noncoercive System of
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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
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U − Ũ ≤ (1 − µ)U.
J. Ineq. Pure and Appl. Math. 3(2) Art. 30, 2002
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
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M. Boulbrachene, M. Haiour and
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W = Û 0 ; W̃ = Ǔ 0 ; γ = 1
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then
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T Û 0 − T Ǔ 0 ≤ (1 − µ)T Û 0 ,
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so
0 ≤ Û 1 − Ǔ 1 ≤ (1 − µ)Û 1 .
Applying (2.20) again, yields
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0 ≤ Û 2 − Ǔ 2 ≤ (1 − µ)2 Û 2
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and quite generally
0 ≤ Û n − Ǔ n ≤ (1 − µ)n Û n ≤ (1 − µ)n Û 0 ≤ (1 − µ)n Û 0 .
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J. Ineq. Pure and Appl. Math. 3(2) Art. 30, 2002
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 < ∞.
On a Noncoercive System of
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Related to Stochastic Control
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M. Boulbrachene, M. Haiour and
B. Chentouf
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3.
The Discrete Problem
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. Boulbrachene, M. Haiour and
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m(h)
(3.3)
∀v ∈ C(Ω) ∩
H01 (Ω) ,
rh v =
X
vl ϕl .
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.
On a Noncoercive System of
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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.
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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
i
i
i

 a (uh , v − u ) = (f , v − uh ) ∀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
On a Noncoercive System of
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these will not be given as they are respectively identical to their continuous
analogue ones.
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:
 i i
 b (ζh , v − ζhi ) = (f i + λwi , v − ζhi ) ∀v ∈ Vh ,
(3.9)
 i
ζh ≤ rh (k + wi+1 ), v ≤ rh (k + wi+1 ).
On a Noncoercive System of
Quasi-Variational Inequalities
Related to Stochastic Control
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M. Boulbrachene, M. Haiour and
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Proposition 3.2. Under the d.m.p, Th is increasing, concave and satisfies Th W ≤
Ûh0 ∀W ∈ L∞ (Ω), W ≤ Ûh0 .
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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:
(3.10)
Ûhn+1 = Th Ûhn
and
(3.11)
Ǔhn+1 = Th Ǔhn
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 
Ûh0 λ Ûh0 + f 0 
∞
∞
Then we have
(3.12)
Th (0) = µÛh0
Proposition 3.4. Let Ch = {W ∈ L∞ (Ω) such that 0 ≤ W ≤ Ûh0 }. Let also
γ ∈]0, 1], W, W̃ ∈ Ch such that:
(3.13)
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W − W̃ ≤ γW .
J. Ineq. Pure and Appl. Math. 3(2) Art. 30, 2002
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).
On a Noncoercive System of
Quasi-Variational Inequalities
Related to Stochastic Control
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M. Boulbrachene, M. Haiour and
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4.
The Finite Element Error Analysis
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:
 i i
b (z , v − z i ) = (F i , v − z i ) ∀v ∈ H01 (Ω)





z i ≤ k + z i+1
(4.1)




 M +1
z
= z1.
n
Denoting by z i = σ(F i , k + z i+1 ), we introduce the sequences Z = (z̄ 1,n , . . . ,
z̄ M,n ) and Z n = (z 1,n , . . . , z M,n ) defined by
z̄ i,n+1 = σ(F i , k + z̄ i+1,n ),
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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.
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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
M
1
M
Proposition 4.2. Let F , . . . , F ; F̃ , . . . , F̃
be two families of right
hands side and Z = z 1 , . . . , z M ; Z̃ = z̃ 1 , . . . , z̃ M be the respective solutions of system (4.1). Then the following holds. If F ≥ F̃ , then Z ≥ Z̃.
0
1,0
,M,0
0
1,0
M,0
Proof. Let Z̄ = z̄ , . . . , z̄
and Z̃ = z̃ , . . . , z̃
such that z̄ i,0 and
i,0
i,0
z̃ are solutions 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
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satisfy the following assertion.
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i
i
If F ≥ F̃ then z̄
i,n
≥ z̃
i,n
∀i = 1, . . . , M.
Indeed, since
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z̄ i,n+1 = σ F i , k + z̄ i+1,n ,
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z̃
i,n+1
= σ F̃ i , k + z̃
i+1,n
,
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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 .
i,n−1
Now assume that z̄ i,,n−1 ≥ z̃
. Then, as F i ≥ F̃ i , applying the same comparison argument as before, we get:
i,n
z̄ i,,n ≥ z̃ .
Finally, by Theorem 4.1, passing to the the limit as n → ∞, we get Z ≥ Z̃.
Remark 4.1. Proposition 4.2 remains true in the discrete case provided the
d.m.p is satisfied.
4.1.
M. Boulbrachene, M. Haiour and
B. Chentouf
A Contraction Associated with System of QVI’s (1.1)
Consider the following mapping
(4.2)
On a Noncoercive System of
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S : L∞ (Ω) → L∞ (Ω)
W → SW = Z = z 1 , . . . , z M ,
where Z is solution to the coercive system of QVI’s below
 i i
b (z , 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)




 M +1
z
= z1.
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By Theorem 4.1, problem (4.3) has one and only one solution.
Proposition 4.3. 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
1 1 i
i
i
Φ=
F − F̃ ; Φ =
F − F̃ λ+β
λ+β
∞
∞
it follows that
On a Noncoercive System of
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Related to Stochastic Control
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M. Boulbrachene, M. Haiour and
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F ≤ F̃ + F i − F̃ i i
i
∞
and
a0 (x) + λ i
F̃ +
F − F̃ ≤ F̃ i + (a0 (x) + λΦ) (because ai0 (x) = β > 0)
λ+β
∞
i
so, by Proposition 4.2, we obtain:
z i ≤ z̃ i + Φi .
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Interchanging the roles of W and W̃ , we similarly get
z̃ i ≤ z i + Φi .
Thus
i
z − z̃ i ∞ ≤ Φi
L (Ω)
which completes the proof.
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.
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:

b(zhi , v − zhi ) = (f + λwi , v − zhi ) ∀v ∈ Vh





zhi ≤ k + zhi+1 ; v ≤ k + zhi+1
(4.5)




 M +1
= zh1 .
zh
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Thanks to [6], [8] system (4.5) has one and only one solution.
Next, making use of Proposition 4.2 and Remark 4.1 we have the contraction
property of Sh .
Proposition 4.4. 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 4.3 and 4.4, 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.
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Related to Stochastic Control
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M. Boulbrachene, M. Haiour and
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4.3.
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An Auxiliary Coercive System of QVI’s
We consider the following coercive system of QVI’s: find Z̄h = z̄h1 , . . . , z̄hM
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 .
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Clearly, (4.6)
is a coercive system whose right hand side depends on U =
1
M
u ,...,u
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 4.5. (cf. [6])
Z̄h − U ≤ Ch2 |Logh|3 .
∞
(4.8)
4.4.
L∞ - Error Estimate For the Noncoercive System of QVI’s
(1.1)
On a Noncoercive System of
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M. Boulbrachene, M. Haiour and
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Let U and Uh be the solutions of system (1.1) and (3.5), respectively. Then we
have:
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Theorem 4.6.
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2
3
kU − Uh k∞ ≤ Ch |Logh| .
Proof. In view of (4.8) and Propositions 4.3 and 4.4, we clearly have
U = SU ; Uh = Sh Uh ; Z̄h = Sh U.
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Then , using estimation (4.8), we have
kSh U − SU k∞ = Z̄h − U ∞
≤ Ch2 |Logh|3 .
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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.
On a Noncoercive System of
Quasi-Variational Inequalities
Related to Stochastic Control
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M. Boulbrachene, M. Haiour and
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References
[1] L.C. EVANS AND A. FRIEDMAN, Optimal stochastic switching and the
Dirichlet Problem for the Bellman equations, Transactions of the American
Mathematical Society, 253 (1979), 365–389.
[2] P.L. LIONS AND J.L MENALDI, Optimal control of stochastic integrals
and Hamilton Jacobi Bellman equations (Part I), SIAM Control and Optimization, 20 (1979).
[3] D. KINDERLEHRER AND G. STAMPACCHIA, An Introduction to Variational Inequalities and their Applications, Academic Press (1980).
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[4] A. BENSOUSSAN AND J.L. LIONS, Applications des Inequations Variationnelles en Controle Stochastique, Dunod, Paris, (1978).
M. Boulbrachene, M. Haiour and
B. Chentouf
[5] A. BENSOUSSAN AND J.L. LIONS, Impulse Control and Quasivariational Inequalities, Gauthier Villars, Paris, (1984).
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[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
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[8] P. CORTEY-DUMONT, Sur l’ analyse numerique des equations de
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J. Ineq. Pure and Appl. Math. 3(2) Art. 30, 2002