Bulletin of Mathematical Analysis and Applications
ISSN: 1821-1291, URL: http://www.bmathaa.org
Volume 4 Issue 4 (2012), Pages 99-108
ON GENERALIZED CONFORMAL MAPS
(COMMUNICATED BY KRISHNAN DUGGAL)
A. M. CHERIF, H. ELHENDI AND M. TERBECHE
Abstract. In this paper, we present some new properties of generalized conformal maps as f-biharmonic between equidimensional Riemannian manifolds.
1. Introduction
Consider a smooth map ϕ : M −→ N between Riemannian manifolds M =
(M m , g) and N = (N n , h) and f : M × N −→ (0, +∞) be a smooth positive
function, then the f -energy functional of ϕ is defined by
Z
1
Ef (ϕ) =
f (x, ϕ(x)) |dx ϕ|2 vg
2 M
(or over any compact subset K ⊂ M ).
A map is called f -harmonic if it is a critical point of the f -energy functional. By
[1] the map ϕ is f -harmonic if and only if
τf (ϕ)
= fϕ τ (ϕ) + dϕ(gradM fϕ ) − e(ϕ)(gradN f ) ◦ ϕ = 0,
(1)
where fϕ : M m −→ (0, +∞) be a smooth positive function defined by
fϕ (x) = f (x, ϕ(x)),
∀x ∈ M,
τ (ϕ) = traceg ∇dϕ is the tension field of ϕ, and e(ϕ) = 21 |dϕ|2 is the energy density
of ϕ.
τf (ϕ) is called the f -tension field of ϕ, and the f -bi-energy functional of ϕ is
defined by
Z
1
|τf (ϕ)|2 vg .
E2,f (ϕ) =
2 M
0
2000 Mathematics Subject Classification: 53A45, 53C20, 58E20.
Keywords and phrases. Conformal maps; f-harmonic maps; f-biharmonic maps.
c 2012 Universiteti i Prishtinës, Prishtinë, Kosovë.
Submitted August 21, 2012. Published November 6, 2012.
99
100
A. M. CHERIF, H. ELHENDI AND M. TERBECHE
A map ϕ is called f -biharmonic if it is a critical point of the f -energy functional.
By [1] the map ϕ is f -bi-harmonic if and only if
τ2,f (ϕ)
= −fϕ traceg RN (τf (ϕ), dϕ)dϕ − traceg ∇ϕ fϕ ∇ϕ τf (ϕ)
N
+e(ϕ)(∇N
τf (ϕ) grad f )
(2)
M
◦ ϕ − dϕ(grad τf (ϕ)(f ))
ϕ
−τf (ϕ)(f )τ (ϕ)+ < ∇ τf (ϕ), dϕ > (gradN f ) ◦ ϕ = 0,
where h , i is the inner product on T ∗ M ⊗ ϕ−1 T N and RN is the curvature tensor
of N .
τ2,f (ϕ) is called the f -bi-tension field of ϕ.
The f -biharmonic concept is a natural generalization of biharmonic maps ([3][4][7][8]).
2. Conformal f -bi-harmonic maps.
Theorem A.
Let ϕ : (M n , g) −→ (N n , h) be a conformal map with dilation λ.
The f -bi-tension field of ϕ is given by
τ2,f (ϕ)
=
F1 dϕ(gradM ln λ) + F2 dϕ(gradM (fN ◦ ϕ))
+2(n − 2)fϕ2 dϕ(RicciM (gradM ln λ)) − 2fϕ dϕ(RicciM (gradM fϕ ))
+nfϕ dϕ(RicciM (gradM (fN ◦ ϕ))) + 2(n − 2)fϕ2 < ∇dϕ, ∇d ln λ >
−2fϕ < ∇dϕ, ∇dfϕ > +nfϕ < ∇dϕ, ∇d(fN ◦ ϕ) >
+(n − 2)fϕ2 dϕ(gradM ∆(ln λ)) − fϕ dϕ(gradM ∆(fϕ ))
n
+ fϕ dϕ(gradM ∆(fN ◦ ϕ)) + 4(n − 2)fϕ ∇ϕ
dϕ(gradM ln λ)
gradM fϕ
2
−(n − 2)2 fϕ2 ∇ϕ
dϕ(gradM ln λ) − ∇ϕ
dϕ(gradM fϕ )
gradM ln λ
gradM fϕ
n(n − 2)
dϕ(gradM ln λ)
f ϕ ∇ϕ
gradM (fN ◦ϕ)
2
n(n − 2) ϕ
n2 ϕ
−
∇gradM ln λ dϕ(gradM (fN ◦ ϕ)) −
∇
dϕ(gradM (fN ◦ ϕ))
M
2
4 grad (fN ◦ϕ)
M
M
M
+(n − 2)dϕ(∇M
ln λ)
gradM ln λ grad (fN ◦ ϕ)) + (n − 2)dϕ(∇gradM (fN ◦ϕ) grad
dϕ(gradM (fN ◦ ϕ)) −
+n∇ϕ
gradM fϕ
M
M
M
−dϕ(∇M
gradM fϕ grad (fN ◦ ϕ)) − dϕ(∇gradM (fN ◦ϕ) grad fϕ )
n
+ dϕ(gradM (|gradM (fN ◦ ϕ)|2 ),
2
where
F1
=
(n − 2)|gradM fϕ |2 + (n − 2)fϕ ∆(fϕ ) − (n − 2)2 g(gradM ln λ, gradM (fN ◦ ϕ))
n(n − 2)
+(n − 2)g(gradM fϕ , gradM (fN ◦ ϕ)) −
|gradM (fN ◦ ϕ)|2 ,
2
and
F2
= −(n − 2)∆(ln λ) + ∆(fϕ ) −
n
∆(fN ◦ ϕ).
2
We need the following lemmas to prove Theorem A.
ON GENERALIZED CONFORMAL MAPS
101
Lemma A. Let ϕ : (M n , g) −→ (N n , h) be a conformal map with dilation λ. The
f -tension field of ϕ is given by
τf (ϕ)
=
(2 − n)fϕ dϕ(gradM ln λ) + dϕ(gradM fϕ )
n
− dϕ gradM (fN ◦ ϕ) ,
2
where, at x ∈ M , fN : N −→ (0, ∞) defined by fN (y) = f (x, y) for all y ∈ N .
Proof. Since τ (ϕ) = (2 − n)dϕ(gradM ln λ), e(ϕ) = n2 λ2 and let {ei }ni=1 be an
orthonormal frame in M , then {fi }ni=1 , where fi ◦ ϕ = λ1 dϕ(ei ) for all i = 1, ..., n,
be an orthonormal frame in N . Then
(gradN f ) ◦ ϕ(x)
(gradN fN )ϕ(x)
=
= fi |ϕ(x) (fN )fi |ϕ(x)
1
=
dϕ(ei )x (fN )dϕ(ei )x
2
λ (x)
1
ei |x (fN ◦ ϕ)dϕ(ei )x
=
λ2 (x)
1
dϕ gradM (fN ◦ ϕ) x ,
=
2
λ (x)
for all x ∈ M .
Lemma B. Let γ : M −→ R be a smooth function, then
traceg (∇ϕ )2 dϕ(gradM γ)
=
dϕ(gradM ln λ) + 2dϕ(RicciM gradM γ)
(2 − n)∇ϕ
gradM γ
−traceg RN (dϕ(gradM γ), dϕ)dϕ + 2 < ∇dϕ, ∇dγ >
+dϕ(gradM ∆(γ)),
M
where < ∇dϕ, ∇dγ >= ∇dϕ(ei , ej )∇dγ(ei , ej ) = ∇dϕ(ei , ej )g(∇M
ei grad γ, ej ).
Proof. Fix a point x0 ∈ M and let {ei }ni=1 be an orthonormal frame, such that
∇M
ei ej = 0 at x0 for all i, j = 1, ..., n. Then calculating at x0 , we have
traceg (∇ϕ )2 dϕ(gradM γ)
=
ϕ
M
∇ϕ
ei ∇ei dϕ(grad γ)
=
M
ϕ
M
M
∇ϕ
ei ∇dϕ(ei , grad γ) + ∇ei dϕ(∇ei grad γ) (3)
=
M
M
M
∇ϕ
ei ∇dϕ(ei , grad γ) + ∇dϕ(ei , ∇ei grad γ)
M
M
+dϕ(∇M
ei ∇ei grad γ)
=
M
∇ϕ
ei ∇dϕ(ei , grad γ)+ < ∇dϕ, ∇dγ >
+dϕ(traceg (∇M )2 gradM γ)
Let ∇2 dϕ be the third fundamental form defined by
ϕ
ϕ
∇2 dϕ(X, Y, Z) = ∇ϕ
X ∇dϕ(Y, Z) − ∇dϕ(∇X Y, Z) − ∇dϕ(Y, ∇X Z),
(4)
for all X, Y, Z ∈ Γ(T M ). And we have
∇2 dϕ(X, Y, Z)
= ∇2 dϕ(Z, Y, X) + dϕ(RM (Z, X)Y )
N
−R (dϕ(Z), dϕ(X))dϕ(Y ).
(5)
102
A. M. CHERIF, H. ELHENDI AND M. TERBECHE
By (4) and (5), the first term of (3) is
M
∇ϕ
ei ∇dϕ(ei , grad γ)
M
= ∇2 dϕ(ei , ei , gradM γ) + ∇dϕ(ei , ∇M
ei grad γ)
= ∇2 dϕ(gradM γ, ei , ei ) + dϕ(RM (gradM γ, ei )ei )
M
−RN (dϕ(gradM γ), dϕ(ei ))dϕ(ei ) + ∇dϕ(ei , ∇M
ei grad γ)
=
∇ϕ
τ (ϕ) + dϕ(RicciM gradM γ)
gradM γ
−traceg RN (dϕ(gradM γ), dϕ)dϕ+ < ∇dϕ, ∇dγ > .
From the equation
traceg (∇M )2 gradM γ = RicciM (gradM γ) + gradM (∆γ),
the Lemma B follows.
Lemma C. Let γ : M −→ R be a smooth function, then
< ∇dϕ(gradM γ), dϕ >= nλ2 g(gradM γ, gradM ln λ) + λ2 ∆(γ).
Proof. Let {ei }ni=1 be an orthonormal frame, then
< ∇dϕ(gradM γ), dϕ >
= h(∇ei dϕ(gradM γ), dϕ(ei ))
= ei (h(dϕ(gradM γ), dϕ(ei )))
−h(dϕ(gradM γ), ∇ei dϕ(ei ))
= ei (λ2 g(gradM γ, ei ))
−(2 − n)h(dϕ(gradM γ), dϕ(gradM ln λ))
= g(gradM γ, gradM λ2 )
+(n − 2)λ2 g(gradM γ, gradM ln λ)
+λ2 ei (g(gradM γ, ei ))
=
2λ2 g(gradM γ, gradM ln λ)
+(n − 2)λ2 g(gradM γ, gradM ln λ)
+λ2 ∆(γ)
= nλ2 g(gradM γ, gradM ln λ) + λ2 ∆(γ).
Proof of Theorem A.
ON GENERALIZED CONFORMAL MAPS
103
Let {ei }ni=1 be an orthonormal frame, such that ∇M
ei ej = 0 at x0 ∈ M for all
i, j = 1, ..., n. Then calculating at x0 , we have
−(2 − n)traceg ∇ϕ fϕ ∇ϕ fϕ dϕ(gradM ln λ)
=
ϕ
M
(n − 2)∇ϕ
ln λ)
ei fϕ ∇ei fϕ dϕ(grad
=
(n − 2)∇ϕ
f dϕ(gradM ln λ)
gradM fϕ ϕ
ϕ
M
+(n − 2)fϕ ∇ϕ
ln λ)
ei ∇ei fϕ dϕ(grad
=
(n − 2)|gradM fϕ |2 dϕ(gradM ln λ)
+(n − 2)fϕ ∇ϕ
dϕ(gradM ln λ)
gradM fϕ
+(n − 2)fϕ ∆(fϕ )dϕ(gradM ln λ)
+2(n − 2)fϕ ∇ϕ
dϕ(gradM ln λ)
gradM fϕ
+(n − 2)fϕ2 traceg (∇ϕ )2 dϕ(gradM ln λ),
and by the lemma B, we obtain
− (2 − n)traceg ∇ϕ fϕ ∇ϕ fϕ dϕ(gradM ln λ)
=
(n − 2)|gradM fϕ |2 dϕ(gradM ln λ)
dϕ(gradM ln λ)
+3(n − 2)fϕ ∇ϕ
gradM fϕ
+(n − 2)fϕ ∆(fϕ )dϕ(gradM ln λ)
n
+(n − 2)fϕ2 (2 − n)∇ϕ
dϕ(gradM ln(6)
λ)
gradM ln λ
+2dϕ(RicciM gradM ln λ)
−traceg RN (dϕ(gradM ln λ), dϕ)dϕ
o
+2 < ∇dϕ, ∇d ln λ > +dϕ(gradM ∆(ln λ)) .
Of the same method, we obtain
− traceg ∇ϕ fϕ ∇ϕ dϕ(gradM fϕ )
=
dϕ(gradM fϕ )
−∇ϕ
gradM fϕ
n
−fϕ (2 − n)∇ϕ
dϕ(gradM ln λ)
gradM fϕ
+2dϕ(RicciM gradM fϕ )
N
(7)
M
−traceg R (dϕ(grad fϕ ), dϕ)dϕ
o
+2 < ∇dϕ, ∇dfϕ > +dϕ(gradM ∆(fϕ )) ,
n
traceg ∇ϕ fϕ ∇ϕ dϕ(gradM (fN ◦ ϕ))
2
=
n ϕ
M
∇
(fN ◦ ϕ))
M f dϕ(grad
ϕ
2 grad
n
n
+ fϕ (2 − n)∇ϕ
dϕ(gradM ln λ)
gradM (fN ◦ϕ)
2
+2dϕ(RicciM gradM (fN ◦ ϕ))
N
(8)
M
−traceg R (dϕ(grad (fN ◦ ϕ)), dϕ)dϕ
o
+2 < ∇dϕ, ∇d(fN ◦ ϕ) > +dϕ(gradM ∆((fN ◦ ϕ))) ,
104
A. M. CHERIF, H. ELHENDI AND M. TERBECHE
by Lemma A we have
N
e(ϕ)∇N
τf (ϕ) grad f
1
n 2n
dϕ(gradM (fN ◦ ϕ))
λ (2 − n)∇ϕ
gradM ln λ λ2
2
1
dϕ(gradM (fN ◦ ϕ))
+∇ϕ
gradM fϕ λ2
o
n
1
M
dϕ(grad
− ∇ϕ
(f
◦
ϕ))
N
M
2 grad (fN ◦ϕ) λ2
n 2 n −2(2 − n)
|gradM ln λ|2 dϕ(gradM (fN ◦ ϕ)) (9)
=
λ
2
λ2
2−n
dϕ(gradM (fN ◦ ϕ))
+ 2 ∇ϕ
gradM ln λ
λ
2
− 2 (gradM fϕ )(ln λ)dϕ(gradM (fN ◦ ϕ))
λ
1
dϕ(gradM (fN ◦ ϕ))
+ 2 ∇ϕ
M
λ grad fϕ
n
+ 2 (gradM (fN ◦ ϕ))(ln λ)dϕ(gradM (fN ◦ ϕ))
λ
o
n
M
− 2 ∇ϕ
dϕ(grad
(f
◦
ϕ))
.
N
M
2λ grad (fN ◦ϕ)
=
The function τf (ϕ)(f ) is given by
τf (ϕ)(f )
= τf (ϕ)(fN )
=
(2 − n)(gradM ln λ)(fN ◦ ϕ) + (gradM fϕ )(fN ◦ ϕ)
n
− |gradM (fN ◦ ϕ)|2 ,
2
then
− dϕ(gradM τf (ϕ)(f ))
=
(n − 2)dϕ(gradM ((gradM ln λ)(fN ◦ ϕ)))
−dϕ(gradM ((gradM fϕ )(fN ◦ ϕ)))
n
+ dϕ(gradM (|gradM (fN ◦ ϕ)|2 ))
2
M
= (n − 2)dϕ(∇M
gradM ln λ grad (fN ◦ ϕ))
+(n −
M
2)dϕ(∇M
gradM (fN ◦ϕ) grad
(10)
ln λ)
M
−dϕ(∇M
gradM fϕ grad (fN ◦ ϕ))
M
−dϕ(∇M
gradM (fN ◦ϕ) grad fϕ )
n
+ dϕ(gradM (|gradM (fN ◦ ϕ)|2 )),
2
− τf (ϕ)(f )τ (ϕ)
=
n
(n − 2) (2 − n)(gradM ln λ)(fN ◦ ϕ) + (gradM fϕ )(fN ◦ ϕ)
o
n
− |gradM (fN ◦ ϕ)|2 dϕ(gradM ln λ).
(11)
2
ON GENERALIZED CONFORMAL MAPS
105
Then by lemma C, we have
< ∇τf (ϕ), dϕ > (gradN f ) ◦ ϕ =
n
(2 − n)nλ2 |gradM ln λ|2 + (2 − n)λ2 ∆(ln λ)
+nλ2 g(gradM fϕ , gradM ln λ) + λ2 ∆(fϕ )
n2 2
λ g(gradM (fN ◦ ϕ), gradM ln λ)
(12)
2
o1
n
− λ2 ∆(fN ◦ ϕ) 2 dϕ(gradM (fN ◦ ϕ)).
2
λ
−
The Theorem A, follows from (2), (6), (7), (8), (9), (10), (11) and (12).
Particular Cases
(1) If f = 1, we obtain
τ2,f (ϕ)
= τ2 (ϕ)
=
2(n − 2)dϕ(RicciM (gradM ln λ)) + 2(n − 2) < ∇dϕ, ∇d ln λ >
+(n − 2)dϕ(gradM ∆(ln λ)) − (n − 2)∇ϕ
dϕ(gradM ln λ).
gradM ln λ
Is the natural bi-tension field of ϕ.
(2) If f1 : M −→ (0, ∞), be a smooth positif function. If f (x, y) = f1 (x) for
all (x, y) ∈ M × N , then the f -bi-tension field of ϕ is given by
τ2,f (ϕ)
=
τ2,f1 (ϕ)
=
(n − 2)|gradM f1 |2 dϕ(gradM ln λ) + (n − 2)f1 ∆(f1 )dϕ(gradM ln λ)
+2(n − 2)f12 dϕ(RicciM (gradM ln λ)) − 2f1 dϕ(RicciM (gradM f1 ))
+2(n − 2)f12 < ∇dϕ, ∇d ln λ > −2f1 < ∇dϕ, ∇df1 >
+(n − 2)f12 dϕ(gradM ∆(ln λ)) − f1 dϕ(gradM ∆(f1 ))
+4(n − 2)f1 ∇ϕ
dϕ(gradM ln λ) − (n − 2)2 f12 ∇ϕ
dϕ(gradM ln λ)
gradM f1
gradM ln λ
−∇ϕ
dϕ(gradM f1 ),
gradM f1
we recover the result obtained in [6].
106
A. M. CHERIF, H. ELHENDI AND M. TERBECHE
Theorem B. Let ϕ : (M n , g) −→ (N n , h) be a conformal map with dilation λ.
Then ϕ is f -bi-harmonic if and only if
0
=
F3 gradM ln λ + F4 gradM (fN ◦ ϕ) + F5 gradM fϕ
+2(n − 2)fϕ2 RicciM (gradM ln λ) − 2fϕ RicciM (gradM fϕ )
M
+nfϕ RicciM (gradM (fN ◦ ϕ)) − 4fϕ ∇M
gradM ln λ grad fϕ
n
+(n − 2)fϕ2 gradM ∆(ln λ) − fϕ gradM ∆(fϕ ) + fϕ gradM ∆(fN ◦ ϕ)
2
n(n − 2)
M
M
M
+4(n − 2)fϕ ∇gradM fϕ grad ln λ −
f ϕ ∇M
ln λ
gradM (fN ◦ϕ) grad
2
M
M
+(n − 2)∇M
ln λ − ∇M
gradM (fN ◦ϕ) grad
gradM (fN ◦ϕ) grad fϕ
(n − 2)(n − 6) 2
fϕ gradM (|gradM ln λ|2 )
2
(n − 2)2 M
∇gradM ln λ gradM (fN ◦ ϕ)
+ 2nfϕ −
2
1
M
− gradM (|gradM fϕ |2 ) + (n − 1)∇M
gradM fϕ grad (fN ◦ ϕ)
2
n(n − 2)
gradM (|gradM (fN ◦ ϕ)|2 ),
−
4
−
where
F3
= F1 − 2(n − 2)fϕ2 ∆(ln λ) + 2fϕ ∆(fϕ ) − nfϕ ∆(fN ◦ ϕ)
−(n − 2)2 fϕ2 |gradM ln λ|2 + |gradM fϕ |2
−ng(gradM fϕ , gradM (fN ◦ ϕ)) +
F4
=
F2 + ng(gradM fϕ , gradM ln λ) −
−
n2
|gradM (fN ◦ ϕ)|2 ,
4
n(n − 2)
(fϕ + 1)|gradM ln λ|2
2
n2
g(gradM (fN ◦ ϕ), gradM ln λ),
2
and
F5
=
4(n − 2)fϕ |gradM ln λ|2 − 2g(gradM fϕ , gradM ln λ)
+ng(gradM ln λ, gradM (fN ◦ ϕ)).
Proof. The Theorem B follows from Theorem A and the following formulae
h(< ∇dϕ, ∇d ln γ >, dϕ(X))
= g(2λ2 ∇gradM ln λ gradM ln γ
2
M
−λ ∆(ln γ)grad
M
h(∇ϕ
ln γ), dϕ(Y ))
X dϕ(grad
(13)
ln λ, X),
= g(λ2 X(ln λ)gradM ln γ − λ2 X(ln γ)gradM ln λ
+
M
λ2 d ln λ(gradM ln γ)X + λ2 ∇M
ln γ, Y ),
X grad
(14)
for all smooth function γ : M −→ R and X, Y ∈ Γ(T M ).
ON GENERALIZED CONFORMAL MAPS
107
Corollary A. Let ϕ : (M n , g) −→ (N n , h) be a conformal map with constant
dilation λ. Then ϕ is f -bi-harmonic if and only if the function f satisfies the
equation
n
0 =
∆(fϕ ) − ∆(fN ◦ ϕ) gradM (fN ◦ ϕ) − 2fϕ RicciM (gradM fϕ )
2
n
+nfϕ RicciM (gradM (fN ◦ ϕ)) − fϕ gradM ∆(fϕ ) + fϕ gradM ∆(fN ◦ ϕ)
2
1
M
M
M
M
2
−∇gradM (fN ◦ϕ) grad fϕ − grad (|grad fϕ | )
2
n(n − 2)
M
+(n − 1)∇M
gradM (|gradM (fN ◦ ϕ)|2 ).
gradM fϕ grad (fN ◦ ϕ) −
4
Corollary B. Let ϕ : (M n , g) −→ (N n , h) be a conformal map with constant
dilation λ. If fϕ = fN ◦ ϕ = γ, then ϕ is f -bi-harmonic if and only if the function
γ satisfies the equation
2−n
∆(γ)gradM γ − (2 − n)γ RicciM (gradM γ)
0 =
2
(2 − n)2
2−n
1
−
γ gradM ∆(γ) −
+
gradM (|gradM γ|2 ).
2
4
2
Example A. Let ϕ : R2 −→ (N 2 , h) be a conformal map with constant dilation λ.
If fϕ = fN ◦ ϕ = γ, then ϕ is f -bi-harmonic if and only if
(
∂γ ∂ 2 γ
∂γ ∂ 2 γ
∂x ∂x2 + ∂y ∂x∂y = 0,
∂γ ∂ 2 γ
∂y ∂y 2
+
∂γ ∂ 2 γ
∂x ∂x∂y
= 0.
Example B. Let ϕ : (M 2 , g) −→ (N 2 , h) be a conformal map with dilation λ. If
fϕ = fN ◦ ϕ = ln λ, then ϕ is f -bi-harmonic if and only if the function λ satisfies
the equation
gradM (|gradM ln λ|2 ) = 0.
Moreover, if M is related, then ϕ is f -bi-harmonic if and only if the function
|gradM ln λ| is constant in M .
Acknowledgements. The authors are highly thankful to the referee for his
valuable suggestions towards the improvement of the paper.
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A. M. CHERIF, H. ELHENDI AND M. TERBECHE
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Ahmed Cherif Mohamed
Laboratory of Geometry, Analysis, Contrle and Applications.
Saida University,BP 138, 200000, Algeria
E-mail address: Ahmedcherif29@hotmail.fr
Elhendi Hichem
Laboratory of Geometry, Analysis, Controle and Applications.
Saida University,BP 138, 200000, Algeria.
E-mail address: elhendihichem@yahoo.fr
Terbeche Mekki
Department of Mathematics, University of Oran Es-Senia, Algeria.
E-mail address: Lgaca Saida2009@hotmail.com