Semilinear Functional Differential Equations of Fractional order with State-Dependent Delay

Mouffak Benchohra, Omar Bennihi, Khalil Ezzinbi

Semilinear Functional Differential Equations of Fractional order with State-Dependent Delay

Abstract. In this paper we provide sufficient conditions for the existence and uni- queness of mild solutions for a class of semilinear functional differential equations of fractional order with state-dependent delay. The nonlinear alternative of Frigon- Granas type for contractions maps in Fr´echet spaces combined with α-resolvent fa- mily is the main tool in our analysis.

2000 Mathematics Subject Classification: 34G20, 34G25, 34H05, 34K09, 34K30.

Key words and phrases: Semilinear functional differential equation, mild solution, resolvent family, fixed point, state-dependent delay, contractive map, Fr´echet space.

1. Introduction. In recent years fractional calculus has found many applica- tions in physics, mechanics, chemistry, porous media, viscoelasticity, electrochemi- stry, electromagnetism, engineering, control, etc. [7, 17, 26, 28]. Recent developments of differential and integral equations of fractional order are reported in the books by Abbas et al. [1], Baleanu et al. [7], Kilbas et al. [19], Lakshmikantham et al. [20], and the references therein.

In this paper we discuss the existence of the unique mild solution defined on the semi-infinite positive real interval [0, +∞) for a class of fractional functional partial differential evolution equations with state dependent delay.

Complicated situations in which the delay depends on the unknown functions

have been proposed in modeling in recent years. These equations are frequently cal-

led equations with state-dependent delay. Existence results and other things were

derived recently from functional differential equations when the solution is depen-

ding on the delay. We refer the reader to the papers by Adimy and Ezzinbi [2],

Agarwal et al. [3], Ait Dads and Ezzinbi [4], and Hernandez et al. [16]. Over the

past several years it has become apparent that equations with state-dependent de-

lay arise also in several areas such as in classical electrodynamics [12], in population

models [8], in models of commodity price fluctuations [9, 23], and in models of blood cell productions [24].

Recently Li and Peng [22] studied a class of abstract homogeneous fractional evolution equation. Baghli et al. [6], have proved global existence and uniqueness results for an initial value problem for functional differential equations of first order with state-dependent delay. Functional differential equations involving the Riemann-Liouville fractional derivative were considered by Benchohra et al. [10].

N’Gu´er´ekata and Mophou [25] studied semilinear neutral fractional functional evo- lution equations with infinite delay using the notion of α - resolvent family.

Motivated by the above papers, in this paper we studied the existence and uniqueness of solutions for semilinear functional differential equations of fractional order with state-dependent delay in a real Banach space (E, |.|) when the delay is infinite. In particular, in Section 3, we consider the following initial value problem

D ^α x(t) = Ax(t) + f (t, x ρ(t,x

) ), t ∈ J := [0, +∞), 0 < α < 1, (1)

x(t) = ϕ(t), t ∈ (−∞, 0], (2)

where A : D(A) ⊂ E → E is the infinitesimal generator of an α-resolvent family (T α (t)) t ­0 defined on a real Banach space E, D ^α is understood here in the Riemann- Liouville sense, f : J × B → E, ρ : J × B → R are appropriate given functions and satisfy some conditions that will be specified later, ϕ belongs to an abstract space denoted B and called phase space, with ϕ(0) = 0. For any function x defined on (−∞, b] and any t ∈ J, we denote by x ^t the element of B defined by

x t (θ) = x(t + θ), θ ∈ (−∞, 0].

the function x t represents the history of the state from −∞ up to the present time t. Our approach is based on the nonlinear alternative of Leray-Schauder type due to Frigon and Granas [14]. These results can be considered as a contribution to this emerging field.

2. Preliminaries. In this section, we collect a few auxiliary results which will be needed in the sequel.

Let b > 0. By C([0, b]; E) we denote the Banach space of continuous functions from [0, b] into E, normed by

kyk ∞ = sup

t∈[0,b] |y(t)|.

B(E) is the space of bounded linear operators from E into E, with the usual su- premum norm

kNk ^B(E) = sup{|N(y)| : |y| = 1}.

A measurable function y : [0, b] → E is Bochner integrable if and only if |y| is Lebes- gue integrable. Let L ¹ ([0, b], E) denotes the Banach space of measurable functions y : [0, b] → E which are Bochner integrable normed by

kyk ^L

= Z b

0 |y(t)|dt.

(For the Bochner integral properties, see the classical monograph of Yosida[29]).

Definition 2.1 The fractional primitive of order α > 0 of a function h : R ⁺ → E is defined by

(3) I ₀ ^α h(t) := 1

Γ(α) Z t

0 (t − s) ^{α −1} h(s)ds,

provided the right side exists pointwise on R ⁺ . Γ is the gamma function.

Definition 2.2 The fractional derivative of order α > 0 of a function h : R ⁺ → E is defined as follow

(4) D ^α ₀ h(t) := 1 Γ(1 − α)

d dt

Z t

0 (t − s) ^−α h(s)ds = d

dt I ₀ ^1−α h(t).

Definition 2.3 The Laplace transform of a function f ∈ L ¹ loc ( R ⁺ , E) is defined by

f (λ) := b Z ∞

0

e ^−λt f (t)dt, Re(λ) > ω, if the integral is absolutely convergent for Re(λ) > ω.

In order to defined the mild solution of the considered problem, we recall the follo- wing definition

Definition 2.4 Let A be a closed and linear operator with Domain D(A) defined on a Banach space E. We call A the generator of an α-resolvent family or solution operator if there exists ω > 0 and a strongly continuous function T α : R ⁺ → L(E) such that

{λ : Re(λ) > ω} ⊂ ρ(A), and

(λ ^α − A) ⁻¹ x = Z ∞

0 e ^−λt T α (t)xdt, Re(λ) > ω, x ∈ E.

In this case, T α (t) is called the solution operator generated by A.

The following result is a direct consequence of ([21], Proposition 3.1 and Lemma 2.2).

Proposition 2.5 Let T α (t) ∈ L(E) be the solution operator with generator A.

Then the following conditions are satisfied:

1. T α (t) is strongly continuous for t ­ 0 and T ^α (0) = I.

2. T α (t)D(A) ⊂ D(A) and AT ^α (t)x = T α (t)Ax for all x ∈ D(A), t ­ 0.

3. For every x ∈ D(A) and t ­ 0,

T α (t)x = x + Z t 0

(t − s) ^{α −1}

Γ(α) AT α (s)xds.

4. Let x ∈ D(A) , then Z t

0

(t − s) ^{α −1}

Γ(α) T α (s)xds ∈ D(A).

and

T α (t)x = x + A Z t 0

(t − s) ^{α −1}

Γ(α) T α (s)xds

Remark 2.6 The concept of a solution operator, as defined above, is closely related to the concept of a resolvent family (see Pr¨uss)[27]. Because of the uniqueness of the Laplace transform, in the border case α = 1, the family T α (t) corresponds to the C 0 - semigroup (see [13]), whereas in the case α = 2 a solution operator corresponds to the concept of cosine family (see [5]).

For more details on α-resolvent family, we refer to [25] and the references therein.

We will define the phase space B axiomatically, using ideas and notations de- veloped by Hale and Kato [15]. More precisely, B will denote the vector space of functions defined from (−∞, 0] into E endowed with a seminorm denoted k.k B and such that the following axioms hold.

• (A ¹ ) If x : (−∞, b) → E, is continuous on [0, b] and x ⁰ ∈ B, then for t ∈ [0, b) the following conditions hold

– (i) x t ∈ B

– (ii)kx ^t k B ¬ K(t) sup{|x(s)| : 0 ¬ s ¬ t} + M(t)kx ⁰ k B , – (iii)|x(t)| ¬ Hkx ^t k B

where H ­ 0 is a constant, K : [0, b) → [0, +∞),

M : [0, + ∞) → [0, +∞) with K continuous and M locally bounded and H, K and M are independent of x(.).

• (A ² ) For the function x in (A 1 ), the function t → x ^t is a B-valued continuous function on [0, b].

• (A ³ ) The space B is complete.

Denote K b = sup{K(t) : t ∈ [0, b]} and M ^b = sup{M(t) : t ∈ [0, b]}.

Remark 2.7 1. [(iii)] is equivalent to |φ(0)| ¬ Hkφk B for every φ ∈ B.

2. Since k · k B is a seminorm, two elements φ, ψ ∈ B can verify kφ − ψk B = 0 without necessarily φ(θ) = ψ(θ) for all θ ¬ 0.

3. From the equivalence of in the first remark, we can see that for all φ, ψ ∈ B

such that kφ − ψk B = 0, we necessarily have that φ(0) = ψ(0).

We now indicate some examples of phase spaces. For other details we refer, for instance to the book by Hino et al. [18].

Example 2.8 Let:

BC the space of bounded continuous functions defined from ( −∞, 0] to E;

BU C the space of bounded uniformly continuous functions defined from ( −∞, 0] to E;

C ^∞ := {φ ∈ BC : lim ^θ →−∞ φ(θ) exist in E } ;

C ⁰ := {φ ∈ BC : lim ^θ →−∞ φ(θ) = 0 } , endowed with the uniform norm kφk = sup{|φ(θ)| : θ ¬ 0}.

We have that the spaces BUC, C ^∞ and C ⁰ satisfy conditions (A 1 )−(A ³ ). However, BC satisfies (A 1 ), (A 3 ) but (A 2 ) is not satisfied.

Example 2.9 The spaces C g , U C g , C _g ^∞ and C _g ⁰ .

Let g be a positive continuous function on (−∞, 0]. We define:

C g := n

φ ∈ C((−∞, 0], E) : ^φ(θ) g(θ) is bounded on (−∞, 0] o

; C _g ⁰ := n

φ ∈ C ^g : lim θ →−∞ φ(θ) g(θ) = 0 o

, endowed with the uniform norm

kφk = sup

 |φ(θ)|

g(θ) : θ ¬ 0

 .

Then we have that the spaces C g and C _g ⁰ satisfy conditions (A 1 )−(A ³ ). We consider the following condition on the function g.

(g 1 ) For all a > 0, sup _0¬t¬a sup n _g(t+θ)

g(θ) : −∞ < θ ¬ −t o

< ∞.

They satisfy conditions (A 1 ) and (A 2 ) if (g 1 ) holds.

Example 2.10 The space C γ . For any real constant γ, we define the functional space C γ by

C γ :=



φ ∈ C((−∞, 0], E) : lim

θ→−∞ e ^γθ φ(θ) exists in E



endowed with the following norm

kφk = sup{e ^γθ |φ(θ)| : θ ¬ 0}.

Then in the space C γ the axioms (A 1 ) − (A ³ ) are satisfied.

Let F = (F, k.k ⁿ ) be a Fr´echet space with a family of semi-norms {k.k ⁿ } n∈N , we say that X is bounded if for every n ∈ N, there exists M ⁿ > 0 such that

kxk ⁿ ¬ M ⁿ f or all x ∈ X.

To F we associate a sequence of Banach spaces {(F ⁿ , k · k ⁿ )} as follows: For every n ∈ N, we consider the equivalence relation ∼ ⁿ defined by : x ∼ ⁿ y if and only if kx − yk ⁿ = 0 for x, y ∈ E. We denote F ⁿ = (F | ∼

, k · k ⁿ ) the quotient space, and we set (F ⁿ , k · k ⁿ ) the completion of F ⁿ with respect to k · k ⁿ . To every X ⊂ F , we associate a sequence {X ⁿ } of subsets X ⁿ ⊂ F ⁿ as follows: For every x ∈ F, we denote [x] _n the equivalence class of x in F ⁿ and we define X ⁿ = {[x] n : x ∈ X}.

We denote X ⁿ , int n (X ⁿ ) and ∂ n X ⁿ , respectively, the closure, the interior and the boundary of X ⁿ with respect to k · k ⁿ in F ⁿ .

We assume that the family of semi-norms {k.k ⁿ } ⁿ ∈N verifies:

kxk ¹ ¬ kxk ² ¬ kxk ³ ¬ . . . for every x ∈ X.

The following definition is the appropriate concept of contraction in F .

Definition 2.11 [14] A function f : F → F is said to be a contraction if for every n ∈ N there exists k ⁿ ∈ [0, 1) such that:

kf(x) − f(y)k ⁿ ¬ k ⁿ kx − yk ⁿ f or all x, y ∈ F.

The corresponding nonlinear alternative result is as follows:

Theorem 2.12 (Nonlinear Alternative) [14]. Let F be a Fr´echet space and X a closed subset of F such that 0 ∈ X and let N : X → F be a contraction map such that N(X) is bounded. Then one of the following statements holds:

• N has a unique fixed point in F .

• There exists 0 ¬ λ < 1, n ∈ N and x ∈ ∂ ⁿ X ⁿ : kx − λN(x)k ⁿ = 0.

3. Main results. Before starting and proving the main results, let us give the definition of mild solution to the semilinear evolution problem (1)-(2) Throughout this work, the function f : J × B → E will be continuous and ϕ(0) = 0.

Definition 3.1 A function x is said to be a mild solution of (1)-(2) if x satisfies

(5) x(t) =

 



ϕ(t), t ∈ (−∞, 0],

R t

0 T α (t − s)f(s, x ρ(s,x

) ) ds, t ∈ J.

Set R(ρ ⁻ ) = {ρ(s, ϕ) : (s, ϕ) ∈ J × B, ρ(s, ϕ) ¬ 0}. We always assume that

ρ : J × B → R is continuous. Additionally, we introduce following hypothesis:

• (H ^ϕ ) The function t → ϕ ^t is continuous from R(ρ ⁻ ) into B and there exists a continuous and bounded function L ^ϕ : R(ρ ⁻ ) → (0, ∞) such that

kϕ ^t k B ¬ L ^ϕ (t)kϕk B for every t ∈ R(ρ ⁻ )

Remark 3.2 The condition (H ϕ ), is frequently verified by continuous and bounded functions. For more details, see for instance ([18], Proposition 7.1.1).

We will need to introduce the following hypotheses which are assumed there- after:

• (H ¹ ) There exists a constant M > 0 such that kT ^α (t)k ^B(E) ¬ c M , t ∈ J.

• (H ² ) There exists a function p ∈ L ¹ loc (J, R ⁺ ) and a continuous nondecreasing function ψ : [0, +∞) → (0, ∞) such that:

|f(t, u)| ¬ p(t)ψ(kuk B ) for a.e. t ∈ J and each u ∈ B.

• (H ³ ) For all n > 0, there exists l n ∈ L ¹ loc (J, R ⁺ ) such that:

|f(t, u) − f(t, v)| ¬ l ^R (t)ku − vk B for all t ∈ [0, n], u, v ∈ B.

Consider the following space

B _+∞ = {x : R → E : x| [0,b] continuous for b > 0 and x 0 ∈ B}, where x| ^[0,b] is the restriction of x to the real compact interval [0, b].

Let us fix r > 1. For every n ∈ N, we define in B +∞ the seminorms by:

kxk ⁿ := sup{e ^−rL

^(t) |x(t)| : t ∈ [0, n]}

where L ^∗ _n (t) = R ^t

0 l n (s)ds, l n (t) = K n M l c n (t) and l n is the function from (H ₃ ) Then B _+∞ is a Fr´echet space with those seminorms family k.k ⁿ .

Lemma 3.3 [16], (Lemma 2.4) If x : (−∞, b] → E is a function such that x ⁰ = ϕ, then

kx ^s k B ¬ (M ^b + L ^ϕ )kϕk B + K b sup{|x(θ)|, θ ∈ [0, max(0, s)]}, s ∈ R(ρ ⁻ ) ∪ J,

where L ^ϕ = sup _{t ∈R(ρ}

) L ^ϕ (t).

Theorem 3.4 Assume that (H ϕ ) and (H 1 ) − (H ³ ) hold and moreover for each n ∈ N

(6) Z +∞

c

ds

ψ(s) > k n M c Z n

0

p(s)ds,

with c n = (M n + L ^ϕ + K n M H) c kϕk B . Then the problem (1)-(2) has a unique mild solution on (−∞, +∞).

Proof We transform the problem (1)-(2) into a fixed point theorem. Consider the operator N : B _+∞ → B +∞ defined by

(7) N (x)(t) =

 



ϕ(t), if t ∈ (−∞, 0],

R t

0 T α (t − s)f(s, x ρ(s,x

) ) ds, if t ∈ J.

Clearly, fixed points of the operator N are mild solutions of the problem (1)-(2).

For ϕ ∈ B, we will define the function y(.) : R → E by

(8) y(t) =

 ϕ(t), if t ∈ (−∞, 0], 0, if t ∈ J.

Then y ₀ = ϕ.

For each function z ∈ B +∞ with z(0) = 0 we denote by z the function defined by

(9) z(t) =

 0, if t ∈ (−∞, 0], z(t), if t ∈ J.

If x(t) satisfies (3.1), we can decompose it as x(t) = y(t) + z(t) for t ­ 0, which implies that x t = y t + z t for every t ­ 0. The function z(.) satisfies

z(t) = Z t

0 T α (t − s)f(s, y ^ρ(s,y

+z

) + z ρ(s,y

+z

) )ds for t ∈ J.

Let

B _+∞ ⁰ = {z ∈ B +∞ : z 0 = 0 ∈ B}.

For any z ∈ B ⁰ _+∞ we have

kzk ∞ = kz ⁰ k B + sup{|z(s)| : 0 ¬ s < +∞} = sup{|z(s)| : 0 ¬ s < +∞}.

Thus (B _+∞ ⁰ , k.k +∞ ) is a Banach space.

We define the operator G : B _+∞ ⁰ → B ⁰ _+∞ by :

G(z)(t) = Z t

0 T α (t − s)f(s, y ^ρ(s,y

+z

) + z ρ(s,y

+z

) )ds for t ∈ J.

Obviously the operator N has a fixed point, is equivalent to G has one, so it turns

to prove that G has a fixed point.

Let z ∈ B +∞ ⁰ be such that z = λG(z) for some λ ∈ [0, 1). By hypotheses (H ¹ ), (H 2 ), (H ϕ ) and (3.3), we have for each t ∈ [0, n]

|z(t)| ¬ Z t

0 kT ^α (t − s)k ^B(E) |f(s, y ^ρ(s,y

+z

) + z ρ(s,y

+z

) )|ds

¬ c M Z t

0 p(s)ψ( ky ρ(s,y

+z

) + z _ρ(s,y

_+z

₎ k B )ds

¬ c M Z t

0 p(s)ψ(K n u(s) + (M n + L ^ϕ + K n M H) c kϕk B )ds, where

u(s) = sup {|z(θ)| : θ ∈ [0, s]}.

Set

c n := (M n + L ^ϕ + K n M H) c kϕk B . Then, for t ∈ [0, n], we have

u(t) ¬ c M Z t

0 p(s)ψ(K n u(s) + c n )ds.

Thus

K n u(t) + c n ¬ c ⁿ + K n M c Z t

0 p(s)ψ(K n u(s) + c n )ds.

Consider now the function µ defined by

µ(t) = sup {K ⁿ u(s) + c n : 0 ¬ s ¬ t}, t ∈ [0, n].

Let t ^∗ ∈ [0, t] be such that

µ(t) = K n u(t ^∗ ) + c n kϕk B . By the previous inequality, we have

µ(t) ¬ c ⁿ + K _n M c Z t

0 p(s)ψ(µ(s))ds, t ∈ [0, n].

Set

v(t) = c n + K n M c Z t

0 p(s)ψ(µ(s))ds.

Then we have µ(t) ¬ v(t) for all t ∈ [0, n].

From the definition of v, we have v(0) = c n and

v ⁰ (t) = K n M p(t)ψ(µ(t)) a.e. t c ∈ [0, n].

Using the nondecreasing character of ψ, we get

v ⁰ (t) ¬ K ⁿ M p(t)ψ(v(t)) a.e. t c ∈ [0, n].

Using condition (6), this implies that for each t ∈ [0, n] we have Z v(t)

c

ds

ψ(s) ¬ K ⁿ M c Z t

0 p(s)ds ¬ K ⁿ M c Z n

0 p(s)ds <

Z _+∞

c

ds ψ(s) .

Thus for t ∈ [0, n] there exists a constant A n such that v(t) ¬ A n and hence µ(t) ¬ A n . Since kzk ⁿ ¬ µ(t), we have kzk ⁿ ¬ A n .

Set

Z = {z ∈ B _+∞ ⁰ : sup

0¬t¬n |z(t)| ¬ A n + 1, for all n ∈ N}.

Clearly Z is closed subset of B _+∞ ⁰ .

We shall show that G : Z → B ⁰ _+∞ is a contraction operator.

Indeed, consider z, z ∈ Z, thus using (H ¹ ) and (H 3 ) for each t ∈ [0, n] and n ∈ N

|G(z)(t) − G(z)(t)| ¬ Z t

0 kT ^α (t − s)k B(E) |f(s, y ρ(s,y

+z

) + z _ρ(s,y

_+z

₎ )

− f(s, y ρ(s,y

+z

) + z _ρ(s,z

_+y

₎ )| ds

¬ Z t

0

M l c n (s)kz ρ(s,y

+z

) − z ρ(s,y

+z

) k B ds.

Using (H ϕ ) and Lemma 3.3, we obtain

|G(z)(t) − G(z)(t)| ¬ Z t

0

M l c n (s)K n |z(s) − z(s)|ds

¬ Z t

0

h l n (s)exp(rl _n ^∗ (s)) ih

exp( −rl n ^∗ (s))|z(s) − z(s)| i ds

¬ Z t

0

h exp(rl ^∗ _n (s)) r

i 0

ds kz − zk ⁿ

¬ 1

r exp(rl _n ^∗ (t))kz − zk ⁿ . Therefore,

kG(z) − G(z)k ⁿ ¬ 1

r kz − zk ⁿ .

So the operator G is a contraction for all n ∈ N. From the choice of Z there is no z ∈ ∂Z such that z = λG(z), λ ∈ (0, 1). Then the second statement in Theorem 2.12 does not hold. The nonlinear alternative of Frigon-Granas shows that the first statement holds. Thus, we deduce that the operator G has a unique fixed-point z ^∗ . Then x ^∗ = y ^∗ + z ^∗ is a fixed point of the operator N, which is the unique mild solution of the problem (1)-(2).

4. Example. To illustrate our results, consider the following system (10) 

 



 



∂^α

∂t^α

(u, ξ) =

u(t, ξ) + R

−∞

a

(s − t)u h

s − ρ

(t)ρ

 R

0

a

(θ)|u(t, θ)|

dθ  , ξ i

ds, t ­ 0, ξ ∈ [0, π], u(t, 0) = u(t, π) = 0, t ­ 0,

u(θ, ξ) = u

(θ, ξ), −∞ < θ ¬ 0, ξ ∈ [0, π].

To represent this system in the abstract form (1)-(2), we choose the space E = L ² ([0, π], R) and the operator A : D(A) ⊂ E → E is given by Aω = ω ⁰⁰ with domain

D(A) := {ω ∈ E : ω ⁰⁰ ∈ E, ω(0) = ω(π) = 0}.

It is well known that A is an infinitesimal generator of an α-resolvent family (T α (t)) t ­0 on E. Furthermore, A has discrete spectrum with eigenvalues −n ² , with n ∈ N and corresponding normalized eigenfunctions given by

z n (ξ) = r 2

π sin(nξ).

In addition, {z ⁿ : n ∈ N} is an orthonormal basis of E and

T α (t)x = X ∞ n=1

exp ⁻ⁿ

^t (x, z n )z n f or x ∈ E and t ­ 0.

It follows from this representation that (T α (t)) _t­0 is compact for every t > 0 and that

kT ^α (t)k ¬ e ^−t for every t ­ 0.

Theorem 4.1 Let B = BUC(R ⁻ ; E) and φ ∈ B. assume that condition (H ^φ ) holds, ρ i : [0, +∞) → [0, ∞), i = 1, 2 are continuous and the functions a ⁱ : R → R are continuous, for i = 1, 2. Then there exists a unique mild solution of (10).

Proof From the assumptions, we have that f (t, ψ)(ξ) =

Z 0

−∞

a 1 (s)ψ(s, ξ)ds,

ρ(s, ψ) = s − ρ ¹ (s)ρ 2

 Z ^π

0 a 2 (θ)|ψ(0, ξ)| ² dθ  ,

are well defined functions, which permit to transform system (10) into the abstract system (1)-(2). Moreover, the function f is linear and bounded. Now, the existence of a mild solution can be deduced from a direct application of Theorem 3.4. From Remark 3.2, we have the following result.

Corollary 4.2 Let φ ∈ B be continuous and bounded. Then there exists a unique mild solution of (10) on (−∞, +∞).

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Mouffak Benchohra

Laboratory of mathematics, University of Sidi Bel Abbes PO Box 89, 22000 Sidi Bel Abbes, Algeria

E-mail: benchohra@univ-sba.dz Omar Bennihi

D´epartement de math´ematiques et informatique, Universit de Saida 20000, Saida, Alg´erie

Khalil Ezzinbi

D´epartement de Math´ematiques, Facult´e des Sciences, Semlalia B.P.2390, Marrakech, Morocco

E-mail: ezzinbi@ucam.ac.ma

(Received: 1.02.2013)