arXiv:1512.00909v1 [math.CA] 3 Dec 2015
This is a preprint of a paper whose final and definite form will be published in Journal of Mathematical Analysis, ISSN: 2217-3412, Volume 7, Issue 1 (2016).
EXISTENCE OF SOLUTION TO A NONLINEAR FIRST-ORDER DYNAMIC EQUATION ON TIME SCALES BENAOUMEUR BAYOUR, AHMED HAMMOUDI, DELFIM F. M. TORRES Abstract. We prove existence of solution to a nonlinear first-order nabla dynamic equation on an arbitrary bounded time scale with boundary conditions, where the right-hand side of the dynamic equation is a continuous function.
1. Introduction In this work we prove existence of solution to the following system: x∇ (t) = f (t, x(t)),
t ∈ Tk ,
x(a) = x(b).
(1.1)
Here T is an arbitrary bounded time scale, where we denote a := min T, b := max T, T◦ = T \ {a}, and f : T◦ × Rn → Rn is a continuous function. Problem (1.1) unifies continuous and discrete problems. We use the notion of tube solution for system (1.1), in the spirit of the works of Gilbert and Frigon [7–9]. This notion is useful to get existence results for systems of differential equations of first order, as a generalization of lower and upper solutions [3, 6, 10, 11]. Our main result provides existence of solution to the nonlinear nabla boundary value problem (1.1). The article is organized as follows. In Section 2 we review some basic definitions and theorems regarding ∇-differentiation and ∇-integration on time scales, and we prove some preliminary results. In Section 3 we introduce the notion of tube solution for system (1.1) and we prove our main result (Theorem 3.3). We end with Section 4, mentioning some directions for future work. 2. Preliminaries A time scale T is defined to be any nonempty closed subset of R. Then the forward and backward jump operators σ, ρ : T → T are defined by σ(t) = inf{s ∈ T : s > t}
and
ρ(t) = sup{t ∈ T : s < t}.
For t ∈ T, we say that t is left-scattered (respectively right-scattered) if ρ(t) < t (respectively σ(t) > t); that t is isolated if it is left-scattered and right-scattered. Similarly, if t > inf(T) and ρ(t) = t, then we say that t is left-dense; if t < sup(T) and σ(t) = t, then we say that t is right-dense. Points that are simultaneously left- and right-dense are called dense. If T has a right-scattered minimum m, then 2010 Mathematics Subject Classification. 34B15; 34N05. Key words and phrases. Calculus on time scales; nabla dynamic equations; nonlinear boundary value problems; existence; tube solution. Submitted Jun 10, 2015. Revised and Accepted Dec 02, 2015. 1
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B. BAYOUR, A. HAMMOUDI, D. F. M. TORRES
we define Tκ := T − {m}; otherwise, we set Tκ := T. The (backward) graininess ν : Tκ → [0, +∞[ is defined by ν(t) := t − ρ(t). Definition 2.1 (See [1, 9]). For f : T → Rn and t ∈ Tκ , the nabla derivative of f at t, denoted by f ∇ (t), is defined to be the number (provided it exists) with the property that given any ǫ > 0 there is a neighborhood U of t such that
f (ρ(t)) − f (s) − f ∇ (t)[ρ(t) − s] ≤ ǫ|ρ(t) − s| for all s ∈ U . If f is ∇-differentiable at t for every t ∈ Tκ , then f : T → Rn is called the ∇-derivative of f on Tκ . Theorem 2.2 (See [5]). Assume f : T → Rn and let t ∈ Tκ . The following holds: (1) If f is ∇-differentiable at t, then f is continuous at t. (2) If f is continuous at the left-scattered point t, then f is ∇-differentiable at t with f (t) − f (ρ(t) . f ∇ (t) = ν(t) (3) If t is left-dense, then f is nabla differentiable at t if and only if the limit lim
s→t
f (t) − f (s) t−s
exists in Rn . In this case, f ∇ (t) = lim
s→t
f (t) − f (s) . t−s
(4) If f is nabla differentiable at t, then f ρ (t) = f (t) − ν(t)f ∇ (t), where f ρ (t) := f (ρ(t)). Theorem 2.3 (See [5]). Assume f, g : T → R are nabla differentiable at t ∈ Tκ . Then, (1) f + g is nabla differentiable at t and (f + g)∇ (t) = f ∇ (t) + g ∇ (t); (2) αf is nabla differentiable at t for every α ∈ R and (αf )∇ (t) = αf ∇ (t); (3) f g is nabla differentiable at t and (f g)∇ (t) = f ∇ (t)g(t) + f ρ (t)g ∇ (t) = f (t)g ∇ (t) + f ∇ (t)g ρ (t); (4) if g(t)g ρ (t) 6= 0, then fg is nabla differentiable at t and ∇ f f ∇ (t)g(t) − f (t)g ∇ (t) (t) = ; g g(t)g ρ (t) (5) if f and f ∇ are continuous, then Z t ∇ Z t f (t, s)∇s = f (ρ(t), t) + f ∇ (t, s)∇s. a
a
n
Theorem 2.4. Let W be an open set of R and t ∈ T be a left-dense point. If g : T → Rn is nabla-differentiable at t and f : W → R is differentiable at g(t) ∈ W , ∇ then f ◦ g is nabla-differentiable at t with (f ◦ g) (t) = hf ′ (g(t)), g ∇ (t)i.
EXISTENCE OF SOLUTION TO A NONLINEAR FIRST-ORDER DYNAMIC EQUATION
3
Proof. Let ǫ > 0. We need to show that there exists a neighborhood U of t such that f (g(t)) − f (g(s)) − hf ′ (g(t)), g ∇ (t)i(t − s) ≤ ǫ|t − s| for all s ∈ U . Let k > 0 be a constant and ǫ′ = kǫ . By hypotheses, there exists a neighborhood U1 of t where
g(t) − g(s) − g ∇ (t)(t − s) ≤ ǫ′ |t − s| for all s ∈ U1 . In addition, there exists a neighborhood V ⊂ W of g(t) such that |f (g(t)) − f (y) − hf ′ (g(t)), g(t) − yi| ≤ ǫ′ |g(t)−y| for all y ∈ V . Since function g is ∇-differentiable at t, it is also continuous at this point, and there exists a neighborhood U2 of t such that g(s) ∈ V for all s ∈ U2 . Let U := U1 ∩ U2 . In this case U is a neighborhood of t and if s ∈ U , then |f (g(t)) − f (g(s)) − hf ′ (g(t)), g ∇ (t)i(t − s)| ≤ |f (g(t)) − f (g(s)) − hf ′ (g(t)), g(t) − g(s)i| + hf ′ (g(t)), g(t) − g(s)i − hf ′ (g(t)), g ∇ (t)i(t − s) ≤ ǫ′ kg(t) − g(s)k + hf ′ (g(t)), g(t) − g(s) − g ∇ (t)(t − s)i ≤ ǫ′ ǫ′ |t − s| + kg ∇ (t)(t − s)k + kf ′ (g(t))k kg(t) − g(s) − g ∇ (t)(t − s)k
≤ ǫ′ (1 + kg ∇ (t)k + kf ′ (g(t))k)|t − s|.
Put k = 1 + kg ∇ (t)k + kf ′ (g(t))k and the theorem is proved.
Example 2.5. Assume x : T → Rn is nabla differentiable at t ∈ T. We know that k · k : Rn \{0} → [0, +∞[ is differentiable if t = ρ(t). It follows from Theorem 2.4 that hx(t), x∇ (t)i kx(t)k∇ = . kx(t)k Definition 2.6. A function f : T → Rn is called ld-continuous provided it is continuous at left-dense points in T and its right-sided limits exist (finite) at rightdense points in T. The set of all ld-continuous functions f : T → Rn is denoted by Cld (T, Rn ). The set of functions f : T → Rn that are nabla-differentiable and 1 whose nabla-derivative is ld-continuous, is denoted by Cld (T, Rn ). It is known that if f is ld-continuous, then there is a function F such that F ∇ = f [4]. In this case, Z b f (t)∇t := F (b) − F (a). a
Theorem 2.7 (See [2]). Assume a, b, c ∈ T. Then Rb Rb Rb (1) a [f (t) + g(t)]∇t = a f (t)∇t + a g(t)∇t; Rb Rb (2) a kf (t)∇t = k a f (t)∇t; Rb Ra (3) a f (t)∇t = − b f (t)∇t; Rb Rc Rb (4) a f (t)∇t = a f (t)∇t + c f (t)∇t; Rb Rb b (5) a f ∇ (t)g(t)∇t = f (t)g(t)|a − a f ρ (t)g ∇ (t)∇t.
Theorem 2.8 (See [2]). The following inequalities hold: Z Z Z b b b f (t)g(t)∇t ≤ |f (t)g(t)|∇t ≤ max |f (t)| |g(t)|∇t. a σ(a)≤t≤b a a
Definition 2.9 (See [5]). For ǫ > 0, the (nabla) exponential function eˆǫ (·, t0 ) : T → R is defined as the unique solution to the initial value problem x∇ (t) = ǫx(t),
x(t0 ) = 1.
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B. BAYOUR, A. HAMMOUDI, D. F. M. TORRES
More explicitly, the exponential function eˆǫ (·, t0 ) : T → R is given by the formula Z t ˆ eˆǫ (t, t0 ) = exp ξǫ (ν(s))∇s , t0
where for h ≥ 0 we define ξˆǫ (h) as ǫ ξˆǫ (h) = − log(1−hǫ) h
if h = 0, otherwise.
Proposition 2.10. If g ∈ C 1 (Tκ , Rn ), then function x : T → Rn defined by " # Z Z eˆ1 (a, b) g(s) g(s) x(t) = eˆ1 (t, b) ∇s − ∇s eˆ1 (a, b) − 1 (a,b]∩T eˆ1 (ρ(s), b) ˆ1 (ρ(s), b) (t,b]∩T e is solution to the problem x∇ (t) − x(t) = g(t),
t ∈ Tκ ,
x(a) = x(b).
(2.1)
Proof. We check (2.1) for each pair (xi , gi ), i ∈ {1, 2, . . . , n}, by direct calculation. To simplify notation, we omit the indices i and we write Z g(s) eˆ1 (a, b) ∇s. k= eˆ1 (a, b) − 1 (a,b]∩T eˆ1 (ρ(s), b) From Theorem 2.3, we have that
+ eˆ1 (ρ(t), b)
g(t) eˆ1 (ρ(t), b)
Z
g(s) ∇s e ˆ (ρ(s), b) 1 (a,b]∩T Z − eˆ1 (t, b)k + eˆ1 (t, b)
x∇ (t) − x(t) = eˆ1 (t, b)k − eˆ1 (t, b)
(a,b]∩T
for all t ∈ Tκ . It is easy to verify that x(a) = x(b).
g(s) ∇s = g(t) eˆ1 (ρ(s), b)
1 Lemma 2.11. Let r ∈ Cld (T, Rn ) be a function such that r∇ (t) < 0 for all t ∈ {t ∈ Tκ ; r(t) > 0}. If r(a) ≥ r(b), then r(t) ≤ 0 for all t ∈ T.
Proof. Suppose that there exists a t ∈ T such that r(t) > 0. Then there exists a t0 ∈ T such that r(t0 ) = maxt∈T (r(t) > 0). If ρ(t0 ) < t0 , then r∇ (t0 ) =
r(ρ(t0 )) − r(t0 ≥ 0, ρ(t0 ) − t0
which contradicts the hypothesis. If t0 > a and t0 = ρ(t0 ), then there exists an interval [t1 , t0 ] such that r(t) > 0 for all t ∈ [t1 , t0 ]. Thus Z t0 r∇ (s)∇s = r(t0 ) − r(t1 ) < 0, t1
which contradicts the maximality of r(t0 ). Finally, if t0 = a, then by hypothesis r(b) ≥ r(a) gives r(a) = r(b). Taking t0 = a, one can check that r(a) ≤ 0 by using previous steps of the proof. The lemma is proved.
EXISTENCE OF SOLUTION TO A NONLINEAR FIRST-ORDER DYNAMIC EQUATION
5
3. Main Result In this section we prove existence of solution to problem (1.1). A solution of 1 this problem is a function x ∈ Cld (T, Rn ) satisfying (1.1). Let us recall that T is bounded with a = min T and b = max T. We introduce the notion of tube solution for problem (1.1) as follows. 1 1 Definition 3.1. Let (v, M ) ∈ Cld (T, Rn ) × Cld (T, [0, +∞[). We say that (v, M ) is a tube solution of (1.1) if (1) hx−v(t), f (t, x(t))−v ∇ (t)i+M (t)kx−v(t)k ≤ M (t)M ∇ (t) for every t ∈ Tκ and for every x ∈ Rn such that kx − v(t)k = M (t); (2) v ∇ (t) = f (t, v(t)) and kx − v(t)k − M ∇ (t) < 0 for every t ∈ Tκ such that M (t) = 0; (3) kv(a) − v(b)k ≤ M (a) − M (b). 1 Let T(v, M ) := {x ∈ Cld (T, Rn ) : kx(t) − v(t)k ≤ M (t) for every t ∈ T}. We consider the following problem:
x∇ (t) − x(t) = f (t, x b(t)) − x b(t),
t ∈ Tκ ,
x(a) = x(b),
where x ˆ(t) =
(
M(t) kx−v(t)k (x(t)
x(t)
− v(t)) + v(t)
(3.1)
if kx − v(t)k > M (t), otherwise.
Let us define the operator Tpˆ : C(T, Rn ) → C(T, Rn ) by " Z f (s, x ˆ(s)) − x ˆ(s) eˆ1 (a, b) ∇s Tpˆ(x)(t) = eˆ1 (t, b) eˆ1 (a, b) − 1 (a,b]∩T eˆ1 (ρ(s), b) # Z f (s, x ˆ(s)) − x ˆ(s) − ∇s . eˆ1 (ρ(s), b) (t,b]∩T 1 1 Proposition 3.2. If (v, M ) ∈ Cld (T, Rn ) × Cld (T, [0, +∞[) is a tube solution of n n (1.1), then Tpˆ : C(T, R ) → C(T, R ) is compact.
Proof. We first prove the continuity of the operator Tpˆ. Let {xn }n∈N be a sequence of C(T, Rn ) converging to x ∈ C(T, Rn ). By Theorem 2.8, kTpˆ(xn )(t) − Tpˆ(x)(t)k
Z
f (s, x ˆn (s)) − f (s, x ˆ(s)) − (ˆ xn (s) − x ˆ(s))
∇s ≤ (1 + c)kˆ e1 (t, b)k
(a,b]∩T eˆ1 (ρ(s), b) ! Z k(1 + c) ≤ kf (s, x ˆn (s)) − f (s, x ˆ(s))k + kˆ xn (s) − x ˆ(s)k ∇s , M (a,b]∩T
1 (a,b) k. Since where k := maxt∈T |ˆ e1 (t, b)|, M := mint∈T (ˆ e1 (t, b)), and c := k eˆ1eˆ(a,b)−1 there is a constant R > 0 such that kˆ xkC(T,Rn) < R, there exists an index N such that kˆ xn kC(T,Rn) < R for all n > N . Thus f is uniformly continuous on Tκ ×BR (0). Therefore, for ǫ > 0 given, there is a δ > 0 such that for all x, y ∈ Rn , where ǫM , kx − yk < δ < 2k(1 + c)(b − a)
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B. BAYOUR, A. HAMMOUDI, D. F. M. TORRES
one has kf (s, y) − f (s, x)k
N such that By assumption, for all s ∈ Tκ it is possible to find an index N ˆ kˆ xn − xˆkC(T,Rn) < δ for n > N . In this case, 2k(1 + c) kTpˆ(xn )(t) − Tpˆ(x)(t)k ≤ M
Z
[a,b)∩T
ǫM ∇s ≤ ǫ. 2k(1 + c)(b − a)
This proves the continuity of Tpˆ. We now show that the set Tpˆ(C(T, Rn )) is relatively compact. Consider a sequence {yn }n∈N of Tpˆ(C(T, Rn )) for all n ∈ N. It exists xn ∈ C(T, Rn ) such that yn = Tpˆ(xn ). From Theorem 2.8 one has k(1 + c) kTpˆ(xn )(t)k ≤ M
Z
kf (s, x ˆn (s))k∇s +
[a,b)∩T
Z
!
kˆ xn (s)k∇s .
[a,b)∩T
By definition, there is an R > 0 such that kˆ xn (s)k ≤ R for all s ∈ T and all n ∈ N. Function f is compact on Tκ × BR (0) and we deduce the existence of a constant A > 0 such that kf (s, x ˆn (s)k ≤ A for all s ∈ Tκ and all n ∈ N. The sequence {yn }n ∈ N is uniformly bounded. Note also that kTpˆ(xn )(t2 ) − Tpˆ(xn )(t1 )k ≤ Bkˆ e1 (t2 , b) − eˆ1 (t1 , b)k
Z
k(A + R) f (s, x ˆn (s)) − x ˆn (s)
∇s < Bkˆ e1 (t2 , b)− eˆ1 (t1 , b)k+ |t2 −t1 | +k
(a,b]∩T eˆ1 (ρ(s), b) M
for t1 , t2 ∈ T, where B is a constant that can be chosen such that it is higher than
Z
eˆ (a, b) Z f (s, x ˆn (s)) − x ˆn (s) f (s, x ˆn (s)) − x ˆn (s)
1 ∇s + ∇s . sup
ˆ1 (a, b) − 1 (a,b]∩T eˆ1 (ρ(s), b) eˆ1 (ρ(s), b) n∈N e (t,b]∩T
This proves that the sequence {yn }n∈N is equicontinuous. It follows from the Arzel` a–Ascoli theorem, adapted to our context, that Tpˆ(C(T, Rn )) is relatively compact. Hence Tpˆ is compact. 1 1 Theorem 3.3. If (v, M ) ∈ Cld (T, [0, +∞[) × Cld (T, Rn ) is a tube solution of (1.1), 1 n then problem (1.1) has a solution x ∈ Cld (T, R ) ∩ T(v, M ).
Proof. By Proposition 3.2, Tpˆ is compact. It has a fixed point by Schauder’s fixed point theorem. Proposition 2.10 implies that this fixed point is a solution to problem (3.1). Then it suffices to show that for every solution x of (3.1) one has x ∈ T(v, M ). Consider the set A = {t ∈ Tκ : kx(t) − v(t)k > M (t)}. If t ∈ A is left dense, then by virtue of Example 2.5 we have ∇
(k x(t) − v(t) k −M (t)) =
hx(t) − v(t), x∇ (t) − v ∇ (t)i − M ∇ (t). kx(t) − v(t)k
EXISTENCE OF SOLUTION TO A NONLINEAR FIRST-ORDER DYNAMIC EQUATION
7
If t ∈ A is left scattered, then (kx(t)−v(t)k − M (t))∇ = kx(t) − v(t)k∇ − M ∇ (t) kx(t) − v(t)k2 − kx(t) − v(t)kkx(ρ(t)) − v(ρ(t))k − M ∇ (t) ν(t)kx(t) − v(t)k hx(t) − v(t), x(t) − v(t) − x(ρ(t)) + v(ρ(t))i − M ∇ (t) ≤ ν(t)kx(t) − v(t)k hx(t) − v(t), [f (t, xˆ(t)) − x ˆ(t) + x(t)] − v ∇ (t)i = − M ∇ (t). kx(t) − v(t)k =
We will show that if t ∈ A, then (kx(t) − v(t)k − M (t))∇ < 0. If t ∈ A and M (t) > 0, then (kx(t)−v(t)k − M (t))∇ = kx(t) − v(t)k∇ − M ∇ (t) kx(t) − v(t)k2 − kx(t) − v(t)kkx(ρ(t)) − v(ρ(t))k − M ∇ (t) ν(t)kx(t) − v(t)k hx(t) − v(t), x(t) − v(t) − x(ρ(t)) + v(ρ(t))i ≤ − M ∇ (t) ν(t)kx(t) − v(t)k hx(t) − v(t), x∇ (t) − v ∇ (t)i − M ∇ (t) = kx(t) − v(t)k hx(t) − v(t), f (t, xˆ(t)) − v ∇ (t)i hx(t) − v(t), −ˆ x(t) + x(t)i = + − M ∇ (t) kx(t) − v(t)k kx(t) − v(t)k hˆ x∇ (t) − v(t), f (t, xˆ(t)) − v ∇ (t)i − M (t) + kx(t) − v(t)k − M ∇ (t) = M (t) M (t)M ∇ (t) − M (t)kx(t) − v(t)k ≤ − M (t) + kx(t) − v(t)k − M ∇ (t) M (t) = −M (t) < 0. =
In addition, if M (t) = 0, then hx(t) − v(t), f (t, xˆ(t)) + [x(t) − x ˆ(t)] − v ∇ (t)i − M ∇ (t) kx(t) − v(t)k hx(t) − v(t), f (t, v(t)) − v ∇ (t)i ≤ + kx(t) − v(t)k − M ∇ (t) < 0. kx(t) − v(t)k
(kx(t)−v(t)k − M (t))∇ =
If we set r(t) := kx(t)−v(t)k−M (t), then r∇ (t) < 0 for every t ∈ {t ∈ Tκ , r(t) ≥ 0}. Moreover, since (v, M ) is a tube solution of (1.1), one has r(a) − r(b) ≤ kv(a) − v(b)k − (M (a) − M (b)) ≤ 0 and thus the hypotheses of Lemma 2.11 are satisfied, which proves the theorem. Example 3.4. Consider the following boundary value problem on time scales: x∇ (t) = a1 kx(t)k2 x(t) − a2 x(t) + a3 ϕ(t), x(a) = x(b),
t ∈ Tκ ,
(3.2)
where a1 , a2 , a3 ≥ 0 are nonnegative real constants chosen such that a2 ≥ a1 +a3 +1 and ϕ : Tκ → Rn is a continuous function satisfying kϕ(t)k = 1 for every t ∈ Tκ . It is easy to check that (v, m) ≡ (0, 1) is a tube solution. By Theorem 3.3, problem (3.2) has a solution x such that kx(t)k ≤ 1 for every t ∈ T.
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B. BAYOUR, A. HAMMOUDI, D. F. M. TORRES
4. Conclusion and Future Work We proved existence of a solution to a nonlinear first-order nabla dynamic equation on time scales. For that the notion of tube solution is used, in the spirit of the works of Frigon and Gilbert [7–9]. Our results can be improved by using ∇Caratheodory functions f on the right-hand side of equation (1.1), which are not necessarily continuous. For that one needs to define a proper Sobolev space and related nabla concepts. This is under investigation and will be addressed elsewhere. Acknowledgments. This research is part of first author’s Ph.D. project, which is carried out at Sidi Bel Abbes University, Algeria. It was essentially finished while Bayour was visiting the Department of Mathematics of University of Aveiro, Portugal, February to April of 2015. The hospitality of the host institution and the financial support of University of Chlef, Algeria, are here gratefully acknowledged. Torres was supported by Portuguese funds through CIDMA and FCT, within project UID/MAT/04106/2013. References [1] D. Anderson, J. Bullock, L. Erbe, A. Peterson and H. Tran, Nabla dynamic equations on time scales, Panamer. Math. J. 13 (2003), no. 1, 1–47. [2] F. M. Atici and G. Sh. Guseinov, On Green’s functions and positive solutions for boundary value problems on time scales, J. Comput. Appl. Math. 141 (2002), no. 1-2, 75–99. [3] C. Bereanu and J. Mawhin, Existence and multiplicity results for periodic solutions of nonlinear difference equations, J. Difference Equ. Appl. 12 (2006), no. 7, 677–695. [4] M. Bohner and A. Peterson, Dynamic equations on time scales, Birkh¨ auser Boston, Boston, MA, 2001. [5] M. Bohner and A. Peterson, Advances in dynamic equations on time scales, Birkh¨ auser Boston, Boston, MA, 2003. [6] Q. Dai and C. C. Tisdell, Existence of solutions to first-order dynamic boundary value problems, Int. J. Difference Equ. 1 (2006), no. 1, 1–17. [7] M. Frigon, Boundary and periodic value problems for systems of nonlinear second order differential equations, Topol. Methods Nonlinear Anal. 1 (1993), no. 2, 259–274. [8] M. Frigon and H. Gilbert, Boundary value problems for systems of second-order dynamic equations on time scales with ∆-Carath´ eodory functions, Abstr. Appl. Anal. 2010, Art. ID 234015, 26 pp. [9] H. Gilbert, Existence theorems for first-order equations on time scales with ∆-Carath´ eodory functions, Adv. Difference Equ. 2010, Art. ID 650827, 20 pp. [10] B. Mirandette, R´ esultats d’existence pour des syst` emes d’´ equations diff´ erentielles du premier ordre avec tube-solution, M.Sc. thesis, University of Montr´ eal, 1996. [11] C. C. Tisdell and A. Zaidi, Basic qualitative and quantitative results for solutions to nonlinear, dynamic equations on time scales with an application to economic modelling, Nonlinear Anal. 68 (2008), no. 11, 3504–3524. Benaoumeur Bayour University of Chlef, B. P. 151, Hay Es-salem Chlef, Algeria E-mail address:
[email protected] Ahmed Hammoudi Laboratoire de Math´ ematiques, Universit´ e de Ain T´ emouchentl B. P. 89, 46000 Ain T´ emouchent, Algeria E-mail address:
[email protected] Delfim F. M. Torres Center for Research and Development in Mathematics and Applications (CIDMA) Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal E-mail address:
[email protected] 
