Electronic Journal of Differential Equations, Vol. 2013 (2013), No. 221, pp. 1–9.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
FINITE FRACTAL DIMENSIONALITY OF ATTRACTORS FOR
NONLOCAL EVOLUTION EQUATIONS
SEVERINO HORÁCIO DA SILVA, FLANK D. M. BEZERRA
Abstract. In this work we consider the Dirichlet problem governed by a non
local evolution equation. We prove the existence of exponential attractors for
the flow generated by this problem, and as a consequence we obtain the finite
dimensionality of the global attractor whose existence was proved in [1].
1. Introduction
Global attractors for dynamical systems generated by non local evolution equations in infinite dimensional Hilbert space have been considered in the literature
within past few years, see [1, 6, 8] and references therein.
In the literature, there are some works on existence of exponential attractors for
the flow governed by non local evolution equations and on the problem of determining upper bounds for the fractal dimension of these attractors (see for instance
[2, 10, 9]).
In this paper, we consider the non linear Dirichlet problem with non local terms
∂t u(x, t) = −u(x, t) + g β(Ku)(x, t) , x ∈ Ω, t > 0
u(x, 0) = u0 (x),
x∈Ω
(1.1)
x 6∈ Ω, t > 0
u(x, t) = 0,
where Ω ⊂ RN (N ≥ 1) is a bounded smooth domain, β > 0 and K is an integral
operator with symmetric kernel
Z
(Ku)(x) :=
J(x, y)u(y)dy.
RN
Here, g : R → R is a non linear real function of class C 1 with g(0) = 0, J is
a
R non negative, symmetric bounded function with bounded derivative, satisfying
J(x, y)dy = 1 and
RN
Z
Z
sup
|∂x J(x, y)|dy ≤ S, sup
|∂x J(x, y)|dx ≤ S,
x∈RN
y∈RN
RN
RN
for some constant 0 < S < ∞.
2000 Mathematics Subject Classification. 34G20, 47H15.
Key words and phrases. Exponential attractor; global attractor; fractal dimension;
non local evolution equations.
c
2013
Texas State University - San Marcos.
Submitted November 1, 2012. Published September 4, 2013.
1
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S. H. DA SILVA, F. D. M. BEZERRA
EJDE-2013/221
For the sake of clarity and future reference, we list the hypotheses on g that were
used in [1].
(H1) The function g : R → R, is globally Lipschitz continuous with constant k1 .
(H2) The function g ∈ C 1 (R) and g 0 is Lipschitz continuous with constant k2 .
(H3) There exists a > 0 such that |g(x)| < a < ∞, for all x ∈ R.
Note that if (H1) and (H2) hold then
|g 0 (x)| ≤ k1 ,
∀x ∈ R.
Remark 1.1. To prevent the flow generated by (1.1) becomes a contraction (see
Theorem 2.2) and, hence, the global attractor be reduced to single point, we assume
k1 β > 1.
In this article, k · k = h·, ·i1/2 denotes the L2 (RN ) norm. We use kJk∞ to denote
kJkL∞ (RN ×RN ;R) and kJ 0 k∞ to denote kJ 0 kL∞ (RN ×RN ;L(RN ×RN ;R)) .
Under hypothesis (H1)–(H3), it was proved in [1] that problem (1.1) has a global
compact
attractor A which is contained in ball centered at the origin of radius
p
a |Ω| in L2 (RN ). Also, under some additional hypotheses on g, the continuity
of the global attractors and the existence of nonhomogeneous equilibria for (1.1)
were proved in [1]. In [2] bi-space global and exponential attractors for the time
continuous dynamical systems are considered and the bounds on their fractal dimension are discussed in the context of the smoothing properties of the system
between appropriately chosen function spaces and applications to the sample problems are given, but no remark is made on problems governed by operators of the
type Hilbert-Schmidt, where the symmetry of the problem is an extra difficulty
inherent in the evolution equations with non local terms.
Our goal is to investigate, under the above conditions, the existence of an exponential attractor for the flow generated by (1.1) and consequently to obtain bounds
on the fractal dimension of the global attractor associated to problem (1.1).
This article is organized as follows. In Section 2 we prove Lipschitz continuity
of the dynamical system generated by (1.1), whose well posedness in X = {u ∈
L2 (RN ) : u(x) = 0, if x ∈
/ Ω} has been established in [1]. In Section 3 we
prove that, in this phase space, the system has an exponential attractor and, as a
consequence , we conclude that the global attractor has finite fractal dimension.
2. Dynamical system generated by (1.1)
It is known from previous work, see [1], that under hypotheses (H1) and (H2)
the function
(
−u(x) + g(β(Ku)(x)), x ∈ Ω
[F (u)](x) =
0,
x∈
/ Ω,
is globally Lipschitz and continuously Fréchet differentiable in X = {u ∈ L2 (RN ) :
u(x) = 0, if x ∈
/ Ω}. Therefore, the autonomous problem
∂t u = F (u)
(2.1)
with initial condition u(x, 0) = u0 (x) generates a C 1 flow in X which is given by
T (t)u0 = u(x, t) where u(x, t) is given by the variation of constants formula by
Z t
−t
u(x, t) = e u(x, 0) +
e−(t−s) g(β(Ku)(x, s))ds.
0
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FINITE FRACTAL DIMENSIONALITY
3
Furthermore, as consequence from Lemma 2.1 below, the problem (2.1) is also well
posed in H 1 .
Lemma 2.1. Under hypotheses (H1), (H2), the subset H 1 of X given by H 1 =
{u ∈ H 1 (RN ) : u(x) = 0, if x ∈
/ Ω} is invariant under the map F .
Proof. If u ∈ H 1 , from hypothesis (H2) it follows that F (u) is differentiable and
∂xi F (u)(x) = −∂xi u(x) + g 0 (β(Ku)(x))β∂xi (Ku)(x)
Z
= −∂xi u(x) + g 0 (β(Ku)(x))β
∂xi J(x, y)u(y)dy.
RN
Using hypotheses (H1) and (H2) and Generalized Young’s Inequality (see [4]), we
obtain
Z
k∂xi F (u)k ≤ k∂xi uk + kg 0 (β(Ku))β
∂xi J(x, y)u(y)dyk
RN
≤ k∂xi uk + k1 βSkuk.
It implies F (u) ∈ H 1 , as claimed.
In the next result we prove the Lipschitz continuity of the flow T (t) generated
by problem (2.1).
Theorem 2.2. Assume hypothesis (H1) holds. Then, for u1 , u2 ∈ X and t ≥ 0,
we have
kT (t)u1 − T (t)u2 k ≤ ect ku1 − u2 k,
(2.2)
for c = k1 β − 1 > 0.
Proof. Let u1 , u2 ∈ X. Suppose that T (t)u1 (x) and T (t)u2 (x) are solutions of (2.1)
with initial conditions u1 and u2 , respectively. Then
Z t
kT (t)u1 −T (t)u2 k ≤ e−t ku1 −u2 k+
e−(t−s) kg(β(KT (s)u1 ))−g(β(KT (s)u2 ))kds,
0
(2.3)
for t ≥ 0. Using (H1) and Young’s inequality, it follows that
kg(β(KT (s)u1 )) − g(β(KT (s)u2 ))k ≤ k1 βkK(T (s)u1 − T (s)u2 )k
≤ k1 βkT (s)u1 − T (s)u2 k.
Then
−t
Z
t
kT (t)u1 − T (t)u2 k ≤ e ku1 − u2 k +
e−(t−s) k1 βkT (s)u1 − T (s)u2 kds.
0
Thus, using Gronwall’s inequality, we obtain
kT (t)u1 − T (t)u2 k ≤ e(k1 β−1)t ku1 − u2 k.
Hence (2.2) is satisfied.
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S. H. DA SILVA, F. D. M. BEZERRA
EJDE-2013/221
3. Existence of an exponential attractor
First we need to introduce some terminology. For a general introduction to
theory of exponential attractors and fractal dimension see, for example [2, 3].
Recall that if B 6= ∅ is a precompact set in the Banach space X then its fractal
dimension is given by
dX
f (B) = lim sup log1/ N (B),
→0
where N (B) denotes the smallest number of -balls in X needed to cover B.
We recall that a set B ⊂ X is an absorbing set for the flow T (t) if, for any
bounded set C in X , there is a t1 = t1 (C) such that T (t)C ⊂ B for any t ≥ t1 (see
[9]).
Let Y and X be Banach spaces such that Y is compactly embedded in X . Recalling the generalization of the notion of an exponential attractor, see [2], we will
say that a nonvoid set M ⊂ Y is called an exponential (Y − X ) attractor for T (t)
if M is positively invariant under T (t), closed in Y, compact in X , dX
f (M) < ∞
and there exists ω > 0 such that for all B bounded in Y,
lim eωt distX (T (t)B, M) = 0.
t→∞
Remark 3.1. If u ∈ X, proceeding as in [7], using Hölder’s inequality, we obtain
Z
|K(u)(x, s)| ≤
J(x, y)|u(y, s)|dy
N
ZR
(3.1)
≤
kJk∞ |u(y, s)|dy
RN
p
≤ kJk∞ |Ω|ku(·, s)k.
Theorem 3.2. Assume that (H1)–(H3) hold. Then ,pfor any ε > 0, the ball
centered at origin and radius ρ = (1 + k1 βkJ 0 k∞ |Ω|)a |Ω| + ε in H 1 = {u ∈
H 1 (RN ) : u(x) = 0, if x 6= Ω} absorbs bounded subsets of H 1 .
Proof. Let u(x, t) be the solution of (2.1) with initial condition u(·, 0) ∈ B, where
B is a bounded subset of H 1 . Then, if x 6∈ Ω we have u(x, t) = 0, and if x ∈ Ω we
obtain, by the variation of constants formula,
Z t
−t
u(x, t) = e u(x, 0) +
e−(t−s) g(β(Ku)(x, s))ds.
(3.2)
0
By (H3) we have
p
ku(·, t)k ≤ e−t ku(·, 0)k + a |Ω|.
Hence, given ε > 0 there exists t1 =
ln( 2ku(·,0)k
)
ε
(3.3)
such that, for t > t1 , we obtain
p
ε
ku(·, t)k < + a |Ω|.
(3.4)
2
Furthermore, using hypothesis (H2), from (3.2) we obtain
Z t
∂x u(x, t) = e−t ∂x u(x, 0) +
e−(t−s) g 0 (β(Ku)(x, s))β∂x (Ku)(x, s)ds.
0
From (H1) and (H2) it follows that |g 0 (x)| ≤ k1 , for all x ∈ R. Then
Z t
−t
|∂x u(x, t)| ≤ e |∂x u(x, 0)| +
e−(t−s) k1 β|∂x (Ku)(x, s)|ds.
0
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FINITE FRACTAL DIMENSIONALITY
But, as in (3.1), using Hölder’s inequality, we obtain
Z
|∂x K(u)(x, s)| ≤
|∂x J(x, y)||u(y, s)|dy
N
ZR
≤
kJ 0 k∞ |u(y, s)|dy
RN
p
≤ kJ 0 k∞ |Ω|ku(·, s)k.
5
(3.5)
Thus, using (3.3) and (3.5), we obtain
|∂x u(x, t)| ≤ e−t |∂x u(x, 0)| + k1 βkJ 0 k∞
p
Z
|Ω|
t
p
e−(t−s) (e−s ku(·, 0)k + a |Ω|)ds
0
−t
0
≤ e |∂x u(x, 0)| + k1 βkJ k∞
p
|Ω|ku(·, 0)ke−t t + k1 βkJ 0 k∞ a|Ω|.
Hence
p
k∂x u(·, t)k ≤ e−t k∂x u(·, 0)k + k1 βkJ 0 k∞ |Ω|ku(·, 0)ke−t t + k1 βkJ 0 k∞ |Ω|a |Ω|.
(3.6)
But, there exists t2 = ln(4k∂x u(·, 0)k/ε) such that for t > t2 we have
e−t k∂x u(·, 0)k <
ε
,
4
(3.7)
and, since limt→∞ e−t t = 0, there exists t3 > 0 such that, for t > t3 , we have
ε
(3.8)
k1 βkJ 0 k∞ |Ω|ku(·, 0)ke−t t < .
4
Then, using (3.6), (3.7) and (3.8), we obtain
k∂x u(·, t)k <
p
ε
+ k1 βkJ 0 k∞ |Ω|a |Ω|.
2
for all t > t∗ := max{t2 , t3 }. It follows that for t > max{t1 , t∗ },
p
ku(·, t)k + k∂x u(·, t)k ≤ (1 + k1 βkJ 0 k∞ |Ω|)a |Ω| + ε.
From this, the result follows immediately.
For the rest of this article, we denote
by B0 the ball in H 1 centered at origin
p
0
and radius ρ = (1 + k1 βkJ k∞ |Ω|)a |Ω| + ε, (with ε > 0 fixed arbitrarily).
Theorem 3.3. Assume (H1)–(H3), and let B0 be the set that absorbs bounded
subsets of H 1 , given in Theorem 3.2. Then, there exists t0 ≥ t1 (B0 ) such that the
following conditions hold: T (t0 ) admits a decomposition
T (t0 ) = P (t0 ) + M (t0 )
2
where P (t0 ) : B0 → X ⊂ L (RN ) is a contraction on B0 ; that is,
kP (t0 )u1 − P (t0 )u2 k ≤ δku1 − u2 k,
∀u1 , u2 ∈ B0 ,
(3.9)
for some 0 ≤ δ < 1/2, and M (t0 ) : B0 → H 1 satisfies
kM (t0 )u1 − M (t0 )u2 kH 1 ≤ kku1 − u2 k,
for some k > 0.
∀u1 , u2 ∈ B0 ,
(3.10)
6
S. H. DA SILVA, F. D. M. BEZERRA
EJDE-2013/221
Proof. Let u(x, t) be the solution of (2.1) with initial condition u, then
Z t
−t
e−(t−s) g(β(KT (s))u)ds.
T (t)u = e u +
0
−t
Write P (t)u = e u and M (t)u =
t0 > ln 2, it follows that
Rt
0
e−(t−s) g(β(KT (s))u)ds. Note that, choosing
kP (t0 )u1 − P (t0 )u2 k ≤ e−t0 ku1 − u2 k,
∀u1 , u2 ∈ B0 ,
−t0
then (3.9) is satisfied with δ = e . Now, from (H3), it follows that
Z t
Z t
p
p
e−(t−s) a |Ω|ds ≤ a |Ω|.
e−(t−s) kg(β(KT (s)u)kds ≤
kM (t)uk ≤
0
0
Using (H1),(H2), and (3.5), we have
Z t
|∂x M (t)u(x)| ≤
e−(t−s) β|g 0 (β(KT (s)u)(x))||∂x (KT (s)u)(x)|ds
0
Z t
p
≤
e−(t−s) k1 βkJ 0 k∞ |Ω|kT (s)ukds.
0
p
Since u ∈ B(0, ρ), it follows that kT (s)uk ≤ ρ + a |Ω|. Hence
p
p
|∂x M (t)u(x)| ≤ k1 βkJ 0 k∞ |Ω|(ρ + a |Ω|).
Therefore, M (t) : B0 → H 1 for all t ≥ 0.
Also, using (H1) and Theorem 2.2, we obtain
Z t
kM (t)u1 − M (t)u2 k ≤
e−(t−s) kg(βKT (s)u1 ) − g(βKT (s)u2 )kds
0
Z t
≤
e−(t−s) βk1 kT (s)u1 − T (s)u2 kds
0
Z t
≤
e−(t−s) βk1 e(k1 β−1)s ku1 − u2 kds
0
Z t
= k1 βku1 − u2 k
e−(t−s) e(k1 β−1)s ds
0
≤ ku1 − u2 ke(k1 β−1)t .
Using hypothesis (H1), (H2), (3.1) and (3.5), it follows that
|∂x M (t)u1 (x) − ∂x M (t)u2 (x)|
Z t
≤
e−(t−s) β|g 0 (β(KT (s)u1 )(x)) − g 0 (β(KT (s)u2 )(x))||∂x (KT (s)u1 )(x)|ds
0
Z t
+
e−(t−s) β|g 0 (β(KT (s)u2 )(x))||∂x [K(T (s)u1 − T (s)u2 )](x)|ds
0
Z t
p
p
≤
e−(t−s) k2 β 2 kJk∞ |Ω|kT (s)u1 − T (s)u2 kkJ 0 k∞ |Ω|kT (s)u1 kds
0
Z t
p
+
e−(t−s) k1 βkJ 0 k∞ |Ω|kT (s)u1 − T (s)u2 kds.
0
Thus, using Theorem 2.2, we have
|∂x M (t)u1 (x) − ∂x M (t)u2 (x)|
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FINITE FRACTAL DIMENSIONALITY
7
t
Z
p
e−(t−s) k2 β 2 kJk∞ |Ω|kJ 0 k∞ (ρ + a |Ω|)e(k1 β−1)s ku1 − u2 kds
≤
0
Z
t
p
|Ω|e(k1 β−1)s ku1 − u2 kds
0
p
p k2 β 2 kJk∞ |Ω|kJ 0 k∞ (ρ + a |Ω|) + k1 βkJ 0 k∞ |Ω| (k1 β−1)t
≤
e
ku1 − u2 k.
k1 β − 1
e−(t−s) k1 βkJ 0 k∞
+
Therefore, (3.10) is satisfied with t0 > ln 2 and
p
p
n
k2 β 2 kJk∞ |Ω|kJ 0 k∞ (ρ + a |Ω|) + k1 βkJ 0 k∞ |Ω| (k1 β−1)t0
k = max e
,
k1 β − 1
(3.11)
o
p
(k1 β−1)t0
.
× |Ω|e
It completes the proof.
The proof of the following lemma is very easy and it will be omitted.
Lemma 3.4. For any bounded interval I, there exist 0 < θ < 1 and c > 0 such
that for t, s ∈ I we have
|e−t − e−s | ≤ c|t − s|θ .
(3.12)
In particular, when I = [t0 , 2t0 ] the inequality above is obtained with c = 1.
Theorem 3.5. Assume (H1) and (H3). Then, for t1 , t2 ∈ [t0 , 2t0 ] and u1 , u2 ∈ B0 ,
T (t) satisfies
kT (t1 )u1 − T (t2 )u2 k ≤ µ(|t1 − t2 |θ + ku1 − u2 k)
(3.13)
with some µ > 0 and 0 < θ < 1.
Proof. Note that
Z t1
T (t1 )u1 − T (t2 )u2 = (e−t1 u1 − e−t2 u2 ) +
e−(t1 −s) g(βKT (s)u1 )ds
0
Z t2
−(t2 −s)
−
e
g(βKT (s)u2 )ds .
0
Using Lemma 3.4, we obtain
ke−t1 u1 − e−t2 u2 k ≤ |e−t1 − e−t2 |ku1 k + e−t2 ku1 − u2 k
≤ ku1 k|t1 − t2 |θ + ku1 − u2 k
= µ1 (|t1 − t2 |θ + ku1 − u2 k),
where µ1 = max{ρ, 1}. Now,
Z t1
Z t2
k
e−(t1 −s) g(βKT (s)u1 )ds −
e−(t2 −s) g(βKT (s)u2 )dsk
0
0
Z t1
Z t1
−(t1 −s)
≤k
e
g(βKT (s)u1 )ds −
e−(t2 −s) g(βKT (s)u2 )dsk
0
0
Z t1
Z t2
−(t2 −s)
+k
e
g(βKT (s)u2 )ds −
e−(t2 −s) g(βKT (s)u2 )dsk.
0
0
Using (H1) and (H3), Young’s inequality, Lemma 3.12 and Theorem 3.4, we have
Z t1
Z t1
−(t1 −s)
k
e
g(βKT (s)u1 )ds −
e−(t2 −s) g(βKT (s)u2 )dsk
0
0
8
S. H. DA SILVA, F. D. M. BEZERRA
t1
Z
≤
EJDE-2013/221
ke−(t1 −s) g(βKT (s)u1 ) − e−(t2 −s) g(βKT (s)u2 )kds
0
t1
Z
≤
|e−(t1 −s) − e−(t2 −s) |kg(βKT (s)u1 )kds
0
t1
Z
e−(t2 −s) kg(βKT (s)u1 ) − g(βKT (s)u2 )kds
Z t1
t1 p
θ
≤
a |Ω|c|t1 − t2 | ds +
e−(t2 −s) k1 βe(k1 β−1)s ku1 − u2 kds
0
0
p
≤ 2t0 a |Ω|c|t1 − t2 |θ + ku1 − u2 kek1 β2t0
+
Z
0
= µ2 (|t1 − t2 |θ + ku1 − u2 k),
p
where µ2 = max{2t0 ac |Ω|, ek1 β2t0 }.
Without loss of generality assuming that t1 < t2 , using hypothesis (H3), we
obtain
Z t1
Z t2
k
e−(t2 −s) g(βKT (s)u2 )ds −
e−(t2 −s) g(βKT (s)u2 )dsk
0
0
Z t2
≤
e−(t2 −s) kg(βKT (s)u2 )kds
t1
t2
Z
p
e−(t2 −s) a |Ω|ds
t1
p
≤ (t2 − t1 )a |Ω|
p
≤ t1−θ
a |Ω|(t2 − t1 )θ
0
≤
= µ3 (t2 − t1 )θ ,
p
where µ3 = t1−θ
a |Ω|. Therefore,
0
kT (t1 )u1 − T (t2 )u2 k ≤ µ(|t2 − t1 |θ + ku1 − u2 k)
for some µ > 0 and 0 < θ < 1.
1,2
2
1
Since W (Ω) ,→ L (Ω) it follows that H ,→ X. Then, from Theorems 3.2, 3.3
and 3.5, it follows that the assumptions in the Proposition 2.7 of [2] are satisfied.
Therefore, we have the following result.
Theorem 3.6. Under the hypotheses (H1)-(H3), for any ν ∈ (0, 21 − δ), there exists
a nonvoid set M = Mν ⊂ B(0, ρ), positively invariant under T (t) and precompact
in X, with the following properties:
(1) there exists ω > 0 such that for any bounded set B ⊂ H 1 we have
lim eωt dist(T (t)B, M) = 0
t→∞
where dist(·, ·) denotes the Hausdorff semi-distance in X (see [5]).
(2) M possesses finite fractal dimension in X; more precisely, we have for any
ν ∈ (0, 21 − δ)
1
1
(1 + log 2(δ+ν)
Nν/k (B(0, 1)),
θ
where df (M) denotes the fractal dimension of M, B(0, 1) denotes the open
ball centered at 0 and radius 1 in H 1 = {u ∈ H 1 (RN ) : u(x) = 0, if x ∈
/ Ω},
dX
f (M) ≤
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FINITE FRACTAL DIMENSIONALITY
9
Nν/k (B) denotes the smallest number of νk −balls in L2 (RN ) needed to cover
B(0, 1) and k is the constant given in (3.11).
Denoting by Mν the closure in X of the set Mν , from [2, Corollary 2.8] we have
the following result.
Corollary 3.7. Under the hypotheses of Theorem 3.6, for any ν ∈ (0, 12 − δ):
(1) Mν is an exponential (H 1 − X) attractor bounded in H 1 ;
(2) there exists a finite dimensional global (H 1 − X) attractor A ⊂ Mν .
Acknowledgments. The authors would like to thank the anonymous referee for
his/her reading of the manuscript and valuable suggestions that have improved the
quality of our initial manuscript. Our gratitude goes also to professors Antônio L.
Pereira from USP and Jesualdo G. das Chagas from UFCG for your suggestions
and discussions.
S. H. da Silva was partially supported by CNPq-Brazil grants Casadinho/Procad552.464/2012-2, INCTMat-5733523/2008-8. F. D. M. Bezerra was partially supported by grant FAPESP 11/04166-5.
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[2] J. Cholewa, R. Czaja, G. Mola; Remarks on the fractal dimension of bi-space global and
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Severino Horácio da Silva
Unidade Acadêmica de Matemática e Estatı́stica, UAME/CCT/UFCG
Rua Aprı́gio Veloso, 882, Bairro Universitário, Campina Grande-PB 58429-900, Brazil
E-mail address: horaciousp@gmail.com, horacio@dme.ufcg.edu.br
Flank D. M. Bezerra
Departamento de Matemática, UFPB, Cidade Universitária, Campus I
João Pessoa-PB 58051-900, Brazil
E-mail address: flank@mat.ufpb.br