Kragujevac Journal of Mathematics
Volume 40(2) (2016), Pages 224–236.
HIGH DIMENSIONAL FRACTIONAL COUPLED SYSTEMS: NEW
EXISTENCE AND UNIQUENESS RESULTS
LOUIZA TABHARIT1 AND ZOUBIR DAHMANI2
Abstract. In this paper, we study a class of high dimensional coupled fractional
differential systems using Caputo approach. We investigate the existence of solutions
using Schaefer fixed point theorem. Moreover, new existence and uniqueness results
are obtained by using the contraction mapping principle. Finally, Some examples
are presented to illustrate our main results.
1. Introduction and Preliminaries
Fractional differential equations have gained a great interest because of their many
applications in modeling of physical and chemical processes and in engineering sciences. For the basic theory of fractional differential equations, see [1–10, 12–14, 17, 22].
Moreover, the nonlinear coupled systems involving fractional derivatives are also very
important, since they occur in various problems of applied mathematics. In the literature, we can find many papers dealing with the existence and uniqueness of solutions.
For more details, we refer the reader to [11, 16, 18–21] and the references therein.
Motivated by cited coupled systems-papers, in this work, we discuss the existence
and uniqueness of solutions for the following problem:
Key words and phrases. Banach contraction principle, Caputo derivative, fixed point, differential
equation, existence, uniqueness.
2010 Mathematics Subject Classification. 30C45, 39B72, 39B82.
Received: October 12, 2015.
Accepted: December 4, 2015.
224
HIGH DIMENSIONAL FRACTIONAL COUPLED SYSTEMS
(1.1)
225

l
X


α
1

fi1 (t, u1 (t) , . . . , um (t) , Dγ1 u1 (t) , . . . , Dγm um (t)) , t ∈ J,
D u1 (t) =




i=1



l

X


α2

u
(t)
=
D
fi2 (t, u1 (t) , . . . , um (t) , Dγ1 u1 (t) , . . . , Dγm um (t)) , t ∈ J,

2



i=1




..

.
l

X


αm


D um (t) =
fim (t, u1 (t) , . . . , um (t) , Dγ1 u1 (t) , . . . , Dγm um (t)) , t ∈ J,



i=1





uk (0)
= ak0 ,




(j)


uk (0)
= 0, j = 1, 2, . . . , n − 2,



 (n−1)
uk
(0) = J rk uk (τk ), rk > 0,
where k = 1, 2, . . . , m. We suppose that n − 1 < αk < n, γk ∈ ]0, n − 1[, k =
1, 2, . . . , m, n ∈ N∗ − {1}, m, l ∈ N∗ , τ ∈]0, 1[, J := [0, 1]. The derivatives Dαk , Dγk ,
k=1,2,...,m
k = 1, 2, . . . , m, are taken in the sense of Caputo. For the functions fik i=1,...,l
:
2m
J × R → R, we will specify them later.
The paper is organised as follows: We begin by introducing some definitions and
lemmas that will be used in the proof of the main results. Then, in the Main Results
Section, we prove the existence of solutions theorems. At the last section, some
illustrative examples are treated. So, let us now present the basic definitions and
lemmas [15].
Definition 1.1. The Riemann-Liouville fractional integral operator of order α > 0,
for a continuous function f on [0, ∞[ is defined as:
Z t

 1
(t − s)α−1 f (s) ds,
α > 0, t ≥ 0,
(1.2)
J α f (t) = Γ (α) 0

f (t),
α = 0, t ≥ 0,
R∞
where Γ (α) := 0 e−x xα−1 dx.
Definition 1.2. The Caputo derivative of order α for a function u : [0, ∞) → R,
which is at least n-times differentiable can be defined as:
Z t
1
α
D u(t) =
(t − s)n−α−1 u(n) (s) ds = J n−α u(n) (t),
Γ (n − α) 0
for n − 1 < α < n, n ∈ N∗ − {1}.
We recall the following lemmas [7, 19].
226
L. TABHARIT AND Z. DAHMANI
Lemma 1.1. For α > 0, the general solution of the fractional differential equation
Dα u(t) = 0, is given by
n−1
X
u(t) =
cj tj ,
j=0
where cj ∈ R, j = 0, . . . , n − 1, n = [α] + 1.
Lemma 1.2. Let α > 0. Then
J α Dα u(t) = u(t) +
n−1
X
cj tj ,
j=0
where cj ∈ R, j = 0, 1, . . . , n − 1, n = [α] + 1.
Lemma 1.3. Let q > p > 0, g ∈ L1 ([a, b]). Then Dp J q f (t) = J q−p f (t), t ∈ [a, b].
Lemma 1.4 (Schaefer fixed point Theorem). Let E be Banach space and T : E → E
is a completely continuous operator. If V = {u ∈ E : u = µT u, 0 < µ < 1} is bounded,
then T has a fixed point in E.
The proof of the following auxiliary lemma is crucial for the problem (1.1).
k=1,...,m
Lemma 1.5. Assume that Qki i=1,...,l ∈ C ([0, 1] , R), m, l ∈ N∗ . And consider the
problem
(1.3)
αk
D uk (t) =
l
X
Qki (t),
t ∈ J, n − 1 < αk < n, n ∈ N∗ − {1} ,
i=1
associated with the conditions:

u (0)
= ak0 ,


 k
(j)
uk (0)
= 0, j = 1, 2, . . . , n − 2,
(1.4)


 (n−1)
uk
(0) = J rk uk (τk ),
where k = 1, 2, . . . , m. Then for all k = 1, 2, . . . , m, we have
l Zt
X
(t − s)αk −1 k
Γ (rk + n) tn−1
uk (t) = −
Qi (s) ds + ak0 +
rk +n−1
Γ
(α
)
(n
−
1)!
τ
−
Γ
(r
+
n)
k
k
k
i=1 0


τk
Z
l
α
+r
−1
r
k
k
k
k
X (τk − s)
a0 τk
.
(1.5)
×
Qki (s) ds −
Γ
(α
+
r
)
Γ
(r
+
1)
k
k
k
i=1
0
Proof. Thanks to (1.2) and by (1.3), we have
t
(1.6)
l Z
X
(t − s)αk −1 k
uk (t) =
Qi (s) ds − ck0 − ck1 t − ck2 t2 − · · · − ckn−1 tn−1 ,
Γ
(α
)
k
i=1
0
HIGH DIMENSIONAL FRACTIONAL COUPLED SYSTEMS
227
where ckj ∈ R, j = 0, 1, 2, . . . , n − 1, k = 1, 2, . . . , m, m ∈ N∗ , and n − 1 < αk < n,
n ∈ N∗ − {1}.
For all k = 1, 2, . . . , m, we observe that

u (0)
= −ck0 ,


 k
(j)
uk (0)
= −j!ckj , j = 1, 2, . . . , n − 2,


 (n−1)
uk
(0) = − (n − 1)!ckn−1 .
The conditions (1.4) will allow us to obtain
 k
c0
= −ak0 ,




ckj
= 0, j = 1, 2, . . . , n − 2,
(1.7)


Γ (rk + n)

ckn−1 =

rk +n−1
Γ (n) τk
− Γ (rk + n)
!
l
rk
X
τ
k
ak0
−
J αk +rk Qki (τk ) ,
Γ (rk + 1) i=1
where k = 1, 2, . . . , m. The values of ckj given by (1.7) replaced in (1.6) imply (1.5).
This completes the proof of Lemma 1.5.
Now, we consider the space:
S := {(u1 , u2 , . . . , un ) : uk ∈ C ([0, 1] , R) , Dγk uk ∈ C ([0, 1] , R) , k = 1, 2, . . . , m} ,
endowed with the norm
k(u1 , u2 , . . . , um )kS = max (kuk k∞ , kDγk uk k∞ ) .
1≤k≤m
This Banach space will be used in the proof of the theorems.
2. Main Results
We impose the following hypotheses:
(H1 ): There exist nonnegative constants
k,=1,2,...,m
µk,
i, i=1,...,l
t ∈ [0, 1] and for all (u1 , u2 , . . . , u2m ),
, such that for all
j, j=1,2,...,2m
(v1 , v2 , . . . , v2m ) ∈ R2m ,
we have
2m
k
X
fi (t, u1 , u2 , . . . , u2m ) − fik (t, v1 , v2 , . . . , v2m ) ≤
µki j |uj − vj | .
j=1
k k=1,2,...,m
: [0, 1] × R2m → R; m, l ∈ N∗ are continuous.
k=1,2...,m
(H3 ): There exist nonnegative functions ωik i=1,...,l ∈ C ([0, 1]), such that: for all
t ∈ [0, 1] and for all (u1 , u2 , . . . , u2m ) ∈ R2m
k
fi (t, u1 , u2 , . . . , u2m ) ≤ ωik (t) ,
(H2 ): The functions fi
i=1,2,...,l
with
sup ωik (t) = Cik .
t∈J
228
L. TABHARIT AND Z. DAHMANI
Setting the following quantities:
Σk =
2m X
l
X
µki
j
,
j=1 i=1
Γ (rk + n) τkαk +rk
zk =
+
,
Γ (αk + 1) (n − 1)! τkrk +n−1 − Γ (rk + n) Γ (αk + rk + 1)
1
z∗k
1
Γ (rk + n) τkαk +rk
:=
+
,
Γ (αk − γk + 1) Γ (n − γk ) τkrk +n−1 − Γ (rk + n) Γ (αk + rk + 1)
and
Γ (rk + n) ak0 τkrk
Wk = ak0 +
,
(n − 1)! τkrk +n−1 − Γ (rk + n) Γ (rk + 1)
k r
a0 τ k
Γ
(r
+
n)
k
k
r +n−1
Wk∗ =
.
Γ (n − γk ) τk k
− Γ (rk + n) Γ (rk + 1)
We now prove the first main result.
Theorem 2.1. Assume that (H1 ) and the following are satisfied
max Σk (zk , z∗k ) < 1.
(2.1)
1≤k≤m
Then, the problem (1.1) has a unique solution on J.
Proof. Let us take the operator P : S → S given by
P (u1 , u2 , . . . , um ) (t) := (P1 (u1 , u2 , . . . , um ) (t) , . . . , Pm (u1 , u2 , . . . , um ) (t)) ,
t ∈ J,
such that
Pk (u1 , u2 , . . . , um ) (t)
l Zt
X
(t − s)αk −1 k
fi (t, u1 (t) , . . . , um (s) , Dγ1 u1 (s) , . . . , Dγm um (s)) ds + ak0
:= −
Γ
(α
)
k
i=1
0
+
Γ (rk + n) tn−1
(n − 1)! τkrk +n−1 − Γ (rk + n)
τ
l Zk
X
(τk − s)αk +rk −1 k
×
fi (t, u1 (t) , . . . , um (s) , Dγ1 u1 (s) , . . . , Dγm um (s)) ds
Γ (αk + rk )
i=1 0
!
ak0 τkrk
−
.
Γ (rk + 1)
We shall prove that P is contractive: Let (u1 , u2 , . . . , um ), (v1 , v2 , . . . , vm ) ∈ S. Then,
for all k = 1, 2, . . . , m and t ∈ J, we get:
|Pk (u1 , u2 , . . . , um ) (t) − Pk (v1 , v2 , . . . , vm ) (t)|
HIGH DIMENSIONAL FRACTIONAL COUPLED SYSTEMS
≤
229
l
X
k
fi s, u1 (s) , . . . , um (s) , Dγ1 u1 (s) , . . . , Dγm um (s)
sup
1
Γ (αk + 1) s∈J
−fik
i=1
γ
s, v1 (s) , . . . , vm (s) , D 1 v1 (s) , . . . , Dγm vm (s) Γ (rk + n) τkαk +rk
+
(n − 1)! τ rk +n−1 − Γ (rk + n) Γ (αk + rk + 1)
k
× sup
l
X
s∈J
−fik
k
fi s, u1 (s) , . . . , um (s) , Dγ1 u1 (s) , . . . , Dγm um (s)
i=1
γ
s, v1 (s) , . . . , vm (s) , D 1 v1 (s) , . . . , Dγm vm (s) .
Using (H1 ), yields
kPk (u1 , u2 , . . . , um ) − Pk (v1 , v2 , . . . , vm )k∞
Γ (r + n) τkαk +rk
r +n−1 k
+
Γ (αk + 1) (n − 1)! τk k
− Γ (rk + n) Γ (αk + rk + 1)
1
≤
×
l
X
µki
+ µki
1
+ · · · + µki
2
i=1
!
max (kuk − vk k∞ , kDγk (uk − vk )k∞ ) .
2m 1≤k≤m
Thus,
kPk (u1 , u2 , . . . , um ) − Pk (v1 , v2 , . . . , vm )k∞
≤ Σk zk k(u1 − v1 , . . . , um − vm , Dγ1 (u1 − v1 ) , . . . , Dγm (um − vm ))kS .
(2.2)
On the other hand, we have
|Dγk Pk (u1 , u2 , . . . , um ) (t) − Dγk Pk (v1 , v2 , . . . , vm ) (t)|
l
X
tαk −γk
fik s, u1 (s) , . . . , um (s) , Dγ1 u1 (s) , . . . , Dγm um (s)
sup
≤
Γ (αk − γk + 1) s∈J i=1
γ
−fik s, v1 (s) , . . . , vm (s) , D 1 v1 (s) , . . . , Dγm vm (s) +
Γ (rk + n) tn−γk −1 τkαk +rk
r +n−1
Γ (n − γk ) τk k
− Γ (r + n) Γ (αk + rk + 1)
l
X
k
fi s, u1 (s) , . . . , um (s) , Dγ1 u1 (s) , . . . , Dγm um (s)
× sup
s∈J
−fik
i=1
γ
s, v1 (s) , . . . , vm (s) , D 1 v1 (s) , . . . , Dγm vm (s) .
Then,
kDγk Pk (u1 , u2 , . . . , um ) − Dγk Pk (v1 , v2 , . . . , vm )k∞
≤
1
Γ (r + n) τkαk +rk
r +n−1 k
+
Γ (αk − γk + 1) Γ (n − γk ) τk k
− Γ (rk + n) Γ (αk + rk + 1)
!
230
L. TABHARIT AND Z. DAHMANI
×
2m X
l
X
µki k(u1 − v1 , . . . , um − vm , Dγ1 (u1 − v1 ) , . . . , Dγm (um − vm ))kS .
i=1 i=1
So,
kDγk Pk (u1 , u2 , . . . , um ) − Dγk Pk (v1 , v2 , . . . , vm )k∞
(2.3)
≤ Σk z∗k k(u1 − v1 , . . . , um − ym , Dγ1 (u1 − v1 ) , . . . , Dγm (um − vm ))kS .
Thanks to (2.3) and (2.2), we obtain
kP (u1 , u2 , . . . , um ) − P (v1 , v2 , . . . , vm )kS
(2.4) ≤ max Σk (zk , z∗k ) k(u1 −v1 , . . . , um −vm , Dγ1 (u1 −v1 ) , . . . , Dγm (um −vm ))kS .
1≤k≤n
By (2.1) the operator P is contractive. Hence, P has a unique fixed point which is a
solution of the system (1.1). This ends the proof.
The following result presents new conditions to the existence of one solution. We
prove the result by using Schaefer theorem:
k=1,2,...,m
Theorem 2.2. Let fik i=1,2,...,l satisfied (H2 ) and (H3 ). Then, the problem (1.1)
has at least one solution on J.
Proof. First, we prove that P is completely continuous. We consider the set
Ωσ := {(u1 , u2 , . . . , um ) ∈ S; k(u1 , u2 , . . . , um )kS ≤ σ, σ > 0} ,
and we show that P maps bounded sets into bounded sets in S.
Let (u1 , u2 , . . . , um ) ∈ Ωσ . The hypothesis (H3 ) implies that
kPk (u1 , u2 , . . . , um )k∞
l
X
tαk
fik s, u1 (s) , . . . , um (s) , Dγ1 u1 (s) , . . . , Dγm um (s) sup
≤
Γ (αk + 1) s∈J i=1
n−1 k rk
k
a0 τk
Γ
(r
k + n) t
r +n−1
+ a0 +
k
(n − 1)! τk
− Γ (rk + n) Γ (rk + 1)
Γ (rk + n) tn−1 τkαk +rk
+
(n − 1)! τkrk +n−1 − Γ (rk + n) Γ (αk + rk + 1)
l
X
k
fi s, u1 (s) , . . . , um (s) , Dγ1 u1 (s) , . . . , Dγm um (s) × sup
s∈J
i=1
(2.5)
Γ (rk + n) τkαk +rk
≤
+
Γ (αk + 1) (n − 1)! τkrk +n−1 − Γ (rk + n) Γ (αk + rk + 1)
k r
k
k
Γ
(r
k + n) a0 τk
r +n−1
+ a0 +
.
(n − 1)! τk k
− Γ (rk + n) Γ (rk + 1)
1
!
sup
s∈J
l
X
i=1
ωik (s)
HIGH DIMENSIONAL FRACTIONAL COUPLED SYSTEMS
231
Therefore,
kPk (u1 , u2 , . . . , um )k∞
Γ (rk + n) τkαk +rk
≤
+
Γ (αk + 1) (n − 1)! τkrk +n−1 − Γ (rk + n) Γ (αk + rk + 1)
k r
l
a0 τ k
X
Γ
(r
+
n)
k
k
r +n−1
×
Cik + ak0 +
(n − 1)! τ k
− Γ (rk + n) Γ (rk + 1)
1
k
i=1
≤ zk
(2.6)
l
X
!
Cik + Wk .
i=1
Furthermore, we get
kDγk Pk (u1 , u2 , . . . , um )k∞
l
X
tαk −γk
fik s, u1 (s) , . . . , um (s) , Dγ1 u1 (s) , . . . , Dγm um (s) ≤
sup
Γ (αk − γk + 1) s∈J i=1
Γ (rk + n) tn−γk −1 ak0 τkrk
+
Γ (n − γk ) τ rk +n−1 − Γ (rk + n) Γ (rk + 1)
k
+
Γ (rk + n) tn−γk −1 τkαk +rk
r +n−1
Γ (n − γk ) τk k
− Γ (rk + n) Γ (αk + rk + 1)
l
X
k
fi s, u1 (s) , . . . , um (s) , Dγ1 u1 (s) , . . . , Dγm um (s) × sup
s∈J
i=1
1
Γ (r + n) τkαk +rk
r +n−1 k
≤
+
Γ (αk − γk + 1) Γ (n − γk ) τk k
− Γ (rk + n) Γ (αk + rk + 1)
l
X
Γ (rk + n) ak0 τkrk
k
×
ωi (t) +
.
Γ (n − γk ) τ rk +n−1 − Γ (rk + n) Γ (rk + 1)
!
k
i=1
Hence,
(2.7)
γk
kD Pk (u1 , u2 , . . . , um )k∞ ≤
z∗k
l
X
Cik + Wk∗ .
i=1
From (2.6) and (2.7), we get
(2.8)
kP (u1 , u2 , . . . , um )kS ≤ max
1≤k≤m
zk
l
X
Cik + Wk , z∗k
i=1
So, P maps bounded sets into bounded sets in S.
l
X
i=1
!
Cik + Wk∗
< ∞.
232
L. TABHARIT AND Z. DAHMANI
The hypothesis (H2 ) implies that P is continuous on S. Now, let t1 , t2 ∈ [0, 1] ;
t1 < t2 , and (u1 , u2 , . . . , um ) ∈ S, we have:
kPk (u1 , u2 , . . . , um ) (t2 ) − Pk (u1 , u2 , . . . , um ) (t1 )k∞
≤
(2.9)
1
αk
Γ (αk + 1)
+
2 (t2 − t1 )
+
tα2 k
−
tαk
1
l
X
Cik
i=1
Γ (rk + n) ak0 τ rk
k
r +n−1
− tn−1
tn−1
1
2
k
(n − 1)! τ
− Γ (rk + n) Γ (rk + 1)
k
m
X k
Γ (rk + n) τkαk +rk
n−1
n−1
r +n−1
+
t
−
t
Ci ,
2
1
(n − 1)! τk k
− Γ (rk + n) Γ (αk + rk + 1)
i=1
and
kDγk Pk (u1 , u2 , . . . , um ) (t2 ) − Dγk Pk (u1 , u2 , . . . , um ) (t1 )k∞
l
2 (t2 − t1 )αk− γk + tα2 k −γk − tα1 k −γk X k
≤
Ci
Γ (αk − γk + 1)
i=1
Γ (rk + n) ak0 τkrk
n−γk −1
n−γk −1
r +n−1
+
t
−
t
2
1
Γ (n − γk ) τ k
− Γ (rk + n) Γ (rk + 1)
k
(2.10)
l
X k
Γ (rk + n) τkαk +rk
n−γk −1
n−γk −1
r +n−1
t
−
t
Ci .
+
2
1
Γ (n − γk ) τk k
− Γ (rk + n) Γ (αk + rk + 1)
i=1
The right-hand sides of the above inequalities (2.9) and (2.10) are independent of
(u1 , u2 , . . . , um ) and tend to zero as t2 − t1 → 0. Then the operator P , is equicontinuous. Hence, the operator P is a completely continuous.
Finally we show that
λ := {(u1 , u2 , . . . , um ) ∈ S, (u1 , u2 , . . . , um ) = ηP (u1 , u2 , . . . , um ) 0 < η < 1 }
is bounded.
For (u1 , u2 , . . . , um ) ∈ λ, we have (u1 , u2 , . . . , um ) (t) = ηP (u1 , u2 , . . . , um ) (t), for
all t ∈ [0, 1]. Using (2.8), we get
!
l
l
X
X
k(u1 , u2 , . . . , um )kS ≤ η max zk
Cik + Wk , z∗k
Cik + Wk∗ < ∞.
1≤k≤m
i=1
i=1
Thus, λ is a bounded. By Lemma 1.4, the operator P has a fixed point which is a
solution of (1.1). Theorem 2.2 is thus proved.
3. Examples
In this section, we give some examples to illustrate our theorems.
HIGH DIMENSIONAL FRACTIONAL COUPLED SYSTEMS
233
Example 3.1. Consider the following system:
(3.1)
8



5
3

2
u
(t)
+
D
u
(t)
D

1
2
11
1


cos (u1 (t)) + cos (u2 (t)) + 
4
D 3 u1 (t) =


3
3

12π 3 (t + 1)

2
u
(t)
+
D
u
(t)
1
+
D
1
2






8
5
|u1 (t) + u2 (t)|
1


+ sin D 3 u1 (t) + sin D 2 u2 (t) ,
+



64π (t + 1) 2π (1 + |u1 (t) + u2 (t)|)




t ∈ [0, 1] ,





8
5
19
|u1 (t) + u2 (t)|
1


5
3
2

D u2 (t) = 14π 3 et 1 + |u (t) + u (t)| + sin D u1 (t) + sin D u2 (t)
1
2
8
5

 

3 u1 (t) + sin D 2 u2 (t) sin
(u
(t))
+
cos
(u
(t))
+
cos
D
2
1
2

t


5
 ,
8
+ 3



6π

3
2
u
(t)
+
sin
D
u
(t)
1
+
sin
(u
(t))
+
cos
(u
(t))
−
cos
D
1
2
1
2






t ∈ [0, 1] ,




√

1
3

(1)
(2)
(3)

,
u1 (0) = 3, u1 (0) = u1 (0) = 0, u1 (0) = J 2



7




√
1
6

(1)
(2)
(3)

3
u2 (0) = 5, u2 (0) = u2 (0) = 0, u2 (0) = J
.
7
For this example, we have: n = 4, l = 2, α1 = 11/3, α2 = 19/5, γ1 = 8/3, γ2 = 5/2,
r1 = 1/2, r2 = 1/3, τ1 = 3/7, τ2 = 6/7, J = [0, 1], and
1
|u3 + u4 |
1
f1 (t, u1 , u2 , u3 , u4 ) =
cos u1 + cos u2 +
,
12π 3 (t + 1)
(1 + |u3 + u4 |)
|u1 + u2 |
1
1
+ sin u3 + sin u4 ,
f2 (t, u1 , u2 , u3 , u4 ) =
64π (t + 1) 2π (1 + |u1 + u2 |)
1
|u1 + u2 |
2
f1 (t, u1 , u2 , u3 , u4 ) =
+ sin u3 + sin u4 ,
14π 3 et 1 + |u1 + u2 |
t2
|sin u1 + cos u2 + cos u3 + sin u4 |
2
f2 (t, u1 , u2 , u3 , u4 ) = 3
.
6π
1 + |sin u1 + cos u2 − cos u3 + sin u4 |
For all t ∈ J and (u1 , u2 , u3 , u4 ), (v1 , v2 , v3 , v4 ) ∈ R4 , we get:
1
f1 (t, u1 , u2 , u3 , u4 ) − f11 (t, v1 , v2 , v3 , v4 )
1
1
1
1
|u1 − v1 | +
|u2 − v2 | +
|u3 − v3 | +
|u4 − v4 | ,
≤
3
3
3
12π
12π 12π 3
12π
f21 (t, u1 , u2 , u3 , u4 ) − f21 (t, v1 , v2 , v3 , v4 )
1
1
1
1
|u1 − v1 | +
|u2 − v2 | +
|u3 − v3 | +
|u4 − v4 | ,
2
2
128π
64π
64π
128π
f12 (t, u1 , u2 , u3 , u4 ) − f12 (t, v1 , v2 , v3 , v4 )
1
1
1
1
≤
|u1 − v1 | +
|u2 − v2 | +
|u3 − v3 | +
|u4 − v4 | ,
3
3
3
14π
14π
14π
14π 3
≤
234
L. TABHARIT AND Z. DAHMANI
2
f2 (t, u1 , u2 , u3 , u4 ) − f22 (t, v1 , v2 , v3 , v4 )
1
1
1
1
|u
−
v
|
+
|u
−
v
|
+
|u
−
v
|
+
|u4 − v4 | .
1
1
2
2
3
3
6π 3
6π 3
6π 3
6π 3
We can take:
1
µ11 1 = µ11 2 = µ11 3 = µ11 4 =
,
12π 3
1
1
1
1
µ12 1 = µ12 2 =
,
,
µ
=
µ
=
2
2
3
4
2
128π
64π
1
µ21 1 = µ21 2 = µ21 3 = µ21 4 =
,
14π 3
1
µ22 1 = µ22 2 = µ22 3 = µ22 4 = 3 .
6π
Indeed,
≤
Σ1 z1 = 0.001517, Σ2 z2 = 0.001820,
Σ1 z∗1 = 0.022304, Σ2 z∗2 = 0.027009.
Thus,
max (Σ1 z1 , Σ2 z2 , Σ1 z∗1 , Σ2 z∗2 ) < 1.
Using Theorem 2.1, system (3.1) has a unique solution on J.
Example 3.2. We consider the following system:
7

4
15
2 u1 (t) + D 3 u2 (t) + D 4 u3 (t)

t
sin
D

19


D 4 u1 (t) =


1 + |sin (u1 (t) + u2 (t) + u3 (t))|





t sin u (t) D 27 u (t)

e

1
1



, t ∈ [0, 1] ,
+

4
15


3 u2 (t) + u3 (t) + D 4 u3 (t) 1
+
cos
u
(t)
+
D

2





17
et sin (u1 (t) + u2 (t) + u3 (t))


7

D 4 u2 (t) =

4
15


2
3
4
2
−
sin
D
u
(t)
+
D
u
(t)
+
D
u
(t)

1
2
3





sin (u2 (t) u3 (t))


, t ∈ [0, 1] ,
+


7
4
15
2 u1 (t) D 3 u2 (t) + D 4 u3 (t)
π
(t
+
1)
+
sin
u
(t)
D
1
(3.2)

7

4
15
 25
t cos D 2 u1 (t)+D 3 u2 (t)+D 4 u3 (t)


D 6 u3 (t) = sin (u1 (t) u2 (t) u3 (t)) e





t sin (u1 (t) u2 (t) u3 (t))



+ 7
, t ∈ [0, 1] ,
4
15


D 2 u1 (t)+D 3 u2 (t)+D 4 u3 (t)

e



√

1
1

(1)
(2)
(3)
(4)

u1 (0) = 3 2, u1 (0) = u1 (0) = u1 (0) = 0, u1 (0) = J 8
,



2




√
1
2

(1)
(2)
(3)
(4)

2

u2 (0) = 7, u2 (0) = u2 (0) = u2 (0) = 0, u2 (0) = J
,


3




√

5
4
(2)
(3)
(4)

u3 (0) = 5, u(1)
u3 (0) = J 8
.
3 (0) = u3 (0) = u3 (0) = 0,
5
HIGH DIMENSIONAL FRACTIONAL COUPLED SYSTEMS
235
We have: n = 5, l = 2, α1 = 19/4, γ1 = 7/2, α2 = 17/4, γ2 = 4/3, α3 = 25/6,
γ3 = 15/4, r1 = 1/8, r2 = 1/2, r3 = 5/8, τ1 = 1/2, τ2 = 2/3, τ3 = 4/5, J = [0, 1].
And, for all (u1 , u2 , u3 , u4 , u5 , u6 ) ∈ R6 , for all t ∈ J, we get
t |sin u4 + u5 + u6 |
≤ C11 = 1,
1 + |sin (u1 + u2 + u3 )|
1
et |sin (u1 u4 )|
f2 (t, u1 , u2 , u3 , u4 , u5 , u6 ) ≤ ω21 (t) =
≤ C21 = e,
1 + |cos (u2 + u5 + u3 + u6 )|
t
2
f1 (t, u1 , u2 , u3 , u4 , u5 , u6 ) ≤ ω12 (t) = e |sin (u1 + u2 + u3 )| ≤ C12 = e,
2 − sin (u4 + u5 + u6 )
2
|sin (u2 u3 )|
1
f2 (t, u1 , u2 , u3 , u4 , u5 , u6 ) ≤ ω22 (t) =
≤ C22 =
,
π (t + 1) + sin (u1 u4 u5 + u6 )
π−1
3
f1 (t, u1 , u2 , u3 , u4 , u5 , u6 ) ≤ ω13 (t) = |sin (u1 u2 u3 )| et cos(u4 +u5 +u6 ) ≤ C13 = e,
1
f1 (t, u1 , u2 , u3 , u4 , u5 , u6 ) ≤ ω11 (t) =
3
f2 (t, u1 , u2 , u3 , u4 , u5 , u6 ) ≤ ω23 (t) = t |sin (u1 (t) u2 (t) u3 (t))| ≤ C23 = 1.
eu4 t+u5 +u6
The functions fik are continuous and bounded on J × R6 . Using Theorem 2.2, the
system (3.2) has at least one solution on J.
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1
Department of Mathematics and Informatics, Faculty SEI,
University of Mostaganem,
Algeria
E-mail address: [email protected]
2
LPAM, Faculty SEI,
UMAB of Mostaganem,
Algeria
E-mail address: [email protected]