International Journal of Analysis and Applications
ISSN 2291-8639
Volume 15, Number 1 (2017), 18-22
http://www.etamaths.com

POSITIVE SOLUTIONS FOR MULTI-ORDER NONLINEAR FRACTIONAL

SYSTEMS

A. GUEZANE-LAKOUD AND R. KHALDI∗

Abstract. In this paper, we study the existence of positive solutions for a class of multi-order

systems of fractional differential equations with nonlocal conditions. The main tool used is Schauder

fixed point theorem and upper and lower solutions method. The results obtained are illustrated by
a numerical example.

1. Introduction

Recently, the investigation of fractional differential equations attracted more attention since it has
many applications in several fields of sciences such as in engineering, physics, chemistry, biology, etc
... [8], [10].

In this work, we use the method of upper and lower solutions to prove the existence of positive
solutions for a system of multi-order fractional differential equations with nonlocal boundary conditions,
where each equation has an order that may be different from the order of the other equations, that is :

(P)

{
Dα

0+
u (t) + f (t,u (t)) = 0, 0 < t < 1,

u (0) = u′ (0) = 0, Au (1) = Bu′ (1) ,

where the function u = (u1,u2, ...,un), ui : [0, 1] → R,

Dα0+u (t) =
(
Dα1

0+
u1 (t) ,D

α2
0+
u2 (t) , ...,D

αn
0+
un (t)

)
,

Dαi
0+

denotes the Reimann-Liouville fractional derivative of order αi, 2 < αi < 3, i ∈ {1, ..,n} , n ≥ 2,
the function f is such that

f (t,u) = (f1 (t,u) , ...,fn (t,u)) ,

u = (u1,u2, ...,un) ,

fi ∈ C ([0, 1] ×Rn,R+) , A = (a1, ...,an) , B = (b1, ...,bn) ∈ Rn.
Fractional differential systems can arise from sciences problems such population problems, dielectric

polarization, electromagnetic waves,...see [3]. Many methods are used for the investigation of fractional
differential equations, such fixed point theory, lower and upper solutions method, Mawhin theory,...see
[1], [2], [4], [5], [6], [7], [9], [11].

This paper is organized as follows: in the second Section, we state some preliminary materials that
will be used later. In section three, we use the upper and lower solutions method to prove the existence
of positive solutions for problem (P). Finally, we give an example illustrating the obtained results.

2. Preliminaries

In this section, we recall the basic definitions and lemmas from fractional calculus theory and the
details can be found in [7], [10].

Received 30th April, 2017; accepted 11th July, 2017; published 1st September, 2017.
2010 Mathematics Subject Classification. 34B10, 26A33, 34B15.
Key words and phrases. fractional Rieman-Liouville derivative; fractional differential equation; upper and lower

solutions method.

c©2017 Authors retain the copyrights of
their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

18



POSITIVE SOLUTIONS FOR MULTI-ORDER NONLINEAR FRACTIONAL SYSTEMS 19

Definition 2.1. The Riemann-Liouville fractional integrals of order α of a function h is defined as

Iα0+h (t) =
1

Γ (α)

∫ t
0

h (s)

(t−s)1−α
ds,

where Γ (α) =
∫∞
0
e−ttα−1dt is the Gamma function, α > 0.

Definition 2.2. The Riemann-Liouville derivative of fractional order α > 0 for a function h is defined
as

Dα0+h (t) =
1

Γ (n−α)

(
d

dt

)n ∫ t
0

(t−s)n−α−1 h (s) ds,

where n = [α] + 1 ([α] denotes the integer part of the real number α).

Lemma 2.1. For α > 0, the solution of the homogeneous equation

Dα0+h (t) = 0,

is given by
h(t) = c1t

α−1 + c2t
α−2 + ··· + cntα−n,

where ci, i = 1, 2, ...,n, are real constants.

Lemma 2.2. Let p, q ≥ 0, h ∈ L1 [0, 1]. Then
I
p
0+
I
q
0+
h (t) = I

p+q
0+

h (t) = I
q
0+
I
p
0+
h (t) .

3. Main results

Lemma 3.1. Let yi ∈ C ([0, 1] ,R) , i ∈{1, ..,n}. Assume that ai > 0 and bi < 0, then for i ∈{1, ..,n},
the linear nonhomogeneous problem

(Si) =




Dαi
0+
ui (t) = −yi (t) , 0 < t < 1,
ui (0) = u

′
i (0) = 0,

aiui (1) = biu
′
i (1) ,

(3.1)

has the following solution

ui (t) =

∫ 1
0

Gi (t,s) yi (s) ds, 0 ≤ t ≤ 1,∀i ∈{1, ..,n} (3.2)

where

Gi (t,s) =



−(t−s)αi−1

Γ (αi)
+

tαi−1 (1 −s)αi−2

(ai − bi (αi − 1)) Γ (αi − 1)

(
ai (1 −s)
αi − 1

− bi
)
,

s ≤ t,
tαi−1 (1 −s)αi−2

(ai − bi (αi − 1)) Γ (αi − 1)

(
ai (1 −s)
αi − 1

− bi
)
,s ≥ t.

Proof. By assuming that ui is a solution of the fractional boundary value problem (P) and using
Lemma 2.1, we obtain

ui (t) = −Iαi0+yi (t) + c1t
αi−1 + c2t

αi−2 + c3t
αi−3, (3.3)

According to conditions ui (0) = 0 and u
′
i (0) = 0, we obtain c2 = c3 = 0. Using the nonlocal condition

aiui (1) = biu
′
i (1), it yields

c1 =
1

ai − bi (αi − 1)
(
aiI

αi
0+
y (1) − biIαi−10+ y (1)

)
. (3.4)

Substituting c1 in Equation 3.3, we get what follows

ui (t) =

∫ 1
0

Gi (t,s) yi (s) ds, (3.5)

where Gi is given above. �

Lemma 3.2. If ai > 0 and bi < 0, i ∈ {1, ..,n} , then the functions Gi are nonnegative, continuous
and

0 ≤ Gi (t,s) ≤
1

Γ (αi)
, 0 ≤ s,t ≤ 1,∀i ∈{1, ..,n} ,



20 GUEZANE-LAKOUD, KHALDI

Proof. The proof is direct, we omit it. �

Let X be the Banach space C ([0, 1] ,R) × ...×C ([0, 1] ,R)︸ ︷︷ ︸
n times

, equipped with the norm

‖u‖ =
∑i=n
i=1 maxt∈[0,1] |ui (t)| .

Define the integral operator T : X → X by Tu = (T1u,T2u,...,Tnu) where

Tiu (t) =

∫ 1
0

Gi (t,s) fi (s,u (s)) ds, ∀i ∈{1, ..,n} (3.6)

Let C = (c1, ...,cn) , D = (d1, ...,dn) ∈ Rn+ such that D > C. We recall that for x = (x1, ...,xn) , y =
(y1, ...,yn) then x ≤ y means xi ≤ yi, for all i ∈{1, ..,n} and [C,D] = {x = (x1, ...,xn) ,ci ≤ xi ≤ di,∀i ∈{1, ..,n}} .
We define the upper and lower control operators U =

(
U1, ...,Un

)
, U = (U1, ...,Un) respectively by

Ui (t,x) = sup{fi (t,y) ,C ≤ y ≤ x} ,
Ui (t,x) = inf {fi (t,y) ,x ≤ y ≤ D} , 0 ≤ t ≤ 1.

From the definition of Ui and Ui we have Ui (t,x) ≤ fi (t,x) ≤ Ui (t,x) , x ∈ [C,D] , 0 ≤ t ≤ 1,
i ∈{1, ..,n} .
Lemma 3.3. The function u ∈ X is a solution of the system (P) if and only if Tiu (t) = ui (t), for all
t ∈ [0, 1] , ∀i ∈{1, ...,n} .

Consequently, the existence of solutions for system (P) can be turned into a fixed point problem in
X for the operator T. Define the cone

K = {u ∈ X,u(t) ≥ 0, 0 ≤ t ≤ 1} .
Let us make the following hypothesis:
(H) There exist two functions θ =

(
θ1, ...,θn

)
, θ = (θ1, ...,θn) ∈ K, such that C ≤ θ (t) ≤ θ (t) ≤ D,

0 ≤ t ≤ 1 and {
θi (t) ≥

∫ 1
0
Gi (t,s) Ui

(
s,θ (s)

)
ds,i ∈{1, ..,n}

θi(t) ≤
∫ 1
0
Gi (t,s) Ui (s,θ (s)) ds,i ∈{1, ..,n}

The functions θ and θ are called respectively upper and lower solutions for problem (P).
Now we are ready to give the main result for problem (P).

Theorem 3.1. Assume that hypothesis (H) holds and f (t, 0) 6= 0, 0 ≤ t ≤ 1, then the fractional
boundary value problem (P) has at least one positive solution u ∈ K satisfying θ (t) ≤ u (t) ≤ θ (t) ,
0 ≤ t ≤ 1.
Proof. Clearly, the continuity of the operator T follows from the continuity of f. Set

Ω =
{
u ∈ K : θ (t) ≤ u (t) ≤ θ (t) , 0 ≤ t ≤ 1

}
,

then Ω is a nonempty, closed and convex subset of X. Firstly, we show that T (Ω) ⊂ Ω. In fact, let
u ∈ Ω, then by the definition of the control functions and hypothesis (H), it yields

Tiu (t) =

∫ 1
0

Gi (t,s) fi (s,u (s)) ds

≤
∫ 1
0

Gi (t,s) Ui
(
s,θ (s)

)
ds ≤ θi (t) , i ∈{1, ..,n}

thus
Tu (t) ≤ θ (t) , 0 ≤ t ≤ 1.

Similarly, we get

Tiu (t) =

∫ 1
0

Gi (t,s) fi (s,u (s)) ds

≥
∫ 1
0

Gi (t,s) Ui (s,θ (s)) ds ≥ θi (t) , i ∈{1, ..,n}



POSITIVE SOLUTIONS FOR MULTI-ORDER NONLINEAR FRACTIONAL SYSTEMS 21

from which follows

Tu (t) ≥ θ (t) , 0 ≤ t ≤ 1
thus T (Ω) ⊂ Ω. Now, we prove that T : Ω → X is completely continuous operator. Set

Mi = max{fi (t,u(t)) , 0 ≤ t ≤ 1,u ∈ Ω} ,

then we have

|Tiu (t)| ≤
∫ 1
0

Gi (t,s) fi (s,u (s)) ds

≤
Mi

Γ (αi)
.

Taking the supremum over [0, 1], then summing the obtained inequalities according to i from 1 to n,
we get

‖Tu‖≤
n∑
i=1

Mi
Γ (αi)

,

which implies that T (Ω) is uniformly bounded.
Let us show that (Tu) is equicontinuous. Indeed, let u ∈ Ω and 0 ≤ t1 < t2 ≤ 1, then

|Tiu (t1) −Tiu (t2)| ≤
∫ 1
0

|Gi (t1,s) −Gi (t2,s)|fi (s,u (s)) ds

≤ Mi
[∫ t1

0

|Gi (t1,s) −Gi (t2,s)|ds

+

∫ t2
t1

|Gi (t1,s) −Gi (t2,s)|ds

+

∫ 1
t2

|Gi (t1,s) −Gi (t2,s)|ds
]

by computation, we get

|Tiu (t1) −Tiu (t2)|

≤ Mi
(

(t2 − t1) (αi − 1)
Γ (αi)

+
(t2 − t1)

αi−1

Γ (αi)
+

3
(
tαi−12 − t

αi−1
1

)
ai − bi (αi − 1)

(
ai

Γ (αi)
+

bi
Γ (αi − 1)

))
.

As t1 → t2, the right-hand side of the above inequality tends to zero. By Ascoli-Arzela theorem, we
conclude that the operator T : Ω → Ω is completely continuous. Finally, Schauder fixed point theorem
implies that T has at least one fixed point u ∈ Ω and then problem (P) has at least one positive
solution in Ω.

As direct consequence of Theorem 3.1, we get the following Corollary. �

Corollary 3.1. Assume that fi are continuous, nonnegative, fi (t, 0) 6= 0, 0 ≤ t ≤ 1 and there exist
two positive constants li and Li such that

0 < li ≤ fi (t,x) ≤ Li, x ≥ 0, 0 ≤ t ≤ 1, i ∈{1, ..,n} , (3.7)

then problem (P) has at at least one positive solution u ∈ X. Furthermore the solution satisfies

0 < li

∫ 1
0

Gi (t,s) ds ≤ ui (t) ≤ Li
∫ 1
0

Gi (t,s) ds,

0 ≤ t ≤ 1, ∀i ∈{1, ..,n} .



22 GUEZANE-LAKOUD, KHALDI

Proof. From equation 3.7 we have

Ui (t,x) ≤ Li, Ui (t,x) ≥ li, 0 ≤ t ≤ 1, x ≥ 0.
Let us choose 



θi (t) = Li
∫ 1
0
Gi (t,s) ds = Li

tαi−1

αiΓ (αi)

(
1 + 1 + 1

αi

)
≥
∫ 1
0
Gi (t,s) Ui

(
s,θ (s)

)
ds, i ∈{1, ..,n}

θi(t) = li
∫ 1
0
Gi (t,s) ds = li

tαi−1

αiΓ (αi)

(
1 + 1 + 1

αi

)
≤
∫ 1
0
Gi (t,s) Ui (s,θ (s)) ds, i ∈{1, ..,n} ,

then the conclusion follows from Theorem 3.1. �

Now, we give an examples to illustrate the usefulness of our main results.

Example 3.1. Consider the following two-dimensional fractional order system

(S) =




D
5
2 u1 (t) +

(
1 + 1

1+u1+u2

)
= 0, D

8
3 u2 (t) + (1 + e

−u1 ) = 0,

u1 (0) = 0, u
′
1 (0) = 0, u1 (1) −u′1 (0) = 0,

u2 (0) = 0, u
′
2 (0) = 0, u2 (1) −u′2 (0) = 0.

We have α =
(
5
2
, 8
3

)
, a1 = a2 = 1, b1 = b2 = −1, f1 (t,u1,u2) = 1 + 11+u1+u2 , f2 (t,u1,u2) = 1 + e

−u1 ,

fi ∈ C
(
[0, 1] ×R2,R+

)
, fi (t, 0) 6= 0, and

1 ≤ fi (t,u1,u2) ≤ 2.
From Corollary 3.1, we conclude the system (S) has at at least one positive solution u ∈ X. Further-
more, the solution u satisfies

0.722 15t
3
2 ≤ u1 (t) ≤ 1.444 3t

3
2

0.591 95t
5
3 ≤ u2 (t) ≤ 1.183 9t

5
3 .

References

[1] B. Ahmad, A. Alsaedi, Existence and uniqueness of solutions for coupled systems of higher-order nonlinear fractional

differential equations. Fixed Point Theory Appl. 2010 (2010), Art. ID 364560.
[2] B. Ahmad, Juan J. Nieto, Riemann–Liouville fractional integro-differential equations with fractional nonlocal integral

boundary conditions. Bound Value Probl. 2011 (2011), Art. ID 36.

[3] Y. Chai, L. Chen, R. Wu, Inverse projective synchronization between two different hyperchaotic systems with frac-
tional order. J. Appl. Math. 2012 (2012), Article ID 762807.

[4] M. Feng, X. Zhang, W. Ge, New existence results for higher-order nonlinear fractional differential equation with

integral boundary conditions . Bound. Value Probl. 2011 (2011), Art. ID 720702.
[5] A. Guezane-Lakoud, A. Ashyralyev, Positive Solutions for a System of Fractional Differential Equations with Nonlocal

Integral Boundary Conditions. Differ. Equ. Dyn. Syst., DOI: 10.1007/s12591-015-0255-9.
[6] J. Henderson, S. K. Ntouyas, I.K. Purnaras, Positive solutions for systems of generalized three-point nonlinear

boundary value problems. Comment. Math. Univ. Carolin. 49 (2008), 79-91.
[7] R. Khaldi, A. Guezane-Lakoud, Upper and lower solutions method for higher order boundary value problems, Progress

in Fractional Differentiation and Applications, Progr. Fract. Differ. Appl. 3 (1) (2017), 53-57.
[8] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier,

Amsterdam 2006.
[9] S. K. Ntouyas, M. Obaid, A coupled system of fractional differential equations with nonlocal integral boundary

conditions. Adv. Differ. Equ. 2012 (2012), Article ID 130.
[10] I. Podlubny, Fractional Differential Equations Mathematics in Sciences and Engineering. Academic Press, New York

1999.
[11] M. Rehman, R. Khan, A note on boundary value problems for a coupled system of fractional differential equations.

Comput. Math. Appl. 61 (2011), 2630-2637.

Laboratory of Advanced Materials, Faculty of Sciences, Badji Mokhtar-Annaba University, P.O. Box

12, 23000 Annaba, Algeria

∗Corresponding author: rkhadi@yahoo.fr


	1. Introduction
	2. Preliminaries
	3. Main results
	References