EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 15, No. 1, 2022, 144-157
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Periodic Solution of Caputo-Fabrizio Fractional
Integro–differential Equation with Periodic and Integral

Boundary Conditions

Ava Sh. Rafeeq

Department of Mathematics, Faculty of Science , University of Zakho, Duhok, Iraq

Abstract. In this paper, we study a new approach of investigation of existence, uniqueness and
stability of the periodic solution of the nonlinear fractional integro-differential equation of type
Caputo-Fabrizio fractional derivative with the initial condition, periodic boundary conditions, and
integral boundary conditions by using successive approximations method and Banach fixed point
theorem. Finally, some examples are present to illustrate the theorems.

2020 Mathematics Subject Classifications: 34A08, 26A33, 34G20, 34C25, 45J05

Key Words and Phrases: Caputo- Fabrizio fractional derivative, integro-differential equation,
periodic and integral boundary conditions, Periodic solutions, successive approximation method,
Banach fixed point theorem

1. Introduction

Fractional differential equations have been recognized in the last decade as important
tools to describe the mathematical modeling of processes in the fields of physics, chem-
istry, engineering, statistics, aerodynamics, control theory, signal and image processing,
etc.[10, 11, 13]. On the other hand, we observe periodic motions in every field of science
and everywhere in real life [6]. The theory and applications of the fractional differential
equations have recently been addressed by several researchers for a variety of problems,
which we refer the reader to [1, 2, 4]. We mention here some of these definitions, such
as Riemann-Liouville, Hadamard, Grünwald-Letnikov, Weyl, Riesz, Erdélyi-Kober, and
Caputo. Compared with an integer order, a significant feature of a fractional order dif-
ferential operator appeared in its hereditary property. In other words, when we describe
a process by a fractional operator, we predict the future state by its current as well as its
past states [14, 16].

However, the new definition suggested by Caputo and Fabrizio [5], which has all the
characteristics of the old definitions, assumes two different representations for the tem-
poral and spatial variables.They claimed that the classical definition given by Caputo

DOI: https://doi.org/10.29020/nybg.ejpam.v15i1.4247

Email address: ava.rafeeq@uoz.edu.krd (A. S. Rafeeq)

http://www.ejpam.com 144 © 2022 EJPAM All rights reserved.



A. S. Rafeeq / Eur. J. Pure Appl. Math, 15 (1) (2022), 144-157 145

appears to be particularly convenient for mechanical phenomena, related to plasticity,
fatigue, damage, and with electromagnetic hysteresis. The main advantage of the Caputo-
Fabrizio approach is that the boundary conditions of the fractional differential equations
with Caputo-Fabrizio derivatives admit the same form as for the integer-order differential
equations. On the other hand, the Caputo-Fabrizio fractional derivative has many signifi-
cant properties, such as its ability in describing matter heterogeneities and configurations
with different scales [12, 20].

In [21], we have the analytic solutions of a viscous fluid with the Caputo and Caputo-
Fabrizio fractional derivatives. In [8], the authors used the fractional derivative with
a nonsingular kernel to model a Maxwell fluid and found semianalytical solutions. In
[22], we found a comparison approach of two latest fractional derivatives models, namely,
Atangana-Baleanu and Caputo- Fabrizio, for a generalized Casson fluid and obtained exact
solutions. Due to the abovementioned applications, the existence of solutions for nonlinear
differential equations is an attractive research topic and has been studied using different
techniques of nonlinear analysis [9, 18]. One of the most important theorems in ordinary
differential equations is Picard’s existence and uniqueness theorem. This theorem, which
is applied on first-order ordinary differential equations, can be generalized to establish
existence and uniqueness results for both higher-order ordinary differential equations and
systems of differential equations [3, 7, 15, 17].

In this paper, we investigate the existence and approximate periodic solution of the
following nonlinear fractional integro-differential equation:

CF
0 Dα

t (u(t)) = h

(
t, u(t),

∫ a(t)

0
g(s, u(s))ds

)
(1.1)

such that t ∈ J = [0, T ], with the initial condition u(0) = u0, where
CF
0 Dα

t denotes
the Caputo-Fabrizio fractional derivative (α ∈ (0, 1]). We extend Picard’s theorem to this
problem, and by the successive approximation method, an iterative process is provided to
obtain the periodic solution.

2. Preliminaries

In this section, we recall some notations and definitions which are needed throughout
this paper. Further, some lemmas and theorems are stated as preparations for the main
results. First, in the following, we provide some basic concepts and definitions in connec-
tion with the new Caputo-Fabrizio derivative.
Let H1(a, b) = {g|g ∈ L2(a, b), g′ ∈ L2(a, b)}, where L2(a, b) is the space of square inte-
grable functions on the interval (a, b).

Definition 1. [10] For a function g : (0,∞) → R, the Caputo derivative of order α > 0
of g is defined by

t
0D

αg(t) =
1

Γ(n− α)

∫ t

0
(t− s)n−α−1g(n)(s)ds (2.1)



A. S. Rafeeq / Eur. J. Pure Appl. Math, 15 (1) (2022), 144-157 146

where n = [α] + 1 and [α] denotes the integer part of α, and Γ(.) denotes the Gamma
function, i.e., Γ(z) =

∫∞
0 e−ttz−1dt

Definition 2. [10] Let g be a function which is defined almost everywhere a.e on [a, b],
for α > 0, we define

b
aD

−αf =
1

Γ(α)

∫ b

a
(b− t)a−1g(t)dt (2.2)

provided that the integral (Lebesgue)exists.

Definition 3. [5] Let g be a given function in H1(a, b). The Caputo-Fabrizio derivative
of fractional order α ∈ (0, 1) is defined as

CF
a Dα

t (g(t)) =

(
N(α)

1− α

)∫ t

a
g′(x) exp

[
−α

t− x

1− α

]
dx (2.3)

where N(α) is a normalization function. Also, if a certain function g does not satisfy in
the restriction g ∈ H1(a, b), then its fractional derivative is redefined as

CF
a Dα

t (g(t)) =
αN(α)

1− α

∫ t

a
(g(t)− g(x)) exp

[
−α

t− x

1− α

]
dx (2.4)

Clearly, if one sets σ = (1− α)/α ∈ (0,∞) and α = 1/(1 + σ) ∈ (0, 1), then the Caputo-
Fabrizio definition becomes

CF
a De

t (g(t)) =
N(σ)

σ

∫ t

a
g′(x) exp

[
− t− x

σ

]
dx (2.5)

where N(0) = N(∞) = 1, and

lim
σ→0

exp

[
− t− x

σ

]
= δ(x− t). (2.6)

Also, the fractional derivative of order (n+α) when n ≥ 1 and α ∈ [0, 1] is defined by the
following

CF
a D

(a+n)
t (g(t)) = αCFD

(a)
t

(
D

(n)
t g(t)

)
(2.7)

Definition 4. [5] Let g ∈ H1(a, b), then its fractional integral of an arbitrary order is
defined as follows:

αC
x
t (g(t)) =

2(1− α)

(2− α)N(α)
g(t) +

2α

(2− α)N(α)

∫ t

a
g(s)ds, t ≥ 0 (2.8)

It is dear, in view of the abowe definition, that the α th Caputo-Fabrizio derivative of
function g is average between g and its first-order integral. Therefore,

2(1− α)

(2− α)N(α)
+

2α

(2− α)N(α)
= 1 (2.9)

So, we arrive at the following

N(α) =
2

2− α
, 0 ≤ α ≤ 1 (2.10)



A. S. Rafeeq / Eur. J. Pure Appl. Math, 15 (1) (2022), 144-157 147

Definition 5. The periodic solution of the fractional integro-differential equation (1.1),
with initial condition u(0) = u0 and periodic boundary condition u(0) = u(T ) are defining
the following integral equation

u (t, u0) = u0 +
2(1−α)

(2−α)N(α)h(t, u(t),
∫ a(t)
0 g(s, u(s))ds)

− ( 2(1−α)
(2−α)N(α)

1
T

∫ T
0 h(s, u(s),

∫ a(s)
0 g(τ, u(τ))dτ)ds+ 2α

(2−α)N(α)∫ t
0 (h(s, u(s),

∫ a(s)
0 g(τ, u(τ))dτ)− 1

T

∫ T
0 h(s, u(s),

∫ a(s)
0 g(τ, u(τ))dτ)ds)ds (2.11)

For all t ∈ J .

Lemma 1. [19] Let g(t) be a vector function which is defined in the interval 0 ≤ t ≤ T ,
then: ∣∣∣∣∫ t

0

(
g(s)− 1

T

∫ T

0
g(s)ds

)
ds

∣∣∣∣ ≤ β(t)M, (2.12)

where M = maxt∈[0,T ] |g(t)| and β(t) = 2t
(
1− t

T

)
,maxt∈[0,T ] |β(t)| ≤ T

2 .
The proof follows directly from the estimate:∣∣∣∣∫ t

0

(
g(s)− 1

T

∫ T

0
g(s)ds

)
ds

∣∣∣∣ ≤ (1− t

T

)∫ t

0
|g(s)|ds+ t

T

∫ T

t
|g(s)|ds

≤ β(t)M

Theorem 1. [10] (Banach fixed point theorem). Let (E, ∥.∥) be a Banach space and
P : E → E be a contraction mapping i.e. Lipchitz continuous with Lipchitz constant
L ∈ [0, 1). Then φ ∈ E has a unique fixed point.

3. Conditions for Convergence of Successive Approximation

Some conditions are needed for investigate of the successive approximation for pe-
riodic solution of the problem (1.1) with u(0) = u0, suppose that the functions h ∈
C ([0, T ]×D1 ×D2,R),g ∈ C ([0, T ]×D1,R) , D1 and D2 are compact subset of R, a(t) is
continuous functions on [0, T ], moreover define |.| = maxt∈[0,T ] |.|, and satisfies the follow-
ing hypothesis.

H1 There exist positive constants M,L, k1, k2,and L1, such that

|h(t, u, z)| ≤ M (3.1)

|g(t, u)| ≤ L (3.2)

|h (t, u1, z1)− h (t, u2, z2)| ≤ k1 |u1 − u2|+ k2 |z1 − z2| (3.3)

|g (t, u1)− g (t, u2)| ≤ L1 |u1 − u2| (3.4)

where zi =
∫ a(t)
0 g (s, ui(s)) ds and for all t ∈ [0, T ], u, u1, u2 ∈ D1 and zi ∈ D2, i = 1, 2

H2 : There exist positive constants aT , such that for t ∈ [0, T ],

|a(t)| ≤ aT (3.5)



A. S. Rafeeq / Eur. J. Pure Appl. Math, 15 (1) (2022), 144-157 148

Define the non-empty set

Dh = D1 −M1 (3.6)

where

M1 =

(
2(1− α) +

αT

2

)
M

Furthermore, we suppose that the following condition is valid:

Λ =

(
2(1− α) +

αT

2

)
(k1 + aTL1k2) < 1 (3.7)

4. Main Results

Our main results separate to the following parts:

4.1. Approximation of Periodic Solution of (1.1)

In this section, we study the periodic approximation solutions of nonlinear fractional
integro-differential equations (1.1) with u(0) = u0. In the beginning, we define the follow-
ing sequence of functions {um+1}∞m=0 given by the iterative formulas

um+1 (t, u0) = u0 +
2(1−α)

(2−α)N(α)h(t, um(t),
∫ a(t)
0 g (s, um(s)) ds)

− 2(1−α)
(2−α)N(α)

1
T

∫ T
0 h(s, um(s),

∫ a(s)
0 g (τ, um(τ))dτ) ds+ 2α

(2−α)N(α)∫ t
0 (h(s, um(s),

∫ a(s)
0 g (τ, um(τ)) dτ)− 1

T

∫ T
0 h(s, um(s),

∫ a(s)
0 g (τ, um(τ)) dτ)ds)ds (4.1)

For all t ∈ J, u0(t) = u0,m = 0, 1, 2, . . .,
then will be introduced by the following theorems.

Theorem 2. If the nonlinear fractional integro-differential equation (1.1) with u(0) = u0
satisfy the conditions H1, and H2, then the sequence of functions (4.1), which are periodic
in t of period T , converges uniformly as m → ∞ on the domain:-

(t,u0) ∈ [0, T ]×D1 (4.2)

to the limit functions uθ defined on the domain (4.2) which is periodic in t of period
T and satisfies the following integral equations:

u (t, u0) = u0 +
2(1−α)

(2−α)N(α)h(t, u(t),
∫ a(t)
0 g(s, u(s))ds)

− ( 2(1−α)
(2−α)N(α)

1
T

∫ T
0 h(s, u(s),

∫ a(s)
0 g(τ, u(τ))dτ)ds+ 2α

(2−α)N(α)∫ t
0 (h(s, u(s),

∫ a(s)
0 g(τ, u(τ))dτ)− 1

T

∫ T
0 h(s, u(s),

∫ a(s)
0 g(τ, u(τ))dτ)ds)ds (4.3)

on the domain (4.2), provided that

|u (t, u0)− um+1 (t, u0)| ≤ Λm(E− Λ)−1M1 (4.4)



A. S. Rafeeq / Eur. J. Pure Appl. Math, 15 (1) (2022), 144-157 149

for all m ≥ 0, u0 ∈ D, and t ∈ J
Proof. Setting m = 0 in the sequence of functions (4.1) and by using Lemma 1, we have

|u1 (t, u0)− u0| ≤
(

4(1− α)

(2− α)N(α)
+

2α

(2− α)N(α)
β(t)

)
M

≤
(
2(1− α) +

αT

2

)
M = M1

for all t ∈ [0, T ], u0 ∈ Dh we get u1 (t, u0) ∈ D1. Thus by mathematical induction, we find
that

|um (t, u0)− u0| ≤ M1 (4.5)

mean that for all t ∈ [0, T ], u0 ∈ Dh we get um (t, u0) ∈ D1,m = 0, 1, 2, ...
Now, we claim that the sequences of functions (4.1) are uniformly convergent on the
domain (4.2). By the inequalities (3.3)-(3.5), we obtain

|um+1 (t, u0)− um (t, u0)| ≤
(
2(1− α) +

αT

2

)
(k1 + aTL1k2)

|um (t, , u0)− um−1 (t, , u0)|
= Λ |um (t, , u0)− um−1 (tr, u0)| (4.6)

By mathematical induction, we obtain that

|um+1 (t,u0)− um (t,u0)| ≤ Λm |u1 (t,u0)− u0| (4.7)

Now from m = 1, 2, . . . and p ≥ 1, we find that

|um+p (t,u0)− um (t,u0)| ≤ Λm(1− Λ)−1

((
2(1− α) +

αT

2

)
M

)
≤ Λm(1− Λ)−1M1, (4.8)

for all t ∈ [0, T], u0 ∈ Dh.
Since Λ =

(
2(1− α) + αT

2

)
(k1 + aTL1k2) < 1 and limm→∞ Λm = 0, so that the right side

of (4.8) tends to zero. Therefore the sequence of functions um (t, u0) ,m = 1, 2, 3, . . . is
converges uniformly on the domain (4.2) to the limit function u (t,u0) which is defined on
the same domain. Let

lim
m→∞

um (t, u0) = uθ (t, u0) (4.9)

Since the sequence of functions (4.1) are periodic in t of period T, then the limiting function
uθ (t, u0 ) is also periodic in t of period T. By using the relation (4.9) and proceeding in
(4.1) to limit, when m → ∞, it is converging that the limiting function u (t, u0) is the
periodic solution of the integral equation (4.3).

Theorem 3. If all assumptions of the Theorem 2 are satisfy, then u (t,u0 ) is a unique
solution of the problem ( 1.1 ) with u(0) = u0.



A. S. Rafeeq / Eur. J. Pure Appl. Math, 15 (1) (2022), 144-157 150

Proof. Assume that û (t, u0) is another solution of the problem (1.1) with u(0) = u0,
as follows

û (t, u0) = u0 +
2(1−α)

(2−α)N(α)h(t, û(t),
∫ a(t)
0 g(s, û(s))ds)

− ( 2(1−α)
(2−α)N(α)

1
T

∫ T
0 h(s, û(s),

∫ a(s)
0 g(τ, û(τ))dτ)ds+ 2α

(2−α)N(α)∫ t
0 (h(s, û(s),

∫ a(s)
0 g(τ, û(τ))dτ)− 1

T

∫ T
0 h(s, û(s),

∫ a(s)
0 g(τ, û(τ))dτ)ds)ds (4.10)

Now, the difference between the two solutions u (t,u0) and û (t, u0),for all t ∈ [0, T]
and u0 ∈ Dh, hence, by the inequalities (3, 3)− (3.5), we get

|u (t, u0)− û (t, u0)| ≤
(
2(1− α) +

αT

2

)
(k1 + aY L1k2) |u (t,u0)− û (t, u0)|

≤ Λ |u (tru0)− û (t, u0)| (4.11)

By mathematical induction, we find that

|u (t, u0)− û (t,u0)| ≤ Λm |u (t, u0)− û (t, u0)| (4.12)

From the condition ( 3.7 ), shows that the solution u (t,u0) = û (t,u0), thus u (t, u0 ) is a
unique periodic solution on the domain (4.2).

4.2. Existence of Periodic Solutions of (1.1)

The problem of the existence of the periodic solution for the problem (1.1) with u(0) =
u0 is uniquely connected with the existence of the zeros of the functions:-

µ (0,u0) =
1

T

∫ T

0
h

(
s, u(s),

∫ a(s)

0
g(τ, u(τ))dτ

)
ds (4.13)

Also, we define the sequences of functions µm (0, u0) are approximately determined by the
following:

µm (0,u0) =
1

T

∫ T

0
h

(
s, um(s),

∫ a(s)

0
g (τ, um(τ)) dτ

)
ds (4.14)

Theorem 4. If the hypotheses and all the conditions of the theorem 2 are given, the
following inequalities are satisfied:-

|µ (0, u0)− µm (0,u0)| ≤ (k1 + aTL1k2) Λ
m(1− Λ)−1M1 (4.15)

holds for all m ≥ 0

Proof. From equations (4.13) to (4.14), we obtain that

|µ (0, u0)− µm (0,u0)| ≤ (k1 + aTL1k2) |u (t,u0)− um (t,u0)|
≤ (k1 + arL1k2) Λ

m(1− Λ)−1M1 (4.16)



A. S. Rafeeq / Eur. J. Pure Appl. Math, 15 (1) (2022), 144-157 151

The inequality (4.15) is hold for all m ≥ 0.

Theorem 5. Let the function h(s, u(s), z(t)) be defined on the intervals [c, d] on R and
periodic in t of period T , suppose that for all m ≥ 0, then the sequences of the functions
µm (0, u0) which are defined in (4.14) satisfy the inequalities:-

minu0∈[c,d] µm (0, u0) ≤ − (k1 + aτL1k2) Λ
m(1− Λ)−1M1

maxu0∈[c,d] µm (0,u0) ≥ (k1 + arL1k2) Λ
m(1− Λ)−1M1

}
(4.17)

Then the problem (1.1) has a periodic solution u (t, u0 ) such that u0 ∈ [c,d] = [c +M1, d−M1]

Proof. Let u1 and u2 be any points belonging to the intervals [c, d], such that

µm (0,u1) = minu0∈[c,d] µm (0, u0)

µm (0,u2) = maxu0∈[c,d] µm (0, u0)

}
(4.18)

By using inequalities (4.15) to (4.18),, the following are obtained:-

µ (0, u1) = µm (0, u1) + (µ (0, u1)− µm (0,u1)) < 0
µ (0, u2) = µm (0, u2) + (µ (0,u2)− µm (0, u2)) > 0

}
(4.19)

and from the continuity of the functions µ (0, u1) , µ (0, u2) and the inequalities (4.19), then
the isolated singular points u0 ∈ [c, d] exist such that µ

(
0, u0

)
= 0. This means that (1.1)

has a periodic solution u (t, u0).

4.3. Stability of Periodic Solution of (1.1)

In this section, we investigate the stability or periodic solution of (1.1).

Theorem 6. Let the function µ (0, u0) be defined by the equation (4.13) where u (t,u0) is
a limit of the sequence of the function (4.1), then the following inequalities yield:-

|µ (0, u0)| ≤ M (4.20)

and ∣∣µ (0,u10)− µ
(
0, u20

)∣∣ ≤ F2F3

∣∣u10 − u20
∣∣ (4.21)

where

F1 = 2(1− α) +
αT

2
, F2 = k1 + aTL1k2, F3 = (1− F1F2)

−1

Proof From the properties of the function u (t, u0) as in the Theorem 2, the function
µ (0, u0) , u0 ∈ D is continuous and bounded by 1−α

αT gT + M in the domain (4.2). From
(4.13), we obtained that

|µ (0, u0)| ≤
1

T

∫ T

0

∣∣∣∣∣h
(
s, u(s),

∫ a(s)

0
g(τ, u(τ))dτ

)∣∣∣∣∣ ds ≤ M (4.22)



A. S. Rafeeq / Eur. J. Pure Appl. Math, 15 (1) (2022), 144-157 152

Next, from inequality (4.13), we get∣∣µ (0,u10)− µ
(
0,u20

)∣∣ ≤ (k1 + aTL1k2)
∣∣u (t, u10)− u

(
t, u20

)∣∣
≤ F2

∣∣u (t, u10)− u
(
t, u20

)∣∣ (4.23)

where the functions u
(
t,u10

)
and u

(
t, u20

)
are solutions of the integral equation:-

u
(
t,uk0

)
= uk0 +

2(1−α)
(2−α)N(α)h

(
t, u
(
t,uk0

)
,
∫ a(t)
0 g

(
s, u
(
s, uk0

))
ds
)
−

2(1−α)
(2−α)N(α)

1
T

∫ T
0 h

(
s, u

(
s, uk0

)
,
∫ a(s)
0 g

(
τ, u

(
τ,uk0

))
dτ
)
ds+

2α
(2−α)N(α)

∫ t
0 (h(s, u

(
s, uk0

)
,
∫ a(s)
0 g

(
τ, u

(
τ,uk0

))
dτ)−

1
T

∫ T
0 h(s, u

(
s, uk0

)
,
∫ a(s)
0 g

(
τ, u

(
τ,uk0

))
dτ)ds)ds (4.24)

where k = 1, 2, from (4.24), we get∣∣u (t, u10)− u
(
t, u20

)∣∣ ≤| u10 − u20
]
+ 4(1−α)

(2−α)N(a) (k1 + aTL1k2)
∣∣u (t,u10)− u

(
t, u20

)∣∣+
αT

(2− α)N(α)
(k1 + aTL1k2)

∣∣u (t, u10)− u
(
t, u20

)∣∣ (4.25)

Therefore, we obtain that∣∣u (t,u10)− u
(
t, u20

)∣∣ ≤ ∣∣u10 − u20
∣∣+ (2(1− α) + αT

2

)
(k1 + aTL1k2)

∣∣u (t,u10)− u
(
t, u20

)∣∣
≤
∣∣u10 − u20

∣∣+ F1F2

∣∣u (t,u10)− u
(
t, u20

)∣∣ (4.26)

From equations (4.26), we have∣∣u (t, u10)− u
(
t, u20

)∣∣ ≤ F3

∣∣u10 − u20
∣∣ (4.27)

Substitutes (4.27) in (4.23), we get that (4.21)

Remark 1. [22]. Theorem 6 confirms the stability of the solution of the problem (1.1),
when a slight change happens in the points u0, then a slight change will happen in the
function µ (0, u0).

4.4. Existence and uniqueness of periodic Solution of (1.1) with integral
boundary condition

In this section, we investigate the periodic solution of the problem (1.1) with integral
boundary conditions:

u(0)− u(T ) =

∫ T

0
H(u(s))ds (4.28)

Where the function H(u(s)) defined and continuous on are compact subset of R and
periodic on t of periodic T.



A. S. Rafeeq / Eur. J. Pure Appl. Math, 15 (1) (2022), 144-157 153

Theorem 7. All assumptions of the Theorem 2 are satisfy, and the function H(u(s))
satisfies

|H (u1)−H (u2)| ≤ L2 |u1 − u2| (4.29)

then the problem (1.1) and integral boundary condition(4.28) has unique solution if

Q =

(
2(1− α) +

αT

2

)
(k1 + aTL1k2) +

(1− α+ αT )

α
L2 < 1 (4.30)

Proof. We define an operator P : C[0, T ] → C[0, T ]

P (u(t)) = u0 − (1−α+αt)
αT

∫ T
0 H(u(s))dt+

2(1−α)
(2−α)N(a) [h(t, u(t),

∫ a(t)
0 g(s, u(s))ds)− 1

T

∫ T
0 h(s, u(s),

∫ a(s)
0 g(τ, u(τ))dτ)ds]+

2α
(2−α)N(α)

∫ t
0 (h(s, u(s),

∫ a(s)
0 g(τ, u(τ))dτ)− 1

T

∫ T
0 h(s, u(s),

∫ a(s)
0 g(τ, u(τ))dτ)ds)ds

Therefore, we get

|P (u(t))− P (w(t))| = ((2(1− α) +
αT

2
) (k1 + aTL1k2) +

(1− α+ αT )

α
L2)|u(t)− w(t)|

From (4.30), the operator P satisfies contraction mapping, hence the problem (1.1) and
(4.28) has unique solution.

Theorem 8. If the hypotheses and all the conditions of the theorem 2 and the inequality
(4.29) are given, the following inequalities are satisfied:-

|σ (0,u0)− σm (0,u0)| ≤
(
k1 + aΥL1k2 +

L1

α

)
Qm(1−Q)−1M3 (4.31)

where

σm (0, u0) =
1

T

∫ T

0
h

(
s, um(s),

∫ a(s)

0
g (τ, um(τ)) dτ

)
ds

+
(2− α)N(α)

2αT

∫ T

0
H (um(s)) ds (4.32)

holds for all m ≥ 0, here

M3 = M1 +
(1− α+ αT )

a
M2 (4.33)

and
M2 ≥ |H(u(t))| (4.34)

The proof of this theorem is direct.



A. S. Rafeeq / Eur. J. Pure Appl. Math, 15 (1) (2022), 144-157 154

Theorem 9. Let the functions h(s, u(s), z(t)) and H(u(t)) be defined on the intervals
[c1, d1] on R and periodic in t of period T, suppose that for all m ≥ 0, then the sequences
of the functions σm (0,u0) which are defined in (4.32) satisfy the inequalities:-

minu0∈[c1,d1] σm (0, u0) ≤ −
(
k1 + aTL1k2 +

L2
α

)
Qm(1−Q)−1M3

maxu0∈[c1,d11 σm (0, u0) ≥
(
k1 + aTL1k2 +

L2
α

)
Qm(1−Q)−1M3

}
(4.35)

Then the problem (1.1) with (4.28) has a periodic solution such that u0 ∈ [c1 +M3, d1 −M3]
where M3 defined in (4.33).

This theorem’s proof was similar to that of theorem 5.

Theorem 10. Let the function σ (0,u0) be defined by the equations (4.32), then the fol-
lowing inequalities yield:-

|σ (0, u0)| ≤ M +
M2

α
(4.36)

and ∣∣σ (0,u10)− σ
(
0, u20

)∣∣ ≤ E2E3

∣∣u10 − u20
∣∣ (4.37)

where

E1 = 2(1− α) +
αT

2
, E2 = k1 + aTL1k2 +

L1

α
, E3 = (1− E1E2)

−1

The proof of this theorem was similar to the proof of theorem 6.

5. Examples

In this section contains two example to illustrate the previous theorems.
Example 5.1. Consider the following fractional integro-differential equation

CF
0 D0.7

t (u(t)) =
1

et + 5
u(t) +

∫ t2

0

1

2(s+ 2)3
sin(u(s))ds (5.1)

such that t ∈ J = [0, 2], with the initial condition u(0) = 1, where CF
0 Dα

t denotes the
fractional Caputo-Fabrizio derivative (a = 0.7 ∈ (0, 1]). Here T = 2, , a(t) = t2,

h(t, u(t), z(t)) =
1

et + 5
u(t) +

∫ t2

0

1

2(s+ 2)3
sin(u(s))ds

g(t, u(t)) =
1

2(s+ 2)3
sin(u(s))

We obtain that k1 = 0.2, k2 = 1, aT = 4, L1 = 0.0625,
so that Λ =

(
2(1− α) + αr

2

)
(k1 + aTL1k2) = 0.585 < 1.

Therefore, by Theorem 2 and Theorem 3, the fractional differential equation (5.1) has



REFERENCES 155

exactly one periodic solution.

Example 5.2. Consider the fractional integro-differential equation (5.1) with integral
boundary conditions

u(0)− u(1) =

∫ 1

0

1

2
cos(u(t))dt (5.2)

such that t ∈ (0, 1], where CF
0 Dα

t denotes the fractional Caputo-Fabrizio derivative (α =
0.7 ∈ [0, 1]). Here T = 1, , a(t), h(t, u(t), z(t)), g(t, u(t)) are defined in previous example
and

H(u(t)) =
1

2
cos(u(t))

we obtain that k1 = 0.2, k2 = 1, aT = 1, L1 = 0.0625, and L2 = 0.5,
so that

Q =

(
2(1− α) +

aT

2

)
(k1 + aTL1k2) +

(1− α+ αT )

a
L2 = 0.9637 < 1

Therefore, by Theorem 7, the boundary value problem ( 5.1 ) and ( 5.2 ) has exactly one
periodic solution.

6. Conclusion

In this paper,we studied the existence, uniqueness, and stability of periodic solutions
of nonlinear fractional integro-differential equation (1.1) where CF

0 Dα
t denotes the frac-

tional Caputo-Fabrizio derivative with the initial condition, periodic boundary conditions,
and integral boundary conditions by using technique successive approximations method
and Banach fixed point theorem. Here conclude that we could investigate the existence,
uniqueness, and stability of periodic solution of Caputo-Fabrizio fractional differential
equation with integral boundary condition

Au(0)−Bu(T ) =
m∑
i=1

Ci

∫ T

0
Hi(u(s))ds,

where A,B and Ci, i = 1, 2, ...,m are constants, and Hi, i = 1, 2, ...,m are defined and
continuous functions on [0, T ]. Finally, some examples are presented to illustrate the
previous theorems.

References

[1] M. Belmekki, J. J. Nieto, and R. R. Lopez. Existence of periodic solution for a
nonlinear fractional differential equation. Boundary Value Problems, 2009:1–18, 2009.

[2] F. E. Bouzenna, M. T. Meftah, and M. Difallah. Application of the caputo–fabrizio
derivative without singular kernel to fractional schrodinger equations. Pramana, 94,
2020.



REFERENCES 156

[3] R. N. Butris and A. Sh. Rafeeq. Solutions for nonlinear system of fractional inte-
gro–differential equations with non-separated integral coupled boundary conditions.
International Journal of Advanced Trends in Computer Science and Engineering, 9,
2020.

[4] A. Cabada and T. Kisela. Existence of positive periodic solutions of some nonlinear
fractional differential equations. Communications in Nonlinear Science and Numeri-
cal Simulation, 50:51–67, 2017.

[5] M. Caputo and Fabrizio. A new definition of fractional derivative without singular
kernel. Progress in Fractional Differentiation and Applications, 2:73–85, 2015.

[6] M. Farkas. Periodic motions. Springer-Verlag, New York, 1994.

[7] M. Feckan and K. Marynets. Approximation approach to periodic bvp for mixed
fractional differential systems. J. Comput. Appl. Math., 339:208–217, 2018.

[8] M. A. Imran, M. B. Riaz, N. A. Shah, and A. A. Zafar. Boundary layer flow of mhd
generalized maxwell fluid over an exponentially accelerated infinite vertical surface
with slip and newtonian heating at the boundary. Results in Physics, 8:1061–1067,
2018.

[9] K. Jothimani, K. Kaliraj, Z. Hammouch, and C. Ravichandran. New results on con-
trollability in the framework of fractional integrodifferential equations with nondense
domain. The European Physical Journal Plus, 134, 2019.

[10] A. A. Kilbas, H. M Srivastava, and J. J. Trujillo. Theory and Applications of Frac-
tional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier,
Amsterdam, 2006.

[11] V. Lakshmikantham, S. Leela, and V. Devi. Theory of Fractional Dynamic Systems.
Cambridge Academic Publishers, Cambridge, 2009.

[12] J. Losada and J. J. Nieto. Properties of the new fractional derivative without singular
kernel. Progress in Fractional Differentiation and Applications, 1:87–92, 2015.

[13] K. S. Miller and B. Ross. An Introduction to the Fractional Calculus and Fractional
Differential Equations. Wiley, New York, 1993.

[14] Sh. A. Murad and S. B. Hadid. Existence and uniquness theorem for fractional
differential equation with integral boundary condition. Journal of Fractional Calculus
and Applications,, 3:1–9, 2012.

[15] Sh. A. Murad and A. Sh. Rafeeq. Existence of solutions of integro-fractional differ-
ential equation when α ∈ (2, 3] through fixed point theorem. J. Math. Comput. Sci.,
11:6392–6402, 2021.



REFERENCES 157

[16] Sh. A. Murad, H. J. Zekri, and S. Hadid. Existence and uniqueness theorem of
fractional mixed volterra-fredholm integrodifferential equation with integral boundary
conditions. International Journal of Differential Equations, 2011:1–16, 2011.

[17] A. Sh. Rafeeq. On the parametrization of nonlinear impulsive fractional inte-
gro–differential system with non-separated integral coupled boundary conditions. Sci-
ence Journal of University of Zakho, 8:160–168, 2020.

[18] C. Ravichandran, K. Logeswari, and F. Jarad. New results on existence in the
framework of atangana–baleanu derivative for fractional integro-differential equations.
Chaos, Solitons and Fractals, 125:194–200, 2019.

[19] M. Ronto and A. M. Samoilenko. Numerical-Analytic Methods in the Theory of
Boundary-Value Problems. World Scientific, Singapore, 2000.

[20] K. Salim and S. Abbas. Boundary value problem for implicit caputo–fabrizio
fractional differential equations. International Journal of Difference Equations,
15:493–510, 2020.

[21] N. A. Shah, M. A. Imran, and F. Miraj. Exact solutions of time fractional free
convection flows of viscous fluid over an isothermal vertical plate with caputo and
caputo-fabrizio derivatives. Journal of Prime Research in Mathematics, 13:56–74,
2017.

[22] N. A. Sheikh, F. Ali, M. Saqib, I. Khan, and S. A. A. Jan. A comparative study of
atangana-baleanu and caputo- fabrizio fractional derivatives to the convective flow of
a generalized casson fluid. The European Physical Journal Plus, 132, 2017.