EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 15, No. 2, 2022, 726-735
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Existence and Uniqueness of Solutions for Nonlinear
Fractional Integro-Differential Equations with Nonlocal

Boundary Conditions

M.J. Mardanov1,2 Y.A. Sharifov1,2,∗, H.N. Aliyev3

1 Institute of Mathematics and Mechanics, ANAS, Baku, Azerbaijan
2 Baku State University Baku, Azerbaijan
3 Baku Engineering University, Khirdalan City, Azerbaijan

Abstract. In this paper, the existence and uniqueness of solutions of fractional integro-differential
equations with nonlocal boundary conditions is investigated. We establish the existence of solution
via Krasnoselskii fixed point theorem; however, the uniqueness results are obtained by applying
the contraction mapping principle.

2020 Mathematics Subject Classifications: AMS 34B37, 34B15

Key Words and Phrases: Nonlocal boundary conditions, Caputo fractional derivative, existence,
uniqueness, fixed point

1. Introduction

Fractional differential equations have been an important tool to describe many prob-
lems and processes in different fields of science. In fact, fractional models are more realistic
than the classical models. Fractional differential equations appear in many fields such as
physics, economics, image processing, blood flow phenomena, aerodynamics, and so on.
For more details about fractional differential equations and their applications, we provide
the following references [10–12, 14–17, 19–21, 25, 27, 33]. Recently, fractional integro-
differential equations were investigated by many researchers in different problems, and a
lot of papers were published in this matter (see, for example, [1, 6, 7]). Furthermore, many
boundary conditions were considered for the fractional-order integro-differential equations;
some of these conditions are the classical, periodic, antiperiodic, nonlocal, multipoint, and
the integral boundary conditions. Boundary value problems with integral boundary con-
ditions constitute a very interesting and important class of problems. They include two,

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v15i2.4366

Email addresses: sharifov22@rambler.ru (Y.A. Sharifov), misirmardanov@yahoo.com (M.J. Mardanov),
hualiyev@beu.edu.az (H.N. Aliyev)

https://www.ejpam.com 726 © 2022 EJPAM All rights reserved.



M.J. Mardanov, H.N. Aliyev, Y.A. Sharifov / Eur. J. Pure Appl. Math, 15 (2) (2022), 726-735 727

three, multipoint and nonlocal boundary value problems as special cases. Integral bound-
ary value problems occur in the mathematical modeling of variety of physics processes
and have recently received considerable attention. For some recent work on boundary
value problems with integral boundary conditions we refer to [2, 3, 8, 9, 18, 30] and the
references cited therein.

On the other hand, many papers have considered the no separated boundary conditions
as they are a very important class of boundary value conditions (we refer the readers to
[4, 5, 13, 22–24, 26, 28, 29, 31, 32]). Motivated by the above discussion, in this paper,
we establish the existence and uniqueness of solutions for a class of fractional integro-
differential equations with no separated boundary value conditions as follows:

In this paper, we study existence and uniqueness of nonlinear fractional integro- dif-
ferential equations of the type

cDα
0+x (t) = f (t, x (t) , φx (t) , ψx (t)) , for t ∈ [0, T ] (1)

subject initial-point and integral boundary conditions

Ax (0) +

∫ T

0
n (t)x (t) dt = C (2)

where 0 < α < 1, cDα
0+ is the Caputo fractional derivatives, A ∈ Rn×n and n (t) :

[0, T ] → Rn×n are given matrices and detN ̸= 0, N =
(
A+

∫ T
0 n (t) dt

)
, φx (t) =∫ t

0 µ (t, s)x (s) ds, ψx (t) =
∫ T
0 γ (t, s)x (s) ds, where µ, γ : [0, T ] × [0, T ] → Rn×n, with

µ0 = max
t,s∈[0,T ]

∥µ (t, s)∥ , γ0 = max
t,s∈[0,T ]

∥γ (t, s)∥ .

The rest of the paper is organized as follows. In Section 2, we give some notations,
recall some concepts, and introduce a concept of a continuous solution for our problem.
In Section 3, we give two main results: the first result based on the Krasnoselskii’s fixed
point theorem and the second result based on the Banach contraction principle.

2. Preliminaries

In this section, we introduce notations, definitions, and preliminary facts that will be
used in the remainder of this paper. By C ([0, T ] ;Rn) we denote the Banach space of all
continuous functions from [0, T ] into Rn with the norm ∥x∥ = max {|x (t)| : t ∈ [0, T ]},
where |·| norm in Rn.

Definition 1. (see [28]). The Riemann–Liouville fractional integral of order α > 0of a
continuous function y : [0,∞) → R, is defined by

Jαy (x) =
1

Γ (α)

∫ x

0
(x− s)α−1 y (s) ds,

provided the right-hand side exists on (0,∞) , where Γ (·) is the Gamma function defined
for any complex number z as

Γ (z) =

∫ ∞

0
tz−1e−tdt.



M.J. Mardanov, H.N. Aliyev, Y.A. Sharifov / Eur. J. Pure Appl. Math, 15 (2) (2022), 726-735 728

Definition 2. (see [28]). The (right-sided) Riemann-Liouville fractional derivative is
defined by

RLDαy = (Dαy) (x) =
d

dxn
(
Jn−αy

)
(x) , x > 0,

where n = [α] + 1, [α]denotes the integer part of real number α, provided the right-hand
side is point-wise defined on (0,∞) .

The Riemann-Liouville fractional derivative is left-inverse (but not right-inverse) of
the Riemann-Liouville fractional integral, which is a natural generalization of the Cauchy
formula for the n-fold primitive of a function y. As to the initial value problems for
fractional differential equations with fractional derivatives in the Riemann-Liouville sense,
they should be given as (bounded) initial values of the fractional integral Jn−α and of its
integer derivatives of order k = 1, 2, ..., n− 1.

Definition 3. (see [28]). The Caputo fractional derivative of order α > 0 of a continuous
function y, is defined by

CDα
0+y = (Dαy) (x) =

(
Jn−αy(n)

)
(x) , n− 1 < α ≤ n, x > 0,

provided the right-hand side is point-wise defined on (a, ∞).

Obviously, this definition allows one to consider the initial-value problems for the
fractional differential equations with initial conditions that are expressed in terms of a
given number of bounded values assumed by the field variable and its derivatives of integer
order.

Remark 1. Under natural conditions on y (x), the Caputo fractional derivative becomes
the conventional integer order derivative of the function y (x) as α→ n.

Remark 2. (see [28]). Let α, β > 0 and n = [α] + 1; then the following relations hold:

cDα
0+t

β = Γ(β)
Γ(β−α) t

β−α, β > n,
cDα

0+t
k = 0, k = 0, 1, ..., n− 1.

Lemma 1. (see [28]). For α > 0, y (t) ∈ C ([0, T ])
⋂
L1 ([0, T ]), the homogeneous frac-

tional differential equation
cDα

0+y (t) = 0,

has a solution
y (t) = c0 + c1t+ c2t

2...+ cn−1t
n−1,

where ci ∈ R, i = 1, 2, ..., n− 1 and n = [α] + 1.

Lemma 2. (see [28]). Assume that y (t) ∈ C ([0, T ])
⋂
L1 ([0, T ]), with derivative of order

n that belongs to C ([0, T ])
⋂
L1 ([0, T ]); then

Iα0+
cDα

0+y (t) = y (t) + c0 + c1t+ c2t
2...+ cn−1t

n−1,

where ci ∈ R, i = 1, 2, ..., n− 1 and n = [α] + 1.



M.J. Mardanov, H.N. Aliyev, Y.A. Sharifov / Eur. J. Pure Appl. Math, 15 (2) (2022), 726-735 729

Lemma 3. (see [28]). Let p, q ≥ 0, f ∈ L1 ([0, T ]) . Then

Ip0+I
q
0+f (t) = Ip+q0+ f (t) = Iq0+I

p
0+f (t)

is satisfied almost everywhere on [0, T ]. Moreover, if f ∈ C ([0, T ]), then (4) is true for
all t ∈ [0, T ].

Lemma 4. (see [28]). If α > 0, f ∈ C ([0, T ]), then cDα
0+I

α
0+f (t) = f (t) for all t ∈ [0, T ] .

We have the following result which is useful in what follows.

Theorem 1. Let y ∈ C ([0, T ] ;Rn). Then the unique solution of the linear boundary value
problem { cDα

0+x (t) = y (t) ,

Ax (0) +
∫ T
0 n (t)x (t) dt = C

(3)

is given by

x (t) = N−1C +
1

Γ (α)

∫ t

0
(t− s)α−1 y (s) ds−

−N−1

Γ (α)

∫ T

0
n (t)

∫ t

0
(t− s)α−1 y (s) dsdt. (4)

Proof. Assume that x is a solution of the boundary value problem (3); then we have

x (t) = x (0) +
1

Γ (α)

∫ t

0
(t− s)α−1 y (s) ds, t ∈ [0, T ] ,

where x (0) is still an arbitrary constant vector.

For determining x (0) we use the boundary value condition Ax (0)+
T∫
0

n (t)x (t) dt = C :

C = Ax (0) +
∫ T
0 n (t)x (t) dt =

(
A+

∫ T
0 n (t) dt

)
x (0)+

+ 1
Γ(α)

∫ T
0 n (t)

∫ t
0 (t− s)α−1 y (s) dsdt.

From here we get

x (0) = N−1C − N−1

Γ (α)

∫ T

0
n (t)

∫ t

0
(t− s)α−1 y (s) dsdt

and consequently for all t ∈ [0, T ] (4) is true.
Lemma 6 (Krasnoselskii’s fixed point theorem, [20])
Let M be a closed, bounded, convex and nonempty subset of a Banach space X.Let

A, B be the operators such that
(a) Ax+By ∈Mx whenever x, y ∈M ;
(b) Ais compact and continuous;
(c)B is a contraction mapping.
Then there exists z ∈M such that z = Az +Bz.



M.J. Mardanov, H.N. Aliyev, Y.A. Sharifov / Eur. J. Pure Appl. Math, 15 (2) (2022), 726-735 730

3. Main results

In this section, the theorems of uniqueness and existence of a solution for problem
(1), (2) will be given. For the forthcoming analysis we impose suitable conditions on
the functions involved in the boundary value problem (1), (2). We assume the following
conditions are met:

(H1) The function f : [0, T ] × Rn → Rn is continuous and satisfies the following
Lipschitz condition

|f (t, x1, x2, x3)− f (t, y1, y2, y.3)| ≤ L (|x1 − y1|+ |x2 − y2|+ |x3 − y3|) ,

xi, yi ∈ Rn, i = 1, 2, 3, t ∈ [0, T ] , L > 0.

(H2) ∥f (t, x, φx, ψx)∥ ≤ G, for all x ∈ Rn, t ∈ [0, T ] , G ≥ 0.

Theorem 2. Assume that f : [0, T ] × Rn → Rn is jointly continuous and satisfies (H1)
and (H2). If

L
∥∥N−1

∥∥ ∥n∥Tα+1

Γ (α+ 2)
(1 + T (φ0 + ψ0)) < 1, (5)

then the fractional integro-differential problem (1), (2) has at least one solution.

Proof. Consider Br = {x ∈ C ([0, T ] ;Rn) : ∥x∥ ≤ r}, where

r ≥ GTα

Γ (α+ 1)
+
∥∥N−1C

∥∥+
G ∥n∥

∥∥N−1
∥∥Tα+1

Γ (α+ 2)
.

Define two mappings A1 and A2 on Br by

(A1x) (t) =
1

Γ (α)

∫ t

0
(t− s)α−1 f (s, x (s) , φx (s) , ψx (s)) ds, (6)

(A2x) (t) = N−1

(
C − 1

Γ (α)

∫ T

0
n (t)

∫ t

0
(t− s)α−1 f (s, x (s) , φx (s) , ψx (s)) dsdt

)
. (7)

For x, y ∈ Br, by (H2), we obtain

∥(A1x) (t) + (A2y) (t)∥ ≤ G

Γ (α)

∫ t

0
(t− s)α−1 ds+

+
∥∥N−1C

∥∥+
G ∥n∥

∥∥N−1
∥∥

Γ (α)

∫ T

0

∫ t

0
(t− s)α−1 dsdt ≤

≤ GTα

Γ (α+ 1)
+
∥∥N−1C

∥∥+
G ∥n∥

∥∥N−1
∥∥Tα+1

Γ (α+ 2)
≤ r.



M.J. Mardanov, H.N. Aliyev, Y.A. Sharifov / Eur. J. Pure Appl. Math, 15 (2) (2022), 726-735 731

This shows that A1x+A2y ∈ Br. Therefore, condition (a) of Lemma 6 holds. It is claimed
that A1 is compact and continuous. Continuity of f implies that (A1x) (t) is continuous.
(A1x) (t) is uniformly bounded on Br as

∥A1x∥ ≤ GTα

Γ (α+ 1)
.

Since f is bounded on the compact set [0, T ]×Br, let sup
[0,T ]×Br

∥f (t, x, y, z)∥ =Mf . Then,

for t1, t2 ∈ [0, T ] , t1 < t2 we get

∥(A1x) (t2)− (A1x) (t1)∥ =

= 1
Γ(α)

∥∥∥∫ t10 (
(t2 − s)α−1 − (t1 − s)α−1

)
f (s, x (s) , φx (s) , ψx (s)) ds+∫ t2

t1
(t2 − s)α−1 f (s, x (s) , φx (s) , ψx (s)) ds ∥ ≤

≤
Mf

Γ (α)

(
tα2
α

− tα1
α

)
,

which is independent of x and tends to zero as t2 → t1. Therefore, A1 is relatively compact
on Br. By Arzela Ascoli’s Theorem, A1 is compact on Br.

For x, y ∈ Br and t ∈ [0, T ], by (H1), we have

∥(A2x) (t)− (A2y) (t)∥ ≤

≤ 1

Γ (α)

∥∥∥∥N−1

∫ T

0
n (t)

∫ t

0
(t− s)α−1 (f (s, x (s) , φx (s) , ψx (s)) −

− f (s, y (s) , φy (s) , ψy (s))) dsdt∥ ≤

≤ L (1 + T (φ0 + ψ0))

∥∥N−1
∥∥ ∥n∥Tα+1

Γ (α+ 1)
∥x− y∥ .

It follows from (5) that A2 is a contraction mapping. Thus, by Krasnoselskii’s fixed point
theorem, boundary value problem (1), (2) has at least one solution.

Theorem 3. Assume that f : [0, T ] × Rn → Rn is a continuous function satisfying the
assumption (A1). Then the problems (3) and (4) has a unique solution on [0, T ] if

L (1 + T (φ0 + ψ0)) Λ < 1,

where Λ = Tα

Γ(α+1) +
∥N−1∥∥n∥Tα+1

Γ(α+2) .

Proof.
Proof. Define a mapping F : C ([0, T ] ;Rn) → ([0, T ] ;Rn) by

(Fx) (t) = 1
Γ(α)

∫ t
0 (t− s)α−1 f (s, x (s) , φx (s) , ψx (s)) ds+

+N−1
(
C − 1

Γ(α)

∫ T
0 n (t)

∫ t
0 (t− s)α−1 f (s, x (s) , φx (s) , ψx (s)) dsdt

)
.

(8)



M.J. Mardanov, H.N. Aliyev, Y.A. Sharifov / Eur. J. Pure Appl. Math, 15 (2) (2022), 726-735 732

Let us first show that FBr ⊂ Br, whereF is the operator defined by (8) and r ≥ MfΛ+∥N−1C∥
1−L(1+T (φ0+ψ0))Λ

with Mf = sup |f (t, 0, 0, 0)|
t∈[0,T ]

. Then, in view of the assumptions (H1) and (H2), we have

|f (t, x, φx, ψx)| ≤ |f (t, x, φx, ψx)− f (t, 0, 0, 0)|+ |f (t, 0, 0, 0)| ≤

≤ L (|x|+ |φx|+ |ψx|) +Mf ≤ L (1 + T (φ0 + ψ0)) r +Mf .

For any x ∈ Br, we have
∥Fx∥ = sup |Fx (t)|

t∈[0,T ]
≤

≤
∥∥N−1C

∥∥+ (L (1 + T (φ0 + ψ0)) r +Mf )

{
Tα

Γ (α+ 1)
+

∥∥N−1
∥∥ ∥n∥Tα+1

Γ (α+ 2)

}
≤ r,

which implies that FBr ⊂ Br. Next, for x, y ∈ C ([0, T ] ;Rn) and for each t ∈ [0, T ], we
obtain

∥Fx− Fy∥ ≤ sup
[0,T ]

1

Γ (α)

∫ t

0
(t− s)α−1 |f (s, x (s) , φx (s) , ψx (s)) −

−f (s, y (s) , φy (s) , ψy (s))| ds+

+

∥∥N−1
∥∥ ∥n∥

Γ (α)

∫ T

0

∫ t

0
(t− s)α−1 |f (s, x (s) , φx (s) , ψx (s))− f (s, y (s) , φy (s) , ψy (s))| ds ≤

≤ L (1 + T (φ0 + ψ0))

{
Tα

Γ (α+ 1)
+

∥∥N−1
∥∥ ∥n∥Tα+1

Γ (α+ 2)

}
∥x− y∥ =

= L (1 + T (φ0 + ψ0)) Λ ∥x− y∥ .
Since L (1 + T (φ0 + ψ0)) Λ < 1, the operator F is a contraction. By Banach contraction
mapping principle the operator F has a unique fixed point, which means that the problems
(1) and (2) has a unique solution for on [0, T ].

4. Conclusion

In the paper a system of fractional integro-differential equations with nonlocal condi-
tions is studied. At first the boundary value problem is reduced to the equivalent integral
equation. Then, using the theorem on fixed points, the condition on the existence and
uniqueness of the solution of the boundary value problem is obtained. The technique used
in this research can be applied to the similar problems for fractional differential equations
subject to multi-point nonlocal conditions

Ex (0) +
J∑
j=1

Bjx (λj) = C,

where E is a unit matrix, Bj ∈ Rn×n are given matrices and

J∑
j=1

∥Bj∥ < 1, 0 < λ1 < λ2 < ... < λJ < T.



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