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How to cite: F. Y. Ishak, “Existence Solution for Nonlinear System of Fractional Integrodifferential 
Equations of Volterra Type with Fractional Boundary Conditions”, JMM, vol. 6, no. 1, pp. 1-12, May 2020. 

 

 Existence Solution for Nonlinear System of Fractional 

Integrodifferential Equations of Volterra Type with Fractional 

Boundary Conditions 
 

 

Faraj Y. Ishak 

University of Duhok Iraq, faraj.ishak@uod.ac  

 
doi: https://doi.org/10.15642/mantik.2020.6.1.1-12 

 

 
Abstrak: Artikel ini menyelidiki eksistensi, keunikan, dan solusi stabil dari sistem persamaan 

diferensial-diferensial Volterra fraksional baru dengan kondisi batas fraksional dengan 

menggunakan teorema eksistensi dan keunikan. Teorema tentang eksistensi dan keunikan dari solusi 

yang ditetapkan di bawah beberapa kondisi yang diperlukan dan cukup pada ruang kompak. Contoh 

sederhana dari hasil aplikasi utama disajikan dalam artikel ini. 

 
Kata kunci: Keberadaan dan keunikan, stabilitas, fractional Integrodifferential equations, masalah 
nilai batas, teorema eksistensi dan keunikan. 

 
Abstract: This article investigates existence, uniqueness and stability solutions of new fractional 

Volterra integro-differential equations system with fractional boundary conditions by using the 

existence and uniqueness theorem. Theorems on existence and uniqueness of solution are 

established under some necessary and sufficient conditions on compact space. A simple example of 
application of the main results of this article is presented. 

 

Keywords: Existence and uniqueness, stability, fractional Integrodifferential equations, boundary 

value problem, existence and uniqueness theorem. 

 

 

 

  

Jurnal Matematika MANTIK 
Vol. 6, No. 1, May 2020, pp. 1-12 

 

ISSN: 2527-3159 (print) 2527-3167 (online) 

mailto:faraj.ishak@uod.ac
http://u.lipi.go.id/1458103791


Jurnal Matematika MANTIK 
Volume 6, Issue. 1, May 2020, pp. 1-12 

2   
 

1. Introduction  

Gottfried Leibniz and Guilliaume L’Hopital sparked initial curiosity into the theory 

of fractional calculus during a 1695 correspondence on the possible value and meaning of 

non-integer-order derivatives. By the late nineteenth century, the combined efforts of a 

number of mathematicians most notably Liouville, Grunwald, Letnikov, and Riemann 
produced a fairly solid theory of fractional calculus for functions of a real variable. 

Though several viable fractional derivatives were proposed, the so-called Riemann-

Liouville and Caputo derivatives are the two most-commonly used today.  
Mathematicians have employed this fractional calculus in recent years to model and solve 

a variety of applied problems. Indeed, as Podlubney outlines in [1]. 

Fractional differential equations have extensive applications in various fields of 
science and engineering. Many phenomena in viscoelasticity, electrochemistry, control 

theory, porous media, electromagnetism, and other fields, can be modelled by fractional 

differential equations. We refer the reader to [2, 4] and references therein for some 

applications. Fractional BVPs defined on intervals have been studied by many authors. 
Many results on the existence, uniqueness, multiplicity, and nonexistence of solutions for 

fractional differential equations subject to various boundary conditions (BCs) have been 

obtained; see for example [9,10,11,12,13,14,15,16]. 
The fractional difference calculus had its origin in the works by Al-Salam [6] and 

Agarwal [7]. More recently, perhaps due to the explosion in research within the fractional 

differential calculus setting, new developments in this theory of fractional difference 

calculus were made, specifically,  analogues of the integral and differential fractional 
operators properties such as the Mittag-Leffler function, the Laplace transform, and 

Taylor’s formula [3,5,8,17,18], just to mention some. 

Butris and Ishak [20], used both methods Picard approximation and Banach fixed 
point theorems for studying the existence and uniqueness solutions to the following 

fractional integral equations:  

 

𝑢(𝑡) = 𝑓(𝑡)+
1

Γ(𝛼)
∫  (𝑡 −𝑠)𝛼−1𝐹(𝑡,𝑠,𝑢(𝑠),𝑤(𝑠))𝑑𝑠
𝑡

𝑎
  

𝑤(𝑡) = 𝑔(𝑡)+
1

Γ(𝛼)
∫  (𝑡 −𝑠)𝛼−1𝐺(𝑡,𝑠,𝑢(𝑠),𝑤(𝑠))𝑑𝑠
𝑏

𝑎
   

 
In this work our aim is to show the existence solutions of the system of integrodifferential 

equations 

𝐷𝛼𝑥(𝑡)+ 𝑓(𝑡,𝑠, [𝜙 𝑥](𝑥)) = 0 
 

𝐷𝛽𝑦(𝑡) +𝑔(𝑡,𝑠, [𝜑 𝑦](𝑦)) = 0     
 

𝐷𝛼−1𝑥(0) = 0,      𝐷𝛼−1𝑥(1) = 𝑏1, 
𝐷𝛽−1𝑦(0) = 0,      𝐷𝛽−1𝑦(1) = 𝑏2 

 Where: 

 

[𝜙 𝑥](𝑥) = ∫ 𝐾(𝑡,𝑠)𝐹(𝑡,𝑠,𝑥(𝑠),𝑦(𝑠))𝑑𝑠
𝑡

−∞
,   [𝜑 𝑦](𝑦) =

∫ 𝐺(𝑡,𝑠)𝐻(𝑡,𝑠,𝑥(𝑠),𝑦(𝑠))𝑑𝑠
𝑡

−∞
  

 

1 < 𝛼,𝛽 ≤ 2,0 ≤ 𝑠 ≤ 𝑡 ≤ 1,𝑏1,𝑏2 ∈ 𝑅, 𝑥 ∈ 𝐷1 ⊆ 𝑅
𝑛 and 𝑦 ∈ 𝐷2 ⊆ 𝑅

𝑚,𝐷1 and 𝐷2 
are compact domain, let the vector functions 𝐹(𝑡,𝑠,𝑥(𝑡),𝑦(𝑡)) , 𝐻(𝑡,𝑠,𝑥(𝑡),𝑦(𝑡)) is 
defined and continuous on the domain: 

 

                           𝐷 = {(𝑡,𝑠,𝑥,𝑦);𝑡,𝑠 𝜖 [0,1] ,𝑥 ∈ 𝐷1 ,𝑦 ∈ 𝐷2}                  (1.2) 

(1.1)  

 

 



F. Y. Ishak 
Existence Solution for Nonlinear System of Fractional Integrodifferential Equations of Volterra Type with 

Fractional Boundary Conditions  

3 
 

Assume that the vector functions 𝐹(𝑡,𝑠,𝑥(𝑡),𝑦(𝑡)), 𝐻(𝑡,𝑠,𝑥(𝑡),𝑦(𝑡)), and kernels 

𝐾(𝑡,𝑠),𝐺(𝑡,𝑠) are satisfying  the following inequalities: 
 

‖𝐹(𝑡,𝑠,𝑥(𝑡),𝑦(𝑡))‖ ≤ 𝑀1  , ‖𝐻(𝑡,𝑠,𝑥(𝑡),𝑦(𝑡))‖ ≤ 𝑀2                       (1.3) 
                                                 
‖𝐹(𝑡,𝑠,𝑥2,𝑦2)− 𝐹(𝑡,𝑠,𝑥1,𝑦1)‖ ≤ 𝐿1(‖𝑥2 −𝑥1‖+ ‖𝑦2 −𝑦1‖)             (1.4) 
 
‖𝐻(𝑡,𝑠,𝑥2,𝑦2)− 𝐻(𝑡,𝑠,𝑥1,𝑦1)‖ ≤ 𝐿2(‖𝑥2 − 𝑥1‖+ ‖𝑦2 −𝑦1‖)             (1.5) 
 

‖𝐾(𝑡,𝑠)‖ ≤ 𝛿1𝑒
−𝜆1(𝑡−𝑠)    ,   ‖𝐺(𝑡,𝑠)‖ ≤ 𝛿2𝑒

−𝜆2(𝑡−𝑠)                              (1.6)  
  

Where 𝑀1,𝑀2,𝐿1,𝐿2,𝜆1,𝜆2,𝛿1,𝛿2,are positive constants 𝑥1,𝑥2 ∈  𝐷 1 𝑦1,𝑦2 ∈ 𝐷2 
𝑡 ,𝑠 𝜖 [0,1]   and  ‖.‖ = max

𝑡∈[0,1]
 |. |, we defined non-empty sets as: 

 

𝐷𝐹 = 𝐷1 −
𝑀1𝛿1(𝛼−1)+𝑏1𝜆1𝛼

𝜆1𝛤(𝛼+1)
  

                                                                

𝐷𝐻 = 𝐷2 −
𝑀2𝛿2(𝛽−1)+𝜆2𝑏2β

𝜆2Γ(𝛽+1)
   

 
As well as we suppose the maximum value of the following matrix: 

  

Δ0 = (

𝐿1𝛿1(𝛼−1)

𝜆1Γ(𝛼+1)

𝐿1𝛿1(𝛼−1)

𝜆1Γ(𝛼+1)

𝐿2𝛿2(𝛽−1)

𝜆2Γ(𝛽+1)

𝐿2𝛿2(𝛽−1)

𝜆2Γ(𝛽+1)

) , less than one i.e. 

. 

𝜆 𝑚𝑎𝑥(Δ0) =
𝐿1𝛿1(𝛼−1)

𝜆1Γ(𝛼+1)
+
𝐿2𝛿2(𝛽−1)

𝜆2Γ(𝛽+1)
< 1          (1.8) 

 

Define a sequence of functions {𝑥𝑚(𝑡,𝑥0)}𝑚=0
∞    , {𝑦𝑚(𝑡,𝑦0)}𝑚=0

∞  as: 
 

 𝑥𝑚+1(𝑡,𝑥0) =
𝑡𝛼−1

Γ(𝛼)
∫  
1

0
∫  𝐾(𝑡,𝑠)𝐹(𝑡,𝑠,𝑥𝑚(𝑠),𝑦𝑚(𝑠))𝑑𝑠𝑑𝑠
𝑡

−∞
−

                                       ∫
(𝑡−𝑠)𝛼−1

Γ(𝛼)

𝑡

0
(∫ 𝐾(𝑡,𝑠)𝐹(𝑡,𝑠,𝑥𝑚(𝑠),𝑦𝑚(𝑠))𝑑𝑠)𝑑𝑠
𝑡

−∞
+
𝑏1𝑡

𝛼−1

Γ(α)
  

        

𝑦𝑚+1(𝑡,𝑦0) =  
𝑡𝛽−1

Γ(𝛽)
∫  
1

0
∫  𝐺(𝑡,𝑠)𝐻(𝑡,𝑠,𝑥𝑚(𝑠),𝑦𝑚(𝑠))𝑑𝑠𝑑𝑠
𝑡

−∞
−

                                          ∫
(𝑡−𝑠)𝛽−1

Γ(𝛽)

𝑡

0
(∫ 𝐺(𝑡,𝑠)𝐻(𝑡,𝑠,𝑥𝑚(𝑠),𝑦𝑚(𝑠))𝑑𝑠)𝑑𝑠
𝑡

−∞
+
𝑏2𝑡

𝛽−1

Γ(β)
  

 

With   𝑥0 = 𝑥(0) = 0,    𝑦0 = 𝑦(0) = 0,          m = 0,1,2… 
 

2. Preliminaries  

Let us recall some basic definitions on fractional calculus, which can be found in 
the literature. 

Definition 2.1 [19] Assume that  𝑓(𝑥,𝑦) is defined on the set (𝑎,𝑏)x𝐺,𝐺 ⊂ 𝑅,𝑓(𝑥,𝑦) is 
said to satisfy Lipschitz condition with respect to the second variable, if for all 𝑥 ∈
(𝑎,𝑏)  and for any 𝑦1 ,𝑦2 ∈ 𝐺  
 

 |𝑓(𝑥,𝑦1)−𝑓(𝑥,𝑦2)| ≤ 𝜉|𝑦1 − 𝑦2| 

(1.9) 

(1.7) 



Jurnal Matematika MANTIK 
Volume 6, Issue. 1, May 2020, pp. 1-12 

4   
 

where 𝜉 > 0 does not depend on 𝑥 ∈ (𝑎,𝑏). 
 

Definition 2.2 [19] The Riemann-Liouville fractional integral of order q is defined by 

𝐼𝑞𝑓(𝑡) =
1

Γ(𝑞)
∫ (𝑡 − 𝑠)𝑞−1
𝑡

0

𝑓(𝑠)𝑑𝑠,       𝑞 > 0 

provided the integral exists. 

 

Definition 2.3 [19] The Riemann-Liouville fractional derivative of order q is defined by 
 

𝐷𝑞𝑓(𝑡) =
1

Γ(𝑛− 𝑞)
(
𝑑

𝑑𝑡
)
𝑛

∫ (𝑡 − 𝑠)𝑛−𝑞−1
𝑡

0

𝑓(𝑠)𝑑𝑠,    𝑛 −1 < 𝑞 ≤ 𝑛, 𝑞 > 0, 

 

Provided the right-hand side is pointwise defined on (0,+∞). 
 

Lemma 2.1 For 𝛼,𝛽 > 0, then the following relation hold: 
 

𝐷𝛼𝑡𝛽 =
Γ(𝛽 + 1)

Γ(𝛽 +1 −𝛼)
𝑡𝛽−𝛼−1,𝛽 > 𝑛 𝑎𝑛𝑑 𝐷𝛼𝑡𝑘 = 0,𝑘 = 0,1,…,𝑛 −1 

 

Lemma 2.2 [3] The equality 𝐷0+
𝛼 𝐼0+

𝛼 𝑓(𝑡) = 𝑓(𝑡),     𝛼 > 0 holds for 𝑓 ∈ 𝐿1(0,1). 
 

Lemma 2.3 Let 𝛼,𝛽 > 0   and let f be a function defined on [0, 1]. Then the following 
formulas hold: 

(i) (𝐼𝑞
𝛽
𝐼𝑞
𝛼𝑓)(𝑥) = (𝐼𝑞

𝛼+𝛽
𝑓)(𝑥) 

(ii) (𝐷𝑞
𝛼𝐼𝑞
𝛼𝑓)(𝑥) = 𝑓(𝑥) 

 

Lemma 2.4 Let 𝛼 > 0 and n be a positive integer. Then, the following equality holds: 
 

(𝐼𝑞
𝛽
𝐷𝑞
𝛼𝑓)(𝑥) = (𝐷𝑞

𝑛𝐼𝑞
𝛼𝑓)(𝑥) −∑

𝑥𝛼−𝑛+𝑘

Γ(α+k−n+1)
(𝐷𝑞

𝑘𝑓)(0)𝑛−1𝑘=0    

 

Lemma 2.5 A functions 𝑥(𝑡),𝑦(𝑡), are solution of system (1.1) if and only if 𝑥(𝑡),𝑦(𝑡) 
have the form: 

 

𝑥 (𝑡) = 
𝑡𝛼−1

Γ(𝛼)
∫ 

1

0

∫ 𝐾(𝑡,𝑠)𝐹(𝑡,𝑠,𝑥 (𝑠),𝑦 (𝑠))𝑑𝑠𝑑𝑠

𝑡

−∞

−∫
(𝑡 −𝑠)𝛼−1

Γ(𝛼)

𝑡

0

(∫ 𝐾(𝑡,𝑠)𝐹(𝑡,𝑠,𝑥 (𝑠),𝑦 (𝑠))𝑑𝑠)𝑑𝑠
𝑡

−∞

+
𝑏1𝑡

𝛼−1

Γ(α)
  

       

𝑦 (𝑡) =
𝑡𝛽−1

𝛤(𝛽)
∫ 

1

0

∫ 𝐺(𝑡,𝑠)𝐻(𝑡,𝑠,𝑥 (𝑠),𝑦 (𝑠))𝑑𝑠𝑑𝑠

𝑡

−∞

−∫
(𝑡 −𝑠)𝛽−1

𝛤(𝛽)

𝑡

0

(∫ 𝐺(𝑡,𝑠)𝐻(𝑡,𝑠,𝑥 (𝑠),𝑦𝑚(𝑠))𝑑𝑠)𝑑𝑠
𝑡

−∞

+
𝑏2𝑡

𝛽−1

𝛤(𝛽)
 

 

Proof: It follows from Lemma 2.4 that the system of fractional differential equation in 

(1.1) is equivalent to the integral equations: 

𝑥(𝑡) = −𝐼𝛼𝑓(𝑡)+ 𝑐1𝑡
𝛼−1 + 𝑐2𝑡

𝛼−2 

𝑦(𝑡) = −𝐼𝛼𝑔(𝑡)+𝑑1𝑡
𝛼−1 +𝑑2𝑡

𝛼−2 

(2.1) 



F. Y. Ishak 
Existence Solution for Nonlinear System of Fractional Integrodifferential Equations of Volterra Type with 

Fractional Boundary Conditions  

5 
 

Where: 

𝑓(𝑡) = ∫  𝐾(𝑡,𝑠)𝐹(𝑡,𝑠,𝑥 (𝑠),𝑦 (𝑠)) 𝑑𝑠
𝑡

−∞
, 𝑔(𝑡) = ∫  𝐺(𝑡,𝑠)𝐻(𝑡,𝑠,𝑥 (𝑠),𝑦 (𝑠))𝑑𝑠

𝑡

−∞
,

𝑐1,𝑐2,𝑑1,𝑑2 ∈ 𝑅      
From the boundary conditions of (1.1) we have 𝑐2 = 0,𝑑2 = 0 and: 
 

𝑐1 =
𝑏1

Γ(𝛼)
+

1

Γ(𝛼)
∫ 𝑓(𝑠)𝑑𝑠
1

0
, 𝑑1 =

𝑏2

Γ(𝛽)
+

1

Γ(𝛽)
∫ 𝑔(𝑠)𝑑𝑠
1

0
  

 
Which is complete the proof. 

 

3. Result and Discussion 

In this section, the theorems of existence, uniqueness, and stability of a solution for 

system (1.1) will be given. 

 
Theorem 3.1: Let the right side of system (1.1) are defined and continuous on domain 

(1.2) .Suppose that the vector functions 𝐹(𝑡,𝑠,𝑥(𝑡),𝑦(𝑡)) , 𝐻(𝑡,𝑠,𝑥(𝑡),𝑦(𝑡)) are 
satisfying the inequalities (1.3)-(1.5) and the conditions (1.6)-(1.9).Then there exist a 

sequences of functions (1.9) converges uniformly as 𝑚 → ∞ on domain (1.2) to the limit 
functions which satisfying integral equations (2.1) Provided that: 
 

‖𝑥∞(𝑡,𝑥0)− 𝑥0‖ ≤
𝑀1𝛿1(𝛼−1)+𝜆1𝑏1𝛼

𝜆1Γ(𝛼+1)
   

 

‖𝑦∞(𝑡,𝑥0)− 𝑦0‖ ≤
𝑀2𝛿2(𝛽−1)+𝜆2𝑏2β

𝜆2Γ(𝛽+1)
       

 

(
‖𝑥𝑚+1(𝑡,𝑥0) −𝑥𝑚(𝑡,𝑥0)‖

‖𝑦m+1(𝑡,𝑦0)−𝑦m(𝑡,𝑦0)‖
) ≤ Δ0

𝑚(𝐼 − Δ0)
−1Φ0   for all 𝑚 ≥ 1   , 𝑡 ∈ [0,1] 

 
Proof: 

By using the sequence of function (1.9) when m=0, we get:  

 

‖𝑥1(𝑡,𝑥0)−𝑥0‖ ≤
𝑡𝛼−1

Γ(𝛼)
∫  
1

0
∫ ‖𝐾(𝑡,𝑠)‖‖𝐹(𝑡,𝑠,𝑥0 (𝑠),𝑦0 (𝑠))‖ 𝑑𝑠𝑑𝑠
𝑡

−∞
−

                                 ∫
(𝑡−𝑠)𝛼−1

Γ(𝛼)

𝑡

0
(∫ ‖𝐾(𝑡,𝑠)‖‖𝐹(𝑡,𝑠,𝑥0 (𝑠),𝑦0 (𝑠))‖𝑑𝑠)𝑑𝑠
𝑡

−∞
+
𝑏1𝑡

𝛼−1

Γ(α)
     

   

≤
𝑡𝛼−1𝑀1
Γ(𝛼)

∫ 

1

0

∫𝛿1𝑒
−𝜆1(𝑡−𝑠)  𝑑𝑠𝑑𝑠

𝑡

−∞

−∫
(𝑡 −𝑠)𝛼−1

Γ(𝛼)

𝑡

0

(∫ 𝛿1𝑒
−𝜆1(𝑡−𝑠)𝑑𝑠)𝑑𝑠

𝑡

−∞

+
𝑏1𝑡

𝛼−1

Γ(α)
 

                                 

≤
𝑀1𝛿1
𝜆1Γ(𝛼)

−
𝑀1𝛿1

𝜆1Γ(𝛼 +1)
+
𝑏1
Γ(α)

≤
𝑀1𝛿1(𝛼− 1)+ 𝑏1𝜆1𝛼

𝜆1𝛤(𝛼 + 1)
 

 

 And by the same we have  

‖𝑦1(𝑡,𝑦0)− 𝑦0‖ ≤
𝑀2𝛿2(𝛽− 1)+ 𝜆2𝑏2β

𝜆2Γ(𝛽 +1)
 

 

That is:    𝑥1(𝑡,𝑥0)𝜖𝐷1  𝑦1(𝑡,𝑦0)𝜖𝐷2, for all  𝑡𝜖[0,1],𝑥0 𝜖 𝐷𝐹,𝑦0 ∈ 𝐷𝐻  

 

(3.1) 



Jurnal Matematika MANTIK 
Volume 6, Issue. 1, May 2020, pp. 1-12 

6   
 

Suppose that  𝑥𝑝(𝑡,𝑥0) 𝜖 𝐷1    ,𝑦𝑝(𝑡,𝑦0) 𝜖 𝐷2  for each 𝑥0 𝜖 𝐷𝐹 ,𝑦0 𝜖 𝐷𝐻   ,𝑝𝜖𝑍
+  , t∈

[0,1] ,by mathematical induction we conclude that: 𝑥𝑚(𝑡,𝑥0) 𝜖 𝐷1    ,𝑦𝑚(𝑡,𝑦0) 𝜖 𝐷2  for 

each 𝑥0 𝜖 𝐷𝐹 ,𝑦0 𝜖 𝐷𝐻    , 𝑚 = 0,1,2,… 
 

To prove that the sequences (1.9) convergence uniformly in domain (1.2): 
 

‖𝑥2(𝑡,𝑥0)−𝑥1(𝑡,𝑥0)‖ ≤
𝑡𝛼−1

Γ(𝛼)
∫  
1

0
∫ ‖𝐾(𝑡,𝑠)‖‖𝐹(𝑡,𝑠,𝑥1 (𝑠),𝑦1 (𝑠))‖ 𝑑𝑠𝑑𝑠
𝑡

−∞
−

                                                      ∫
(𝑡−𝑠)𝛼−1

Γ(𝛼)

𝑡

0
(∫ ‖𝐾(𝑡,𝑠)‖‖𝐹(𝑡,𝑠,𝑥1 (𝑠),𝑦1 (𝑠))‖𝑑𝑠)𝑑𝑠
𝑡

−∞
−

                                                       
𝑡𝛼−1

Γ(𝛼)
∫  
1

0
∫ ‖𝐾(𝑡,𝑠)‖‖𝐹(𝑡,𝑠,𝑥0 (𝑠),𝑦0 (𝑠))‖ 𝑑𝑠𝑑𝑠
𝑡

−∞
+

                                                      ∫
(𝑡−𝑠)𝛼−1

Γ(𝛼)

𝑡

0
(∫ ‖𝐾(𝑡,𝑠)‖‖𝐹(𝑡,𝑠,𝑥0 (𝑠),𝑦0 (𝑠))‖𝑑𝑠)𝑑𝑠
𝑡

−∞
          

       

≤
𝑡𝛼−1

Γ(𝛼)
∫  
1

0
∫ 𝛿1𝑒

−𝜆1(𝑡−𝑠) ‖𝐹(𝑡,𝑠,𝑥1 (𝑠),𝑦1 (𝑠))− 𝐹(𝑡,𝑠,𝑥0 (𝑠),𝑦0 (𝑠))‖ 𝑑𝑠𝑑𝑠
𝑡

−∞
      

−∫
(𝑡−𝑠)𝛼−1

Γ(𝛼)

𝑡

0
(∫ 𝛿1𝑒

−𝜆1(𝑡−𝑠) ‖𝐹(𝑡,𝑠,𝑥1 (𝑠),𝑦1 (𝑠))− 𝐹(𝑡,𝑠,𝑥0 (𝑠),𝑦0 (𝑠))‖𝑑𝑠)𝑑𝑠
𝑡

−∞
 

 

≤
𝐿1𝑡

𝛼−1

Γ(𝛼)
(‖𝑥1 −𝑥0‖ +‖𝑦1 −𝑦0‖)∫  

1

0
∫ 𝛿1𝑒

−𝜆1(𝑡−𝑠) 𝑑𝑠𝑑𝑠
𝑡

−∞
−   

                               𝐿1(‖𝑥1 − 𝑥0‖+ ‖𝑦1 − 𝑦0‖)∫
(𝑡−𝑠)𝛼−1

Γ(𝛼)

𝑡

0
(∫ 𝛿1𝑒

−𝜆1(𝑡−𝑠)
𝑡

−∞
𝑑𝑠)𝑑𝑠  

 

≤
𝐿1𝛿1(𝛼𝑡

𝛼−1 −𝑡𝛼)

𝜆1Γ(𝛼 +1)
(‖𝑥1 − 𝑥0‖+ ‖𝑦1 −𝑦0‖) 

   And by the same 

‖𝑦2(𝑡,𝑦0)−𝑦1(𝑡,𝑦0)‖ ≤
𝐿2𝛿2(𝛽𝑡

𝛽−1 − 𝑡𝛽)

𝜆2𝛤(𝛽 +1)
−(‖𝑥1 −𝑥0‖ +‖𝑦1 − 𝑦0‖) 

 

By the mathematical induction the following inequalities hold: 

  

‖𝑥𝑚+1(𝑡,𝑥0)−𝑥𝑚(𝑡,𝑥0)‖ ≤
𝐿1𝛿1(𝛼𝑡

𝛼−1−𝑡𝛼)

𝜆1Γ(𝛼+1)
(‖𝑥𝑚 − 𝑥𝑚−1‖+ ‖𝑦𝑚 − 𝑦𝑚−1‖)    

    

‖𝑦𝑚+1(𝑡,𝑦0)−𝑦𝑚(𝑡,𝑦0)‖ ≤
𝐿2𝛿2(𝛽𝑡

𝛽−1−𝑡𝛽)

𝜆2Γ(𝛽+1)
(‖𝑥𝑚 − 𝑥𝑚−1‖+ ‖𝑦𝑚 −𝑦𝑚−1‖)    

 

Rewrite (3.2) with vector form: 

 

(
‖𝑥𝑚+1(𝑡,𝑥0)− 𝑥𝑚(𝑡,𝑥0)‖

‖𝑦m+1(𝑡,𝑦0)− 𝑦m(𝑡,𝑦0)‖
) ≤

(

 
 

𝐿1𝛿1(𝛼𝑡
𝛼−1 − 𝑡𝛼)

𝜆1Γ(𝛼 +1)

𝐿1𝛿1(𝛼𝑡
𝛼−1 −𝑡𝛼)

𝜆1Γ(𝛼 +1)

𝐿2𝛿2(𝛽𝑡
𝛽−1 − 𝑡𝛽)

𝜆2Γ(𝛽+1)

𝐿2𝛿2(𝛽𝑡
𝛽−1 −𝑡𝛽)

𝜆2Γ(𝛽+ 1) )

 
 
(
‖𝑥𝑚 − 𝑥𝑚−1‖

‖𝑦𝑚 − 𝑦𝑚−1‖
)   

      

That is:                Φ𝑚+1(𝑡,𝑥0,𝑦0) ≤  Δ(𝑡)Φ𝑚(𝑡,𝑥0,𝑦0)                           (3.3) 
 
Where: 

(3.2) 



F. Y. Ishak 
Existence Solution for Nonlinear System of Fractional Integrodifferential Equations of Volterra Type with 

Fractional Boundary Conditions  

7 
 

Δ(𝑡) = (

𝐿1𝛿1(𝛼𝑡
𝛼−1−𝑡𝛼)

𝜆1Γ(𝛼+1)

𝐿1𝛿1(𝛼𝑡
𝛼−1−𝑡𝛼)

𝜆1Γ(𝛼+1)

𝐿2𝛿2(𝛽𝑡
𝛽−1−𝑡𝛽)

𝜆2Γ(𝛽+1)

𝐿2𝛿2(𝛽𝑡
𝛽−1−𝑡𝛽)

𝜆2Γ(𝛽+1)

),Φ𝑚+1 =

(
‖𝑥𝑚+1(𝑡,𝑥0) −𝑥𝑚(𝑡,𝑥0)‖

‖𝑦m+1(𝑡,𝑦0)−𝑦m(𝑡,𝑦0)‖
)  ,Φ𝑚 = (

‖𝑥𝑚 −𝑥𝑚−1‖

‖𝑦𝑚 − 𝑦𝑚−1‖
)      

 
Take the maximum value for both sides of (3.3): 

 

Φ 𝑚+1 ≤ Δ0 Φ𝑚                      (3.4) 
 

where                                Δ0 = max
𝑡𝜖[0,1]

 Δ (𝑡),         Δ0 = (

𝐿1𝛿1(𝛼−1)

𝜆1Γ(𝛼+1)

𝐿1𝛿1(𝛼−1)

𝜆1Γ(𝛼+1)

𝐿2𝛿2(𝛽−1)

𝜆2Γ(𝛽+1)

𝐿2𝛿2(𝛽−1)

𝜆2Γ(𝛽+1)

) 

By repletion of (3.4) we obtain: 

Φ𝑚+1 ≤ Λ0
𝑚 Φ1 

 

Φ1 ≤ (

𝑀1𝛿1(𝛼−1)+𝜆1𝑏1𝛼

𝜆1Γ(𝛼+1)

𝑀2𝛿2(𝛽−1)+𝜆2𝑏2β

𝜆2Γ(𝛽+1)

),and also we get: ∑ Φ𝑖
𝑚
𝑖=1 ≤ ∑ Δ0

𝑖−1Φ1
𝑚
𝑖=1    (3.5)                         

Since the matrix Δ0 has eigenvalue 𝜆1 = 0, 
 

 

𝜆2 = 𝜆𝑚𝑎𝑥(Δ0) =
𝑀1𝛿1(𝛼 −1) +𝜆1𝑏1𝛼

𝜆1Γ(𝛼+ 1)
+
𝑀2𝛿2(𝛽 − 1)+ 𝜆2𝑏2β

𝜆2Γ(𝛽 +1)
< 1 

 

the series (3.5) is uniformly convergent, i.e. 

 

lim
𝑚→∞

∑ Δ0
𝑖−1Φ1

𝑚
𝑖=1 = ∑ Δ0

𝑖−1Φ1
∞
𝑖=1 = (𝐼 −Δ0)

−1Φ1              (3.6) 

 

Thus the limiting relation (3.6) signifies uniform convergence of sequences: 
{𝑥𝑚(𝑡,𝑥0)}𝑚=0

∞    , {𝑦𝑚(𝑡,𝑦0)}𝑚=0
∞   ,that is: 

 

lim
𝑚→∞

𝑥𝑚(𝑡,𝑥0) = 𝑥 (𝑡,𝑥0)  ,and   lim
𝑚→∞

𝑦𝑚(𝑡,𝑦0) = 𝑦 (𝑡,𝑦0)     

 

By all conditions and inequalities of the theorem the estimate 
  

(
‖𝑥𝑚+1(𝑡,𝑥0) −𝑥𝑚(𝑡,𝑥0)‖

‖𝑦m+1(𝑡,𝑦0)−𝑦m(𝑡,𝑦0)‖
) ≤ Δ0

𝑚(𝐼 − Δ0)
−1Φ1             

 

Is hold for all m=1, 2, … 

 

To prove that 𝑥 (𝑡,𝑥0)𝜖𝐷1  and 𝑦(𝑡,𝑦0)𝜖𝐷2  we prove that: 

 

lim
𝑚→∞

(
𝑡𝛼−1

Γ(𝛼)
∫ 

1

0

∫ 𝐾(𝑡,𝑠)𝐹(𝑡,𝑠,𝑥𝑚(𝑠),𝑦𝑚(𝑠))𝑑𝑠𝑑𝑠

𝑡

−∞

−∫
(𝑡 −𝑠)𝛼−1

Γ(𝛼)

𝑡

0

(∫ 𝐾(𝑡,𝑠)𝐹(𝑡,𝑠,𝑥𝑚(𝑠),𝑦𝑚(𝑠))𝑑𝑠)𝑑𝑠
𝑡

−∞

+
𝑏1𝑡

𝛼−1

Γ(α)
) = 



Jurnal Matematika MANTIK 
Volume 6, Issue. 1, May 2020, pp. 1-12 

8   
 

𝑡𝛼−1

Γ(𝛼)
∫  
1

0
∫  𝐾(𝑡,𝑠)𝐹(𝑡,𝑠,𝑥 (𝑠),𝑦 (𝑠))𝑑𝑠𝑑𝑠
𝑡

−∞
−

∫
(𝑡−𝑠)𝛼−1

Γ(𝛼)

𝑡

0
(∫ 𝐾(𝑡,𝑠)𝐹(𝑡,𝑠,𝑥 (𝑠),𝑦 (𝑠))𝑑𝑠)𝑑𝑠
𝑡

−∞
 + 

𝑏1𝑡
𝛼−1

Γ(α)
    (3.7) 

 

lim
𝑚→∞

( 
𝑡𝛽−1

Γ(𝛽)
∫ 

1

0

∫ 𝐺(𝑡,𝑠)𝐻(𝑡,𝑠,𝑥𝑚(𝑠),𝑦𝑚(𝑠))𝑑𝑠𝑑𝑠

𝑡

−∞

−∫
(𝑡 −𝑠)𝛽−1

Γ(𝛽)

𝑡

0

(∫ 𝐺(𝑡,𝑠)𝐻(𝑡,𝑠,𝑥𝑚(𝑠),𝑦𝑚(𝑠))𝑑𝑠)𝑑𝑠
𝑡

−∞

+
𝑏2𝑡

𝛽−1

Γ(β)
= 

 
𝑡𝛽−1

Γ(𝛽)
∫  
1

0
∫  𝐺(𝑡,𝑠)𝐻(𝑡,𝑠,𝑥 (𝑠),𝑦 (𝑠))𝑑𝑠𝑑𝑠
𝑡

−∞
−

∫
(𝑡−𝑠)𝛽−1

Γ(𝛽)

𝑡

0
(∫ 𝐺(𝑡,𝑠)𝐻(𝑡,𝑠,𝑥 (𝑠),𝑦 (𝑠))𝑑𝑠)𝑑𝑠
𝑡

−∞
+ 

𝑏2𝑡
𝛽−1

Γ(β)
    (3.8) 

 

We have:  

            

‖
𝑡𝛼−1

Γ(𝛼)
∫ 

1

0

∫ 𝐾(𝑡,𝑠)𝐹(𝑡,𝑠,𝑥𝑚(𝑠),𝑦𝑚(𝑠))𝑑𝑠𝑑𝑠

𝑡

−∞

−∫
(𝑡 −𝑠)𝛼−1

Γ(𝛼)

𝑡

0

(∫ 𝐾(𝑡,𝑠)𝐹(𝑡,𝑠,𝑥𝑚(𝑠),𝑦𝑚(𝑠))𝑑𝑠)𝑑𝑠
𝑡

−∞

+ 
𝑏1𝑡

𝛼−1

Γ(α)

− 
𝑡𝛼−1

Γ(𝛼)
∫ 

1

0

∫ 𝐾(𝑡,𝑠)𝐹(𝑡,𝑠,𝑥 (𝑠),𝑦 (𝑠))𝑑𝑠𝑑𝑠

𝑡

−∞

+ ∫
(𝑡 − 𝑠)𝛼−1

Γ(𝛼)

𝑡

0

(∫ 𝐾(𝑡,𝑠)𝐹(𝑡,𝑠,𝑥 (𝑠),𝑦 (𝑠))𝑑𝑠)𝑑𝑠
𝑡

−∞

−
𝑏1𝑡

𝛼−1

Γ(α)
‖ 

≤ 
𝑡𝛼−1

Γ(𝛼)
∫ ∫𝛿1𝑒

−𝜆1(𝑡−𝑠)‖𝐹(𝑡,𝑠,𝑥𝑚(𝑠),𝑦𝑚(𝑠))−𝐹(𝑡,𝑠,𝑥 (𝑠),𝑦 (𝑠))‖𝑑𝑠𝑑𝑠

𝑡

−∞

 

1

0

−∫
(𝑡 −𝑠)𝛼−1

Γ(𝛼)

𝑡

0

(∫ 𝛿1𝑒
−𝜆1(𝑡−𝑠)‖𝐹(𝑡,𝑠,𝑥𝑚(𝑠),𝑦𝑚(𝑠))

𝑡

−∞

−𝐹(𝑡,𝑠,𝑥 (𝑠),𝑦 (𝑠))‖𝑑𝑠)𝑑𝑠 
 

≤
𝐿1𝛿1(𝛼𝑡

𝛼−1 − 𝑡𝛼)

𝜆1Γ(𝛼+ 1)
(‖𝑥𝑚 − 𝑥 ‖+ ‖𝑦𝑚 − 𝑦 ‖) 

 

And for the function y(t,𝑦0) we have 

‖
𝑡𝛽−1

Γ(𝛽)
∫ 

1

0

∫ 𝐺(𝑡,𝑠)𝐻(𝑡,𝑠,𝑥𝑚(𝑠),𝑦𝑚(𝑠))𝑑𝑠𝑑𝑠

𝑡

−∞

−∫
(𝑡 −𝑠)𝛽−1

Γ(𝛽)

𝑡

0

(∫ 𝐺(𝑡,𝑠)𝐻(𝑡,𝑠,𝑥𝑚(𝑠),𝑦𝑚(𝑠))𝑑𝑠)𝑑𝑠
𝑡

−∞

+
𝑏2𝑡

𝛽−1

Γ(β)

−
𝑡𝛽−1

Γ(𝛽)
∫ 

1

0

∫ 𝐺(𝑡,𝑠)𝐻(𝑡,𝑠,𝑥 (𝑠),𝑦 (𝑠))𝑑𝑠𝑑𝑠

𝑡

−∞

+∫
(𝑡 −𝑠)𝛽−1

Γ(𝛽)

𝑡

0

(∫ 𝐺(𝑡,𝑠)𝐻(𝑡,𝑠,𝑥 (𝑠),𝑦 (𝑠))𝑑𝑠)𝑑𝑠
𝑡

−∞

−
𝑏2𝑡

𝛽−1

Γ(β)
‖  



F. Y. Ishak 
Existence Solution for Nonlinear System of Fractional Integrodifferential Equations of Volterra Type with 

Fractional Boundary Conditions  

9 
 

≤
𝐿2𝛿2(𝛽𝑡

𝛽−1 − 𝑡𝛽)

𝜆2Γ(𝛽+ 1)
(‖𝑥𝑚 − 𝑥 ‖+ ‖𝑦𝑚 −𝑦 ‖) 

         

And since the sequences: {𝑥𝑚(𝑡,𝑥0)}𝑚=0
∞    , {𝑦𝑚(𝑡,𝑦0)}𝑚=0

∞   uniformly convergence to 

𝑥 (𝑡,𝑥0), 𝑦 (𝑡,𝑦0) respectively on the interval [0,1] ,that is (3.7),(3.8) satisfies. 
 

Theorem (3.2): If all conditions and assumptions of theorem (3.1) satisfied, then the 

functions  𝑥 (𝑡,𝑥0), 𝑦(𝑡,𝑦0) are unique solution for system (1.1) on domain (1.2). 
 

Proof: let  
 

(
𝑢 (𝑡,𝑢0)

𝑤 (𝑡,𝑤0)
) =      

    (

𝑡𝛼−1

Γ(𝛼)
∫  
1

0
∫  𝐾(𝑡,𝑠)𝐹(𝑡,𝑠,𝑢 (𝑠),𝑤 (𝑠))𝑑𝑠𝑑𝑠
𝑡

−∞
− ∫

(𝑡−𝑠)𝛼−1

Γ(𝛼)

𝑡

0
(∫ 𝐾(𝑡,𝑠)𝐹(𝑡,𝑠,𝑢 (𝑠),𝑤 (𝑠))𝑑𝑠)𝑑𝑠
𝑡

−∞
 +

𝑏1𝑡
𝛼−1

Γ(α)

 
𝑡𝛽−1

Γ(𝛽)
∫  
1

0
∫  𝐺(𝑡,𝑠)𝐻(𝑡,𝑠,𝑢 (𝑠),𝑤 (𝑠))𝑑𝑠𝑑𝑠
𝑡

−∞
− ∫

(𝑡−𝑠)𝛽−1

Γ(𝛽)

𝑡

0
(∫ 𝐺(𝑡,𝑠)𝐻(𝑡,𝑠,𝑢 (𝑠),𝑤 (𝑠))𝑑𝑠)𝑑𝑠
𝑡

−∞
+
𝑏2𝑡

𝛽−1

Γ(β)

)  

 

be another solution for system (1.1) then: 

 

‖𝑥 (𝑡,𝑥0)−𝑢(𝑡,𝑢0)‖ ≤
𝑡𝛼−1

Γ(𝛼)
∫  
1

0
∫  ‖𝐾(𝑡,𝑠)‖‖𝐹(𝑡,𝑠,𝑥 (𝑠),𝑦 (𝑠))−
𝑡

−∞

𝐹(𝑡,𝑠,𝑢 (𝑠),𝑤 (𝑠))‖𝑑𝑠𝑑𝑠− ∫
(𝑡−𝑠)𝛼−1

Γ(𝛼)

𝑡

0
(∫ ‖𝐾(𝑡,𝑠)‖‖𝐹(𝑡,𝑠,𝑥 (𝑠),𝑦 (𝑠))−
𝑡

−∞

𝐹(𝑡,𝑠,𝑢 (𝑠),𝑤 (𝑠))‖𝑑𝑠)𝑑𝑠  
 

≤
𝐿1𝛿1(𝛼𝑡

𝛼−1 −𝑡𝛼)

𝜆1Γ(𝛼 +1)
(‖𝑥 − 𝑢 ‖ +‖𝑦 −𝑤 ‖)  

       

And  

‖𝑦 (𝑡,𝑦0)−𝑤(𝑡,𝑤0)‖ ≤
𝑡𝛼−1

Γ(𝛼)
∫  
1

0
∫  ‖𝐺(𝑡,𝑠)‖‖𝐻(𝑡,𝑠,𝑥 (𝑠),𝑦 (𝑠))−
𝑡

−∞

𝐻(𝑡,𝑠,𝑢 (𝑠),𝑤 (𝑠))‖𝑑𝑠𝑑𝑠− ∫
(𝑡−𝑠)𝛼−1

Γ(𝛼)

𝑡

0
(∫ ‖𝐺(𝑡,𝑠)‖‖𝐻(𝑡,𝑠,𝑥 (𝑠),𝑦 (𝑠))−
𝑡

−∞

𝐻(𝑡,𝑠,𝑢 (𝑠),𝑤 (𝑠))‖𝑑𝑠)𝑑𝑠      
     

≤
𝐿2𝛿2(𝛽𝑡

𝛽−1 −𝑡𝛽)

𝜆2Γ(𝛽 + 1)
(‖𝑥 −𝑢 ‖+ ‖𝑦− 𝑤 ‖)  

Rewrite in vector form: 
 

(
‖𝑥 (𝑡,𝑥0)−𝑢(𝑡,𝑢0)‖

‖𝑦 (𝑡,𝑦0)−w(𝑡,𝑤0)‖
) ≤ Δ(𝑡)(

‖𝑥(𝑡)−𝑢(𝑡)‖

‖𝑦 (𝑡)− 𝑤(𝑡)‖
)                        (3.9) 

 
By take the maximum value for both sides of (3.9) and reputation it we get: 

 

(
‖𝑥 (𝑡,𝑥0)−𝑢(𝑡,𝑢0)‖

‖𝑦 (𝑡,𝑦0)−w(𝑡,𝑤0)‖
) ≤ Δ0

𝑚 (
‖𝑥(𝑡)−𝑢(𝑡)‖

‖𝑦 (𝑡)− 𝑢(𝑡)‖
)                        (3.10) 

 

From (3.9) and condition (1.8) we have 𝐴0
𝑚 → 0 when 𝑚 → ∞ that is: 

 

𝑥 (𝑡,𝑥0) = 𝑢(𝑡,𝑢0) and 𝑦 (𝑡,𝑦0)− w(𝑡,𝑤0) 
 
Therefor 𝑥 (𝑡,𝑥0), 𝑦(𝑡,𝑦0) is a unique solution for system (1.1). 



Jurnal Matematika MANTIK 
Volume 6, Issue. 1, May 2020, pp. 1-12 

10   
 

Theorem (3.3): Under the hypothesis and conditions of theorem (3.1) if  

�̃�(𝑡,𝑥0), �̃�(𝑡,𝑦0) is any other solution of system (1.1), then the solution is stable if 
satisfies the inequality:  
  

(
‖𝑥 (𝑡,𝑥0)−�̃�(𝑡,𝑥0)‖

‖𝑦 (𝑡,𝑦0)−�̃�(𝑡,𝑦0)‖
) ≤ (

𝜖1
𝜖2
) 

Where: 

�̃�(𝑡,𝑥0) = 
𝑡𝛼−1

Γ(𝛼)
∫  
1

0
∫  𝐾(𝑡,𝑠)𝐹(𝑡,𝑠, �̃� (𝑠),�̃� (𝑠))𝑑𝑠𝑑𝑠
𝑡

−∞
−

                                                         ∫
(𝑡−𝑠)𝛼−1

Γ(𝛼)

𝑡

0
(∫ 𝐾(𝑡,𝑠)𝐹(𝑡,𝑠, �̃� (𝑠), �̃� (𝑠))𝑑𝑠)𝑑𝑠
𝑡

−∞
 +

𝑏1𝑡
𝛼−1

Γ(α)
                                                 

   �̃�(𝑡,𝑦0) =
𝑡𝛽−1

Γ(𝛽)
∫  
1

0
∫  𝐺(𝑡,𝑠)𝐻(𝑡,𝑠, �̃� (𝑠), �̃�(𝑠))𝑑𝑠𝑑𝑠
𝑡

−∞
−

                                                        ∫
(𝑡−𝑠)𝛽−1

Γ(𝛽)

𝑡

0
(∫ 𝐺(𝑡,𝑠)𝐻(𝑡,𝑠, �̃� (𝑠), �̃� (𝑠))𝑑𝑠)𝑑𝑠
𝑡

−∞
+

𝑏2𝑡
𝛽−1

Γ(β)
    

Proof:  

                        ‖𝑥(𝑡,𝑥0) −�̃�(𝑡,𝑥0)‖ ≤
𝐿1𝛿1(𝛼𝑡

𝛼−1−𝑡𝛼)

𝜆1Γ(𝛼+1)
(‖𝑥 −�̃�‖+ ‖𝑦 −�̃�‖)  

 

                          ‖𝑦(𝑡,𝑦0)−�̃� (𝑡,𝑦0)‖ ≤
𝐿2𝛿2(𝛽𝑡

𝛽−1−𝑡𝛽)

𝜆2𝛤(𝛽+1)
(‖𝑥 −�̃�‖+ ‖𝑦− �̃� ‖) 

 

Rewrite (3.11), (3.12) in victor form we get: 

 

(
‖𝑥(𝑡,𝑥0)− �̃�(𝑡,𝑥0)‖

‖𝑦(𝑡,𝑦0)−�̃� (𝑡,𝑦0)‖
) ≤ Δ(𝑡)(

‖𝑥(𝑡) −�̅�(𝑡)‖

‖𝑦(𝑡) −�̅�(𝑡)‖
) 

 

By condition (1.8) and for 𝜖1,𝜖2 ≥ 0 we have: 
 

(
‖𝑥(𝑡,𝑥0)− �̃�(𝑡,𝑥0)‖

‖𝑦(𝑡,𝑦0)− �̃� (𝑡,𝑦0)‖
) ≤ (

𝜖1
𝜖2
)                             (3.13) 

 

By the definition of stability, we find that �̃�(𝑡,𝑥0), �̃� (𝑡,𝑦0) is stable solution for system 
(1.1) 

 
Example (3.1): consider the following system of fractional integrodifferential equations: 

 

𝐷4 3⁄ 𝑥(𝑡) = ∫ (4𝑒2𝑠 +1)
𝑡

−∞

𝑥(𝑠)

𝑦(𝑠)
𝑑𝑠 

𝐷3 2⁄ 𝑦(𝑡) = ∫ 3cos (2𝑠)(𝑦(𝑠)
𝑡

−∞

+ 𝑠𝑖𝑛(𝑥(𝑠)))𝑑𝑠 

𝐷1 3⁄ 𝑥(0) = 0,         𝐷1 3⁄ 𝑥(1) = 2,       𝐷1 2⁄ 𝑥(0) = 0,         𝐷1 2⁄ 𝑥(1) = 3 
 

Comparing (3.14) and (1.1) we see that, 𝛼 = 4 3⁄ ,𝛽 = 3 2⁄ ,𝑘(𝑡,𝑠) = (4𝑒2𝑠 +
1),𝐺(𝑡,𝑠) = 3cos (2𝑠) 
𝐹(𝑡,𝑠,𝑥(𝑠),𝑦(𝑠)) = 𝑥(𝑠) 𝑦(𝑠)⁄ ,𝐻(𝑡,𝑠,𝑥(𝑠),𝑦(𝑠)) = 𝑦(𝑠) +sin(𝑥(𝑠)) , if we choose 𝑀1 = 1,   

𝑀2 = 1,𝐿1 = 2𝑀1,𝐿2 = 2𝑀2,   𝛿1 = 1,𝛿2 = 2,𝜆1 = 2,𝜆2 = 3,then (1.3)− (1.6) holds and ∶ 
 

(3.11) 

(3.12)

(3.14) 



F. Y. Ishak 
Existence Solution for Nonlinear System of Fractional Integrodifferential Equations of Volterra Type with 

Fractional Boundary Conditions  

11 
 

𝜆 𝑚𝑎𝑥(Δ0) =
𝐿1𝛿1(𝛼 −1)

𝜆1Γ(𝛼 + 1)
+
𝐿2𝛿2(𝛽−1)

𝜆2Γ(𝛽+1)
=
(2)(1)(4 3⁄ −1)

(2)( 1.1906)
+
(2)(2)(3 2− 1)⁄

(3)( 1.3293)
< 1 

 

Thus, by Theorems (3.1) -(3.3), we obtain that (1.1) has a unique stability solution.  
 

4. Conclusions 

The article presented some existence and uniqueness results for a boundary value 

problem of fractional integro-differential system. The prove of the theorems based on two 

basic conditions (1.8), (3.1). The basic of fractional differentiation were used to find the 

solution formula. The idea of existence and uniqueness theorem is the basis for finding 
results. The present work can be extended to boundary value problem with nonlocal and 

nonseparated fractional boundary conditions. 

 

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