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How to cite: R. I. Hasan, “Existence, Uniqueness, and Stability Solutions of Non-Linear System of Integral 

Equations”, J. Mat. Mantik, vol. 6, no. 2, pp. 76-82, October 2020. 

 

Existence, Uniqueness, and Stability Solutions of Nonlinear  

System of Integral Equations  
 

 

Rizgar Issa Hasan 

  Duhok Polytechnic University, rizqar.issa@dpu.edu.krd 

 
doi: https://doi.org/10.15642/mantik.2020.6.2.76-82 

 

 
Abstrak. Tujuan dari penelitian ini adalah mempelajari keberadaan, keunikan dan solusi stabilitas 

sistem baru nonlinier persamaan integral dengan menggunakan metode pendekatan Picard 

(pendekatan berurutan) dan teorema titik tetap Banach. Studi tentang persamaan integral nonlinier 

semacam itu bersifat lebih umum dan mendorong kita untuk memperbaiki guna memperluas hasil 

Butris. Teorema tentang keberadaan dan keunikan solusi ditetapkan dalam beberapa kondisi yang 

diperlukan dan cukup pada domain tertutup dan terbatas (ruang kompak). 

 

Kata kunci: Metode pendekatan Picard; Teorema titik tetap Banach; Persamaan integral 

 
Abstract. The aim of this work is to study the existence, uniqueness, and stability solutions of a new 
nonlinear system of integral equation by using Picard approximation (successive approximation) 

method and Banach fixed point theorem. The study of such nonlinear integral equations is more 

general and leads us to improve to extend the result of Butris. Theorems on the existence and 

uniqueness of a solution are established under some necessary and sufficient conditions on closed and 

bounded domains (compact spaces). 
 

Keywords: Picard approximation method; Banach fixed point theorem; Integral equation 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 
  

Jurnal Matematika MANTIK 

Vol. 6, No. 2, October 2020, pp. 76-82 

 

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

http://u.lipi.go.id/1458103791


Rizgar Issa Hasan 

Existence, Uniqueness, and Stability Solutions of Nonlinear System of Integral Equations 

77 

1. Introduction  

Integral equation has been arisen in many mathematical and engineering field, so that 

solving this kind of problems are more efficient and useful in many research branches. 

Analytical solution of this kind of equation is not accessible in general form of equation 

and we can only get an exact solution only in special cases. But in industrial problems we 

have not spatial cases so that we try to solve this kind of equations numerically in general 

format. Many numerical schemes are employed to give an approximate solution with 

sufficient accuracy [3,4,6,10]. Many authors create and develop Successive approximation 

method and Banach fixed point theorem [1, 2,5,7,8,9] and schemes to investigate the 

solution of integral equations describing many applications in mathematical and 

engineering field. 

Burris [1] has been used Picard approximation (successive approximation) method 

and Banach fixed point theorem. which were introduced by Rama [6] to study the solution 

of Volterra integral equation of the second kind which has the form: 

  𝑢(𝑡) =  𝑓(𝑡) +  ∫ 𝐹 (
𝑡 

𝑎
𝑡, 𝑠)𝓊(𝑠 )𝑑𝑠. 

In this equation the functions f(t ) and F(t, s ) are continuous on the interval  0 ≤ t ≤
a and square region  0 ≤ t , s ≤ b. 

Definition 1. Let 
0

{ ( )}
m m

f t


=
 be a sequence of functions defined on a set. E ⊆ 𝑅1 We 

say that 
0

{ ( )}
m m

f t


=
 converges uniformly to the limit function f on E if , given   > 0 

there exists  a positive integer N  such that : 

( ) ( )
m

f t f t− <  ,    ( ,m N t E  ) 

Theorem1. If f is continuous on [ , ]a b  and if ( ) ( )
x

a

F x f t dt=  , a x b  , then ( )F x  is 

also continuous on [ , ]a b   

Definition 2. Let 𝑓 be a continuous function defined on a domain 𝐺 = {(𝑡, 𝑥): 𝑎 ≤ 𝑡 ≤
𝑏, 𝑐 ≤ 𝑥 ≤ 𝑑} . Then 𝑓 is said to satisfy a Lipschitz condition in the variable 𝑥 on 𝐺, 
provided that a constant 𝐿 > 0 exists with the property that 
|𝑓(𝑡, 𝑥1) − 𝑓(𝑡, 𝑥2)| ≤ 𝐿|𝑥1 − 𝑥2|    , 
for all (𝑡, 𝑥1), (𝑡, 𝑥2 ) ∈ 𝐺. The constant 𝐿 is called a Lipschitz constant for 𝑓.   
Definition 3. A solution 𝑥(𝑡) is said to be stable if for each > 0 , 
There exists a 𝛿 > 0 such that any solution �̅�(𝑡) which satisfies  
‖�̅�(𝑡0) − 𝑥(𝑡0)‖ < 𝛿     for some 𝑡0 , also satisfies 
‖�̅�(𝑡) − 𝑥(𝑡)‖ <          for all 𝑡 ≥ 𝑡0 . 
Definition 4. Let E be a vector space a real-valued function ‖. ‖ of  𝐸 into 𝑅1 called a norm 
if satisfies: 

I. ‖𝑥‖ ≥ 0   for all 𝑥 ∈ 𝐸, 
II. ‖𝑥‖ = 0   if and only if  𝑥 = 0, 
III. ‖𝑥 + 𝑦‖ ≤ ‖𝑥‖ + ‖𝑦‖  for all 𝑥, 𝑦 ∈ 𝐸, 
IV. ‖𝛼 𝑥‖ = |𝛼| ‖𝑥‖ for all 𝑥 ∈ 𝐸 and 𝛼 ∈ 𝑅. 

Definition 5. A linear space 𝐸 with a norm defined on it is called a normed space. 
Definition 6. A normed linear space  𝐸  is called complete if every Cauchy sequence in 𝐸 
converges to an element in. 

Definition 7. A complete normed linear space is a Banach space. 

Definition 8. if  𝑇 maps 𝐸 into itself and 𝑧 is a point of 𝐸 such that 𝑇𝑧 = 𝑧 , then 𝑧 is a 
fixed point of  𝑇 . 
Definition 9. Let (C [0, T], ‖. ‖ ) be a norm space if 𝑇 maps into itself we say that 𝑇 is a 
contraction mapping on C [0,T] if there exists  𝛼 ∈ 𝑅 with 0 < 𝛼 < 1  such that  
 ‖𝑇𝑥 − 𝑇𝑦‖ ≤ 𝛼‖𝑥 − 𝑦‖,        (𝑥, 𝑦) ∈  C [0, T].                              



Jurnal Matematika MANTIK 

Volume 6, No. 2, October 2020, pp. 76-82 

78 

Theorem 2. Let 𝐸 be a Banach space, if  𝑇 is a contraction mapping on 𝐸 then 𝑇 has one 
and only one fixed point in. 

(For the definitions and theorems see [1,5,6]). 

Our work is extending some results of Butris [1], by using the same method above. 

Consider the following integral equation: 

𝑢(𝑡) =  𝑢0 +  ∫  𝐹(𝑡, 𝑠)𝑔(𝑠, 𝑢 (𝑠))𝑑𝑠 +
𝑡

0
   ∫ 𝐾 (

𝑏(𝑡)

𝛼(𝑡)
𝑡, 𝑠)𝓊(𝑠)𝑑𝑠   (1) 

Where t∈ 𝐷 ⊆ 𝑅1,D is a compact set. 
Suppose the functions 𝑔(t, u), 𝑎(t) and 𝑏(t) are defined, continuous on the domain:- 
𝐷= {(t,s) : 0 ≤s≤ t ≤ b }         (2) 
Also the function 𝑔(t, u) and kernels  F(t, s) and K(t,s) satisfied  the following inequalities: 
‖𝑔(𝑡, 𝓊)‖ ≤ M          (3)                                 
‖𝑔(𝑡, 𝓊1) –  𝑔(𝑡, 𝓊2)‖ ≤  𝐿 ‖𝓊1 –  𝓊2‖       (4)               

 ∫  
𝑏(𝑡)

𝛼(𝑡)
‖ K(t, s)‖ds ≤ Kh , h = ‖𝑏(𝑡) − 𝑎(𝑡)‖      (5)                                                

‖ F(t, s)‖   ≤ 𝑁         (6) 
where F(t, s) and K(t,s) are kernels  of the integral  equation (1) and M, L,K , 𝑁 > 0.  
      Define a sequence of functions {𝑢𝑚(𝑡)}𝑚=0

∞  by 

𝑢𝑚+1 (𝑡 )=𝓊0  + ∫  𝐹(𝑡, 𝑠)𝑔(𝑠, 𝑢𝑚 (𝑠))𝑑𝑠
𝑡

0
+ ∫ 𝐾(𝑡, 𝑠)𝑢𝑚(𝑠)𝑑𝑠

𝑏(𝑡)

𝑎(𝑡)
   (7) 

with 

 𝓊0(0) = 𝓊0 , m =0, 1, 2,  ... . 
Also define a non-empty set as follows: 

𝐷𝑓 = D - (MNb + Kh δ0), ‖𝑢0‖ ≤ δ0       (8)                     
 

2. Existence and uniqueness of solution of (1) 

In this section, we study the existence and uniqueness of equation (1) by using Picard 

approximation method (Successive approximation). 

Theorem 3.   Suppose that the integral equation (1) satisfying the inequalities (2), (3), (4), 

(5), (6), and relation (8). Then there exists a sequence of functions (7) converges uniformly 

to the limit functions u = u(t) which is define by the integral equation  

𝑢(𝑡) =  𝑢0  +  ∫  𝐹(𝑡, 𝑠) 𝑔 (𝑡, 𝑢(𝑠))𝑑𝑠 + ∫ 𝐾 (𝑡, 𝑠)𝑢(𝑠)𝑑𝑠
𝑏(𝑡)

𝑎(𝑡)

𝑡

0
    (9)                                                    

Which is a unique solution of (1). 

 Proof for m= 0 in (7), we have  

‖𝑢1(𝑡) − 𝑢0‖  ≤  ∫   ‖𝑔(𝑠, 𝑢0)‖

𝑡

0

 ‖𝐹(𝑡, 𝑠)‖ 𝑑𝑠 + 𝐾ℎ 𝛿0  

 ≤  𝑀𝑁𝑏  + 𝐾ℎ 𝛿0 
So that 
‖𝑢1(𝑡) −  𝑢0‖  ≤ 𝑀𝑁𝑏  + 𝐾ℎ 𝛿0                         
for all t∈ [0, 𝑏], i. e.  
𝑢1(𝑡) ∈ 𝐷 for all  and 𝑢0  𝜖 𝐷𝑓 . 

By mathematical induction, we can prove that 𝑢m(t) ∈  D , for all [0, 𝑏]  and 𝑢0  𝜖 𝐷𝑓 , m = 

1,2,3,....  

That is 

   ‖𝑢𝑚(𝑡) −  𝑢0‖  ≤ 𝑀𝑁𝑏  + 𝐾ℎ 𝛿0       (10) 
for all 𝑡𝜖  [0, 𝑏]  and 𝑢0  𝜖 𝐷𝑓. 

Next, we shall prove that (7) convergent uniformly on D. 

For m = 1 in (7), we have  
‖𝑢2(𝑡) −  𝑢1(𝑡)‖ ≤ (𝐿𝑁𝑏 + 𝐾ℎ)‖𝑢1(𝑡) −  𝑢0‖ 
For m = 2 in (8), we have  



Rizgar Issa Hasan 

Existence, Uniqueness, and Stability Solutions of Nonlinear System of Integral Equations 

79 

‖𝑢3(𝑡) −  𝑢2(𝑡)‖ ≤ (𝐿𝑁𝑏 + 𝐾ℎ)
2

𝑜
‖𝑢1(𝑡) −  𝑢0‖ 

And so on, by mathematical induction, we have 

‖𝑢𝑚+1(𝑡) −  𝑢𝑚(𝑡)‖ ≤ (𝐿𝑁𝑏 + 𝐾ℎ)
𝑚

𝑜
‖𝑢1(𝑡) −  𝑢0‖ 

Suppose that 𝛿 =( 𝐿𝑁𝑏 + 𝐾ℎ) < 1. Then 

∑‖𝑢𝑚+1(𝑡) −  𝑢𝑚 (𝑡)‖  ≤ (1 +  𝛿 + 𝛿
2 + 𝛿 3 + ⋯ 𝛿 𝑚 + ⋯ ) ‖𝑢1(𝑡) − 𝑢0‖  ≤  

 1 

1 − 𝛿
 ‖𝑢1 − 𝑢0‖

𝑘

𝑖=1

 

Therefore, the sequence of functions {𝑢𝑚(𝑡)}𝑚=0
∞  converges uniformly on the domain D. 

Now, we shall prove that 𝑢(t) ∈C(D). Taking  

𝑢(𝑡) = 𝑢0 +  ∫  𝐹 (𝑡, 𝑠)𝑔(𝑠, 𝑢(𝑠))𝑑𝑠 + ∫ 𝐾(𝑡, 𝑠)𝑢(𝑠)𝑑𝑠

𝑏(𝑡)

𝑎(𝑡)

 
𝑡

0

 

And  

𝑢𝑚(𝑡) = 𝑢0 +  ∫ 𝐹 (𝑡, 𝑠) 𝑔(𝑠, 𝑢𝑚(𝑠)) 𝑑𝑠 + ∫ 𝐾(𝑡, 𝑠)𝑢𝑚−1(𝑠)𝑑𝑠

𝑏(𝑡)

𝑎(𝑡)

 
𝑡

0

 

Therefore, 

  

‖𝑢 (𝑡) −  𝑢𝑚(𝑡)‖ ≤ (𝐿𝑁𝑏 + 𝐾ℎ)
𝑚

𝑜
‖𝑢𝑚(𝑡) − 𝑢 (𝑡)‖ 

 

Since {𝑢𝑚(𝑡)}𝑚=0
∞ , is convergent uniformly, then lim

𝑚→∞
𝑢𝑚(𝑡) = 𝑢(𝑡). 

i.e. ‖𝑢𝑚(𝑡) − 𝑢(𝑡)‖ ≤∈1 Choosing ∈1=  
∈

𝐿𝑁𝑏+𝐾ℎ
 , we get  

‖𝑢𝑚(𝑡) − 𝑢(𝑡)‖  < ∈ 
and hence  

u(t) ∈ C(D), for all 𝑡𝜖  [0, 𝑏]  and 𝑢0  𝜖 𝐷𝑓 . 

Finally, we shall prove that 𝑢(t) is a unique solution of (1). Suppose that  

�̅�(𝑡) =  𝑢0  + ∫ 𝐹(𝑡, 𝑠) 𝑔(𝑠, �̅�(𝑠))𝑑𝑠 +  ∫ 𝐾(𝑡, 𝑠)�̅�(𝑠)𝑑𝑠
𝑏(𝑡)

𝑎(𝑡)

𝑡

0

 

is another solution of (1). 

For m = 1 in (8), we have 

  
‖𝑢1(𝑡) − �̅�(𝑡)‖ ≤ (𝐿𝑁𝑏 + 𝐾ℎ)‖�̅�(𝑡) − 𝑢0‖ 
  

By mathematical induction, we have  

‖𝑢𝑚(𝑡) − �̅�(𝑡)‖ ≤  𝛿 
𝑚

𝑜
‖�̅�(𝑡) − 𝑢0‖ ≤

  1  

1 − 𝛿
 ‖𝑢1 − 𝑢0‖ 

Since   𝛿 < 1, then 
lim𝑚→∞ 𝑢𝑚 = �̅�(𝑡) = 𝑢(𝑡) 
Thus �̅�(t) = 𝑢(t)  
And hence 𝑢(t) is a unique solution of (1). 
 

3. Stability solution of integral equation (1) 

In this section we study the stability solution of (1) by the following theorem: 

Theorem 4. If the inequalities (3), (4), (5)  and relation (7) are satisfied  and 𝑤(𝑡) is another 
solution of (1), then the solution u(t) is stable if satisfies the following inequality: 
‖u(t) –  ω(t)‖ <∈, for all t >  0 and ∈ >  0 
where  



Jurnal Matematika MANTIK 

Volume 6, No. 2, October 2020, pp. 76-82 

80 

𝜔(𝑡) = 𝜔0  + ∫ 𝐹(𝑡, 𝑠) 𝑔(𝑠, 𝜔(𝑠))𝑑𝑠 + ∫ 𝐾(𝑡, 𝑠)𝜔(𝑠)𝑑𝑠

𝑏(𝑡)

𝑎(𝑡)

 
𝑡

0

 

Proof. Consider  

‖𝑢(𝑡) –  𝜔(𝑡)‖  ≤  ‖𝑢0 − 𝜔0‖ + ∫  𝐿𝑁
𝑡

0
‖𝑢(𝑠) − 𝜔(𝑠)‖𝑑𝑠 + ∫  

𝑏(𝑡)

𝑎(𝑡)
𝐾‖𝑢(𝑠) − 𝜔(𝑠)‖ds 

 

≤  ‖𝑢0 − 𝜔0‖ + ( 𝐿𝑁𝑏 +Kh) ‖𝑢(𝑡) − 𝜔(𝑡)‖ 
 So 

‖𝑢(𝑡) –  𝜔(𝑡)‖  ≤  
  1  

1 − 𝛿
 || 𝑢0 –  𝜔0||   

Putting ∈2   =
  ∈   

|| 𝑢0 – 𝜔0|| ( 1−𝛿)
 ,then 

 ‖𝑢(𝑡) –  𝜔(𝑡)‖ ≤ ∈. 
  So that the solution 𝑢(𝑡) is stable for all 𝑡  ≥ 0. 

 

4. Another method of a solution of the integral equation (1) 

In this section, we study the existence and uniqueness solution of (1) by using Banach 
fixed point theorem. 

Theorem 5.  Let all assumptions and conditions of Theorem 4 be satisfied. Then the 

solution u(t) is a unique of (1) . 
Proof. Let (C[0, b], ‖. ‖ ) be Banach space. Define a mapping   T on C[0, b] by: 

T 𝑢(t) = 𝑢0  + ∫  𝐹(𝑡, 𝑠)𝑔(𝑠, 𝑢(𝑠))𝑑𝑠 + ∫ 𝐾(𝑡, 𝑠)𝑢(𝑠)𝑑𝑠
𝑏(𝑡)

𝑎(𝑡)
 

𝑡

0
 

Since 𝑔(t,𝑢) and  the kernels F(t, s) and K(t,s) are continuous on the domain(2),then 

∫  𝐹(𝑡, 𝑠)𝑑𝑔(𝑠, 𝑢(𝑢))𝑠 + ∫ 𝐾(𝑡, 𝑠)𝑢(𝑠)𝑑𝑠
𝑏(𝑡)

𝑎(𝑡)
)

𝑡

0
 is also continuous on the same domain.  

i.e. T: C[0, b]  →  C[0, b]. 
Now, we claim that T is a contraction mapping on C[0, b]. 
Let 𝑢(t) and �̅�(𝑡) ∈ D, then  

‖T𝑢(𝑡) − T�̅�(𝑡)‖  ≤ ∫ 𝐿𝑁 ‖𝑢(𝑠) − �̅�(𝑠)‖𝑑𝑠
𝑡

0
+∫ 𝐾‖𝑢(𝑠) −  �̅�(𝑠)‖𝑑𝑠 

𝑏(𝑡)

𝑎(𝑡)
 

                                   ≤  𝛿‖𝑢(𝑠) −  �̅�(𝑠)‖ 
Thus  
‖T𝑢(𝑡) − T�̅�(𝑡)‖ ≤ 𝛿‖𝑢(𝑡) − �̅�(𝑡)‖ 
Since 𝛿 <  1, therefore T is a contraction mapping on C[0, b].   
Then T𝑢(t) = 𝑢(t) and  

𝑢(𝑡) = 𝑢0  + ∫ 𝐹(𝑡, 𝑠) 𝑔(𝑠, 𝑢(𝑠))𝑑𝑠 + ∫ 𝐾(𝑡, 𝑠)𝑢(𝑠)𝑑𝑠

𝑏(𝑡)

𝑎(𝑡)

 
𝑡

0

 

Hence 𝑢(𝑡) is a unique continuous solution of (1).  
Remark. The Picard approximation method given global solution but  Banach fixed point 

theorem  give us the local  solution of the integral equation(1). 

 

5. Some examples only in successive approximation. 

Example A. Consider the following system of integral equation: 

u(t) =  u0 +  ∫  F(t, s)g(s, u (s))ds +
t

0
   ∫ K (

b(t)

α(t)
t, s)𝓊(s)ds   

with    

𝑢0 = 0.1 , I = (0,2), s =3.22      
F(t,s)=2.23ln(t+0.55), g(s,u(s))= u(s)+4.22s 

and 



Rizgar Issa Hasan 

Existence, Uniqueness, and Stability Solutions of Nonlinear System of Integral Equations 

81 

K(t,s)=2.23(2𝑡−1-6.23), a(t)=t, b(t)=t+h,h=0.01 

 
Figure 1. Successive approximation of global solution of (1) 

 

Example B. Consider the same system of integral equation but local interval and another 

functions. 

u(t) =  u0 +  ∫  F(t, s)g(s, u (s))ds +
t

0

   ∫ K (
b(t)

α(t)

t, s)𝓊(s)ds 

with 

u0 = 0.1 ,  I= [0,2] ,s=3.22 
and 

F(t,s)=5.23e−s
2
, g(s,u(s))= 𝑠2-3s 

 

K(t,s)= 5.23tan(t + 2) − 6.23, a(t)=t , b(t)=t+h ,h=0.001. 

 
Figure 2. Successive approximation of local solution of (1) 

Remark.  Successive approximation using MATLAB are show in figure (A) and (B). 



Jurnal Matematika MANTIK 

Volume 6, No. 2, October 2020, pp. 76-82 

82 

6. Conclusions 

         This paper provided the existence, uniqueness, and stability solution for non-linear 

system of integral equations.  Picard approximation (Successive approximation) method 

and Banach fixed point theorem have been used in this study which were introduced by [7]. 

Thus, the non-linear integral equations that we have introduced in the study become more 

general and detailed than those introduced by Butris [1]. 

 

7. Acknowledgement 

The author is grateful to the reviewers for their suggestions and Prof. Dr. Raad N. 

Butris for his scientific orientations, advices, and support 

 

References 

[1] Butris, R. N., Solutions for the Volterra integral equations of the second kind, thesis, 
M.Sc., university of Mosul, Mosul, Iraq, 1984. 

[2] Butris, R. N., and Rafeq, A. Sh., Existence and uniqueness solution for nonlinear 
Volterra Integral   Equation, J. Duhok Univ. Vol. 14, No. 1, (Pure and Eng. Sciences), 

2011. 

[3] Butris, R. N. and Hussen Abdul-Qader, M. A.  Some results in theory integro-
differential equation of fractional order”, Iraq, Mosul, J. of Educ. And sci, Vol. 49, 

2001. 

[4] Goma, I.A. Method of successive approximations in a two-point boundary problem 
with parameter. Ukr Math J 29, 594–599, 1977. https://doi.org/10.1007/BF01085968 

[5] Plaat, O., “Ordinary differential equations”, Holden Day, Inc. San Francisco, 
Cambridge, London, Amsterdam, 1971 

[6] Rama, M. M., “Ordinary differential equations theory and applications”, United 
Kingdom, 1981. 

[7] Manouchehr Kazemi & Reza Ezzati, Existence of solutions for some nonlinear 
Volterra integral equations via Petryshyn’s fixed point theorem, Int. J. Nonlinear 

Anal. Appl., Vol. 9 No. 1, 2018. 

[8] Pakhshan   Hasan & Nejmaddin Abdulla, Existence and uniqueness of solution for 
linear mixed Volterra-Fredholmintegral equations in Banach Space, American 

Journal of Computational and Applied Mathematics, 9(1), 2019. 

[9] Struble, R. A., “Nonlinear differential equations”, McGraw-Hill Book Company, Inc, 
New York, 1962. 

[10] Tricomi, F. G., Integral Equations, Turin University, Turin, Italy, 1965.