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Existence and Uniqueness Solution for System of Nonlinear Fractional
Integro-Differential Equation
Azhaar H. Sallo
[email protected]
Department of Mathematics, Faculty of Science, University of Duhok,
Kurdistan Region, Iraq.
Abstract
This article investigates a system of nonlinear fractional integro – differential equations with initial
conditions. Some new existence and uniqueness results are obtained by applying Picard approximation
method.
Keywords: Nonlinear Fractional Integro- Differential Equation, Existence and Uniqueness Solution, Caputo Fractional
Derivative, Picard Approximation Method.
1 Introduction
In the past few years, fractional order models are found to be more adequate than integer order models
for some real word problems. Fractional derivatives provide an excellent tool for the description of memory
and hereditary properties of various materials and processes. Integro-differential equations arise in many
engineering and scientific disciplines often as approximation to partial differential equations, which represent
much of the continuum phenomena. The existence and uniqueness problems of fractional nonlinear
differential and integro-differential equations as a basic theoretical part of some applications are investigated
by many authors (see for examples [3], [6], [5]and [7]).
Most of the practical systems are integro-differential equations in nature and hence the study of integro-
differential equations is very important . Exact solutions of integral equations play an important role in the
proper understanding of qualitative features of many phenomena and processes in various areas of natural
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science, it can be used to verify the consistency and estimate errors of various numerical, asymptotic, and
approximate methods. the fractional integro-differential equations have been studied by many authors (
[1],[2],[4],[9],[10],and [11]].
In this work, we considered the following system of nonlinear fractional integro-differential equations
of the form:
1
0
1
2
0
2
, , , , ,
01
, , , , ,
(0)
b
b
x t f t x t y t k t s x s y s ds
x c
y t g t x t y t k t s x s y s ds
y c
Where 0 1, 0 s t b , 1 2,c c R .
Let the vectors functions , ,f t x t y t , , ,g t x t y t , 1 , , ,k t s x t y t and
2 , , ,k t s x t y t
are defined and continuous in the domain
* *
1 2, , , 0, 0, (2)t s x y b b D D
Where *
1D and *
2D are closed and bounded domains subsets of Euclidean space nR . Suppose that the vectors
functions satisfying the following inequalities:
1A There exists constants 1 2,M M such that 1 1 2, , , , ,f t x y M and k t s x y M
2A There exists constants 3 4,M M such that 3 2 4, , , , ,g t x y M and k t s x y M
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3A There exists constants 1 2,L L such that 2 2 1 1 1 2 1 2 1, , , ,f t x y f t x y L x x y y
1 2 2 1 1 1 2 2 1 2 1, , , , , ,and k t s x y k t s x y L x x y y
4A There exists constants 3 4,L L such that 2 2 1 1 3 2 1 2 1, , , ,g t x y g t x y L x x y y
2 2 2 2 1 1 4 2 1 2 1, , , , , ,and k t s x y k t s x y L x x y y
for all *
1 2 10, , , ,t b x x x D and *
1 2 1, ,y y y D , where 1 2 3 4, , ,M M M M are positive constant
vectors 1 2 3, , ,L L L and 4L are positive constant matrices , and also 0,
. max .t b
.
We define the non- empty sets:
1 2
1 1
3 4
2 2
13
1
M M b bD D
M M b bD D
Furthermore, we suppose that the greatest eigenvalue max of the matrix:
1 2 1 2
3 4 3 4
1 1
1 1
L bL t L bL t
tL bL t L bL t
, does not exceed unity, i.e.:
1 2 3 4
max 1 (4)1 1
L bL b L bL bt
2 Preliminaries
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In this section, we introduce some definitions, and some fundamental facts of Caputo’s derivative of
fractional order (see [13], [14]).
Definition 2.1. Caputo's derivative for a function : 0,f R can been written as
( )
1
0
1 ( ), 1
( )
t nt
o n
f t dtD f t n
n t s
Where denotes the integer part of real number .
Definition 2.2. The integral
1
0
1 ( ), 0
( )
tt
o s
f tI f t dt t
t s
Where 0, is called Riemann-Liouville fractional integral of order .
Definition 2.3. A sequence 1n
f of functions defined on a set D is said to converge uniformly on D if given
0, there is a positive integer number 0N such that for all t D and all 0 ,n N we have
( ) ( )nf t f t .
Lemma 2.1. Let 0, , then:
2 1
0 1 2 1( ) ( ) ... ,t s n
a a nD D y t y t c c t c t c t for some ic R , i=0, 1, 2, …, n-1, 1n .
3 The Main Results.
The study of the existence and uniqueness solution for the system of nonlinear fractional integro-
differential equations (1) will be introduced by the theorems:
Theorem 3.1. (Existence theorem)
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Let 1 2( , , ) , ( , , ), ( , , , ) ( , , , )f t x y g t x y k t s x y and k t s x y be vector functions which are defined and
continues on the domains (2). If the hypotheses 1A -
4A are satisfied, then the system of nonlinear
fractional integro-differential equations (1) has a solution 0 0
0 0
, ,
, ,
x t x y
y t x y which defined as:
1
0 0 1 0 0 0 0 1 0 0 0 0
0 0
1
0 0 2 0 0 0 0 2 0 0 0 0
0 0
1, , , , , , , , , , , , , , ,
1, , , , , , , , , , , , , , ,
t b
t b
x t x y c t s f s x s x y y s x y k s x x y y x y d ds
y t x y c t s f s x s x y y s x y k s x x y y x y d ds
Proof: Consider the sequences of the functions 0 0 0
( , , )m mx t x y and
0 0 0( , , )m m
y t x y are continuous and
defined by:
1
1 0 0 1 0 0 0 0
0
1
1 0 0 0 0
0 0
1, , , , , , , ,
1, , , , , , , (5)
t
m m m
t b
m m
x t x y c t s f s x s x y y s x y ds
t s k s x x y y x y d ds
1
1 0 0 1 0 0 0 0
0
1
2 0 0 0 0
0 0
1, , , , , , , ,
1, , , , , , , (6)
t
m m m
t b
m m
y t x y c t s g s x s x y y s x y ds
t s k s x x y y x y d ds
The proof will be given in several steps.
Step 1: We shall to prove *
0 0 1( , , )mx t x y D and *
0 0 2( , , )my t x y D for all 0,t b , 0 1x D and 0 2y D
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When m= 1 and using1A ,
1A , we have:
1
1 0 0 1 0 0 0 0 0 0
0
1
1 0 0 0 0 0 0
0 0
1( , , ) , , , , , ,
1, , , , , , ,
t
t b
x t x y c t s f s x s x y y s x y ds
t s k s x x y y x y d ds
1 2
1
M bM t
that is ,
1 2
1 0 0 1( , , )1
M bM bx t x y c
therefore
1
1 0 0 2 0 0 0 0 0 0
0
1
2 0 0 0 0 0 0
0 0
1( , , ) , , , , , ,
1, , , , , , ,
t
t b
y t x y c t s g s x s x y y s x y ds
t s k s x x y y x y d ds
3 4
1
M bM t
that is, 3 4
1 0 0 2( , , )1
M bM by t x y c
so that, *
1 0 0 1( , , )x t x y D and
*
1 0 0 2( , , )y t x y D for all 0 1 0 2, , 0,x D y D t b .
Therefore by mathematical induction, we get
1 2
0 0 1( , , )1
m
M bM bx t x y c and 3 4
0 0 1( , , )1
m
M bM by t x y c
In the others words
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*
0 0 1 0 1
*
0 0 1 0 2
( , , ) , , 0,
( , , ) , , 0, , 0,1, 2,...
m
m
x t x y D x D t b
y t x y D y D t b m
Step 2: we claim that the sequences of functions (5) and (6) are uniformly convergent on the domain (2).
For m=1 and using3A ,
4A . We find that:
1
2 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0
0
1( , , ) ( , , ) , , , , , , , , , , , ,
t
x t x y x t x y t s f s x s x y y s x y f s x s x y y s x y ds
1
1 1 0 0 1 0 0 1 0 0 0 0 0 0
0 0
1, , , , , , , , , , , , , ,
t b
t s k s x x y y x y k s x x y y x y d ds
1
1 1 0 0 0 0 0 1 0 0 0 0 0
0
1, , , , , , , ,
t
t s L x s x y x s x y y s x y y s x y ds
1
2 1 0 0 0 0 0 1 0 0 0 0 0
0 0
1, , , , , , , ,
t b
t s L x x y x x y y x y y x y d ds
1 2
2 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0( , , ) ( , , ) , , , , , , , ,1
L bL tx t x y x t x y x s x y x s x y y s x y y s x y
and
1
2 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0
0
1( , , ) ( , , ) , , , , , , , , , , , ,
t
y t x y y t x y t s g s x s x y y s x y g s x s x y y s x y ds
1
2 1 0 0 1 0 0 2 0 0 0 0 0 0
0 0
1, , , , , , , , , , , , , ,
t b
t s k s x x y y x y k s x x y y x y d ds
1
3 1 0 0 0 0 0 1 0 0 0 0 0
0
1, , , , , , , ,
t
t s L x s x y x s x y y s x y y s x y ds
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1
4 1 0 0 0 0 0 1 0 0 0 0 0
0 0
1, , , , , , , ,
t b
t s L x x y x x y y x y y x y d ds
3 4
2 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0( , , ) ( , , ) , , , , , , , ,1
L bL ty t x y y t x y x s x y x s x y y s x y y s x y
So that
1 2 1 2
1 0 0 0 0 0 0 1 0 0 0 0 1 0 0( , , ) ( , , ) , , , , , , , , (7)1 1
m m m m m m
L bL t L bL tx t x y x t x y x s x y x s x y y s x y y s x y
3 4 3 4
1 0 0 0 0 0 0 1 0 0 0 0 1 0 0( , , ) ( , , ) , , , , , , , , (8)1 1
m m m m m m
L bL t L bL ty t x y y t x y x s x y x s x y y s x y y s x y
Rewrite the inequalities (7) and (8) as a vectors:
1 2 1 2
0 0 1 0 0
1 0 0 0 0
1 0 0 0 0 3 4 3 4
0 0 1 0 0
, , , ,( , , ) ( , , ) 1 1
( , , ) ( , , ), , , ,
1 1
m m
m m
m m
m m
L bL t L bL tx s x y x s x y
x t x y x t x y
y t x y y t x y L bL t L bL ty s x y y s x y
or
( 1) 0 0 0 0, , , , 9m mV t x y t V t x y
1 0 0 0 0
( 1) 0 0
1 0 0 0 0
( , , ) ( , , ), ,
( , , ) ( , , )
m m
m
m m
x t x y x t x yV t x y
y t x y y t x y
1 2 1 2
3 4 3 4
1 110
1 1
L bL t L bL t
tL bL t L bL t
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0 0 1 0 0
0 0
0 0 1 0 0
, , , ,
, ,
, , , ,
m m
m
m m
x s x y x s x y
V t x y
y s x y y s x y
from (9) we get:
( 1) 0 ( ) 11m mV t t V t
Where 00,
max ( )t b
t .To prove the sequences (5) and (6) uniformly convergent , we must find the
eigenvalue of the matrix (10) as the follows:
1 2 1 2
3 4 3 4
1 10
1 1
L bL b L bL b
L bL b L bL b
2
1 2 3 4 1 2 3 4
20
1 1 1
L bL b L bL b L bL L bL b
or 3 4 1 2
21 1
L bL b L bL b
By iteration we find:
1 2
( 1) 0 0 0
3 4
1, 12
1
m
m
M bM b
V V VM bM b
hence we obtain ,
1
0 0
1 1
(13)m m
i
i
i i
V V
1 0
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Since the matrix 0 has greatest eigenvalue max 0
and by (4) we find the sequences (5) and (6)
uniformly convergent to functions 0 0, ,x t x y and
0 0, ,y t x y .i.e. :
0 0 0 0
0 0 0 0
lim , , , ,
lim , , , ,
mm
mm
x t x y x t x y
y t x y y t x y
Step 3: We have to show that *
0 0 1, ,x t x y D and *
0 0 2, ,y t x y D , 0 1 0 2, , 0,x D y D t b
Indeed, it is enough to show that
1
0 0 0 0 1 0 0 0 0
0 0
1
0 0 0 0 2 0 0 0 0
0 0
1, , , , , , , , , , , , ,
lim1
, , , , , , , , , , , , ,
t b
m m m m
t bm
m m m m
t s f s x s x y y s x y k s x x y y x y d ds
t s g s x s x y y s x y k s x x y y x y d ds
1
0 0 0 0 1 0 0 0 0
0 0
1
0 0 0 0 2 0 0 0 0
0 0
1, , , , , , , , , , , , ,
1, , , , , , , , , , , , ,
t b
t b
t s f s x s x y y s x y k s x x y y x y d ds
t s g s x s x y y s x y k s x x y y x y d ds
1
0 0 0 0 0 0 0 0
0
1, , , , , , , , , , , ,
t
m mt s f s x s x y y s x y f s x s x y y s x y ds
1
1 0 0 0 0 1 0 0 0 0
0 0
1, , , , , , , , , , , , , ,
t b
m mt s k s x x y y x y k s x x y y x y d ds
1
0 0 0 0 0 0 0 0
0
1, , , , , , , , , , , ,
t
m mt s f s x s x y y s x y f s x s x y y s x y ds
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1
1 0 0 0 0 1 0 0 0 0
0 0
1, , , , , , , , , , , , , ,
t b
m mt s k s x x y y x y k s x x y y x y d ds
11 2
0 0 0 0 0 0 0 0
0
, , , , , , , ,
t
m m
L bLt s x s x y x s x y y s x y y s x y ds
1 2
0 0 0 0 0 0 0 0, , , , , , , ,m m
L bL tx s x y x s x y y s x y y s x y
Similarly
1
0 0 0 0 0 0 0 0
0
1, , , , , , , , , , , ,
t
m mt s g s x s x y y s x y g s x s x y y s x y ds
1
2 0 0 0 0 2 0 0 0 0
0 0
1, , , , , , , , , , , , , ,
t b
m mt s k s x x y y x y k s x x y y x y d ds
3 4
0 0 0 0 0 0 0 0, , , , , , , ,m m
L bL tx s x y x s x y y s x y y s x y
Since the sequences 0 0 0, ,m m
x t x y and 0 0 0, ,m m
y t x y uniformly convergent to the functions 0 0, ,x t x y
and 0 0, ,y t x y respectively on the interval 0,b , then
1
0 0 0 0 1 0 0 0 0
0 0
1, , , , , , , , , , , , ,
t b
unif
m m m mt s f s x s x y y s x y k s x x y y x y d ds
1
0 0 0 0 1 0 0 0 0
0 0
1, , , , , , , , , , , , ,
t b
t s f s x s x y y s x y k s x x y y x y d ds
1
0 0 0 0 2 0 0 0 0
0 0
1, , , , , , , , , , , , ,
t b
unif
m m m mt s g s x s x y y s x y k s x x y y x y d ds
1
0 0 0 0 2 0 0 0 0
0 0
1, , , , , , , , , , , , ,
t b
t s g s x s x y y s x y k s x x y y x y d ds
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Therefore
1
0 0 0 0 1 0 0 0 0
0 0
1
0 0 0 0 2 0 0 0 0
0 0
1, , , , , , , , , , , , ,
lim1
, , , , , , , , , , , , ,
t b
m m m m
t bm
m m m m
t s f s x s x y y s x y k s x x y y x y d ds
t s g s x s x y y s x y k s x x y y x y d ds
1
0 0 0 0 1 0 0 0 0
0 0
1
0 0 0 0 2 0 0 0 0
0 0
1, , , , , , , , , , , , ,
1, , , , , , , , , , , , ,
t b
t b
t s f s x s x y y s x y k s x x y y x y d ds
t s g s x s x y y s x y k s x x y y x y d ds
We deduce that 0 0
0 0
, ,
, ,
x t x y
y t x y is a solution for the system of nonlinear fractional integro-differential
equations (1).
Theorem 3.2. (Uniqueness theorem)
Assume that the hypotheses 1 4A A are satisfying. Then the system of nonlinear fractional integro-differential
equations (1) has a unique solution.
Proof : Let us consider 0 0
0 0
, ,
, ,
p t x y
q t x y to be another solution of the system (1), that is
1
0 0 1 0 0 0 0 1 0 0 0 0
0 0
1
0 0 2 0 0 0 0 2 0 0 0 0
0 0
1, , , , , , , , , , , , , , ,
1, , , , , , , , , , , , , , ,
t b
t b
p t x y c t s f s p s x y q s x y k s p x y q x y d ds
q t x y c t s f s p s x y q s x y k s p x y q x y d ds
hence we have
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1
0 0 0 0 0 0 0 0 0 0 0 0
0
1 0 0 0 0 1 0 0 0 0
0
1, , , , , , , , , , , , , , , ,
, , , , , , , , , , , , , ,
t
b b
o
x t x y p t x y t s f s x s x y y s x y f s p s x y q s x y
k s x x y y x y d k s p x y q x y d ds
11 2
0 0 0 0 0 0 0 0 0 0 0 0
0
, , , , , , , , , , , ,
tL bLx t x y p t x y t s x s x y p s x y y s x y q x y ds
0 0 0 0 0 0 0 0, , , , , , , ,x s x y p s x y y s x y q x y
1 2
0 0 0 0 0 0 0 0 0 0 0 0, , , , , , , , , , , ,1
L bL tx t x y p t x y x s x y p s x y y s x y q x y
Similarly, we get:
3 4
0 0 0 0 0 0 0 0 0 0 0 0, , , , , , , , , , , ,1
L bL ty t x y q t x y x s x y p s x y y s x y q x y
So that
0 0 0 0 0 0 0 0
0
0 0 0 0 0 0 0 0
, , , , , , , ,(15)
, , , , , , , ,
x s x y p s x y x s x y p s x y
y s x y q x y y s x y q x y
By iteration we have:
0 0 0 0 0 0 0 0
0
0 0 0 0 0 0 0 0
, , , , , , , ,
, , , , , , , ,
mx s x y p s x y x s x y p s x y
y s x y q x y y s x y q x y
But From the condition (4), we have 0 0m when m , hence we obtain that:
0 0 0 0, , , ,x t x y p t x y and 0 0 0 0, , , ,y t x y q t x y . i.e.
0 0
0 0
, ,
, ,
x t x y
y t x y is a unique solution for the
system of nonlinear fractional integro-differential equations (1) on the domain (2).
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