International Journal of Mechanical Engineering and Technology (IJMET)
Volume 10, Issue 04, April 2019, pp. 786–800, Article ID: IJMET_10_04_078
Available online at http://www.iaeme.com/ijmet/issues.asp?JType=IJMET&VType=10&IType=4
ISSN Print: 0976-6340 and ISSN Online: 0976-6359
© IAEME Publication
Scopus Indexed
PERIODIC SOLUTION OF INTEGRODIFFERENTIAL EQUATIONS DEPENDED ON
SPECIAL FUNCTION WITH SINGULAR
KERNELS
Raad N. Butris and Raveen F. Taher
University of Duhok, College of Basic Education, Department of Mathematics,
Kurdistan Region, Iraq
ABSTRACT
In this paper, we investigate the existence, uniqueness and stability of periodic
solution of new integro-differential equations depended on special function with
singular kernels. The numerical-analytic method has been used to study the periodic
solutions for the ordinary differential equations that were introduced by Samoilenko.
Also these investigation lead us to the improving and extending the results of Butris
and extended Samoilenko method.
Key words: Numerical-analytic methods, existence, uniqueness and stability, periodic
solution, integro-differential equations, special function, singular kernels
Cite this Article: Raad N. Butris and Raveen F. Taher, Periodic Solution of IntegroDifferential Equations Depended on Special Function with Singular Kernels,
International Journal of Mechanical Engineering and Technology 10(4), 2019, pp.
786–800.
http://www.iaeme.com/IJMET/issues.asp?JType=IJMET&VType=10&IType=4
1. INTRODUCTION
Many results about the existence and approximation of periodic solutions of integrodifferential equations have been obtained by the numerical-analytic methods that were
proposed by Samoilenko[18] which had been later applied in many subjects in physics and
technology using mathematical methods that depends on the linear and nonlinear integrodifferential equations, and it became clear that the existence of periodic solutions and it is
algorithm structure from more important problems, to present time where many of studies and
researches [9,14,16,17] dedicates for treatment the autonomous and no autonomous periodic
systems and specially with integro-differential equations. Numerical-analytic method
[11,15,18,20, 21] owing to the great possibilities of exploiting computers are becoming
versatile means of the finding and approximate construction of periodic solutions of integrodifferential equations. Samoilenko [1,3,4,5,6,7,8,12,19] assumes the numerical-analytic
method to study the periodic solutions for ordinary differential equations and it is algorithm
structure and this method include uniformly sequences of periodic functions and the results of
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Raad N. Butris and Raveen F. Taher
that study is using of the periodic solutions on wide range in the difference of new processes
industry and technology as in the some studies [1,2,3,12,13].
Burris[3] has been used the numerical-analytic method of periodic solution for ordinary
differential equations which were introduced by Samoilenko [18] to study the periodic
solution of the system, nonlinear integro-differential equation which has the form
(
.
)
(
∫
(
) ( ))
/
where
is a closed and bounded domain.
The vector functions ( ( ) ) and ( ( ) ) are defined on the domain :( ( ) )
,
(
) ,
where
is a bounded domain subset of Euclidean space .
Our work is studying the existence, uniqueness and stability of periodic solution of
integro- differential equations which has the form :(
(
) ( )
(
∫
)( ( )
( ))
(
(
(
) ( )
(
) ( )
(
∫
)( ( )
( ))
)
}
where
∫
∫
and
(
) ( )
where
and
are compact domains.
) and ( ( ) ( )
Let the vector functions ( ( ) ( )
) are
defined and continuous on the domain :( ( )
)
(
)
}
( )
( ( )
)
(
)
where
and
are bounded domains subset of Euclidean space
Also
(
∫
)( ( )
,
- (
and periodic in
( ( )
)
( ( )
)
(
(
)
(
(
(
∫
)( ( )
( ))
,
-.
and satisfy the following inequalities :
‖ ( )‖‖ (
)‖
}
‖ ( )‖
(
)
)
(
( ))
)
(
(
)
)
(
(
)
)
(
)
(
)
)
(
)
)
(
For all
)
and
are belong to
and
respectively, where
and
are positive constants.
(
)
(
)
( ) The singular
Also
is said to be special function provided that
kernels ( ) and ( ) satisfying the following conditions :(
(
)
)
(
)
(
)
}
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(
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)
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Periodic Solution of Integro-Differential Equations Depended on Special Function with Singular
Kernels
where
and are positive constants.
We defined the non-empty sets as follows:(
}
)
Furthermore, we suppose that the largest eigen-value of the matrix
(
, is less than one, i.e.
√
( )
(
)
(
.
where
(
*(
)
(
)
with (
where
∫ (
)
(
(
(
)
∫ [ (
(
)
(
∫
(
(
)
)
)
(
)
(
∫ [ (
∫
(
/ and
*
Define a sequence of functions *
the following :(
/.
)
(
∫
)+
)
)
(
)
(
(
)
∫
(
)(
(
)
(
))
∫
(
)(
(
)
(
))
) by
]
(
)
)
)
(
on the domains (
)
)
)
)
(
)
]
(
)
)
(
)
By using lemma3.1 [18], we can state and proof the following lemma:
) and ( ( )
) be vectors which are defined
Lemma 1.1. Let ( ( )
on the interval ,
-, then
( )
(
)
(
) (
)
(
)
( )
(
)
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Raad N. Butris and Raveen F. Taher
satisfies for
where ( )= (
(
).
)
/ for all
(
∫ [ (
∫
(
( )
and
)
)
)
)
) (
(
(
-,
) (
(
∫ [ (
∫
) (
(
(
,
)
) (
)
]
)
]
)
)
Proof.
(
)
(
)∫ ‖ (
‖ (
∫
(
)‖ ‖ (
)‖ ‖ (
)
(
)
)‖
)‖
( )
So that
(
)
( )
(
)
and
(
)
( )
(
)
From (
) and (
) we conclude that the inequality (
Approximation solution of (
) holds.
)
The investigation of approximation periodic solution of ( ) will be introduced by the
following theorem .
) and ( ( )
Theorem 2.1. Let the vector functions ( ( )
) are
defined, continuous on the domain ( ) and periodic in of period . Suppose the above
functions satisfy the inequalities ( ) to ( ) and the conditions ( ) ( ). Then there
) converges uniformly on the domain :exist a sequence of functions ( ) and (
(
) ,
( )
) and (
to the limit functions (
) defined in the domain ( ) which is
periodic in of period and satisfies the following vector form :(
)
(
∫ [ (
∫
(
(
) (
) (
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)
)
)
)
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]
(
)
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Periodic Solution of Integro-Differential Equations Depended on Special Function with Singular
Kernels
(
)
(
∫ [ (
∫
) (
(
(
) (
and it's a unique solution of (
(
)
)
(
)
)
(
)
(
(
)
)
)
)
)
]
(
)
) Provided that :-
(
,
(
)
(
)
and
)
(
)
(
)
(
(
)
)
)
in (
Proof. Setting
(
) and by using Lemma
(
)∫ ‖ (
‖ (
∫
, we have
(
)‖ ‖ (
)‖ ‖ (
)
(
)
)‖
)‖
( )
and hence
(
)
( )
(
so that
)
Also from (
(
, when
(
)
(
)
we have
( )
)
)
.
), and by using Lemma
)
(
i.e.
for all
(
for all
.
Then by mathematical induction we can prove that
(
)
}
(
)
(
i.e.
)
(
)
, for all
,
Rewrite ( ) by the vector from , we get ( )
Now, we shall prove that the sequence of functions ( ) and (
uniformly on the domain ( )
By using Lemma
and from ( ) when
in ( ) we find that
(
)
(
)
(
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*∫ ‖ (
790
-
) convergent
)‖
editor@iaeme.com
Raad N. Butris and Raveen F. Taher
‖ (
[
)
,‖ (
)
‖ (
∫
(
‖ (
therefore
(
)
)
(
‖
‖
(
‖ (
)‖ (
)
‖ (
( )
(
)
)
,‖ (
)
(
)
( )
‖
)
‖
‖ (
‖
)
‖
and
‖
(
)
‖
( )
‖ (
Rewrite inequalities ( ) and (
( )
( )
where
(
)
(
(
(
)
(
)
)
‖
) ,we have
‖ (
)
)
( )
(
‖
‖
Then by mathematical induction we can prove that
(
)
(
)
‖ (
( )
)
‖
and
and from (
)
( )
(
and
(
)
‖-]
)
‖ (
( )
where
‖ (
)‖ [
( )
And by lemma
)
(
)‖ (
‖-]
)
where
(
)
(
(
)
)
(
(
)‖
)‖
(
)‖
)‖
(
(
)
) by vector form, i.e.
(
)
)
)
)
(
)
(
)
(
)
(
)
and
( )
( )
( )
(
)
( )
( )
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Periodic Solution of Integro-Differential Equations Depended on Special Function with Singular
Kernels
Now, we take the maximum value of the both sides of the (
where
,
-
(
and by repetition of (
) we get
(
)
( ) we obtain
)
) we find that
(
)
(
)
where
(
∑
, and also we get
∑
By using (
∑
Let
(
(
(
) then the series (
∑
)
) is uniformly convergent that is
(
)
(
)
)
(
)
)
)
(
(
(
)
)
Since the sequence of functions (
) and (
(
) then the limiting vector function (
(
) are defined and continuous in the
)
domain (
) is also defined , continuous
)
(
)
and periodic on the same domain and hence the vector function (
)
solution
(
)
of ( ).
(
)
) is a unique solution of (
Finally, we prove that (
(
(
)
)
) be another solution of (
Let (
(
).
)
)
where
(
)
∫ [ (
(
)
(
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Raad N. Butris and Raveen F. Taher
∫
(
(
)
(
∫ [ (
∫
(
(
)
(
(
)
)
)
)
(
]
)
(
(
)
(
)
)
)
)
]
where
∫ (
) (
)
∫
(
) (
∫
(
)( (
)
(
))
∫
(
)( (
)
(
))
)
(
)
Now
(
)
‖ (
)
‖
(
)‖
(
)
(
)‖
(
)
(
)‖
(
)
and also
(
)
(
)
‖ (
)
‖
Rewrite the inequalities (
(
)
(
(
(
)
(
) and (
)
)
)
(
)‖
(
)
) in a vector form :(
)
(
(
(
)
(
)
)
(
)
)
Then by the condition ( ) we have
(
)
(
)
(
) ( +
(
)
(
)
That is
(
)
(
)
(
)
(
)
(
)
(
(
)
)
)is a unique solution of (
and hence (
(
)
)
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Periodic Solution of Integro-Differential Equations Depended on Special Function with Singular
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Existence of periodic solution of (
)
The problem of existence of periodic solution of (
existence of zero of the vector functions
(
(
∫
)
(
) (
) is uniquely connected with the
)
)
)
(
(
(
)
(
∫
(
(
) (
(
)
)
)
)
)
) is approximately determined by the following vector
the vector function (
(
)
sequence :(
)
(
)
(
∫
(
)
(
)
)
)
(
(
( ( )
(
)
)
∫
(
)
Theorem 3.1. Under the conditions and hypothesis of theorem (
inequality:
(
)
(
)
)
(
⟨(
(
*
)
)
(
)
(
*
)
⟩ and
⟩
Proof . From the equations (
(
)
(
)
is satisfied where
⟨(
) then the following
)
(
(
)
) and (
(
) we have
)
⟨(
(
*
)
⟩
(
)
and similarly
(
)
(
)
⟨(
*
(
)
⟩
(
)
From ( ) and ( ), we get ( ).
Now, we prove the following theorem taking into account that the inequality ( ) will
be satisfied for all
) and ( ( )
Theorem 3.2. Let the vector functions ( ( )
)
- and , - on
be defined on the intervals ,
and periodic in of period suppose that
for all
the sequences of functions
(
) and
(
) which are
defined in ( ) and satisfy the inequalities :-
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Raad N. Butris and Raveen F. Taher
(
)
(
)
(
)
(
)
}
(
)
}
(
)
(
Then (
)
) has a periodic solution (
) such that :
(
0
)
1 and
Proof. Let
such that
(
and
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
,
-
be any points belonging on the intervals
and
respectively,
}
}
By using the inequalities (
(
)
(
)
)( )(
( (
)(
)
) and (
(
(
)
(
)
(
(
)
(
))
(
)
(
)
(
(
)
(
))
(
)
(
)
(
(
)
(
))
(
)
(
)
) we obtains
))
}
}
(
)
(
)
(
) and
and from the continuity of the functions
(
) and the
) and (
,
- and
inequalities(
), then there exist and isolated points
,
- such that
(
) 0 and
(
)
. This means that ( ) has a
periodic solution
(
) and
(
)
Stability of solution of (
).
In this section, we study the stability periodic solution of ( ).
(
) and
Theorem 4.1. Let the vector functions
(
) are defined by
) is a limit of the sequence of the functions ( ) the
equations ( ) where (
), then the following
function (
) is the limit of the sequence of the functions (
inequalities yields :(
)
(
) (
)
( )
(
)
And
(
)
(
)
(
)
(
)
(
)
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(
)(
)
(
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)
Periodic Solution of Integro-Differential Equations Depended on Special Function with Singular
Kernels
.
where
.
/
/0
1
Proof. From the properties of the functions (
(
)and
(
, the functions
and bounded by
in the domain ( ).
) and (
) and
)
(
) as in the theorem
are continuous
) we get
(
By using (
)
(
In similar way, we have
)
(
)
(
)
then we rewrite ( ) and ( ) by the vector from, we get ( )
By using ( ) and also from inequality ( ) and ( ) we get
(
)
(
)
‖ (
)
(
)
‖ (
Similarly
(
)
(
)
)
‖ (
)
(
‖ (
where the functions (
solutions of the equation :(
)
(
(
∫ [ (
∫
)
) (
(
(
(
∫ [ (
)
) (
∫
)
(
(
(
)
(
)
(
)
) and (
(
)
) are
)
)
) (
)
)
)
) (
) (
(
)
]
(
)
)
) (
)
)
]
where
∫ (
) (
)
∫
(
) (
∫
(
)( (
)
(
))
∫
(
)( (
)
(
))
)
where
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Raad N. Butris and Raveen F. Taher
From (
(
) we get
)
(
)
‖
‖
‖ (
(
)
)
‖ (
)
(
)
(
)
Therefore
(
)
(
)
) ‖
(
(
Also from (
(
)
‖
‖ (
)
)
(
)
) we have
(
)
‖
‖
‖ (
(
)
)
‖ (
)
(
)
and hence
(
)
(
)
) ‖
(
(
(
)
) in (
(
)
)
(
)
(
) we get
(
) ‖
‖
,(
*(
*- ‖
‖
,(
*(
*- ‖ (
/.
/-
,.
Putting
‖ (
)
Now, by substituting (
‖
)
(
)
and substituting in the last inequality, we obtain
(
)
(
)
) ‖
(
‖
as the
.
(
)
/
(
‖
‖
.
)
‖ (
)
(
)
)
(
)
/
*‖
(
‖
‖
‖
‖ (
This implies that
(
)
(
‖
)
‖
(
*(
(
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)
)
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‖
‖
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)
Periodic Solution of Integro-Differential Equations Depended on Special Function with Singular
Kernels
.
Putting
/
and substituting in the last inequality, we obtain
(
)
(
)
(
)
(
)
‖
‖
‖
) in (
Also, substituting the inequalities (
(
*‖
(
)
) we find that
) ‖
‖
)
,
(
(
‖
*‖
(
‖
‖-
and hence
(
)
(
.
)
‖
1‖
/0
So, substituting inequalities (
(
)
(
)
‖
Therefore
(
(
‖
)
(
‖
(
) and (
) in inequality (
‖
, ‖
) we get
‖-
‖
‖
,
)
,
‖
‖
‖
And by the same techniques, substituting inequalities (
) we get
(
)
(
)
‖
, ‖
‖
So
(
)
(
‖)
)
,
,
‖
‖
‖
‖
‖
‖
‖
‖
) and (
‖
‖
(
)
) in inequality
‖
(
)
) by the vector form, we get ( )
rewrite (
) and (
Remark 4.1. Theorem ( ), confirms the stability of the solution of ( ), that is when a
slight change happens in the point
, then a slight change will happen in the function
(
) For this remark see [10].
Remark 4.2. In the general case the initial values of the periodic solutions of ( ) should be
sought by a numerical method. It is possible to use the properties of the
constant
expressed by the following theorem.
Theorem 4.2. Suppose that the system ( ) be defined on the domain ( ). Then there
exists a sequence of functions
(
) and
(
) which are defined by ( )
and ( ) the following inequality holds :(
)
) (
) (
(
)
(
)
(
)
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798
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Raad N. Butris and Raveen F. Taher
for all
Proof. From (
(
)
) we get
(
)
By using the inequalities (
(
)
⟨(
And from (
(
)
*
(
)
⟨(
*
Rewrite the inequalities (
(
(
) and (
)
)
)
)
(
(
) and (
)
*
(
(
)
(
), we get
(
)
)
(
) )
from the inequality (
(
⟨(
*
(
)
(
)
(
)
)
)
(
⟩
)
) in a vector form as :(
)
⟩
(
(
)
⟩
) and (
⟨(
(
)
)
(
), we get
(
) we get
(
By using the inequalities (
(
(
) we get (
)
⟩
)
)
)
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[1]
Aziz, M. A., Periodic solutions for some systems of non-linear ordinary differential
equations, M. Sc. Thesis, College of Education, University of Mosul, (2006).
[2]
Butris, R. N., Ava, SH. R. and Hewa, S. F., Existence, uniqueness and stability of periodic
solution for nonlinear system of integro-differential equations science Journal of
University of Zakho Vol. 5.No. 1 pp. 120-127, March- (2017)..
[3]
Butris, R. N., Periodic solution of non-linear system of integro-differential equations
depending the Gamma distribution, India, Gen. Math. Notes V0l 1.15, No. 1, (2013).
[4]
Kigurodze and Puza, M., On periodic solutions of non-linear functional differential
equations, Georgian Math. J. Vol. 6, No. 1.(1999).
[5]
Narjanov, O. D., On a Periodic solution for integro-differential equation, Math. J. , Kiev,
Ukraine, Tom.2. (1977).
[6]
Naima, D. M., Periodic solution of non-autonomous second–order differential equations
and boundary value problem, India, Issue. 6. (1999).
[7]
Perestyuk, N. A., The periodic solutions for nonlinear systems of differential equations.
Maths. And Meca J., Univ. of Kiev, Ukraine No. 5, (1967).
[8]
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