International Journal of Engineering Inventions
e-ISSN: 2278-7461, p-ISSN: 2319-6491
Volume 3, Issue 8 (April 2014) PP: 01-16
www.ijeijournal.com Page | 1
Periodic Solutions for Non-Linear Systems of Integral Equations
Prof. Dr. Raad. N. Butris1, Baybeen S. Fars
2
1, 2
Department of Mathematics, Faculty of Science, University of Zakho
Abstract: The aim of this paper is to study the existence and approximation of periodic solutions for non-linear
systems of integral equations, by using the numerical-analytic method which were introduced by Samoilenko[
10, 11]. The study of such nonlinear integral equations is more general and leads us to improve and extend the
results of Butris [2].
Keyword And Phrases: Numerical-analytic methods, existence and approximation of periodic solutions,
nonlinear system, integral equations.
I. 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, ,12,13,14,15]. Many author create and develop numerical-analytic methods [1, 2,5,
7,8,9] and schemes to investigate periodic solution of integral equations describing many applications in
mathematical and engineering field.
Consider the following system of non-linear integral equations which has the form:
π₯ π‘, π₯0 , π¦0 = πΉ0 π‘ + π1(π , π₯ π , π₯0, π¦0 , π¦ π , π₯0 , π¦0 ,
π‘
0
, πΊ1 π , π π1(π, π₯ π, π₯0 , π¦0 , π¦ π, π₯0 , π¦0 )
π
ββ
ππ,
, π1(π, π₯ π, π₯0 , π¦0 , π¦ π, π₯0 , π¦0 )πππ π
π π )ππ β―( I1 )
π¦ π‘, π₯0, π¦0 = πΊ0 π‘ + π2(π , π₯ π , π₯0 , π¦0 ,π¦ π , π₯0 , π¦0 ,
π‘
0
, πΊ2 π , π π2(π, π₯ π, π₯0 , π¦0 , π¦ π, π₯0, π¦0 )
π
ββ
ππ,
, π2(π, π₯ π, π₯0 , π¦0 , π¦ π, π₯0 , π¦0 )ππ
π π
π π
)ππ β― ( I2 )
where π₯ β π· β π π ,π· is closed and bounded domain subset of Euclidean space π π . Let the vector functions
π1 π‘, π₯,π¦, π§,π€ = π11 π‘, π₯, π¦, π§,π€ , π12 π‘, π₯, π¦, π§,π€ ,β¦ , π1π π‘, π₯, π¦, π§,π€ ,
π2 π‘,π₯, π¦,π’, π£ = π21 π‘, π₯, π¦,π’, π£ , π22 π‘, π₯, π¦,π’, π£ ,β¦ , π2π π‘, π₯, π¦,π’, π£ ,
π1 π‘, π₯, π¦ = π11 π‘, π₯, π¦ ,π12 π‘, π₯, π¦ ,β¦ ,π1π π‘, π₯, π¦ ,
π2 π‘, π₯, π¦ = π21 π‘, π₯, π¦ ,π22 π‘, π₯, π¦ ,β¦ ,π2π π‘, π₯, π¦ ,
πΉ0 π‘ = (πΉ01 π‘ ,πΉ02 π‘ ,β¦ ,πΉ0π π‘ ),
and
πΊ0 π‘ = (πΊ01 π‘ ,πΊ02 π‘ ,β― ,πΊ0π π‘ ) are defined and continuous on the domains:
π‘, π₯, π¦, π§,π€ β π 1 Γ π· Γ π·1 Γ π·2 = ββ,β Γ π· Γ π·1 Γ π·2 Γ π·3
π‘, π₯, π¦,π’, π£ β π 1 Γ π· Γ π·1 Γ π·2 = ββ,β Γ π· Γ π·1 Γ π·2 Γ π·3
β― 1.1
and periodic in t of period T, Also a(t) and b(t) are continuous and periodic in t of period π. Suppose that the functions π1 π‘, π₯, π¦, π§,π€ , π2 π‘, π₯, π¦,π’, π£ , π1 π‘, π₯, π¦
and π2 π‘, π₯, π¦ satisfies the following inequalities:
Periodic Solutions for Non-Linear Systems of Integral Equations
www.ijeijournal.com Page | 2
π1 π‘, π₯, π¦, π§,π€ β€ π1 , π2 π‘, π₯, π¦,π’, π£ β€ π2 , π1 ,π2 > 0
π1 π‘, π₯, π¦ β€ π1 , π1 π‘, π₯, π¦ β€ π1 , π1,π 2 > 0 β― 1.2
π1 π‘, π₯1 , π¦1 , π§1 ,π€1 β π1 π‘, π₯2 , π¦2 , π§2 ,π€2 β€ πΎ1 π₯1 β π₯2 + πΎ2 π¦1βπ¦2 +
+πΎ3 π§1βπ§2 + πΎ4 π€1βπ€2 β― 1.3 π2 π‘, π₯1, π¦1 ,π’1, π£1 β π2 π‘, π₯2 , π¦2 ,π’2, π£2 β€ πΏ1 π₯1 β π₯2 + πΏ2 π¦1βπ¦2 +
+πΏ3 π’1βπ’2 + πΏ4 π£1βπ£2 , β― 1.4 π1 π‘, π₯1 , π¦1 β π1 π‘, π₯2 , π¦2 β€ π 1 π₯1 β π₯2 + π 2 π¦1 β π¦2 , β― 1.5 π2 π‘, π₯1 , π¦1 β π2 π‘, π₯2 , π¦2 β€ π»1 π₯1 β π₯2 + π»2 π¦1 β π¦2 , β― 1.6 for all π‘ β π 1 , π₯, π₯1 , π₯2 β π· , π¦ , π¦1 , π¦2 β π·1 , π§, π§1, π§2 ,π’,π’1,π’2 β π·2 , and π€,π€1,π€2 , π£, π£1 , π£2 β π·3 , where π1 = π11 ,π12 ,β¦ ,π1π , π2 = π21 ,π22 ,β¦ ,π2π , π1 = (π11 ,π12 ,β¦ ,π1π) and
π2 = π21 ,π22 ,β¦ ,π2π are positive constant vectors and πΎ1 = (πΎ1ππ), πΎ2 = (πΎ2ππ
), πΎ3 = (πΎ3 ππ), πΎ4 = (πΎ4 ππ
),
πΏ1 = (πΏ1ππ), πΏ2 = (πΏ2ππ
),
πΏ3 = ( πΏ3 ππ ), πΏ4 = (πΏ4 ππ
), π 1 = (π 1ππ), π 2 = (π 2ππ
), π»1 = (π»1 ππ), and
π»2 = (π»2 ππ) are positive constant matrices.
Also πΊ1 π‘, π and πΊ2 π‘, π are (nΓn) continuous positive matrix and
periodic in t, s of period T in the domain π 1 Γ π 1 and satisfy the following conditions:
πΊ1 π‘, π β€ πΎ πβπ1 π‘βπ , πΎ, π1 > 0 β― 1.7 and
πΊ2(π‘, π ) β€ πΏ πβπ2 π‘βπ , πΏ, π2 > 0 β― 1.8 where ββ < 0 β€ π β€ π‘ β€ π < β , π, π = 1,2,β― ,π, and π = πππ₯π‘β 0,π π π‘ β π π‘ , . = πππ₯π‘β 0,π .
Now define a non-empty sets as follows:
π·π1= π· β
π
2π 1 ,
π·1π2= π·1 β
π
2π 2 ,
π·2π1= π·2 β
π
2 πΎ
π1
π 1π1 + π 2π2 ,
π·3π1= π·3 β
π
2π π 1π1 + π 2π2 ,
π·2π2= π·2 β
π
2 πΏ
π2
π»1π1 + π»2π2 ,
π·3π2= π·3 β
π
2π π»1π1 + π»2π2 .
β― 1.9
Forever, we suppose that the greatest eigen-value ππππ₯ of the following
matrix:
Q0 =
π
2[πΎ1 + π 1(
πΎ
π1
πΎ3 + ππΎ4)] π
2[πΎ2 + π 2(
πΎ
π1
πΎ3 + ππΎ4)]
π
2[πΏ1 + π»1(πΏ3
πΏ
π2
+ ππΏ4)] π
2[πΏ2 + π»2(
πΏ
π2
πΏ3 + ππΏ4)]
β― 1.10
is less than unity, i.e.
ππππ₯ π0 =Ο1 + π1
2 + 4(π2 β π3)
2< 1 , β― 1.11
where Ο1 =π
2[πΎ1 + π 1(
πΎ
π1πΎ3 + ππΎ4)] +
π
2[πΏ2 + π»2(
πΏ
π2πΏ3 + ππΏ4)],
Ο2 = (π
2[πΎ2 + π 2(
πΎ
π1πΎ3 + ππΎ4)])(
π
2[πΏ1 + π»1(
πΏ
π2πΏ3 + ππΏ4)]) and
π3 = (π
2[πΎ1 + π 1(
πΎ
π1πΎ3 + ππΎ4)])(
π
2[πΏ2 + π»2(
πΏ
π2πΏ3 + ππΏ4)]).
By using lemma 3.1[10],we can state and prove the following lemma:
Lemma 1.1. Let π1(π‘, π₯, π¦, π§,π€) and π2(π‘, π₯, π¦,π’, π£) be a vector functions which are defined and continuous in
the interval 0,π , then the inequality:
Periodic Solutions for Non-Linear Systems of Integral Equations
www.ijeijournal.com Page | 3
πΏ1 π‘, π₯0 , π¦0
πΏ2(π‘, π₯0 , π¦0) β€
π1πΌ π‘
π2πΌ π‘ β― 1.12
satisfies for 0 β€ π‘ β€ π and πΌ π‘ β€π
2 ,
where πΌ π‘ = 2π‘(1 βπ‘
π) for all π‘ β 0,π ,
πΏ1 π‘, π₯0 , π¦0 = [π1(π , π₯0 , π¦0 ,
π‘
0
πΊ1 π , π π1(π, π₯0, π¦0)
π
ββ
ππ,
, π1(π, π₯0 , π¦0)ππ
π π
π π
) β1
π π1(π , π₯0 , π¦0 ,
π
0
, πΊ1 π , π π1(π, π₯0 , π¦0)
π
ββ
ππ, π1(π, π₯0, π¦0)ππ
π π
π π
)ππ ]ππ
πΏ2 π‘, π₯0 , π¦0 = [π2(π , π₯0 , π¦0 ,
π‘
0
πΊ2 π , π π2(π, π₯0 , π¦0)
π
ββ
ππ,
, π2(π, π₯0, π¦0)ππ
π π
π π
) β1
π π2(π , π₯0 , π¦0 ,
π
0
, πΊ2 π , π π2(π, π₯0 , π¦0)
π
ββ
ππ, π2(π, π₯0 , π¦0)ππ
π π
π π
)ππ ]ππ
Proof.
πΏ1 π‘, π₯0 , π¦0 β€ (1 βπ‘
π) π1(π , π₯0 , π¦0 , πΊ1
π
ββ
π , π π1 π, π₯0, π¦0 ππ,
π‘
0
, π1 π, π₯0 , π¦0 ππ
π(π )
π(π )
) ππ +
+π‘
π π1(π , π₯0 , π¦0 , πΊ1
π
ββ
π , π π1 π, π₯0, π¦0 ππ,
π
π‘
π1 π, π₯0, π¦0 ππ)
π(π )
π(π )
ππ
β€ (1 βπ‘
π) π1
π‘
0
ππ +π‘
π π1
π
π‘
ππ
β€ π1[(1 βπ‘
π)π‘ +
π‘
π(π β π‘)]
so that:
πΏ1 π‘, π₯0 , π¦0 β€ π1πΌ π‘ β― 1.13 And similarly, we get also
πΏ2 π‘, π₯0 , π¦0 β€ π2πΌ π‘ β― 1.14
From ( 1.13) and ( 1.14), then the inequality ( 1.12) satisfies for all
0 β€ π‘ β€ π and πΌ π‘ β€π
2 . β
II. Approximation Solution Of (I1) And (I2) In this section, we study the approximate periodic solution of (I1) and
(I2) by proving the following theorem:
Theorem 2.1. If the system (I1) and (I2) satisfies the inequalities ( 1.2), ( 1.3), 1.4), (1.5),(1.6) and
conditions(1.7),(1.8) has periodic solutions π₯ = π₯ π‘, π₯0 , π¦0 and π¦ = π¦(π‘, π₯0 , π¦0), then the sequence of
functions:
π₯π+1 π‘, π₯0 , π¦0 = πΉ0 π‘ + [π1(π , π₯π π , π₯0 , π¦0 , π¦π π , π₯0 , π¦0 ,
π‘
0
Periodic Solutions for Non-Linear Systems of Integral Equations
www.ijeijournal.com Page | 4
, πΊ1 π , π π1(π, π₯π π, π₯0 , π¦0 , π¦π π, π₯0, π¦0 )
π
ββ
ππ,
, π1(π, π₯π π, π₯0 , π¦0 , π¦π π, π₯0, π¦0 )ππ)
π π
π π
β1
π π1(π , π₯π π , π₯0 , π¦0 ,
π
0
, π¦π π , π₯0 , π¦0 , πΊ1 π , π π1(π, π₯π π, π₯0 , π¦0 , π¦π π, π₯0 , π¦0 )
π
ββ
ππ,
, π1(π, π₯π π, π₯0 , π¦0 , π¦π π, π₯0 , π¦0 )ππ)ππ ]ππ
π π
π π
β― 2.1
with
π₯0 π‘, π₯0 , π¦0 = πΉ0 π‘ = π₯0 , for all m = 0, 1,2,β―
π¦π+1 π‘, π₯0 , π¦0 = πΊ0 π‘ + [π2(π , π₯π π , π₯0 , π¦0 , π¦π π , π₯0 ,π¦0 ,
π‘
0
, πΊ2 π , π π2(π, π₯π π, π₯0, π¦0 , π¦π π, π₯0 , π¦0 )
π
ββ
ππ,
, π2(π, π₯π π, π₯0 , π¦0 , π¦π π, π₯0 , π¦0 )ππ)
π π
π π
β1
π π2(π , π₯π π , π₯0 , π¦0 ,
π
0
, π¦π π , π₯0, π¦0 , πΊ2 π , π π2(π, π₯π π, π₯0 , π¦0 , π¦π π, π₯0 , π¦0 )
π
ββ
ππ,
, π2(π,π₯π π, π₯0, π¦0 , π¦π π, π₯0 , π¦0 )ππ)ππ ]ππ
π π
π π
β― 2.2
with
π¦0 π‘, π₯0 , π¦0 = πΊ0 π‘ = π¦0 , for all m = 0, 1,2,β―
periodic in t of period T, and uniformly converges as π β β in the domain:
π‘, π₯0 , π¦0 β 0,π Γ π·π1 Γ π·1π2
β― 2.3
to the limit functions π₯0 π‘, π₯0 , π¦0 , π¦0 π‘, π₯0 , π¦0 defined in the domain ( 2.3) which are periodic in t of period T
and satisfying the system of integral equations:
π₯ π‘, π₯0 , π¦0 = πΉ0 π‘ + [π1(π , π₯ π , π₯0 , π¦0 ,π¦ π , π₯0 , π¦0 ,
π‘
0
, πΊ1 π , π π1(π, π₯ π, π₯0, π¦0 , π¦ π, π₯0 , π¦0 )
π
ββ
ππ,
, π1(π, π₯ π, π₯0 , π¦0 , π¦ π, π₯0 , π¦0 )ππ)
π π
π π
β1
π π1(π , π₯ π , π₯0 , π¦0 ,
π
0
, π¦ π , π₯0 ,π¦0 , πΊ1 π , π π1(π, π₯ π, π₯0, π¦0 , π¦ π, π₯0 , π¦0 )
π
ββ
ππ,
, π1(π, π₯ π, π₯0 , π¦0 , π¦ π, π₯0 , π¦0 )ππ)ππ ]ππ
π π
π π
β― 2.4
and
π¦ π‘, π₯0, π¦0 = πΊ0 π‘ + [π2(π , π₯ π , π₯0 , π¦0 , π¦ π , π₯0, π¦0 ,
π‘
0
Periodic Solutions for Non-Linear Systems of Integral Equations
www.ijeijournal.com Page | 5
, πΊ2 π , π π2(π, π₯ π, π₯0 , π¦0 , π¦ π, π₯0, π¦0 )
π
ββ
ππ,
, π2(π, π₯ π, π₯0 , π¦0 , π¦ π, π₯0, π¦0 )ππ)
π π
π π
β1
π π2(π , π₯ π , π₯0 , π¦0 ,
π
0
, π¦ π , π₯0, π¦0 , πΊ2 π , π π2(π, π₯ π, π₯0 , π¦0 , π¦ π, π₯0, π¦0 )
π
ββ
ππ,
, π2(π, π₯ π, π₯0 , π¦0 , π¦ π, π₯0, π¦0 )ππ)ππ ]ππ π π
π π β― 2.5
provided that:
π₯0 π‘, π₯0 , π¦0 β π₯π π‘, π₯0 , π¦0
π¦0 π‘, π₯0 , π¦0 β π¦π π‘, π₯0 , π¦0 β€ π0
π (πΈ β π0)β1πΆ0 β― 2.6
for all π β₯ 0 , where
πΆ0 =
π
2π1
π
2π2
, πΈ is identity matrix.
Proof. Consider the sequence of functions π₯1 π‘, π₯0 , π¦0 , π₯2 π‘, π₯0 , π¦0 ,β―, π₯π π‘, π₯0 , π¦0 ,β―, and π¦1 π‘, π₯0, π¦0 , π¦2 π‘, π₯0 , π¦0 ,β― , π¦π π‘, π₯0, π¦0 ,β―, defined
by the recurrences relations ( 2.1) and ( 2.2). Each of these functions of sequence defined and continuous in the
domain ( 1.1) and periodic in π‘ of period π.
Now, by the lemma 1.1, and using the sequence of functions (2.1),
when π = 0, we get:
By mathematical induction and lemma1.1 , we have the following inequality:
π₯π π‘, π₯0 , π¦0 β π₯0 β€π
2 π1 β― 2.13
i.e. π₯π π‘,π₯0 , π¦0 β π·, for all π‘ β π 1 , π₯0 β π·π1, π¦0 β π·1π2
, π = 0,1,2,β―,
and
π¦π π‘, π₯0, π¦0 β π¦0 β€π
2 π2 β― 2.14
i.e. π¦π π‘, π₯0 , π¦0 β π·, for all π‘ β π 1 , π₯0 β π·π1, π¦0 β π·1π2
,π = 0,1,2,β―,
Now, from ( 2.13), we get:
π§π π‘, π₯0 , π¦0 β π§0(π‘, π₯0 , π¦0) β€π
2(πΎ
π1
) π 1π1 + π 2π2 β― 2.15
and
π€π π‘, π₯0 , π¦0 β π€0(π‘,π₯0 , π¦0) β€π
2π π 1π1 + π 2π2 β― 2.16
for all π‘ β π 1 , π₯0 β π·π1, π¦0 β π·1π2
, π§0 π‘, π₯0 , π¦0 β π·2π1and
π€0 π‘,π₯0 , π¦0 β π·3π1,
i.e. π§π π‘, π₯0 , π¦0 β π·2 and π€π π‘, π₯0 , π¦0 β π·3 , for all π₯0 β π·π1, π¦0 β π·1π2
,
where
π§π π‘, π₯0 , π¦0 = πΊ1 π‘, π π1 π , π₯π π , π₯0 , π¦0 , π¦π π , π₯0 , π¦0 ππ
π‘
ββ
and
π€π π‘, π₯0 , π¦0 = π1 π , π₯π π , π₯0 , π¦0 , π¦π π , π₯0 , π¦0 ππ
π(π‘)
π(π‘)
, π = 0,1,2,β― ,
Also from (4.2.14), we get:
π’π π‘, π₯0 , π¦0 β π’0(π‘, π₯0, π¦0) β€π
2(πΏ
π2
) π»1π1 + π»2π2 β― 2.17
and
ππ π‘, π₯0 , π¦0 β π0(π‘, π₯0, π¦0) β€π
2π π»1π1 + π»2π2 β― 2.18
Periodic Solutions for Non-Linear Systems of Integral Equations
www.ijeijournal.com Page | 6
for all π‘ β π 1 , π₯0 β π·π1, π¦0 β π·1π2
, π’0 π‘, π₯0, π¦0 β π·2π2and
π£0 π‘, π₯0 , π¦0 β π·3π2,
i.e. π’π π‘, π₯0 , π¦0 β π·2 and π£π π‘, π₯0, π¦0 β π·3 , for all π₯0 β π·π1, π¦0 β π·1π2
,
where
π’π π‘, π₯0 , π¦0 = πΊ2 π‘, π π2 π , π₯π π , π₯0 , π¦0 , π¦π π , π₯0, π¦0 ππ
π‘
ββ
and
π£π π‘, π₯0 , π¦0 = π2 π , π₯π π , π₯0 , π¦0 , π¦π π , π₯0 , π¦0 ππ
π(π‘)
π(π‘)
, π = 0,1,2,β― ,
Next, we claim that two sequences π₯π (π‘, π₯0, π¦0)}π=0β , π¦π (π‘, π₯0, π¦0) }π=0
β
are convergent uniformly to the limit functions π₯, π¦ on the domain ( 2.3).
By mathematical induction and lemma 1.1, we find that:
π₯π+1 π‘, π₯0 , π¦0 β π₯π π‘, π₯0 , π¦0 β€
β€ πΌ(π‘)[πΎ1 + π 1(πΎ
π1
πΎ3 + ππΎ4)] π₯π π‘, π₯0 , π¦0 β π₯πβ1 π‘, π₯0 , π¦0
+πΌ(π‘)[πΎ2 + π 2(πΎ
π1
πΎ3 + ππΎ4)] π¦π π‘, π₯0, π¦0 β π¦πβ1 π‘, π₯0 , π¦0
β― 2.19 and
π¦π+1 π‘, π₯0 , π¦0 β π¦π π‘, π₯0, π¦0 β€
β€ πΌ(π‘)[πΏ1 + π»1(πΏ
π2
πΏ3 + ππΏ4)] π₯π π‘, π₯0, π¦0 β π₯πβ1 π‘, π₯0, π¦0 +
+πΌ(π‘)[πΏ2 + π»2(πΏ
π2
πΏ3 + ππΏ4)] π¦π π‘, π₯0 , π¦0 β π¦πβ1 π‘, π₯0 , π¦0
β― 2.20 We can write the inequalities ( 2.19) and ( 2.20) in a vector form:
πΆπ+1 π‘, π₯0 , π¦0 β€ π π‘ πΆπ π‘, π₯0, π¦0 β― 2.21 where
πΆπ+1 π‘, π₯0, π¦0 = π₯π+1 π‘, π₯0 , π¦0 β π₯π π‘, π₯0 , π¦0
π¦π+1 π‘, π₯0, π¦0 β π¦π π‘, π₯0 , π¦0 ,
and
π π‘ =
πΌ(π‘)[πΎ1 + π 1(
πΎ
π1
πΎ3 + ππΎ4)] πΌ(π‘)[πΎ2 + π 2(πΎ
π1
πΎ3 + ππΎ4)]
πΌ(π‘)[πΏ1 + π»1(πΏ
π2
πΏ3 + ππΏ4)] πΌ(π‘)[πΏ2 + π»2(πΏ
π2
πΏ3 + ππΏ4)]
,
πΆπ π‘, π₯0 , π¦0 = π₯π π‘, π₯0 ,π¦0 β π₯πβ1 π‘, π₯0 , π¦0
π¦π π‘, π₯0 , π¦0 β π¦πβ1 π‘, π₯0 , π¦0
Now, we take the maximum value to both sides of the inequality ( 2.21),
for all 0 β€ π‘ β€ π and πΌ π‘ β€π
2 , we get:
πΆπ+1 β€ π0πΆπ , β― 2.22 where π0 = πππ₯π‘β 0,π π(π‘) .
Q0 =
π
2[πΎ1 + π 1(
πΎ
π1
πΎ3 + ππΎ4)] π
2[πΎ2 + π 2(
πΎ
π1
πΎ3 + ππΎ4)]
π
2[πΏ1 + π»1(
πΏ
π2
πΏ3 + ππΏ4)] π
2[πΏ2 + π»2(
πΏ
π2
πΏ3 + ππΏ4)]
By iterating the inequality (4.2.22), we find that:
πΆπ+1 β€ π0ππΆ0 , β― 2.23
which leads to the estimate:
πΆπ β€ π0πβ1
π
π=1
πΆ0
π
π=1
β― 2.24
Since the matrix π0 has maximum eigen-values of (4.1.13) and the series
(4.2.24) is uniformly convergent, i. e.
Periodic Solutions for Non-Linear Systems of Integral Equations
www.ijeijournal.com Page | 7
limπββ
π0πβ1
π
π=1
πΆ0 = π0πβ1
β
π=1
πΆ0 = (πΈ β π0)β1πΆ0 β― ( 2.25)
The limiting relation (4.2.25) signifies a uniform convergence of the sequence
π₯π (π‘, π₯0, π¦0) , π¦π (π‘, π₯0 , π¦0) π=0β in the domain (4.2.3) as π β β.
Let
limπββ
π₯π π‘, π₯0 , π¦0 = π₯0 π‘, π₯0 , π¦0 ,
limπββ
π¦π π‘, π₯0 , π¦0 = π¦0 π‘, π₯0 , π¦0 . β― ( 2.26)
Finally, we show that π₯ π‘, π₯0 , π¦0 β‘ π₯0 π‘, π₯0 ,π¦0 β π· and
π¦ π‘, π₯0 , π¦0 β‘ π¦0(π‘, π₯0 , π¦0) β π·1, for all π₯0 β π·π1 and π¦0 β π·1π2
.
By using inequalities ( 2.1) and ( 2.4) and lemma 1.1, such that:
[π1(π , π₯π (π , π₯0 , π¦0), π¦π (π , π₯0 , π¦0),
π‘
0
π§π π , π₯0, π¦0 ,π€π (π , π₯0 , π¦0)) β
β1
π π1(π , π₯π (π , π₯0 , π¦0), π¦π (π , π₯0 , π¦0),
π
0
π§π π , π₯0, π¦0 ,π€π (π , π₯0 , π¦0))ππ ]ππ β
β [π1(π , π₯(π , π₯0 , π¦0), π¦(π , π₯0, π¦0),
π‘
0
π§ π , π₯0, π¦0 ,π€(π , π₯0 , π¦0)) β
β1
π π1(π , π₯(π , π₯0, π¦0), π¦(π , π₯0 , π¦0),
π
0
π§ π , π₯0 , π¦0 ,π€(π , π₯0, π¦0))ππ ]ππ
β€ πΌ(π‘) [πΎ1 + π 1(πΎ
π1
πΎ3 + ππΎ4)] π₯π π , π₯0 , π¦0 β π₯ π , π₯0, π¦0 +
+[πΎ2 + π 2(πΎ
π1πΎ3 + ππΎ4)] π¦π π , π₯0, π¦0 β π¦ π , π₯0 , π¦0 β€
β€π
2 [πΎ1 + π 1(
πΎ
π1
πΎ3 + ππΎ4)] π₯π π‘, π₯0, π¦0 β π₯ π‘, π₯0 , π¦0 +
+[πΎ2 + π 2(πΎ
π1πΎ3 + ππΎ4)] π¦π π‘, π₯0 , π¦0 β π¦ π‘, π₯0 , π¦0 .
From inequality ( 2.26) and on the other hand suppose that:.
π₯π π‘, π₯0 , π¦0 β π₯ π‘, π₯0 , π¦0 β€ π1 , π¦π π‘, π₯0 , π¦0 β π¦ π‘, π₯0 , π¦0 β€ π2 .
Thus
[π1(π , π₯π (π , π₯0 , π¦0), π¦π (π , π₯0 , π¦0),
π‘
0
π§π π , π₯0, π¦0 ,π€π (π , π₯0 , π¦0)) β
β1
π π1(π , π₯π (π , π₯0 , π¦0), π¦π (π , π₯0 , π¦0),
π
0
π§π π , π₯0, π¦0 ,π€π (π , π₯0 , π¦0))ππ ]ππ β
β [π1(π , π₯(π , π₯0 , π¦0), π¦(π , π₯0, π¦0),
π‘
0
π§ π , π₯0, π¦0 ,π€(π , π₯0 , π¦0)) β
β1
π π1(π , π₯(π , π₯0 , π¦0), π¦(π ,π₯0 , π¦0),
π
0
π§ π , π₯0, π¦0 ,π€(π , π₯0 , π¦0))ππ ]ππ
β€π
2[πΎ1 + π 1(
πΎ
π1
πΎ3 + ππΎ4)]π1 +π
2[πΎ2 + π 2(
πΎ
π1
πΎ3 + ππΎ4)]π2
Putting π1 =π3
π
2[πΎ1+π 1(
πΎ
π1πΎ3+ππΎ4)]
and π2 =π4
π
2[πΎ2+π 2(
πΎ
π1πΎ3+ππΎ4)]
and substituting in the last equation, we have:
[π1(π , π₯π (π , π₯0 , π¦0), π¦π (π , π₯0 , π¦0),
π‘
0
π§π π , π₯0, π¦0 ,π€π (π , π₯0 , π¦0)) β
Periodic Solutions for Non-Linear Systems of Integral Equations
www.ijeijournal.com Page | 8
β1
π π1(π , π₯π (π , π₯0 , π¦0), π¦π (π , π₯0 , π¦0),
π
0
π§π π , π₯0, π¦0 ,π€π (π , π₯0 , π¦0))ππ ]ππ β
β [π1(π , π₯(π , π₯0 , π¦0), π¦(π , π₯0, π¦0),
π‘
0
π§ π , π₯0, π¦0 ,π€(π , π₯0 , π¦0)) β
β1
π π1(π , π₯(π , π₯0 , π¦0), π¦(π , π₯0, π¦0),
π
0
π§ π , π₯0, π¦0 ,π€(π , π₯0 , π¦0))ππ ]ππ
β€π
2[πΎ1 + π 1(
πΎ
π1
πΎ3 + ππΎ4)]π3
π2
[πΎ1 + π 1(πΎπ1πΎ3 + ππΎ4)]
+
+π
2[πΎ2 + π 2(
πΎ
π1
πΎ3 + ππΎ4)]π4
π2
[πΎ2 + π 2(πΎπ1πΎ3 + ππΎ4)]
β€ π3 + π4, and choosing π3 + π4 = π , we get:
[π1(π , π₯π (π , π₯0, π¦0), π¦π (π , π₯0 , π¦0),
π‘
0
π§π π , π₯0 , π¦0 ,π€π (π , π₯0, π¦0)) β
β1
π π1(π , π₯π (π , π₯0 , π¦0), π¦π (π , π₯0 , π¦0),
π
0
π§π π , π₯0, π¦0 ,π€π (π , π₯0 , π¦0))ππ ]ππ β
β [π1(π , π₯(π , π₯0 , π¦0), π¦(π , π₯0, π¦0),
π‘
0
π§ π , π₯0, π¦0 ,π€(π , π₯0 , π¦0)) β
β1
π π1(π , π₯(π ,π₯0 , π¦0), π¦(π , π₯0 , π¦0),
π
0
π§ π , π₯0 , π¦0 ,π€(π , π₯0 , π¦0))ππ ]ππ β€ π
for all π β₯ 0,
i. e. limπββ
[π1(π , π₯π (π , π₯0, π¦0), π¦π (π , π₯0 , π¦0),
π‘
0
π§π π , π₯0 , π¦0 ,π€π (π , π₯0, π¦0)) β
β1
π π1(π , π₯π (π , π₯0 , π¦0), π¦π (π , π₯0 , π¦0),
π
0
π§π π , π₯0, π¦0 ,π€π (π , π₯0 , π¦0))ππ ]ππ =
β [π1(π , π₯(π , π₯0 , π¦0), π¦(π , π₯0, π¦0),
π‘
0
π§ π , π₯0, π¦0 ,π€(π , π₯0 , π¦0)) β
β1
π π1(π , π₯(π , π₯0, π¦0), π¦(π , π₯0 , π¦0),
π
0
π§ π , π₯0 , π¦0 ,π€(π , π₯0, π¦0))ππ ]ππ .
So π₯ π‘, π₯0 , π¦0 β π·, and π₯ π‘, π₯0, π¦0 β‘ π₯0(π‘, π₯0, π¦0) is a periodic solution of ( I1),
Also by using the same method above we can prove π¦ π‘, π₯0, π¦0 β π·1 , and π¦ π‘, π₯0 , π¦0 β‘ π¦0 π‘, π₯0 , π¦0 is
also periodic solution of (πΌ2 ).
Theorem 2.2. With the hypotheses and all conditions of the theorem 2.1, the periodic solution of integral
equations ( πΌ1 ) and ( πΌ2 ) are a unique on the domain (1.3).
Proof. Suppose that π₯ π‘, π₯0 , π¦0 and π¦ π‘, π₯0 , π¦0 be another periodic solutions
for the systems (πΌ1) and ( πΌ2 ) defined and continuous and periodic in π‘ of period π, this means that:
π₯ π‘, π₯0 , π¦0 = πΉ0 π‘ + [π1(π , π₯ π , π₯0 , π¦0 ,π¦ π , π₯0 , π¦0 ,
π‘
0
, πΊ1 π , π π1(π, π₯ π, π₯0 , π¦0 , π¦ π, π₯0, π¦0 )
π
ββ
ππ,
Periodic Solutions for Non-Linear Systems of Integral Equations
www.ijeijournal.com Page | 9
, π1(π, π₯ π, π₯0 , π¦0 , π¦ π, π₯0 , π¦0 )ππ)
π π
π π
β1
π π1(π , π₯ π , π₯0 , π¦0 ,
π
0
, π¦ π , π₯0 , π¦0 , πΊ1 π , π π1(π, π₯ π, π₯0 , π¦0 , π¦ π, π₯0 , π¦0 )
π
ββ
ππ,
, π1(π, π₯ π, π₯0 , π¦0 , π¦ π, π₯0, π¦0 )ππ)ππ ]ππ
π π
π π
β― 2.27
and
π¦ π‘, π₯0, π¦0 = πΊ0 π‘ + [π2(π , π₯ π , π₯0 , π¦0 , π¦ π , π₯0, π¦0 ,
π‘
0
, πΊ2 π , π π2(π, π₯ π, π₯0, π¦0 , π¦ π, π₯0 , π¦0 )
π
ββ
ππ,
, π2(π, π₯ π, π₯0 , π¦0 , π¦ π, π₯0, π¦0 )ππ)
π π
π π
β1
π π2(π , π₯ π , π₯0 , π¦0 ,
π
0
, π¦ π , π₯0 , π¦0 , πΊ2 π , π π2(π, π₯ π, π₯0 , π¦0 , π¦ π, π₯0, π¦0 )
π
ββ
ππ,
, π2(π, π₯ π, π₯0 , π¦0 , π¦ π, π₯0, π¦0 )ππ)ππ ]ππ
π π
π π
β― 2.28
For their difference, we should obtain the inequality:
π₯ π‘, π₯0 , π¦0 β π₯ π‘, π₯0, π¦0 β€
β€π
2 [πΎ1 + π 1(
πΎ
π1
πΎ3 + ππΎ4)] π₯ π‘, π₯0 , π¦0 β π₯ π‘,π₯0 , π¦0 +
+π
2[πΎ2 + π 2(
πΎ
π1
πΎ3 + ππΎ4)] π¦ π‘, π₯0 , π¦0 β π¦ π‘, π₯0 , π¦0 β― 2.29
And also
π¦ π , π₯0 , π¦0 β π¦ π , π₯0, π¦0 β€
β€ (1 βπ‘
π) [πΏ1 π₯ π , π₯0 , π¦0 β π₯ π , π₯0, π¦0 + πΏ2 π¦ π , π₯0 , π¦0 β π¦ π , π₯0, π¦0
π‘
0
+πΏ
π2
πΏ3 π»1 π₯ π , π₯0, π¦0 β π₯ π , π₯0 , π¦0 + π»2 π¦ π , π₯0 , π¦0 β π¦ π , π₯0 , π¦0 +
+ππΏ4(π»1 π₯ π , π₯0 , π¦0 β π₯ π , π₯0 , π¦0 + π»2 π¦ π , π₯0 , π¦0 β π¦ π , π₯0 , π¦0 )]ππ
+π‘
π πΏ1 π₯ π , π₯0 , π¦0 β π₯ π , π₯0 , π¦0 + πΏ2 π¦ π , π₯0 ,π¦0 β π¦ π , π₯0, π¦0 +
π
π‘
+πΏ
π2
πΏ3 π»1 π₯ π , π₯0 , π¦0 β π₯ π , π₯0 , π¦0 + π»2 π¦ π , π₯0 , π¦0 β π¦ π , π₯0, π¦0 +
+ππΏ4(π»1 π₯ π , π₯0 , π¦0 β π₯ π , π₯0 , π¦0 + π»2 π¦ π , π₯0 , π¦0 β π¦ π , π₯0 , π¦0 )]ππ
β€ πΌ(π‘)[πΏ1 + π»1(πΏ
π2
πΏ3 + ππΏ4)] π₯ π‘, π₯0 , π¦0 β π₯ π‘, π₯0, π¦0 +
+πΌ(π‘)[πΏ2 + π»2(πΏ
π2πΏ3 + ππΏ4)] π¦ π‘, π₯0 , π¦0 β π¦ π‘, π₯0 , π¦0
so that:
π¦ π‘, π₯0, π¦0 β π¦ π‘, π₯0 , π¦0 β€
β€π
2[πΏ1 + π»1(
πΏ
π2
πΏ3 + ππΏ4)] π₯ π‘, π₯0 , π¦0 β π₯ π‘, π₯0 , π¦0 +
+π
2[πΏ2 + π»2(
πΏ
π2πΏ3 + ππΏ4)] π¦ π‘, π₯0 , π¦0 β π¦ π‘, π₯0, π¦0 β― 2.30
Periodic Solutions for Non-Linear Systems of Integral Equations
www.ijeijournal.com Page | 10
and the inequalities (4.2.29) and (4.2.30) would lead to the estimate:
π₯ π‘, π₯0 , π¦0 β π₯ π‘, π₯0 , π¦0
π¦ π‘, π₯0 , π¦0 β π¦ π‘, π₯0 , π¦0 β€ π0
π₯ π‘, π₯0 , π¦0 β π₯ π‘, π₯0, π¦0
π¦ π‘, π₯0 , π¦0 β π¦ π‘, π₯0 , π¦0
β― 2.31 By iterating the inequality (4.2.27), which should find:
π₯ π‘, π₯0 , π¦0 β π₯ π‘, π₯0 , π¦0
π¦ π‘, π₯0 , π¦0 β π¦ π‘, π₯0 , π¦0 β€ π0
π π₯ π‘, π₯0 , π¦0 β π₯ π‘, π₯0, π¦0
π¦ π‘, π₯0 , π¦0 β π¦ π‘, π₯0 , π¦0 .
But π0π β 0 as π β β, so that proceeding in the last inequality which is contradict the supposition It follows
immediately π₯ π‘, π₯0 , π¦0 = π₯ π‘, π₯0, π¦0 and
π¦ π‘, π₯0 , π¦0 = π¦ π‘, π₯0 , π¦0 .
III. Existence of Solution of (π°π) and (π°π) The problem of existence of periodic solution of period T of the system
( πΌ1) and ( πΌ2) are uniquely connected with the existence of zero of the functions β1 0, π₯0, π¦0 = β1 and
β2 0, π₯0, π¦0 = β2 which has the form:
β1:π·π1Γ π·1π2
β π π
β1 0, π₯0, π¦0 = 1
π π1(π‘, π₯0 π‘, π₯0 , π¦0 , π¦
0 π‘, π₯0 , π¦0 ,
π
0
, πΊ1 π‘, π π1 π , π₯0 π , π₯0 , π¦0 , π¦0 π , π₯0, π¦0
π‘
ββ
ππ
, π1 π , π₯0 π , π₯0 , π¦0 , π¦0 π , π₯0, π¦0 ππ )ππ‘
π π‘
π π‘
β― 3.1
β2:π·π1Γ π·1π2
β π π
β2 0, π₯0, π¦0 = 1
π π2(π‘, π₯0 π‘, π₯0 , π¦0 , π¦
0 π‘, π₯0 , π¦0 ,
π
0
, πΊ2 π‘, π π2 π , π₯0 π , π₯0 , π¦0 , π¦0 π , π₯0 , π¦0
π‘
ββ
ππ
, π2 π , π₯0 π , π₯0 ,π¦0 , π¦0 π , π₯0 , π¦0 ππ )ππ‘
π π‘
π π‘
β― 3.2
where the function π₯0 π‘, π₯0 , π¦0 is the limit of the sequence of the functions π₯π (π‘, π₯0 , π¦0) and the function
π¦0 π‘, π₯0 , π¦0 is the limit of the sequence of the functions π¦π π‘, π₯0 , π¦0 . Since this two functions are approximately determined from the sequences of functions:
β1π :π·π1Γ π·1π2
β π π
β1π 0, π₯0 , π¦0 = 1
π π1(π‘, π₯π π‘, π₯0 , π¦0 , π¦π π‘, π₯0 , π¦0 ,
π
0
, πΊ1 π‘, π π1 π , π₯π π , π₯0 , π¦0 , π¦π π , π₯0 , π¦0
π‘
ββ
ππ
, π1 π , π₯π π , π₯0 , π¦0 , π¦π π , π₯0 , π¦0 ππ )ππ‘
π π‘
π π‘
β― 3.3
β2π :π·π1Γ π·1π2
β π π
β2π 0, π₯0 , π¦0 = 1
π π2(π‘, π₯π π‘, π₯0 , π¦0 , π¦π π‘, π₯0, π¦0 ,
π
0
, πΊ2 π‘, π π2 π , π₯π π , π₯0 , π¦0 , π¦π π , π₯0 , π¦0
π‘
ββ
ππ
Periodic Solutions for Non-Linear Systems of Integral Equations
www.ijeijournal.com Page | 11
, π2 π , π₯π π , π₯0 , π¦0 , π¦π π , π₯0 , π¦0 ππ )ππ‘
π π‘
π π‘
β― 3.4
for all π = 0,1,2,β―
Theorem 3.1. If the hypotheses and all conditions of the theorem 2.1 and 2.2 are satisfied, then the following
inequality satisfied:
β1 0, π₯0, π¦0 β β1π 0, π₯0 , π¦0 β€ ππ β― 3.5 β2 0, π₯0, π¦0 β β2π 0, π₯0, π¦0 β€ ππ β― 3.6 satisfied for all π β₯ 0 , π₯0 β π·π1 and π¦0 β π·1π2
,
where
ππ = [πΎ1 + π 1(πΎ
π1
πΎ3 + ππΎ4)] [πΎ2 + π 2(πΎ
π1
πΎ3 + ππΎ4)] ,
,π0π+1 πΈ β π0
β1πΆ0 and
ππ = [πΏ1 + π»1(πΏ
π2πΏ3 + ππΏ4)] [πΏ2 + π»2(
πΏ
π2πΏ3 + ππΏ4)] ,
, π0π+1 πΈ β π0
β1πΆ0 . Proof. By using the relation ( 3.1) and ( 3.3), we have:
β1 0, π₯0 , π¦0 β β1π 0, π₯0 , π¦0
β€ [πΎ1 + π 1(πΎ
π1
πΎ3 + ππΎ4)] π₯0 π‘, π₯0 , π¦0 β π₯π π‘, π₯0 , π¦0 +
+[πΎ2 + π 2(πΎ
π1
πΎ3 + ππΎ4)] π¦0 π‘, π₯0 , π¦0 β π¦π π‘, π₯0 , π¦0
β€ [πΎ1 + π 1(πΎ
π1
πΎ3 + ππΎ4)] [πΎ2 + π 2(πΎ
π1
πΎ3 + ππΎ4)] ,
,π0π+1 πΈ β π0
β1πΆ0 = ππ And also by using the relation ( 3.2) and ( 3.4), we get:
β2 0, π₯0, π¦0 β β2π 0, π₯0, π¦0 β€
β€ [πΏ1 + π»1(πΏ
π2
πΏ3 + ππΏ4)] [πΏ2 + π»2(πΏ
π2
πΏ3 + ππΏ4)] ,
, π0π+1 πΈ β π0
β1πΆ0 = ππ
where . denotes the ordinary scalar product in the space π π . β
Theorem 3.2. Let the vector functions π1 π‘, π₯,π¦, π§,π€ , π2 π‘, π₯, π¦,π’, π£ , π1 π‘, π₯, π¦ and π2 π‘, π₯, π¦ be defined on the domain:
πΊ = 0 β€ π β€ π‘ β€ π, π β€ π₯, π¦ β€ π , π β€ π§,π’ β€ π, π β€ π€, π£ β€ π} β π 1 , and periodic in t of period T.
Assume that the sequence of functions (4.3.3) and (4.3.4) satisfies the
inequalities:
mina+
π2π1β€x0β€bβ
π2π1
c+π2π2β€y0β€dβ
π2π2
β1m 0, x0, y0 β€ β πm ,
maxa+
π2π1β€x0β€bβ
π2π1
c+π2π2β€y0β€dβ
π2π2
β1m 0, x0, y0 β₯ πm ,
β― 3.7
mina+
π2π1β€x0β€bβ
π2π1
c+π2π2β€y0β€dβ
π2π2
β2m 0, x0, y0 β€ β Ξ·m
,
maxa+
π2π1β€x0β€bβ
π2π1
c+π2π2β€y0β€dβ
π2π2
β2m 0, x0 , y0 β₯ Ξ·m
,
β― 3.8
for all π β₯ 0, where
ππ = [πΎ1 + π 1(πΎ
π1
πΎ3 + ππΎ4)] [πΎ2 + π 2(πΎ
π1
πΎ3 + ππΎ4)] ,
,π0π+1 πΈ β π0
β1πΆ0 and
Periodic Solutions for Non-Linear Systems of Integral Equations
www.ijeijournal.com Page | 12
ππ = [πΏ1 + π»1(πΏ
π2
πΏ3 + ππΏ4)] [πΏ2 + π»2(πΏ
π2
πΏ3 + ππΏ4)] ,
, π0π+1 πΈ β π0
β1πΆ0 Then the system (4.1.1) and (4.1.2) has periodic solution of period T
π₯ = π₯(π‘, π₯0 , π¦0) and π¦ = π¦(π‘, π₯0 , π¦0) for which π₯0 β [π +π
2π1 , π β
π
2π1] and
π¦0 β [π +π
2π2,π β
π
2π2].
Proof. Let π₯1 , π₯2 be any two points in the interval [π +π
2π1 , π β
π
2π1] and π¦1 , π¦2 be any two points in the
interval [π +π
2π2,π β
π
2π2],
such that:
Ξ1m 0, π₯1 , π¦1 = mina+
π2π1β€x0β€bβ
π2π1
c+π2π2β€y0β€dβ
π2π2
β1m 0, π₯0 , π¦0 ,
Ξ1m 0, π₯2 , π¦2 = maxa+
π2π1β€x0β€bβ
π2π1
c+π2π2β€y0β€dβ
π2π2
β1π 0,π₯0 , π¦0 ,
β― 3.9
Ξ2m 0, π₯1, π¦1 = mina+
π2π1β€x0β€bβ
π2π1
c+π2π2β€y0β€dβ
π2π2
β2m 0, x0, y0 ,
Ξ2m 0, π₯2 , π¦2 = maxa+
π2π1β€x0β€bβ
π2π1
c+π2π2β€y0β€dβ
π2π2
β2m 0, x0, y0 ,
β― 3.10
By using the inequalities ( 3.5), ( 3.6), ( 3.7) and ( 3.8), we have:
β1 0, π₯1 , π¦1 = β1π 0, π₯1 , π¦1 + β1 0, π₯1 , π¦1 β β1π (0, π₯1 , π¦1) β€ 0 ,
β1 0, π₯2, π¦2 = β1π 0, π₯2 , π¦2 + β1 0, π₯2 , π¦2 β β1π(0, π₯2 , π¦2) β€ 0 .
β― 3.11
β2 0, π₯1 , π¦1 = β2π 0, π₯1 , π¦1 + β2 0, π₯1, π¦1 β β2π (0, π₯1 , π¦1) β€ 0 ,
β2 0, π₯2, π¦2 = β2π 0, π₯2 , π¦2 + β2 0, π₯2, π¦2 β β2π (0, π₯2, π¦2) β€ 0 .
β― 3.12 It follows from the inequalities ( 3.11) and ( 3.12) in virtue of the
continuity of the functions β1(0, π₯0 , π¦0) and β1(0, π₯0 , π¦0) that there exists an isolated singular point (π₯0, π¦0) =(π₯0, π¦0) , π₯0 β π₯1 , π₯2 and π¦0 β π¦1 , π¦2 , so that β1(0, π₯0 , π¦0) = 0 and β2(0, π₯0, π¦0) = 0. This means that the system ( 3.1) and (4.3.2) has a periodic
solutions π₯ π‘, π₯0 , π¦0 , π¦ π‘, π₯0 , π¦0 for which
π₯0 β [π +π
2π1 , π β
π
2π1] and π¦0 β [π +
π
2π2,π β
π
2π2]. β
Remark 3.1. Theorem 3.2 is proved when π π = π 1, on the other hand as π₯0, π¦0 are a scalar singular point
which should be isolated (For this remark, see [5]).
IV. Stability Theorem Of Solution (π°π) And (π°π) In this section, we study theorem on stability of a periodic solution for the integral equations ( πΌ1) and ( πΌ2).
Theorem 4.1. If the function β1 0, π₯0, π¦0 ,Ξ2(0, π₯0 , π¦0) are defined by
equations ( 3.1) and ( 3.2), where the function π₯0(π‘, π₯0, π¦0) is a limit of the sequence of the functions ( 2.1) , the
function π¦0(π‘, π₯0 , π¦0) is the limit of the sequence of the functions ( 2.2) , Then the following inequalities yields:
β1 0, π₯0 , π¦0 β€ π1 β― 4.1 β2 0, π₯0, π¦0 β€ π2 β― 4.2
and
β2 0, π₯01, π¦0
1 β β2 0, π₯02 , π¦0
2 β€ πΉ1πΉ2πΈ3 πΉ01 π‘ β πΉ0
2 π‘ +
+π
2πΉ1πΉ2πΈ2( πΈ3 + πΈ4 + πΉ1πΈ4(1 β
π
2πΈ1)) πΊ0
1 π‘ β πΊ02(π‘)
β― 4.4
for all π₯0, π₯01 , π₯0
2 β π·π1, π¦0 , π¦0
1 , π¦02 β π·1π2
, and πΌ π‘ = 2π‘(1 βπ‘
π) β€
π
2 ,
where πΈ1 = [πΎ1 + π 1(πΎ
π1πΎ3 + ππΎ4)],πΈ2 = [πΎ2 + π 2(
πΎ
π1πΎ3 + ππΎ4)],
Periodic Solutions for Non-Linear Systems of Integral Equations
www.ijeijournal.com Page | 13
πΈ3 = [πΏ1 + π»1(πΏ
π2πΏ3 + ππΏ4)], πΈ4 = [πΏ2 + π»2(
πΏ
π2πΏ3 + ππΏ4)],
πΉ1 = [(1 βπ
2πΈ1)(1 β
π
2πΈ4)]β1 and πΉ2 = (1 β
π2
4π2π3πΉ1)β1
Proof. From the properties of the functions π₯0 π‘, π₯0 , π¦0 and π¦0 π‘, π₯0 , π¦0 as in the theorem 4.2.1, the
functions Ξ1 = β1 π₯0, π¦0 , Ξ1 = β1 π₯0 , π¦0 , π₯0 β π·π1
, π¦0 β π·1π2 are continuous and bounded by π1,π2 in the domain ( 1.3).
From relation ( 3.1), we find:
β1 (0, π₯0 , π¦0 β€1
π π1(π‘, π₯0 π‘, π₯0 , π¦0 , π¦
0 π‘, π₯0 , π¦0 ,
π
0
, πΊ1 π‘, π π1 π , π₯0 π , π₯0 , π¦0 , π¦0 π , π₯0 , π¦0 ππ ,
π‘
ββ
, π1 π , π₯0 π , π₯0 , π¦0 , π¦0 π , π₯0, π¦0 π π )
π(π‘)
π(π‘)
ππ‘
by using the Lemma 4.1.1, gives:
β1(0, π₯0 , π¦0) β€ π1 . And from relation ( 3.2), we get:
β2 (0, π₯0 , π¦0 β€1
π π2(π‘, π₯0 π‘, π₯0 , π¦0 , π¦
0 π‘, π₯0 , π¦0 ,
π
0
, πΊ2 π‘, π π2 π , π₯0 π , π₯0 , π¦0 , π¦0 π , π₯0 , π¦0 ππ ,
π‘
ββ
, π2 π , π₯0 π , π₯0, π¦0 , π¦
0 π , π₯0 , π¦0 ππ )
π(π‘)
π(π‘)
ππ‘
and using Lemma 4.1.1, we have:
β2(0, π₯0, π¦0) β€ π2
By using equation ( 3.1) and lemma 1.1, we get:
β1 0, π₯01, π¦0
1 β β1 0, π₯02 , π¦0
2
β€1
π [πΎ1 π₯
0 π‘, π₯01 , π¦0
1 β π₯0 π‘, π₯02, π¦0
2 +
π
0
+πΎ2 π¦0 π‘, π₯0
1 , π¦01 β π¦0 π‘, π₯0
2, π¦02 +
+πΎ
π1
πΎ3(π 1 π₯0 π‘, π₯0
1, π¦01 β π₯0 π‘, π₯0
2, π¦02 +
+π 2 π¦0 π‘, π₯0
1, π¦01 β π¦0 π‘, π₯0
2, π¦02 ) +
+ ππΎ4(π 1 π₯0 π‘, π₯0
1 , π¦01 β π₯0 π‘, π₯0
2, π¦02 +
+π 2 π¦0 π‘, π₯0
1, π¦01 β π¦0 π‘, π₯0
2, π¦02 )]ππ‘
β€ [πΎ1 + π 1(πΎ
π1
πΎ3 + ππΎ4)] π₯0 π‘, π₯01, π¦0
1 β π₯0 π‘, π₯02, π¦0
2 +
+[πΎ2 + π 2(πΎ
π1
πΎ3 + ππΎ4)] π¦0 π‘, π₯01 , π¦0
1 β π¦0 π‘, π₯02, π¦0
2
so,
β1 0, π₯01, π¦0
1 β β1 0, π₯02 , π¦0
2 β€ πΈ1 π₯0 π‘, π₯0
1 , π¦01 β π₯0 π‘, π₯0
2, π¦02 +
+πΈ2 π¦0 π‘, π₯01, π¦0
1 β π¦0 π‘, π₯02, π¦0
2 β― 4.5 And also by using equation ( 3.2) and lemma 1.1, gives:
β2 0, π₯01, π¦0
1 β β2 0, π₯02 , π¦0
2
β€1
π [πΏ1 π₯
0 π‘, π₯01 , π¦0
1 β π₯0 π‘, π₯02, π¦0
2 +
π
0
+πΏ2 π¦0 π‘, π₯0
1 , π¦01 β π¦0 π‘, π₯0
2, π¦02 +
Periodic Solutions for Non-Linear Systems of Integral Equations
www.ijeijournal.com Page | 14
+πΏ
π2
πΏ3(π»1 π₯0 π‘, π₯0
1, π¦01 β π₯0 π‘, π₯0
2, π¦02 +
+π»2 π¦0 π‘, π₯0
1 , π¦01 β π¦0 π‘, π₯0
2, π¦02 ) +
+ ππΏ4(π»1 π₯0 π‘, π₯0
1, π¦01 β π₯0 π‘, π₯0
2, π¦02 +
+π»2 π¦0 π‘, π₯0
1, π¦01 β π¦0 π‘, π₯0
2, π¦02 )]ππ‘
β€ [πΏ1 + π»1(πΏ
π2
πΏ3 + ππΏ4)] π₯0 π‘, π₯01 , π¦0
1 β π₯0 π‘, π₯02, π¦0
2 +
+[πΏ2 + π»2(πΏ
π2πΏ3 + ππΏ4)] π¦0 π‘, π₯0
1, π¦01 β π¦0 π‘, π₯0
2, π¦02 .
Therefore,
β2 0, π₯01, π¦0
1 β β2 0, π₯02 , π¦0
2 β€ πΈ3 π₯0 π‘, π₯0
1, π¦01 β π₯0 π‘, π₯0
2 , π¦02 +
+πΈ4 π¦0 π‘, π₯01, π¦0
1 β π¦0 π‘, π₯02, π¦0
2 β― 4.6
where the functions π₯0 π‘, π₯01, π¦0
1 , π¦0 π‘, π₯01 , π¦0
1 , π₯0 π‘, π₯02, π¦0
2 and
π¦0 π‘, π₯02, π¦0
2 are solutions of the equation:
π₯ π‘, π₯0π , π¦0
π = πΉ0π π‘ + [π1(π , π₯(π , π₯0
π , π¦0π), π¦(π , π₯0
π , π¦0π),
π‘
0
, πΊ1 π , π π1(π, π₯(π, π₯0π , π¦0
π), π¦(π, π₯0π , π¦0
π))
π
ββ
ππ,
, π1(π, π₯(π, π₯0π , π¦0
π), π¦(π, π₯0π , π¦0
π))ππ
π π
π π
) β
β1
π π1(π , π₯(π , π₯0
π , π¦0π), π¦(π , π₯0
π , π¦0π)
π
0
,
, πΊ1 π , π π1(π, π₯(π, π₯0π , π¦0
π), π¦(π, π₯0π , π¦0
π))
π
ββ
ππ,
, π1(π, π₯(π, π₯0π , π¦0
π), π¦(π, π₯0π , π¦0
π))ππ
π π
π π
)ππ ]ππ β― ( 4.7)
and
π¦ π‘, π₯0π , π¦0
π = πΊ0π π‘ + [π2(π , π₯(π , π₯0
π , π¦0π), π¦(π , π₯0
π , π¦0π),
π‘
0
, πΊ2 π , π π2(π, π₯(π, π₯0π , π¦0
π), π¦(π, π₯0π , π¦0
π))
π
ββ
ππ,
, π2(π, π₯(π, π₯0π , π¦0
π), π¦(π, π₯0π , π¦0
π))ππ
π π
π π
) β
β1
π π2(π , π₯(π , π₯0
π , π¦0π), π¦(π , π₯0
π , π¦0π)
π
0
,
, πΊ2 π , π π2(π, π₯(π, π₯0π , π¦0
π), π¦(π, π₯0π , π¦0
π))
π
ββ
ππ,
, π2(π, π₯(π, π₯0π , π¦0
π), π¦(π, π₯0π , π¦0
π))ππ
π π
π π
)ππ ]ππ β― ( 4.8)
where π = 1, 2. From (4.4.7), we get:
π₯0 π‘, π₯01 , π¦0
1 βπ₯0 π‘, π₯02, π¦0
2 β€ πΉ0
1 π‘ β πΉ02 π‘ +
Periodic Solutions for Non-Linear Systems of Integral Equations
www.ijeijournal.com Page | 15
+πΌ(π‘)[πΎ1 + π 1(πΎ
π1
πΎ3 + ππΎ4)] π₯0 π‘, π₯01, π¦0
1 β π₯0 π‘, π₯02, π¦0
2 +
+πΌ π‘ [πΎ2 + π 2(πΎ
π1
πΎ3 + ππΎ4)] π¦0 π‘, π₯01, π¦0
1 β π¦0 π‘, π₯02, π¦0
2
β€ πΉ01 π‘ β πΉ0
2 π‘ +
+π
2[πΎ1 + π 1(
πΎ
π1
πΎ3 + ππΎ4)] π₯0 π‘, π₯01 , π¦0
1 β π₯0 π‘, π₯02, π¦0
2 +
+π
2[πΎ2 + π 2(
πΎ
π1
πΎ3 + ππΎ4)] π¦0 π‘, π₯01 , π¦0
1 β π¦0 π‘, π₯02, π¦0
2
such that:
π₯0 π‘, π₯01, π¦0
1 βπ₯0 π‘, π₯02, π¦0
2 β€ πΉ01 π‘ β πΉ0
2 π‘ +
+π
2πΈ1 π₯
0 π‘, π₯01, π¦0
1 β π₯0 π‘, π₯02 ,π¦0
2 +
+π
2πΈ2 π¦0 π‘, π₯0
1 , π¦01 β π¦0 π‘, π₯0
2, π¦02
therefore:
π₯0 π‘, π₯01, π¦0
1 βπ₯0 π‘, π₯02, π¦0
2 β€ (1 βπ
2πΈ1)β1 πΉ0
1 π‘ β πΉ02 π‘ +
+π
2πΈ2(1 β
π
2πΈ1)β1 π¦0 π‘, π₯0
1, π¦01 β π¦0 π‘, π₯0
2, π¦02
β― 4.9 And also from relation ( 4.8), we have:
π¦0 π‘, π₯01 , π¦0
1 βπ¦0 π‘, π₯02, π¦0
2 β€ β€ πΊ0
1 π‘ β πΊ02 π‘ +
+πΌ(π‘)[πΏ1 + π»1(πΏ
π2
πΏ3 + ππΏ4)] π₯0 π‘, π₯01, π¦0
1 β π₯0 π‘, π₯02, π¦0
2 +
+πΌ(π‘)[πΏ2 + π»2(πΏ
π2
πΏ3 + ππΏ4)] π¦0 π‘, π₯01 , π¦0
1 β π¦0 π‘, π₯02, π¦0
2 β€
β€ πΊ01 π‘ β πΊ0
2 π‘ +
+π
2[πΏ1 + π»1(
πΏ
π2
πΏ3 + ππΏ4)] π₯0 π‘, π₯01, π¦0
1 β π₯0 π‘, π₯02, π¦0
2 +
+π
2[πΏ2 + π»2(
πΏ
π2
πΏ3 + ππΏ4)] π¦0 π‘, π₯01, π¦0
1 β π¦0 π‘, π₯02, π¦0
2
so that:
π¦0 π‘, π₯01 , π¦0
1 βπ¦0 π‘, π₯02, π¦0
2 β€ πΊ01 π‘ β πΊ0
2 π‘ +
+π
2πΈ3 π₯
0 π‘, π₯01, π¦0
1 β π₯0 π‘, π₯02, π¦0
2 +
+π
2πΈ4 π¦0 π‘, π₯0
1, π¦01 β π¦0 π‘, π₯0
2, π¦02
and hence:
π¦0 π‘, π₯01 , π¦0
1 βπ¦0 π‘, π₯02, π¦0
2 β€ (1 βπ
2πΈ4)β1 πΊ0
1 π‘ β πΊ02 π‘ +
+π
2πΈ3(1 β
π
2πΈ4)β1 π₯0 π‘, π₯0
1, π¦01 β π₯0 π‘, π₯0
2 , π¦02
β― 4.10 Now, by substituting inequality ( 4.10) in ( 4.9), we get:
π₯0 π‘, π₯01, π¦0
1 β π₯0 π‘, π₯02 , π¦0
2 β€ (1 βπ
2πΈ1)β1 πΉ0
1 π‘ β πΉ02 π‘ +
+π
2πΈ2(1 β
π
2πΈ1)β1[(1 β
π
2πΈ4)β1 πΊ0
1 π‘ β πΊ02 π‘ +
+π
2πΈ3(1 β
π
2πΈ4)β1 π₯0 π‘, π₯0
1 , π¦01 β π₯0 π‘, π₯0
2, π¦02 ]
β€ (1 βπ
2πΈ1)β1 πΉ0
1 π‘ β πΉ02 π‘ +
+π
2πΈ2[(1 β
π
2πΈ1)(1 β
π
2πΈ4)]β1 πΊ0
1 π‘ β πΊ02 π‘ +
+π2
4πΈ2πΈ3[(1 β
π
2πΈ1)(1 β
π
2πΈ4)]β1 π₯0 π‘, π₯0
1, π¦01 β π₯0 π‘, π₯0
2, π¦02 ]
putting πΉ1 = [(1 βπ
2πΈ1)(1 β
π
2πΈ4)]β1,
and substituting in the last inequality, we obtain:
Periodic Solutions for Non-Linear Systems of Integral Equations
www.ijeijournal.com Page | 16
π₯0 π‘, π₯01, π¦0
1 β π₯0 π‘, π₯02 , π¦0
2 β€ (1 βπ
2πΈ1)β1 πΉ0
1 π‘ β πΉ02 π‘ +
+π
2πΈ2πΉ1 πΊ0
1 π‘ β πΊ02 π‘ + +
π2
4πΈ2πΈ3πΉ1 π₯
0 π‘, π₯01, π¦0
1 β π₯0 π‘, π₯02, π¦0
2 ]
as the πΉ1(1 βπ
2πΈ4) = (1 β
π
2πΈ1)β1
π₯0 π‘, π₯01, π¦0
1 β π₯0 π‘, π₯02 , π¦0
2 β€ πΉ1(1 βπ
2πΈ4) πΉ0
1 π‘ β πΉ02 π‘ +
+π
2πΈ2πΉ1 πΊ0
1 π‘ β πΊ02 π‘ +
π2
4πΈ2πΈ3πΉ1 π₯
0 π‘, π₯01 , π¦0
1 β π₯0 π‘, π₯02, π¦0
2 ]
which implies that:
π₯0 π‘, π₯01, π¦0
1 β π₯0 π‘, π₯02 , π¦0
2 β€β€ πΉ1(1 βπ
2πΈ4)(1 β
π2
4πΈ2πΈ3πΉ1)β1 πΉ0
1 π‘ β πΉ02 π‘ +
+π
2πΈ2πΉ1(1 β
π2
4πΈ2πΈ3πΉ1)β1 πΊ0
1 π‘ β πΊ02 π‘
putting πΉ2 = (1 βπ2
4πΈ2πΈ3πΉ1)β1
and substituting in the last inequality, we obtain:
π₯0 π‘, π₯01, π¦0
1 β π₯0 π‘, π₯02 , π¦0
2 β€ πΉ1πΉ2(1 βπ
2πΈ4) πΉ0
1 π‘ β πΉ02 π‘ +
+π
2πΈ2πΉ1πΉ2 πΊ0
1 π‘ β πΊ02 π‘ β― 4.11
Also, substituting the inequalities ( 4.11) in ( 4.10), we find that:
π¦0 π‘, π₯01 , π¦0
1 βπ¦0 π‘, π₯02, π¦0
2 β€ (1 βπ
2πΈ4)β1 πΊ0
1 π‘ β πΊ02 π‘ +
+π
2πΈ3(1 β
π
2πΈ4)β1[πΉ1πΉ2(1 β
π
2πΈ4) πΉ0
1 π‘ β πΉ02 π‘ +
+π
2πΈ2πΉ1πΉ2 πΊ0
1 π‘ β πΊ02 π‘ ]
and hence
π¦0 π‘, π₯01 , π¦0
1 βπ¦0 π‘, π₯02, π¦0
2 β€π
2πΈ3πΉ1πΉ2 πΉ0
1 π‘ β πΉ02 π‘ +
+[πΉ1(1 βπ
2πΈ1) +
π
2πΉ1πΉ2πΈ2] πΊ0
1 π‘ β πΊ02 π‘
β― 4.12 so, substituting inequalities ( 4.11) and ( 4.12) in inequality ( 4.5), we get the inequality ( 4.3).
and the same, substituting inequalities ( 4.11) and ( 4.12) in inequality
( 4.6), gives the inequality ( 4.4). β
REFERENCES [1] Aziz, M.A., (2006), Periodic solutions for some systems of non-linear ordinary differential equations, M. Sc. Thesis, college of
Education, University of Mosul.
[2] Butris, R. N. and Rafeq, A. Sh., (2011), Existence and Uniqueness, Solution for Non-linear Volterra Integral Equation, J. Duhok Univ. Vol.No. 1, (Pure and Eng. Sciences).
[3] Jaswon, M. A. and Symm, G. T., (1977), Integral Equations Methods in Potential Theory and Elastostatics, A subsidiary of
Hart court brace Jovanovich Publishers, Academic press, London.
[4] Korol, I. I., (2005), On periodic solutions of one class of systems of differential equations, Ukraine, Math. J. Vol. 57, No. 4.
[5] Mitropolsky, Yu. A. and Martynyuk, D. I., (1979), For Periodic Solutions for the Oscillations System with Retarded Argument,
Kiev, Ukraine. [6] Rama, M. M., (1981), Ordinary Differential Equations Theory and Applications, Britain.
[7] Rafeq, A. Sh., (2009), Periodic solutions for some classes of non-linear systems of integro-differential equations, M. Sc. Thesis,
college of Education, University of Duhok. [8] Perestyuk, N. A., (1971), The periodic solutions for non-linear systems of differential equations, Math. and Meca. J., Univ. of Kiev,
Kiev, Ukraine,5, 136-146.
[9] Perestyuk, N. A. and Martynyuk, D. I., (1976), Periodic solutions of a certain class systems of differential equations, Math. J., Univ. of Kiev , Ukraine,No 3.
[10] Samoilenko, A. M. and Ronto, N. I., (1976), A Numerical β Analytic Methods for Investigating of Periodic Solutions, Kiev,
Ukraine. [11] Samoilenko, A. M., (1966), A numerical β analytic methods for investigations of periodic systems of ordinary differential equations
II, Kiev, Ukraine, Math. J.,No 5.
[12] Shestopalov, Y. V. and Smirnov, Y. G., (2002), Integral Equations, Karlstad University. [13] Struble, R. A., (1962), Non-Linear Differential Equations, Mc Graw-Hall Book Company Inc., New York.
[14] Tarang, M., (2004), Stability of the spline collocation method for Volterra integro-differential equations, Thesis, University of
Tartu. [15] Tricomi, F. G., (1965), Integral Equations, Turin University, Turin, Italy.