periodic solutions for non-linear systems of integral equations
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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:
ð¥ ð¡, ð¥0 , ðŠ0 = ð¹0 ð¡ + ð1(ð , ð¥ ð , ð¥0, ðŠ0 , ðŠ ð , ð¥0 , ðŠ0 ,
ð¡
0
, ðº1 ð , ð ð1(ð, ð¥ ð, ð¥0 , ðŠ0 , ðŠ ð, ð¥0 , ðŠ0 )
ð
ââ
ðð,
, ð1(ð, ð¥ ð, ð¥0 , ðŠ0 , ðŠ ð, ð¥0 , ðŠ0 )ððð ð
ð ð )ðð â¯( I1 )
ðŠ ð¡, ð¥0, ðŠ0 = ðº0 ð¡ + ð2(ð , ð¥ ð , ð¥0 , ðŠ0 ,ðŠ ð , ð¥0 , ðŠ0 ,
ð¡
0
, ðº2 ð , ð ð2(ð, ð¥ ð, ð¥0 , ðŠ0 , ðŠ ð, ð¥0, ðŠ0 )
ð
ââ
ðð,
, ð2(ð, ð¥ ð, ð¥0 , ðŠ0 , ðŠ ð, ð¥0 , ðŠ0 )ðð
ð ð
ð ð
)ðð ⯠( I2 )
where ð¥ â ð· â ð ð ,ð· is closed and bounded domain subset of Euclidean space ð ð . Let the vector functions
ð1 ð¡, ð¥,ðŠ, ð§,ð€ = ð11 ð¡, ð¥, ðŠ, ð§,ð€ , ð12 ð¡, ð¥, ðŠ, ð§,ð€ ,⊠, ð1ð ð¡, ð¥, ðŠ, ð§,ð€ ,
ð2 ð¡,ð¥, ðŠ,ð¢, ð£ = ð21 ð¡, ð¥, ðŠ,ð¢, ð£ , ð22 ð¡, ð¥, ðŠ,ð¢, ð£ ,⊠, ð2ð ð¡, ð¥, ðŠ,ð¢, ð£ ,
ð1 ð¡, ð¥, ðŠ = ð11 ð¡, ð¥, ðŠ ,ð12 ð¡, ð¥, ðŠ ,⊠,ð1ð ð¡, ð¥, ðŠ ,
ð2 ð¡, ð¥, ðŠ = ð21 ð¡, ð¥, ðŠ ,ð22 ð¡, ð¥, ðŠ ,⊠,ð2ð ð¡, ð¥, ðŠ ,
ð¹0 ð¡ = (ð¹01 ð¡ ,ð¹02 ð¡ ,⊠,ð¹0ð ð¡ ),
and
ðº0 ð¡ = (ðº01 ð¡ ,ðº02 ð¡ ,⯠,ðº0ð ð¡ ) are defined and continuous on the domains:
ð¡, ð¥, ðŠ, ð§,ð€ â ð 1 à ð· à ð·1 à ð·2 = ââ,â à ð· à ð·1 à ð·2 à ð·3
ð¡, ð¥, ðŠ,ð¢, ð£ â ð 1 à ð· à ð·1 à ð·2 = ââ,â à ð· à ð·1 à ð·2 à ð·3
⯠1.1
and periodic in t of period T, Also a(t) and b(t) are continuous and periodic in t of period ð. Suppose that the functions ð1 ð¡, ð¥, ðŠ, ð§,ð€ , ð2 ð¡, ð¥, ðŠ,ð¢, ð£ , ð1 ð¡, ð¥, ðŠ
and ð2 ð¡, ð¥, ðŠ satisfies the following inequalities:
Periodic Solutions for Non-Linear Systems of Integral Equations
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ð1 ð¡, ð¥, ðŠ, ð§,ð€ †ð1 , ð2 ð¡, ð¥, ðŠ,ð¢, 𣠆ð2 , ð1 ,ð2 > 0
ð1 ð¡, ð¥, ðŠ †ð1 , ð1 ð¡, ð¥, ðŠ †ð1 , ð1,ð 2 > 0 ⯠1.2
ð1 ð¡, ð¥1 , ðŠ1 , ð§1 ,ð€1 â ð1 ð¡, ð¥2 , ðŠ2 , ð§2 ,ð€2 †ðŸ1 ð¥1 â ð¥2 + ðŸ2 ðŠ1âðŠ2 +
+ðŸ3 ð§1âð§2 + ðŸ4 ð€1âð€2 ⯠1.3 ð2 ð¡, ð¥1, ðŠ1 ,ð¢1, ð£1 â ð2 ð¡, ð¥2 , ðŠ2 ,ð¢2, ð£2 †ð¿1 ð¥1 â ð¥2 + ð¿2 ðŠ1âðŠ2 +
+ð¿3 ð¢1âð¢2 + ð¿4 ð£1âð£2 , ⯠1.4 ð1 ð¡, ð¥1 , ðŠ1 â ð1 ð¡, ð¥2 , ðŠ2 †ð 1 ð¥1 â ð¥2 + ð 2 ðŠ1 â ðŠ2 , ⯠1.5 ð2 ð¡, ð¥1 , ðŠ1 â ð2 ð¡, ð¥2 , ðŠ2 †ð»1 ð¥1 â ð¥2 + ð»2 ðŠ1 â ðŠ2 , ⯠1.6 for all ð¡ â ð 1 , ð¥, ð¥1 , ð¥2 â ð· , ðŠ , ðŠ1 , ðŠ2 â ð·1 , ð§, ð§1, ð§2 ,ð¢,ð¢1,ð¢2 â ð·2 , and ð€,ð€1,ð€2 , ð£, ð£1 , ð£2 â ð·3 , where ð1 = ð11 ,ð12 ,⊠,ð1ð , ð2 = ð21 ,ð22 ,⊠,ð2ð , ð1 = (ð11 ,ð12 ,⊠,ð1ð) and
ð2 = ð21 ,ð22 ,⊠,ð2ð are positive constant vectors and ðŸ1 = (ðŸ1ðð), ðŸ2 = (ðŸ2ðð
), ðŸ3 = (ðŸ3 ðð), ðŸ4 = (ðŸ4 ðð
),
ð¿1 = (ð¿1ðð), ð¿2 = (ð¿2ðð
),
ð¿3 = ( ð¿3 ðð ), ð¿4 = (ð¿4 ðð
), ð 1 = (ð 1ðð), ð 2 = (ð 2ðð
), ð»1 = (ð»1 ðð), and
ð»2 = (ð»2 ðð) are positive constant matrices.
Also ðº1 ð¡, ð and ðº2 ð¡, ð are (nÃn) continuous positive matrix and
periodic in t, s of period T in the domain ð 1 Ã ð 1 and satisfy the following conditions:
ðº1 ð¡, ð †ðŸ ðâð1 ð¡âð , ðŸ, ð1 > 0 ⯠1.7 and
ðº2(ð¡, ð ) †ð¿ ðâð2 ð¡âð , ð¿, ð2 > 0 ⯠1.8 where ââ < 0 †ð †ð¡ †ð < â , ð, ð = 1,2,⯠,ð, and ð = ððð¥ð¡â 0,ð ð ð¡ â ð ð¡ , . = ððð¥ð¡â 0,ð .
Now define a non-empty sets as follows:
ð·ð1= ð· â
ð
2ð 1 ,
ð·1ð2= ð·1 â
ð
2ð 2 ,
ð·2ð1= ð·2 â
ð
2 ðŸ
ð1
ð 1ð1 + ð 2ð2 ,
ð·3ð1= ð·3 â
ð
2ð ð 1ð1 + ð 2ð2 ,
ð·2ð2= ð·2 â
ð
2 ð¿
ð2
ð»1ð1 + ð»2ð2 ,
ð·3ð2= ð·3 â
ð
2ð ð»1ð1 + ð»2ð2 .
⯠1.9
Forever, we suppose that the greatest eigen-value ðððð¥ of the following
matrix:
Q0 =
ð
2[ðŸ1 + ð 1(
ðŸ
ð1
ðŸ3 + ððŸ4)] ð
2[ðŸ2 + ð 2(
ðŸ
ð1
ðŸ3 + ððŸ4)]
ð
2[ð¿1 + ð»1(ð¿3
ð¿
ð2
+ ðð¿4)] ð
2[ð¿2 + ð»2(
ð¿
ð2
ð¿3 + ðð¿4)]
⯠1.10
is less than unity, i.e.
ðððð¥ ð0 =Ï1 + ð1
2 + 4(ð2 â ð3)
2< 1 , ⯠1.11
where Ï1 =ð
2[ðŸ1 + ð 1(
ðŸ
ð1ðŸ3 + ððŸ4)] +
ð
2[ð¿2 + ð»2(
ð¿
ð2ð¿3 + ðð¿4)],
Ï2 = (ð
2[ðŸ2 + ð 2(
ðŸ
ð1ðŸ3 + ððŸ4)])(
ð
2[ð¿1 + ð»1(
ð¿
ð2ð¿3 + ðð¿4)]) and
ð3 = (ð
2[ðŸ1 + ð 1(
ðŸ
ð1ðŸ3 + ððŸ4)])(
ð
2[ð¿2 + ð»2(
ð¿
ð2ð¿3 + ðð¿4)]).
By using lemma 3.1[10],we can state and prove the following lemma:
Lemma 1.1. Let ð1(ð¡, ð¥, ðŠ, ð§,ð€) and ð2(ð¡, ð¥, ðŠ,ð¢, ð£) be a vector functions which are defined and continuous in
the interval 0,ð , then the inequality:
Periodic Solutions for Non-Linear Systems of Integral Equations
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ð¿1 ð¡, ð¥0 , ðŠ0
ð¿2(ð¡, ð¥0 , ðŠ0) â€
ð1ðŒ ð¡
ð2ðŒ ð¡ ⯠1.12
satisfies for 0 †ð¡ †ð and ðŒ ð¡ â€ð
2 ,
where ðŒ ð¡ = 2ð¡(1 âð¡
ð) for all ð¡ â 0,ð ,
ð¿1 ð¡, ð¥0 , ðŠ0 = [ð1(ð , ð¥0 , ðŠ0 ,
ð¡
0
ðº1 ð , ð ð1(ð, ð¥0, ðŠ0)
ð
ââ
ðð,
, ð1(ð, ð¥0 , ðŠ0)ðð
ð ð
ð ð
) â1
ð ð1(ð , ð¥0 , ðŠ0 ,
ð
0
, ðº1 ð , ð ð1(ð, ð¥0 , ðŠ0)
ð
ââ
ðð, ð1(ð, ð¥0, ðŠ0)ðð
ð ð
ð ð
)ðð ]ðð
ð¿2 ð¡, ð¥0 , ðŠ0 = [ð2(ð , ð¥0 , ðŠ0 ,
ð¡
0
ðº2 ð , ð ð2(ð, ð¥0 , ðŠ0)
ð
ââ
ðð,
, ð2(ð, ð¥0, ðŠ0)ðð
ð ð
ð ð
) â1
ð ð2(ð , ð¥0 , ðŠ0 ,
ð
0
, ðº2 ð , ð ð2(ð, ð¥0 , ðŠ0)
ð
ââ
ðð, ð2(ð, ð¥0 , ðŠ0)ðð
ð ð
ð ð
)ðð ]ðð
Proof.
ð¿1 ð¡, ð¥0 , ðŠ0 †(1 âð¡
ð) ð1(ð , ð¥0 , ðŠ0 , ðº1
ð
ââ
ð , ð ð1 ð, ð¥0, ðŠ0 ðð,
ð¡
0
, ð1 ð, ð¥0 , ðŠ0 ðð
ð(ð )
ð(ð )
) ðð +
+ð¡
ð ð1(ð , ð¥0 , ðŠ0 , ðº1
ð
ââ
ð , ð ð1 ð, ð¥0, ðŠ0 ðð,
ð
ð¡
ð1 ð, ð¥0, ðŠ0 ðð)
ð(ð )
ð(ð )
ðð
†(1 âð¡
ð) ð1
ð¡
0
ðð +ð¡
ð ð1
ð
ð¡
ðð
†ð1[(1 âð¡
ð)ð¡ +
ð¡
ð(ð â ð¡)]
so that:
ð¿1 ð¡, ð¥0 , ðŠ0 †ð1ðŒ ð¡ ⯠1.13 And similarly, we get also
ð¿2 ð¡, ð¥0 , ðŠ0 †ð2ðŒ ð¡ ⯠1.14
From ( 1.13) and ( 1.14), then the inequality ( 1.12) satisfies for all
0 †ð¡ †ð and ðŒ ð¡ â€ð
2 . â
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 ð¥ = ð¥ ð¡, ð¥0 , ðŠ0 and ðŠ = ðŠ(ð¡, ð¥0 , ðŠ0), then the sequence of
functions:
ð¥ð+1 ð¡, ð¥0 , ðŠ0 = ð¹0 ð¡ + [ð1(ð , ð¥ð ð , ð¥0 , ðŠ0 , ðŠð ð , ð¥0 , ðŠ0 ,
ð¡
0
Periodic Solutions for Non-Linear Systems of Integral Equations
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, ðº1 ð , ð ð1(ð, ð¥ð ð, ð¥0 , ðŠ0 , ðŠð ð, ð¥0, ðŠ0 )
ð
ââ
ðð,
, ð1(ð, ð¥ð ð, ð¥0 , ðŠ0 , ðŠð ð, ð¥0, ðŠ0 )ðð)
ð ð
ð ð
â1
ð ð1(ð , ð¥ð ð , ð¥0 , ðŠ0 ,
ð
0
, ðŠð ð , ð¥0 , ðŠ0 , ðº1 ð , ð ð1(ð, ð¥ð ð, ð¥0 , ðŠ0 , ðŠð ð, ð¥0 , ðŠ0 )
ð
ââ
ðð,
, ð1(ð, ð¥ð ð, ð¥0 , ðŠ0 , ðŠð ð, ð¥0 , ðŠ0 )ðð)ðð ]ðð
ð ð
ð ð
⯠2.1
with
ð¥0 ð¡, ð¥0 , ðŠ0 = ð¹0 ð¡ = ð¥0 , for all m = 0, 1,2,â¯
ðŠð+1 ð¡, ð¥0 , ðŠ0 = ðº0 ð¡ + [ð2(ð , ð¥ð ð , ð¥0 , ðŠ0 , ðŠð ð , ð¥0 ,ðŠ0 ,
ð¡
0
, ðº2 ð , ð ð2(ð, ð¥ð ð, ð¥0, ðŠ0 , ðŠð ð, ð¥0 , ðŠ0 )
ð
ââ
ðð,
, ð2(ð, ð¥ð ð, ð¥0 , ðŠ0 , ðŠð ð, ð¥0 , ðŠ0 )ðð)
ð ð
ð ð
â1
ð ð2(ð , ð¥ð ð , ð¥0 , ðŠ0 ,
ð
0
, ðŠð ð , ð¥0, ðŠ0 , ðº2 ð , ð ð2(ð, ð¥ð ð, ð¥0 , ðŠ0 , ðŠð ð, ð¥0 , ðŠ0 )
ð
ââ
ðð,
, ð2(ð,ð¥ð ð, ð¥0, ðŠ0 , ðŠð ð, ð¥0 , ðŠ0 )ðð)ðð ]ðð
ð ð
ð ð
⯠2.2
with
ðŠ0 ð¡, ð¥0 , ðŠ0 = ðº0 ð¡ = ðŠ0 , for all m = 0, 1,2,â¯
periodic in t of period T, and uniformly converges as ð â â in the domain:
ð¡, ð¥0 , ðŠ0 â 0,ð à ð·ð1 à ð·1ð2
⯠2.3
to the limit functions ð¥0 ð¡, ð¥0 , ðŠ0 , ðŠ0 ð¡, ð¥0 , ðŠ0 defined in the domain ( 2.3) which are periodic in t of period T
and satisfying the system of integral equations:
ð¥ ð¡, ð¥0 , ðŠ0 = ð¹0 ð¡ + [ð1(ð , ð¥ ð , ð¥0 , ðŠ0 ,ðŠ ð , ð¥0 , ðŠ0 ,
ð¡
0
, ðº1 ð , ð ð1(ð, ð¥ ð, ð¥0, ðŠ0 , ðŠ ð, ð¥0 , ðŠ0 )
ð
ââ
ðð,
, ð1(ð, ð¥ ð, ð¥0 , ðŠ0 , ðŠ ð, ð¥0 , ðŠ0 )ðð)
ð ð
ð ð
â1
ð ð1(ð , ð¥ ð , ð¥0 , ðŠ0 ,
ð
0
, ðŠ ð , ð¥0 ,ðŠ0 , ðº1 ð , ð ð1(ð, ð¥ ð, ð¥0, ðŠ0 , ðŠ ð, ð¥0 , ðŠ0 )
ð
ââ
ðð,
, ð1(ð, ð¥ ð, ð¥0 , ðŠ0 , ðŠ ð, ð¥0 , ðŠ0 )ðð)ðð ]ðð
ð ð
ð ð
⯠2.4
and
ðŠ ð¡, ð¥0, ðŠ0 = ðº0 ð¡ + [ð2(ð , ð¥ ð , ð¥0 , ðŠ0 , ðŠ ð , ð¥0, ðŠ0 ,
ð¡
0
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, ðº2 ð , ð ð2(ð, ð¥ ð, ð¥0 , ðŠ0 , ðŠ ð, ð¥0, ðŠ0 )
ð
ââ
ðð,
, ð2(ð, ð¥ ð, ð¥0 , ðŠ0 , ðŠ ð, ð¥0, ðŠ0 )ðð)
ð ð
ð ð
â1
ð ð2(ð , ð¥ ð , ð¥0 , ðŠ0 ,
ð
0
, ðŠ ð , ð¥0, ðŠ0 , ðº2 ð , ð ð2(ð, ð¥ ð, ð¥0 , ðŠ0 , ðŠ ð, ð¥0, ðŠ0 )
ð
ââ
ðð,
, ð2(ð, ð¥ ð, ð¥0 , ðŠ0 , ðŠ ð, ð¥0, ðŠ0 )ðð)ðð ]ðð ð ð
ð ð ⯠2.5
provided that:
ð¥0 ð¡, ð¥0 , ðŠ0 â ð¥ð ð¡, ð¥0 , ðŠ0
ðŠ0 ð¡, ð¥0 , ðŠ0 â ðŠð ð¡, ð¥0 , ðŠ0 †ð0
ð (ðž â ð0)â1ð¶0 ⯠2.6
for all ð ⥠0 , where
ð¶0 =
ð
2ð1
ð
2ð2
, ðž is identity matrix.
Proof. Consider the sequence of functions ð¥1 ð¡, ð¥0 , ðŠ0 , ð¥2 ð¡, ð¥0 , ðŠ0 ,â¯, ð¥ð ð¡, ð¥0 , ðŠ0 ,â¯, and ðŠ1 ð¡, ð¥0, ðŠ0 , ðŠ2 ð¡, ð¥0 , ðŠ0 ,⯠, ðŠð ð¡, ð¥0, ðŠ0 ,â¯, 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 ð¡ of period ð.
Now, by the lemma 1.1, and using the sequence of functions (2.1),
when ð = 0, we get:
By mathematical induction and lemma1.1 , we have the following inequality:
ð¥ð ð¡, ð¥0 , ðŠ0 â ð¥0 â€ð
2 ð1 ⯠2.13
i.e. ð¥ð ð¡,ð¥0 , ðŠ0 â ð·, for all ð¡ â ð 1 , ð¥0 â ð·ð1, ðŠ0 â ð·1ð2
, ð = 0,1,2,â¯,
and
ðŠð ð¡, ð¥0, ðŠ0 â ðŠ0 â€ð
2 ð2 ⯠2.14
i.e. ðŠð ð¡, ð¥0 , ðŠ0 â ð·, for all ð¡ â ð 1 , ð¥0 â ð·ð1, ðŠ0 â ð·1ð2
,ð = 0,1,2,â¯,
Now, from ( 2.13), we get:
ð§ð ð¡, ð¥0 , ðŠ0 â ð§0(ð¡, ð¥0 , ðŠ0) â€ð
2(ðŸ
ð1
) ð 1ð1 + ð 2ð2 ⯠2.15
and
ð€ð ð¡, ð¥0 , ðŠ0 â ð€0(ð¡,ð¥0 , ðŠ0) â€ð
2ð ð 1ð1 + ð 2ð2 ⯠2.16
for all ð¡ â ð 1 , ð¥0 â ð·ð1, ðŠ0 â ð·1ð2
, ð§0 ð¡, ð¥0 , ðŠ0 â ð·2ð1and
ð€0 ð¡,ð¥0 , ðŠ0 â ð·3ð1,
i.e. ð§ð ð¡, ð¥0 , ðŠ0 â ð·2 and ð€ð ð¡, ð¥0 , ðŠ0 â ð·3 , for all ð¥0 â ð·ð1, ðŠ0 â ð·1ð2
,
where
ð§ð ð¡, ð¥0 , ðŠ0 = ðº1 ð¡, ð ð1 ð , ð¥ð ð , ð¥0 , ðŠ0 , ðŠð ð , ð¥0 , ðŠ0 ðð
ð¡
ââ
and
ð€ð ð¡, ð¥0 , ðŠ0 = ð1 ð , ð¥ð ð , ð¥0 , ðŠ0 , ðŠð ð , ð¥0 , ðŠ0 ðð
ð(ð¡)
ð(ð¡)
, ð = 0,1,2,⯠,
Also from (4.2.14), we get:
ð¢ð ð¡, ð¥0 , ðŠ0 â ð¢0(ð¡, ð¥0, ðŠ0) â€ð
2(ð¿
ð2
) ð»1ð1 + ð»2ð2 ⯠2.17
and
ðð ð¡, ð¥0 , ðŠ0 â ð0(ð¡, ð¥0, ðŠ0) â€ð
2ð ð»1ð1 + ð»2ð2 ⯠2.18
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for all ð¡ â ð 1 , ð¥0 â ð·ð1, ðŠ0 â ð·1ð2
, ð¢0 ð¡, ð¥0, ðŠ0 â ð·2ð2and
ð£0 ð¡, ð¥0 , ðŠ0 â ð·3ð2,
i.e. ð¢ð ð¡, ð¥0 , ðŠ0 â ð·2 and ð£ð ð¡, ð¥0, ðŠ0 â ð·3 , for all ð¥0 â ð·ð1, ðŠ0 â ð·1ð2
,
where
ð¢ð ð¡, ð¥0 , ðŠ0 = ðº2 ð¡, ð ð2 ð , ð¥ð ð , ð¥0 , ðŠ0 , ðŠð ð , ð¥0, ðŠ0 ðð
ð¡
ââ
and
ð£ð ð¡, ð¥0 , ðŠ0 = ð2 ð , ð¥ð ð , ð¥0 , ðŠ0 , ðŠð ð , ð¥0 , ðŠ0 ðð
ð(ð¡)
ð(ð¡)
, ð = 0,1,2,⯠,
Next, we claim that two sequences ð¥ð (ð¡, ð¥0, ðŠ0)}ð=0â , ðŠð (ð¡, ð¥0, ðŠ0) }ð=0
â
are convergent uniformly to the limit functions ð¥, ðŠ on the domain ( 2.3).
By mathematical induction and lemma 1.1, we find that:
ð¥ð+1 ð¡, ð¥0 , ðŠ0 â ð¥ð ð¡, ð¥0 , ðŠ0 â€
†ðŒ(ð¡)[ðŸ1 + ð 1(ðŸ
ð1
ðŸ3 + ððŸ4)] ð¥ð ð¡, ð¥0 , ðŠ0 â ð¥ðâ1 ð¡, ð¥0 , ðŠ0
+ðŒ(ð¡)[ðŸ2 + ð 2(ðŸ
ð1
ðŸ3 + ððŸ4)] ðŠð ð¡, ð¥0, ðŠ0 â ðŠðâ1 ð¡, ð¥0 , ðŠ0
⯠2.19 and
ðŠð+1 ð¡, ð¥0 , ðŠ0 â ðŠð ð¡, ð¥0, ðŠ0 â€
†ðŒ(ð¡)[ð¿1 + ð»1(ð¿
ð2
ð¿3 + ðð¿4)] ð¥ð ð¡, ð¥0, ðŠ0 â ð¥ðâ1 ð¡, ð¥0, ðŠ0 +
+ðŒ(ð¡)[ð¿2 + ð»2(ð¿
ð2
ð¿3 + ðð¿4)] ðŠð ð¡, ð¥0 , ðŠ0 â ðŠðâ1 ð¡, ð¥0 , ðŠ0
⯠2.20 We can write the inequalities ( 2.19) and ( 2.20) in a vector form:
ð¶ð+1 ð¡, ð¥0 , ðŠ0 †ð ð¡ ð¶ð ð¡, ð¥0, ðŠ0 ⯠2.21 where
ð¶ð+1 ð¡, ð¥0, ðŠ0 = ð¥ð+1 ð¡, ð¥0 , ðŠ0 â ð¥ð ð¡, ð¥0 , ðŠ0
ðŠð+1 ð¡, ð¥0, ðŠ0 â ðŠð ð¡, ð¥0 , ðŠ0 ,
and
ð ð¡ =
ðŒ(ð¡)[ðŸ1 + ð 1(
ðŸ
ð1
ðŸ3 + ððŸ4)] ðŒ(ð¡)[ðŸ2 + ð 2(ðŸ
ð1
ðŸ3 + ððŸ4)]
ðŒ(ð¡)[ð¿1 + ð»1(ð¿
ð2
ð¿3 + ðð¿4)] ðŒ(ð¡)[ð¿2 + ð»2(ð¿
ð2
ð¿3 + ðð¿4)]
,
ð¶ð ð¡, ð¥0 , ðŠ0 = ð¥ð ð¡, ð¥0 ,ðŠ0 â ð¥ðâ1 ð¡, ð¥0 , ðŠ0
ðŠð ð¡, ð¥0 , ðŠ0 â ðŠðâ1 ð¡, ð¥0 , ðŠ0
Now, we take the maximum value to both sides of the inequality ( 2.21),
for all 0 †ð¡ †ð and ðŒ ð¡ â€ð
2 , we get:
ð¶ð+1 †ð0ð¶ð , ⯠2.22 where ð0 = ððð¥ð¡â 0,ð ð(ð¡) .
Q0 =
ð
2[ðŸ1 + ð 1(
ðŸ
ð1
ðŸ3 + ððŸ4)] ð
2[ðŸ2 + ð 2(
ðŸ
ð1
ðŸ3 + ððŸ4)]
ð
2[ð¿1 + ð»1(
ð¿
ð2
ð¿3 + ðð¿4)] ð
2[ð¿2 + ð»2(
ð¿
ð2
ð¿3 + ðð¿4)]
By iterating the inequality (4.2.22), we find that:
ð¶ð+1 †ð0ðð¶0 , ⯠2.23
which leads to the estimate:
ð¶ð †ð0ðâ1
ð
ð=1
ð¶0
ð
ð=1
⯠2.24
Since the matrix ð0 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
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limðââ
ð0ðâ1
ð
ð=1
ð¶0 = ð0ðâ1
â
ð=1
ð¶0 = (ðž â ð0)â1ð¶0 ⯠( 2.25)
The limiting relation (4.2.25) signifies a uniform convergence of the sequence
ð¥ð (ð¡, ð¥0, ðŠ0) , ðŠð (ð¡, ð¥0 , ðŠ0) ð=0â in the domain (4.2.3) as ð â â.
Let
limðââ
ð¥ð ð¡, ð¥0 , ðŠ0 = ð¥0 ð¡, ð¥0 , ðŠ0 ,
limðââ
ðŠð ð¡, ð¥0 , ðŠ0 = ðŠ0 ð¡, ð¥0 , ðŠ0 . ⯠( 2.26)
Finally, we show that ð¥ ð¡, ð¥0 , ðŠ0 â¡ ð¥0 ð¡, ð¥0 ,ðŠ0 â ð· and
ðŠ ð¡, ð¥0 , ðŠ0 â¡ ðŠ0(ð¡, ð¥0 , ðŠ0) â ð·1, for all ð¥0 â ð·ð1 and ðŠ0 â ð·1ð2
.
By using inequalities ( 2.1) and ( 2.4) and lemma 1.1, such that:
[ð1(ð , ð¥ð (ð , ð¥0 , ðŠ0), ðŠð (ð , ð¥0 , ðŠ0),
ð¡
0
ð§ð ð , ð¥0, ðŠ0 ,ð€ð (ð , ð¥0 , ðŠ0)) â
â1
ð ð1(ð , ð¥ð (ð , ð¥0 , ðŠ0), ðŠð (ð , ð¥0 , ðŠ0),
ð
0
ð§ð ð , ð¥0, ðŠ0 ,ð€ð (ð , ð¥0 , ðŠ0))ðð ]ðð â
â [ð1(ð , ð¥(ð , ð¥0 , ðŠ0), ðŠ(ð , ð¥0, ðŠ0),
ð¡
0
ð§ ð , ð¥0, ðŠ0 ,ð€(ð , ð¥0 , ðŠ0)) â
â1
ð ð1(ð , ð¥(ð , ð¥0, ðŠ0), ðŠ(ð , ð¥0 , ðŠ0),
ð
0
ð§ ð , ð¥0 , ðŠ0 ,ð€(ð , ð¥0, ðŠ0))ðð ]ðð
†ðŒ(ð¡) [ðŸ1 + ð 1(ðŸ
ð1
ðŸ3 + ððŸ4)] ð¥ð ð , ð¥0 , ðŠ0 â ð¥ ð , ð¥0, ðŠ0 +
+[ðŸ2 + ð 2(ðŸ
ð1ðŸ3 + ððŸ4)] ðŠð ð , ð¥0, ðŠ0 â ðŠ ð , ð¥0 , ðŠ0 â€
â€ð
2 [ðŸ1 + ð 1(
ðŸ
ð1
ðŸ3 + ððŸ4)] ð¥ð ð¡, ð¥0, ðŠ0 â ð¥ ð¡, ð¥0 , ðŠ0 +
+[ðŸ2 + ð 2(ðŸ
ð1ðŸ3 + ððŸ4)] ðŠð ð¡, ð¥0 , ðŠ0 â ðŠ ð¡, ð¥0 , ðŠ0 .
From inequality ( 2.26) and on the other hand suppose that:.
ð¥ð ð¡, ð¥0 , ðŠ0 â ð¥ ð¡, ð¥0 , ðŠ0 †ð1 , ðŠð ð¡, ð¥0 , ðŠ0 â ðŠ ð¡, ð¥0 , ðŠ0 †ð2 .
Thus
[ð1(ð , ð¥ð (ð , ð¥0 , ðŠ0), ðŠð (ð , ð¥0 , ðŠ0),
ð¡
0
ð§ð ð , ð¥0, ðŠ0 ,ð€ð (ð , ð¥0 , ðŠ0)) â
â1
ð ð1(ð , ð¥ð (ð , ð¥0 , ðŠ0), ðŠð (ð , ð¥0 , ðŠ0),
ð
0
ð§ð ð , ð¥0, ðŠ0 ,ð€ð (ð , ð¥0 , ðŠ0))ðð ]ðð â
â [ð1(ð , ð¥(ð , ð¥0 , ðŠ0), ðŠ(ð , ð¥0, ðŠ0),
ð¡
0
ð§ ð , ð¥0, ðŠ0 ,ð€(ð , ð¥0 , ðŠ0)) â
â1
ð ð1(ð , ð¥(ð , ð¥0 , ðŠ0), ðŠ(ð ,ð¥0 , ðŠ0),
ð
0
ð§ ð , ð¥0, ðŠ0 ,ð€(ð , ð¥0 , ðŠ0))ðð ]ðð
â€ð
2[ðŸ1 + ð 1(
ðŸ
ð1
ðŸ3 + ððŸ4)]ð1 +ð
2[ðŸ2 + ð 2(
ðŸ
ð1
ðŸ3 + ððŸ4)]ð2
Putting ð1 =ð3
ð
2[ðŸ1+ð 1(
ðŸ
ð1ðŸ3+ððŸ4)]
and ð2 =ð4
ð
2[ðŸ2+ð 2(
ðŸ
ð1ðŸ3+ððŸ4)]
and substituting in the last equation, we have:
[ð1(ð , ð¥ð (ð , ð¥0 , ðŠ0), ðŠð (ð , ð¥0 , ðŠ0),
ð¡
0
ð§ð ð , ð¥0, ðŠ0 ,ð€ð (ð , ð¥0 , ðŠ0)) â
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â1
ð ð1(ð , ð¥ð (ð , ð¥0 , ðŠ0), ðŠð (ð , ð¥0 , ðŠ0),
ð
0
ð§ð ð , ð¥0, ðŠ0 ,ð€ð (ð , ð¥0 , ðŠ0))ðð ]ðð â
â [ð1(ð , ð¥(ð , ð¥0 , ðŠ0), ðŠ(ð , ð¥0, ðŠ0),
ð¡
0
ð§ ð , ð¥0, ðŠ0 ,ð€(ð , ð¥0 , ðŠ0)) â
â1
ð ð1(ð , ð¥(ð , ð¥0 , ðŠ0), ðŠ(ð , ð¥0, ðŠ0),
ð
0
ð§ ð , ð¥0, ðŠ0 ,ð€(ð , ð¥0 , ðŠ0))ðð ]ðð
â€ð
2[ðŸ1 + ð 1(
ðŸ
ð1
ðŸ3 + ððŸ4)]ð3
ð2
[ðŸ1 + ð 1(ðŸð1ðŸ3 + ððŸ4)]
+
+ð
2[ðŸ2 + ð 2(
ðŸ
ð1
ðŸ3 + ððŸ4)]ð4
ð2
[ðŸ2 + ð 2(ðŸð1ðŸ3 + ððŸ4)]
†ð3 + ð4, and choosing ð3 + ð4 = ð , we get:
[ð1(ð , ð¥ð (ð , ð¥0, ðŠ0), ðŠð (ð , ð¥0 , ðŠ0),
ð¡
0
ð§ð ð , ð¥0 , ðŠ0 ,ð€ð (ð , ð¥0, ðŠ0)) â
â1
ð ð1(ð , ð¥ð (ð , ð¥0 , ðŠ0), ðŠð (ð , ð¥0 , ðŠ0),
ð
0
ð§ð ð , ð¥0, ðŠ0 ,ð€ð (ð , ð¥0 , ðŠ0))ðð ]ðð â
â [ð1(ð , ð¥(ð , ð¥0 , ðŠ0), ðŠ(ð , ð¥0, ðŠ0),
ð¡
0
ð§ ð , ð¥0, ðŠ0 ,ð€(ð , ð¥0 , ðŠ0)) â
â1
ð ð1(ð , ð¥(ð ,ð¥0 , ðŠ0), ðŠ(ð , ð¥0 , ðŠ0),
ð
0
ð§ ð , ð¥0 , ðŠ0 ,ð€(ð , ð¥0 , ðŠ0))ðð ]ðð †ð
for all ð ⥠0,
i. e. limðââ
[ð1(ð , ð¥ð (ð , ð¥0, ðŠ0), ðŠð (ð , ð¥0 , ðŠ0),
ð¡
0
ð§ð ð , ð¥0 , ðŠ0 ,ð€ð (ð , ð¥0, ðŠ0)) â
â1
ð ð1(ð , ð¥ð (ð , ð¥0 , ðŠ0), ðŠð (ð , ð¥0 , ðŠ0),
ð
0
ð§ð ð , ð¥0, ðŠ0 ,ð€ð (ð , ð¥0 , ðŠ0))ðð ]ðð =
â [ð1(ð , ð¥(ð , ð¥0 , ðŠ0), ðŠ(ð , ð¥0, ðŠ0),
ð¡
0
ð§ ð , ð¥0, ðŠ0 ,ð€(ð , ð¥0 , ðŠ0)) â
â1
ð ð1(ð , ð¥(ð , ð¥0, ðŠ0), ðŠ(ð , ð¥0 , ðŠ0),
ð
0
ð§ ð , ð¥0 , ðŠ0 ,ð€(ð , ð¥0, ðŠ0))ðð ]ðð .
So ð¥ ð¡, ð¥0 , ðŠ0 â ð·, and ð¥ ð¡, ð¥0, ðŠ0 â¡ ð¥0(ð¡, ð¥0, ðŠ0) is a periodic solution of ( I1),
Also by using the same method above we can prove ðŠ ð¡, ð¥0, ðŠ0 â ð·1 , and ðŠ ð¡, ð¥0 , ðŠ0 â¡ ðŠ0 ð¡, ð¥0 , ðŠ0 is
also periodic solution of (ðŒ2 ).
Theorem 2.2. With the hypotheses and all conditions of the theorem 2.1, the periodic solution of integral
equations ( ðŒ1 ) and ( ðŒ2 ) are a unique on the domain (1.3).
Proof. Suppose that ð¥ ð¡, ð¥0 , ðŠ0 and ðŠ ð¡, ð¥0 , ðŠ0 be another periodic solutions
for the systems (ðŒ1) and ( ðŒ2 ) defined and continuous and periodic in ð¡ of period ð, this means that:
ð¥ ð¡, ð¥0 , ðŠ0 = ð¹0 ð¡ + [ð1(ð , ð¥ ð , ð¥0 , ðŠ0 ,ðŠ ð , ð¥0 , ðŠ0 ,
ð¡
0
, ðº1 ð , ð ð1(ð, ð¥ ð, ð¥0 , ðŠ0 , ðŠ ð, ð¥0, ðŠ0 )
ð
ââ
ðð,
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, ð1(ð, ð¥ ð, ð¥0 , ðŠ0 , ðŠ ð, ð¥0 , ðŠ0 )ðð)
ð ð
ð ð
â1
ð ð1(ð , ð¥ ð , ð¥0 , ðŠ0 ,
ð
0
, ðŠ ð , ð¥0 , ðŠ0 , ðº1 ð , ð ð1(ð, ð¥ ð, ð¥0 , ðŠ0 , ðŠ ð, ð¥0 , ðŠ0 )
ð
ââ
ðð,
, ð1(ð, ð¥ ð, ð¥0 , ðŠ0 , ðŠ ð, ð¥0, ðŠ0 )ðð)ðð ]ðð
ð ð
ð ð
⯠2.27
and
ðŠ ð¡, ð¥0, ðŠ0 = ðº0 ð¡ + [ð2(ð , ð¥ ð , ð¥0 , ðŠ0 , ðŠ ð , ð¥0, ðŠ0 ,
ð¡
0
, ðº2 ð , ð ð2(ð, ð¥ ð, ð¥0, ðŠ0 , ðŠ ð, ð¥0 , ðŠ0 )
ð
ââ
ðð,
, ð2(ð, ð¥ ð, ð¥0 , ðŠ0 , ðŠ ð, ð¥0, ðŠ0 )ðð)
ð ð
ð ð
â1
ð ð2(ð , ð¥ ð , ð¥0 , ðŠ0 ,
ð
0
, ðŠ ð , ð¥0 , ðŠ0 , ðº2 ð , ð ð2(ð, ð¥ ð, ð¥0 , ðŠ0 , ðŠ ð, ð¥0, ðŠ0 )
ð
ââ
ðð,
, ð2(ð, ð¥ ð, ð¥0 , ðŠ0 , ðŠ ð, ð¥0, ðŠ0 )ðð)ðð ]ðð
ð ð
ð ð
⯠2.28
For their difference, we should obtain the inequality:
ð¥ ð¡, ð¥0 , ðŠ0 â ð¥ ð¡, ð¥0, ðŠ0 â€
â€ð
2 [ðŸ1 + ð 1(
ðŸ
ð1
ðŸ3 + ððŸ4)] ð¥ ð¡, ð¥0 , ðŠ0 â ð¥ ð¡,ð¥0 , ðŠ0 +
+ð
2[ðŸ2 + ð 2(
ðŸ
ð1
ðŸ3 + ððŸ4)] ðŠ ð¡, ð¥0 , ðŠ0 â ðŠ ð¡, ð¥0 , ðŠ0 ⯠2.29
And also
ðŠ ð , ð¥0 , ðŠ0 â ðŠ ð , ð¥0, ðŠ0 â€
†(1 âð¡
ð) [ð¿1 ð¥ ð , ð¥0 , ðŠ0 â ð¥ ð , ð¥0, ðŠ0 + ð¿2 ðŠ ð , ð¥0 , ðŠ0 â ðŠ ð , ð¥0, ðŠ0
ð¡
0
+ð¿
ð2
ð¿3 ð»1 ð¥ ð , ð¥0, ðŠ0 â ð¥ ð , ð¥0 , ðŠ0 + ð»2 ðŠ ð , ð¥0 , ðŠ0 â ðŠ ð , ð¥0 , ðŠ0 +
+ðð¿4(ð»1 ð¥ ð , ð¥0 , ðŠ0 â ð¥ ð , ð¥0 , ðŠ0 + ð»2 ðŠ ð , ð¥0 , ðŠ0 â ðŠ ð , ð¥0 , ðŠ0 )]ðð
+ð¡
ð ð¿1 ð¥ ð , ð¥0 , ðŠ0 â ð¥ ð , ð¥0 , ðŠ0 + ð¿2 ðŠ ð , ð¥0 ,ðŠ0 â ðŠ ð , ð¥0, ðŠ0 +
ð
ð¡
+ð¿
ð2
ð¿3 ð»1 ð¥ ð , ð¥0 , ðŠ0 â ð¥ ð , ð¥0 , ðŠ0 + ð»2 ðŠ ð , ð¥0 , ðŠ0 â ðŠ ð , ð¥0, ðŠ0 +
+ðð¿4(ð»1 ð¥ ð , ð¥0 , ðŠ0 â ð¥ ð , ð¥0 , ðŠ0 + ð»2 ðŠ ð , ð¥0 , ðŠ0 â ðŠ ð , ð¥0 , ðŠ0 )]ðð
†ðŒ(ð¡)[ð¿1 + ð»1(ð¿
ð2
ð¿3 + ðð¿4)] ð¥ ð¡, ð¥0 , ðŠ0 â ð¥ ð¡, ð¥0, ðŠ0 +
+ðŒ(ð¡)[ð¿2 + ð»2(ð¿
ð2ð¿3 + ðð¿4)] ðŠ ð¡, ð¥0 , ðŠ0 â ðŠ ð¡, ð¥0 , ðŠ0
so that:
ðŠ ð¡, ð¥0, ðŠ0 â ðŠ ð¡, ð¥0 , ðŠ0 â€
â€ð
2[ð¿1 + ð»1(
ð¿
ð2
ð¿3 + ðð¿4)] ð¥ ð¡, ð¥0 , ðŠ0 â ð¥ ð¡, ð¥0 , ðŠ0 +
+ð
2[ð¿2 + ð»2(
ð¿
ð2ð¿3 + ðð¿4)] ðŠ ð¡, ð¥0 , ðŠ0 â ðŠ ð¡, ð¥0, ðŠ0 ⯠2.30
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and the inequalities (4.2.29) and (4.2.30) would lead to the estimate:
ð¥ ð¡, ð¥0 , ðŠ0 â ð¥ ð¡, ð¥0 , ðŠ0
ðŠ ð¡, ð¥0 , ðŠ0 â ðŠ ð¡, ð¥0 , ðŠ0 †ð0
ð¥ ð¡, ð¥0 , ðŠ0 â ð¥ ð¡, ð¥0, ðŠ0
ðŠ ð¡, ð¥0 , ðŠ0 â ðŠ ð¡, ð¥0 , ðŠ0
⯠2.31 By iterating the inequality (4.2.27), which should find:
ð¥ ð¡, ð¥0 , ðŠ0 â ð¥ ð¡, ð¥0 , ðŠ0
ðŠ ð¡, ð¥0 , ðŠ0 â ðŠ ð¡, ð¥0 , ðŠ0 †ð0
ð ð¥ ð¡, ð¥0 , ðŠ0 â ð¥ ð¡, ð¥0, ðŠ0
ðŠ ð¡, ð¥0 , ðŠ0 â ðŠ ð¡, ð¥0 , ðŠ0 .
But ð0ð â 0 as ð â â, so that proceeding in the last inequality which is contradict the supposition It follows
immediately ð¥ ð¡, ð¥0 , ðŠ0 = ð¥ ð¡, ð¥0, ðŠ0 and
ðŠ ð¡, ð¥0 , ðŠ0 = ðŠ ð¡, ð¥0 , ðŠ0 .
III. Existence of Solution of (ð°ð) and (ð°ð) The problem of existence of periodic solution of period T of the system
( ðŒ1) and ( ðŒ2) are uniquely connected with the existence of zero of the functions â1 0, ð¥0, ðŠ0 = â1 and
â2 0, ð¥0, ðŠ0 = â2 which has the form:
â1:ð·ð1Ã ð·1ð2
â ð ð
â1 0, ð¥0, ðŠ0 = 1
ð ð1(ð¡, ð¥0 ð¡, ð¥0 , ðŠ0 , ðŠ
0 ð¡, ð¥0 , ðŠ0 ,
ð
0
, ðº1 ð¡, ð ð1 ð , ð¥0 ð , ð¥0 , ðŠ0 , ðŠ0 ð , ð¥0, ðŠ0
ð¡
ââ
ðð
, ð1 ð , ð¥0 ð , ð¥0 , ðŠ0 , ðŠ0 ð , ð¥0, ðŠ0 ðð )ðð¡
ð ð¡
ð ð¡
⯠3.1
â2:ð·ð1Ã ð·1ð2
â ð ð
â2 0, ð¥0, ðŠ0 = 1
ð ð2(ð¡, ð¥0 ð¡, ð¥0 , ðŠ0 , ðŠ
0 ð¡, ð¥0 , ðŠ0 ,
ð
0
, ðº2 ð¡, ð ð2 ð , ð¥0 ð , ð¥0 , ðŠ0 , ðŠ0 ð , ð¥0 , ðŠ0
ð¡
ââ
ðð
, ð2 ð , ð¥0 ð , ð¥0 ,ðŠ0 , ðŠ0 ð , ð¥0 , ðŠ0 ðð )ðð¡
ð ð¡
ð ð¡
⯠3.2
where the function ð¥0 ð¡, ð¥0 , ðŠ0 is the limit of the sequence of the functions ð¥ð (ð¡, ð¥0 , ðŠ0) and the function
ðŠ0 ð¡, ð¥0 , ðŠ0 is the limit of the sequence of the functions ðŠð ð¡, ð¥0 , ðŠ0 . Since this two functions are approximately determined from the sequences of functions:
â1ð :ð·ð1Ã ð·1ð2
â ð ð
â1ð 0, ð¥0 , ðŠ0 = 1
ð ð1(ð¡, ð¥ð ð¡, ð¥0 , ðŠ0 , ðŠð ð¡, ð¥0 , ðŠ0 ,
ð
0
, ðº1 ð¡, ð ð1 ð , ð¥ð ð , ð¥0 , ðŠ0 , ðŠð ð , ð¥0 , ðŠ0
ð¡
ââ
ðð
, ð1 ð , ð¥ð ð , ð¥0 , ðŠ0 , ðŠð ð , ð¥0 , ðŠ0 ðð )ðð¡
ð ð¡
ð ð¡
⯠3.3
â2ð :ð·ð1Ã ð·1ð2
â ð ð
â2ð 0, ð¥0 , ðŠ0 = 1
ð ð2(ð¡, ð¥ð ð¡, ð¥0 , ðŠ0 , ðŠð ð¡, ð¥0, ðŠ0 ,
ð
0
, ðº2 ð¡, ð ð2 ð , ð¥ð ð , ð¥0 , ðŠ0 , ðŠð ð , ð¥0 , ðŠ0
ð¡
ââ
ðð
Periodic Solutions for Non-Linear Systems of Integral Equations
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, ð2 ð , ð¥ð ð , ð¥0 , ðŠ0 , ðŠð ð , ð¥0 , ðŠ0 ðð )ðð¡
ð ð¡
ð ð¡
⯠3.4
for all ð = 0,1,2,â¯
Theorem 3.1. If the hypotheses and all conditions of the theorem 2.1 and 2.2 are satisfied, then the following
inequality satisfied:
â1 0, ð¥0, ðŠ0 â â1ð 0, ð¥0 , ðŠ0 †ðð ⯠3.5 â2 0, ð¥0, ðŠ0 â â2ð 0, ð¥0, ðŠ0 †ðð ⯠3.6 satisfied for all ð ⥠0 , ð¥0 â ð·ð1 and ðŠ0 â ð·1ð2
,
where
ðð = [ðŸ1 + ð 1(ðŸ
ð1
ðŸ3 + ððŸ4)] [ðŸ2 + ð 2(ðŸ
ð1
ðŸ3 + ððŸ4)] ,
,ð0ð+1 ðž â ð0
â1ð¶0 and
ðð = [ð¿1 + ð»1(ð¿
ð2ð¿3 + ðð¿4)] [ð¿2 + ð»2(
ð¿
ð2ð¿3 + ðð¿4)] ,
, ð0ð+1 ðž â ð0
â1ð¶0 . Proof. By using the relation ( 3.1) and ( 3.3), we have:
â1 0, ð¥0 , ðŠ0 â â1ð 0, ð¥0 , ðŠ0
†[ðŸ1 + ð 1(ðŸ
ð1
ðŸ3 + ððŸ4)] ð¥0 ð¡, ð¥0 , ðŠ0 â ð¥ð ð¡, ð¥0 , ðŠ0 +
+[ðŸ2 + ð 2(ðŸ
ð1
ðŸ3 + ððŸ4)] ðŠ0 ð¡, ð¥0 , ðŠ0 â ðŠð ð¡, ð¥0 , ðŠ0
†[ðŸ1 + ð 1(ðŸ
ð1
ðŸ3 + ððŸ4)] [ðŸ2 + ð 2(ðŸ
ð1
ðŸ3 + ððŸ4)] ,
,ð0ð+1 ðž â ð0
â1ð¶0 = ðð And also by using the relation ( 3.2) and ( 3.4), we get:
â2 0, ð¥0, ðŠ0 â â2ð 0, ð¥0, ðŠ0 â€
†[ð¿1 + ð»1(ð¿
ð2
ð¿3 + ðð¿4)] [ð¿2 + ð»2(ð¿
ð2
ð¿3 + ðð¿4)] ,
, ð0ð+1 ðž â ð0
â1ð¶0 = ðð
where . denotes the ordinary scalar product in the space ð ð . â
Theorem 3.2. Let the vector functions ð1 ð¡, ð¥,ðŠ, ð§,ð€ , ð2 ð¡, ð¥, ðŠ,ð¢, ð£ , ð1 ð¡, ð¥, ðŠ and ð2 ð¡, ð¥, ðŠ be defined on the domain:
ðº = 0 †ð †ð¡ †ð, ð †ð¥, ðŠ †ð , ð †ð§,𢠆ð, ð †ð€, 𣠆ð} â ð 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+
ð2ð1â€x0â€bâ
ð2ð1
c+ð2ð2â€y0â€dâ
ð2ð2
â1m 0, x0, y0 †â ðm ,
maxa+
ð2ð1â€x0â€bâ
ð2ð1
c+ð2ð2â€y0â€dâ
ð2ð2
â1m 0, x0, y0 ⥠ðm ,
⯠3.7
mina+
ð2ð1â€x0â€bâ
ð2ð1
c+ð2ð2â€y0â€dâ
ð2ð2
â2m 0, x0, y0 †â ηm
,
maxa+
ð2ð1â€x0â€bâ
ð2ð1
c+ð2ð2â€y0â€dâ
ð2ð2
â2m 0, x0 , y0 ⥠ηm
,
⯠3.8
for all ð ⥠0, where
ðð = [ðŸ1 + ð 1(ðŸ
ð1
ðŸ3 + ððŸ4)] [ðŸ2 + ð 2(ðŸ
ð1
ðŸ3 + ððŸ4)] ,
,ð0ð+1 ðž â ð0
â1ð¶0 and
Periodic Solutions for Non-Linear Systems of Integral Equations
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ðð = [ð¿1 + ð»1(ð¿
ð2
ð¿3 + ðð¿4)] [ð¿2 + ð»2(ð¿
ð2
ð¿3 + ðð¿4)] ,
, ð0ð+1 ðž â ð0
â1ð¶0 Then the system (4.1.1) and (4.1.2) has periodic solution of period T
ð¥ = ð¥(ð¡, ð¥0 , ðŠ0) and ðŠ = ðŠ(ð¡, ð¥0 , ðŠ0) for which ð¥0 â [ð +ð
2ð1 , ð â
ð
2ð1] and
ðŠ0 â [ð +ð
2ð2,ð â
ð
2ð2].
Proof. Let ð¥1 , ð¥2 be any two points in the interval [ð +ð
2ð1 , ð â
ð
2ð1] and ðŠ1 , ðŠ2 be any two points in the
interval [ð +ð
2ð2,ð â
ð
2ð2],
such that:
Î1m 0, ð¥1 , ðŠ1 = mina+
ð2ð1â€x0â€bâ
ð2ð1
c+ð2ð2â€y0â€dâ
ð2ð2
â1m 0, ð¥0 , ðŠ0 ,
Î1m 0, ð¥2 , ðŠ2 = maxa+
ð2ð1â€x0â€bâ
ð2ð1
c+ð2ð2â€y0â€dâ
ð2ð2
â1ð 0,ð¥0 , ðŠ0 ,
⯠3.9
Î2m 0, ð¥1, ðŠ1 = mina+
ð2ð1â€x0â€bâ
ð2ð1
c+ð2ð2â€y0â€dâ
ð2ð2
â2m 0, x0, y0 ,
Î2m 0, ð¥2 , ðŠ2 = maxa+
ð2ð1â€x0â€bâ
ð2ð1
c+ð2ð2â€y0â€dâ
ð2ð2
â2m 0, x0, y0 ,
⯠3.10
By using the inequalities ( 3.5), ( 3.6), ( 3.7) and ( 3.8), we have:
â1 0, ð¥1 , ðŠ1 = â1ð 0, ð¥1 , ðŠ1 + â1 0, ð¥1 , ðŠ1 â â1ð (0, ð¥1 , ðŠ1) †0 ,
â1 0, ð¥2, ðŠ2 = â1ð 0, ð¥2 , ðŠ2 + â1 0, ð¥2 , ðŠ2 â â1ð(0, ð¥2 , ðŠ2) †0 .
⯠3.11
â2 0, ð¥1 , ðŠ1 = â2ð 0, ð¥1 , ðŠ1 + â2 0, ð¥1, ðŠ1 â â2ð (0, ð¥1 , ðŠ1) †0 ,
â2 0, ð¥2, ðŠ2 = â2ð 0, ð¥2 , ðŠ2 + â2 0, ð¥2, ðŠ2 â â2ð (0, ð¥2, ðŠ2) †0 .
⯠3.12 It follows from the inequalities ( 3.11) and ( 3.12) in virtue of the
continuity of the functions â1(0, ð¥0 , ðŠ0) and â1(0, ð¥0 , ðŠ0) that there exists an isolated singular point (ð¥0, ðŠ0) =(ð¥0, ðŠ0) , ð¥0 â ð¥1 , ð¥2 and ðŠ0 â ðŠ1 , ðŠ2 , so that â1(0, ð¥0 , ðŠ0) = 0 and â2(0, ð¥0, ðŠ0) = 0. This means that the system ( 3.1) and (4.3.2) has a periodic
solutions ð¥ ð¡, ð¥0 , ðŠ0 , ðŠ ð¡, ð¥0 , ðŠ0 for which
ð¥0 â [ð +ð
2ð1 , ð â
ð
2ð1] and ðŠ0 â [ð +
ð
2ð2,ð â
ð
2ð2]. â
Remark 3.1. Theorem 3.2 is proved when ð ð = ð 1, on the other hand as ð¥0, ðŠ0 are a scalar singular point
which should be isolated (For this remark, see [5]).
IV. Stability Theorem Of Solution (ð°ð) And (ð°ð) In this section, we study theorem on stability of a periodic solution for the integral equations ( ðŒ1) and ( ðŒ2).
Theorem 4.1. If the function â1 0, ð¥0, ðŠ0 ,Î2(0, ð¥0 , ðŠ0) are defined by
equations ( 3.1) and ( 3.2), where the function ð¥0(ð¡, ð¥0, ðŠ0) is a limit of the sequence of the functions ( 2.1) , the
function ðŠ0(ð¡, ð¥0 , ðŠ0) is the limit of the sequence of the functions ( 2.2) , Then the following inequalities yields:
â1 0, ð¥0 , ðŠ0 †ð1 ⯠4.1 â2 0, ð¥0, ðŠ0 †ð2 ⯠4.2
and
â2 0, ð¥01, ðŠ0
1 â â2 0, ð¥02 , ðŠ0
2 †ð¹1ð¹2ðž3 ð¹01 ð¡ â ð¹0
2 ð¡ +
+ð
2ð¹1ð¹2ðž2( ðž3 + ðž4 + ð¹1ðž4(1 â
ð
2ðž1)) ðº0
1 ð¡ â ðº02(ð¡)
⯠4.4
for all ð¥0, ð¥01 , ð¥0
2 â ð·ð1, ðŠ0 , ðŠ0
1 , ðŠ02 â ð·1ð2
, and ðŒ ð¡ = 2ð¡(1 âð¡
ð) â€
ð
2 ,
where ðž1 = [ðŸ1 + ð 1(ðŸ
ð1ðŸ3 + ððŸ4)],ðž2 = [ðŸ2 + ð 2(
ðŸ
ð1ðŸ3 + ððŸ4)],
Periodic Solutions for Non-Linear Systems of Integral Equations
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ðž3 = [ð¿1 + ð»1(ð¿
ð2ð¿3 + ðð¿4)], ðž4 = [ð¿2 + ð»2(
ð¿
ð2ð¿3 + ðð¿4)],
ð¹1 = [(1 âð
2ðž1)(1 â
ð
2ðž4)]â1 and ð¹2 = (1 â
ð2
4ð2ð3ð¹1)â1
Proof. From the properties of the functions ð¥0 ð¡, ð¥0 , ðŠ0 and ðŠ0 ð¡, ð¥0 , ðŠ0 as in the theorem 4.2.1, the
functions Î1 = â1 ð¥0, ðŠ0 , Î1 = â1 ð¥0 , ðŠ0 , ð¥0 â ð·ð1
, ðŠ0 â ð·1ð2 are continuous and bounded by ð1,ð2 in the domain ( 1.3).
From relation ( 3.1), we find:
â1 (0, ð¥0 , ðŠ0 â€1
ð ð1(ð¡, ð¥0 ð¡, ð¥0 , ðŠ0 , ðŠ
0 ð¡, ð¥0 , ðŠ0 ,
ð
0
, ðº1 ð¡, ð ð1 ð , ð¥0 ð , ð¥0 , ðŠ0 , ðŠ0 ð , ð¥0 , ðŠ0 ðð ,
ð¡
ââ
, ð1 ð , ð¥0 ð , ð¥0 , ðŠ0 , ðŠ0 ð , ð¥0, ðŠ0 ð ð )
ð(ð¡)
ð(ð¡)
ðð¡
by using the Lemma 4.1.1, gives:
â1(0, ð¥0 , ðŠ0) †ð1 . And from relation ( 3.2), we get:
â2 (0, ð¥0 , ðŠ0 â€1
ð ð2(ð¡, ð¥0 ð¡, ð¥0 , ðŠ0 , ðŠ
0 ð¡, ð¥0 , ðŠ0 ,
ð
0
, ðº2 ð¡, ð ð2 ð , ð¥0 ð , ð¥0 , ðŠ0 , ðŠ0 ð , ð¥0 , ðŠ0 ðð ,
ð¡
ââ
, ð2 ð , ð¥0 ð , ð¥0, ðŠ0 , ðŠ
0 ð , ð¥0 , ðŠ0 ðð )
ð(ð¡)
ð(ð¡)
ðð¡
and using Lemma 4.1.1, we have:
â2(0, ð¥0, ðŠ0) †ð2
By using equation ( 3.1) and lemma 1.1, we get:
â1 0, ð¥01, ðŠ0
1 â â1 0, ð¥02 , ðŠ0
2
â€1
ð [ðŸ1 ð¥
0 ð¡, ð¥01 , ðŠ0
1 â ð¥0 ð¡, ð¥02, ðŠ0
2 +
ð
0
+ðŸ2 ðŠ0 ð¡, ð¥0
1 , ðŠ01 â ðŠ0 ð¡, ð¥0
2, ðŠ02 +
+ðŸ
ð1
ðŸ3(ð 1 ð¥0 ð¡, ð¥0
1, ðŠ01 â ð¥0 ð¡, ð¥0
2, ðŠ02 +
+ð 2 ðŠ0 ð¡, ð¥0
1, ðŠ01 â ðŠ0 ð¡, ð¥0
2, ðŠ02 ) +
+ ððŸ4(ð 1 ð¥0 ð¡, ð¥0
1 , ðŠ01 â ð¥0 ð¡, ð¥0
2, ðŠ02 +
+ð 2 ðŠ0 ð¡, ð¥0
1, ðŠ01 â ðŠ0 ð¡, ð¥0
2, ðŠ02 )]ðð¡
†[ðŸ1 + ð 1(ðŸ
ð1
ðŸ3 + ððŸ4)] ð¥0 ð¡, ð¥01, ðŠ0
1 â ð¥0 ð¡, ð¥02, ðŠ0
2 +
+[ðŸ2 + ð 2(ðŸ
ð1
ðŸ3 + ððŸ4)] ðŠ0 ð¡, ð¥01 , ðŠ0
1 â ðŠ0 ð¡, ð¥02, ðŠ0
2
so,
â1 0, ð¥01, ðŠ0
1 â â1 0, ð¥02 , ðŠ0
2 †ðž1 ð¥0 ð¡, ð¥0
1 , ðŠ01 â ð¥0 ð¡, ð¥0
2, ðŠ02 +
+ðž2 ðŠ0 ð¡, ð¥01, ðŠ0
1 â ðŠ0 ð¡, ð¥02, ðŠ0
2 ⯠4.5 And also by using equation ( 3.2) and lemma 1.1, gives:
â2 0, ð¥01, ðŠ0
1 â â2 0, ð¥02 , ðŠ0
2
â€1
ð [ð¿1 ð¥
0 ð¡, ð¥01 , ðŠ0
1 â ð¥0 ð¡, ð¥02, ðŠ0
2 +
ð
0
+ð¿2 ðŠ0 ð¡, ð¥0
1 , ðŠ01 â ðŠ0 ð¡, ð¥0
2, ðŠ02 +
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+ð¿
ð2
ð¿3(ð»1 ð¥0 ð¡, ð¥0
1, ðŠ01 â ð¥0 ð¡, ð¥0
2, ðŠ02 +
+ð»2 ðŠ0 ð¡, ð¥0
1 , ðŠ01 â ðŠ0 ð¡, ð¥0
2, ðŠ02 ) +
+ ðð¿4(ð»1 ð¥0 ð¡, ð¥0
1, ðŠ01 â ð¥0 ð¡, ð¥0
2, ðŠ02 +
+ð»2 ðŠ0 ð¡, ð¥0
1, ðŠ01 â ðŠ0 ð¡, ð¥0
2, ðŠ02 )]ðð¡
†[ð¿1 + ð»1(ð¿
ð2
ð¿3 + ðð¿4)] ð¥0 ð¡, ð¥01 , ðŠ0
1 â ð¥0 ð¡, ð¥02, ðŠ0
2 +
+[ð¿2 + ð»2(ð¿
ð2ð¿3 + ðð¿4)] ðŠ0 ð¡, ð¥0
1, ðŠ01 â ðŠ0 ð¡, ð¥0
2, ðŠ02 .
Therefore,
â2 0, ð¥01, ðŠ0
1 â â2 0, ð¥02 , ðŠ0
2 †ðž3 ð¥0 ð¡, ð¥0
1, ðŠ01 â ð¥0 ð¡, ð¥0
2 , ðŠ02 +
+ðž4 ðŠ0 ð¡, ð¥01, ðŠ0
1 â ðŠ0 ð¡, ð¥02, ðŠ0
2 ⯠4.6
where the functions ð¥0 ð¡, ð¥01, ðŠ0
1 , ðŠ0 ð¡, ð¥01 , ðŠ0
1 , ð¥0 ð¡, ð¥02, ðŠ0
2 and
ðŠ0 ð¡, ð¥02, ðŠ0
2 are solutions of the equation:
ð¥ ð¡, ð¥0ð , ðŠ0
ð = ð¹0ð ð¡ + [ð1(ð , ð¥(ð , ð¥0
ð , ðŠ0ð), ðŠ(ð , ð¥0
ð , ðŠ0ð),
ð¡
0
, ðº1 ð , ð ð1(ð, ð¥(ð, ð¥0ð , ðŠ0
ð), ðŠ(ð, ð¥0ð , ðŠ0
ð))
ð
ââ
ðð,
, ð1(ð, ð¥(ð, ð¥0ð , ðŠ0
ð), ðŠ(ð, ð¥0ð , ðŠ0
ð))ðð
ð ð
ð ð
) â
â1
ð ð1(ð , ð¥(ð , ð¥0
ð , ðŠ0ð), ðŠ(ð , ð¥0
ð , ðŠ0ð)
ð
0
,
, ðº1 ð , ð ð1(ð, ð¥(ð, ð¥0ð , ðŠ0
ð), ðŠ(ð, ð¥0ð , ðŠ0
ð))
ð
ââ
ðð,
, ð1(ð, ð¥(ð, ð¥0ð , ðŠ0
ð), ðŠ(ð, ð¥0ð , ðŠ0
ð))ðð
ð ð
ð ð
)ðð ]ðð ⯠( 4.7)
and
ðŠ ð¡, ð¥0ð , ðŠ0
ð = ðº0ð ð¡ + [ð2(ð , ð¥(ð , ð¥0
ð , ðŠ0ð), ðŠ(ð , ð¥0
ð , ðŠ0ð),
ð¡
0
, ðº2 ð , ð ð2(ð, ð¥(ð, ð¥0ð , ðŠ0
ð), ðŠ(ð, ð¥0ð , ðŠ0
ð))
ð
ââ
ðð,
, ð2(ð, ð¥(ð, ð¥0ð , ðŠ0
ð), ðŠ(ð, ð¥0ð , ðŠ0
ð))ðð
ð ð
ð ð
) â
â1
ð ð2(ð , ð¥(ð , ð¥0
ð , ðŠ0ð), ðŠ(ð , ð¥0
ð , ðŠ0ð)
ð
0
,
, ðº2 ð , ð ð2(ð, ð¥(ð, ð¥0ð , ðŠ0
ð), ðŠ(ð, ð¥0ð , ðŠ0
ð))
ð
ââ
ðð,
, ð2(ð, ð¥(ð, ð¥0ð , ðŠ0
ð), ðŠ(ð, ð¥0ð , ðŠ0
ð))ðð
ð ð
ð ð
)ðð ]ðð ⯠( 4.8)
where ð = 1, 2. From (4.4.7), we get:
ð¥0 ð¡, ð¥01 , ðŠ0
1 âð¥0 ð¡, ð¥02, ðŠ0
2 †ð¹0
1 ð¡ â ð¹02 ð¡ +
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+ðŒ(ð¡)[ðŸ1 + ð 1(ðŸ
ð1
ðŸ3 + ððŸ4)] ð¥0 ð¡, ð¥01, ðŠ0
1 â ð¥0 ð¡, ð¥02, ðŠ0
2 +
+ðŒ ð¡ [ðŸ2 + ð 2(ðŸ
ð1
ðŸ3 + ððŸ4)] ðŠ0 ð¡, ð¥01, ðŠ0
1 â ðŠ0 ð¡, ð¥02, ðŠ0
2
†ð¹01 ð¡ â ð¹0
2 ð¡ +
+ð
2[ðŸ1 + ð 1(
ðŸ
ð1
ðŸ3 + ððŸ4)] ð¥0 ð¡, ð¥01 , ðŠ0
1 â ð¥0 ð¡, ð¥02, ðŠ0
2 +
+ð
2[ðŸ2 + ð 2(
ðŸ
ð1
ðŸ3 + ððŸ4)] ðŠ0 ð¡, ð¥01 , ðŠ0
1 â ðŠ0 ð¡, ð¥02, ðŠ0
2
such that:
ð¥0 ð¡, ð¥01, ðŠ0
1 âð¥0 ð¡, ð¥02, ðŠ0
2 †ð¹01 ð¡ â ð¹0
2 ð¡ +
+ð
2ðž1 ð¥
0 ð¡, ð¥01, ðŠ0
1 â ð¥0 ð¡, ð¥02 ,ðŠ0
2 +
+ð
2ðž2 ðŠ0 ð¡, ð¥0
1 , ðŠ01 â ðŠ0 ð¡, ð¥0
2, ðŠ02
therefore:
ð¥0 ð¡, ð¥01, ðŠ0
1 âð¥0 ð¡, ð¥02, ðŠ0
2 †(1 âð
2ðž1)â1 ð¹0
1 ð¡ â ð¹02 ð¡ +
+ð
2ðž2(1 â
ð
2ðž1)â1 ðŠ0 ð¡, ð¥0
1, ðŠ01 â ðŠ0 ð¡, ð¥0
2, ðŠ02
⯠4.9 And also from relation ( 4.8), we have:
ðŠ0 ð¡, ð¥01 , ðŠ0
1 âðŠ0 ð¡, ð¥02, ðŠ0
2 ††ðº0
1 ð¡ â ðº02 ð¡ +
+ðŒ(ð¡)[ð¿1 + ð»1(ð¿
ð2
ð¿3 + ðð¿4)] ð¥0 ð¡, ð¥01, ðŠ0
1 â ð¥0 ð¡, ð¥02, ðŠ0
2 +
+ðŒ(ð¡)[ð¿2 + ð»2(ð¿
ð2
ð¿3 + ðð¿4)] ðŠ0 ð¡, ð¥01 , ðŠ0
1 â ðŠ0 ð¡, ð¥02, ðŠ0
2 â€
†ðº01 ð¡ â ðº0
2 ð¡ +
+ð
2[ð¿1 + ð»1(
ð¿
ð2
ð¿3 + ðð¿4)] ð¥0 ð¡, ð¥01, ðŠ0
1 â ð¥0 ð¡, ð¥02, ðŠ0
2 +
+ð
2[ð¿2 + ð»2(
ð¿
ð2
ð¿3 + ðð¿4)] ðŠ0 ð¡, ð¥01, ðŠ0
1 â ðŠ0 ð¡, ð¥02, ðŠ0
2
so that:
ðŠ0 ð¡, ð¥01 , ðŠ0
1 âðŠ0 ð¡, ð¥02, ðŠ0
2 †ðº01 ð¡ â ðº0
2 ð¡ +
+ð
2ðž3 ð¥
0 ð¡, ð¥01, ðŠ0
1 â ð¥0 ð¡, ð¥02, ðŠ0
2 +
+ð
2ðž4 ðŠ0 ð¡, ð¥0
1, ðŠ01 â ðŠ0 ð¡, ð¥0
2, ðŠ02
and hence:
ðŠ0 ð¡, ð¥01 , ðŠ0
1 âðŠ0 ð¡, ð¥02, ðŠ0
2 †(1 âð
2ðž4)â1 ðº0
1 ð¡ â ðº02 ð¡ +
+ð
2ðž3(1 â
ð
2ðž4)â1 ð¥0 ð¡, ð¥0
1, ðŠ01 â ð¥0 ð¡, ð¥0
2 , ðŠ02
⯠4.10 Now, by substituting inequality ( 4.10) in ( 4.9), we get:
ð¥0 ð¡, ð¥01, ðŠ0
1 â ð¥0 ð¡, ð¥02 , ðŠ0
2 †(1 âð
2ðž1)â1 ð¹0
1 ð¡ â ð¹02 ð¡ +
+ð
2ðž2(1 â
ð
2ðž1)â1[(1 â
ð
2ðž4)â1 ðº0
1 ð¡ â ðº02 ð¡ +
+ð
2ðž3(1 â
ð
2ðž4)â1 ð¥0 ð¡, ð¥0
1 , ðŠ01 â ð¥0 ð¡, ð¥0
2, ðŠ02 ]
†(1 âð
2ðž1)â1 ð¹0
1 ð¡ â ð¹02 ð¡ +
+ð
2ðž2[(1 â
ð
2ðž1)(1 â
ð
2ðž4)]â1 ðº0
1 ð¡ â ðº02 ð¡ +
+ð2
4ðž2ðž3[(1 â
ð
2ðž1)(1 â
ð
2ðž4)]â1 ð¥0 ð¡, ð¥0
1, ðŠ01 â ð¥0 ð¡, ð¥0
2, ðŠ02 ]
putting ð¹1 = [(1 âð
2ðž1)(1 â
ð
2ðž4)]â1,
and substituting in the last inequality, we obtain:
Periodic Solutions for Non-Linear Systems of Integral Equations
www.ijeijournal.com Page | 16
ð¥0 ð¡, ð¥01, ðŠ0
1 â ð¥0 ð¡, ð¥02 , ðŠ0
2 †(1 âð
2ðž1)â1 ð¹0
1 ð¡ â ð¹02 ð¡ +
+ð
2ðž2ð¹1 ðº0
1 ð¡ â ðº02 ð¡ + +
ð2
4ðž2ðž3ð¹1 ð¥
0 ð¡, ð¥01, ðŠ0
1 â ð¥0 ð¡, ð¥02, ðŠ0
2 ]
as the ð¹1(1 âð
2ðž4) = (1 â
ð
2ðž1)â1
ð¥0 ð¡, ð¥01, ðŠ0
1 â ð¥0 ð¡, ð¥02 , ðŠ0
2 †ð¹1(1 âð
2ðž4) ð¹0
1 ð¡ â ð¹02 ð¡ +
+ð
2ðž2ð¹1 ðº0
1 ð¡ â ðº02 ð¡ +
ð2
4ðž2ðž3ð¹1 ð¥
0 ð¡, ð¥01 , ðŠ0
1 â ð¥0 ð¡, ð¥02, ðŠ0
2 ]
which implies that:
ð¥0 ð¡, ð¥01, ðŠ0
1 â ð¥0 ð¡, ð¥02 , ðŠ0
2 â€â€ ð¹1(1 âð
2ðž4)(1 â
ð2
4ðž2ðž3ð¹1)â1 ð¹0
1 ð¡ â ð¹02 ð¡ +
+ð
2ðž2ð¹1(1 â
ð2
4ðž2ðž3ð¹1)â1 ðº0
1 ð¡ â ðº02 ð¡
putting ð¹2 = (1 âð2
4ðž2ðž3ð¹1)â1
and substituting in the last inequality, we obtain:
ð¥0 ð¡, ð¥01, ðŠ0
1 â ð¥0 ð¡, ð¥02 , ðŠ0
2 †ð¹1ð¹2(1 âð
2ðž4) ð¹0
1 ð¡ â ð¹02 ð¡ +
+ð
2ðž2ð¹1ð¹2 ðº0
1 ð¡ â ðº02 ð¡ ⯠4.11
Also, substituting the inequalities ( 4.11) in ( 4.10), we find that:
ðŠ0 ð¡, ð¥01 , ðŠ0
1 âðŠ0 ð¡, ð¥02, ðŠ0
2 †(1 âð
2ðž4)â1 ðº0
1 ð¡ â ðº02 ð¡ +
+ð
2ðž3(1 â
ð
2ðž4)â1[ð¹1ð¹2(1 â
ð
2ðž4) ð¹0
1 ð¡ â ð¹02 ð¡ +
+ð
2ðž2ð¹1ð¹2 ðº0
1 ð¡ â ðº02 ð¡ ]
and hence
ðŠ0 ð¡, ð¥01 , ðŠ0
1 âðŠ0 ð¡, ð¥02, ðŠ0
2 â€ð
2ðž3ð¹1ð¹2 ð¹0
1 ð¡ â ð¹02 ð¡ +
+[ð¹1(1 âð
2ðž1) +
ð
2ð¹1ð¹2ðž2] ðº0
1 ð¡ â ðº02 ð¡
⯠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). â
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