Periodic Solutions for Nonlinear Systems of Integro-Differential Equations of Operators with Impulsive Action
1. International Journal of Engineering Inventions
e-ISSN: 2278-7461, p-ISSN: 2319-6491
Volume 2, Issue 2 (January 2013) PP: 57-64
www.ijeijournal.com P a g e | 57
Periodic Solutions for Nonlinear Systems of Integro-
Differential Equations of Operators with Impulsive Action
Dr. Raad N. Butris
Assistant Prof. Department of Mathematics Faculty of Science University of Zakho
Iraq-Duhok-Kurdistan Region
Abstract:- In this paper we investigate the existence and approximation of the construction of periodic solutions
for nonlinear systems of integro-differential equation of operators with impulsive action, by using the
numerical-analytic method for periodic solutions which is given by Samoilenko. This investigation leads us to
the improving and extending to the above method and expands the results gained by Butris.
Keyword and Phrases:- Numerical-analytic methods existence of periodic solutions, nonlinear integro-
differential equations, subjected to impulsive action of operators.
I. INTRODUCTION
They are many subjects in mechanics, physics, biology, and other subjects in science and engineering
requires the investigation of periodic solution of integro-differential equations with impulsive action of various
types and their systems in particular, they are many works devoted to problems of the existence of periodic
solutions of integro-differential equations with impulsive action . samoilenko[5,6] assumed a numerical analytic
method to study to periodic solution for ordinary differential equation and this method include uniformly
sequences of periodic functionsโ as Intel studies [2,3,4,7,8].
The present work continues the investigations in this direction. These investigations lead us to the
improving and extending the above method and expand the results gained by Butris [1].
๐๐ฅ
๐๐ก
= ๐(๐ก, ๐ฅ, ๐ด๐ฅ, ๐ต๐ฅ, ๐(๐ก, ๐ฅ ๐ , ๐ด๐ฅ ๐ , ๐ต๐ฅ(๐ ))๐๐
๐ก
๐กโ๐
), ๐ก โ ๐ก๐
ฮ๐ฅ ๐ก=๐ก ๐
= ๐ผ๐(๐ฅ, ๐ด๐ฅ, ๐ต๐ฅ, ๐ ๐ , ๐ฅ ๐ , ๐ด๐ฅ ๐ , ๐ต๐ฅ ๐ ๐๐
๐ก
๐กโ๐
)
โฏ(1.1)
Where ๐ฅ โ ๐ท โ ๐ ๐
, ๐ท is closed bounded domain.
The vector functions ๐(๐ก, ๐ฅ, ๐ฆ, ๐ง, ๐ค) and ๐(๐ก, ๐ฅ, ๐ฆ, ๐ง) are defined on the domain
๐ก, ๐ฅ, ๐ฆ, ๐ง, ๐ค โ ๐ 1
ร ๐ท ร ๐ท1 ร ๐ท2 ร ๐ท3 = โโ,โ ร ๐ท ร ๐ท1 ร ๐ท2 ร ๐ท3 โฏ 1.2
Which are piecewise continuous functions in ๐ก, ๐ฅ, ๐ฆ, ๐ง, ๐ค and periodic in ๐ก of period ๐, where ๐ท1, ๐ท2 and ๐ท3 are
bounded domain subset of Euclidean spaces ๐ ๐
.
Let ๐ผ๐ ๐ฅ, ๐ฆ, ๐ง, ๐ค be continuous vector functions, defined on the domain (1.2) and
๐ผ๐+๐ ๐ฅ, ๐ฆ, ๐ง, ๐ค = ๐ผ๐ ๐ฅ, ๐ฆ, ๐ง, ๐ค , ๐ก๐+๐ = ๐ก๐ + ๐ โฏ 1.3
For all ๐ โ ๐ง+
, ๐ฅ โ ๐ท, ๐ฆ โ ๐ท1, ๐ง โ ๐ท2, ๐ค โ ๐ท3 and for some number ๐ .
Let the operators A and B transform any piecewise continuous functions from the domain ๐ท to the piecewise
contiuous function in the domain ๐ท1 and ๐ท2 respectively. Moreover ๐ด๐ฅ ๐ก + ๐ = ๐ด๐ฅ(๐ก) and ๐ต๐ฅ ๐ก + ๐ =
๐ต๐ฅ ๐ก .
Suppose that ๐ ๐ก, ๐ฅ, ๐ฆ, ๐ง, ๐ค , ๐ ๐ก, ๐ฅ, ๐ฆ, ๐ง , ๐ผ๐(๐ก, ๐ฅ, ๐ฆ, ๐ง, ๐ค) and the operators A, B satisfy the following
inequalities:
๐ ๐ก, ๐ฅ, ๐ฆ, ๐ง, ๐ค โค ๐, ๐ผ๐(๐ก, ๐ฅ, ๐ฆ, ๐ง, ๐ค) โค ๐, โฏ 1.4
๐ ๐ก, ๐ฅ1, ๐ฆ1, ๐ง1, ๐ค1 โ ๐ ๐ก, ๐ฅ2, ๐ฆ2, ๐ง2, ๐ค2 โค ๐พ( ๐ฅ1 โ ๐ฅ2 + ๐ฆ1 โ ๐ฆ2 +
๐ง1 โ ๐ง2 + ๐ค1 โ ๐ค2 )
๐ ๐ก, ๐ฅ1, ๐ฆ1, ๐ง1 โ ๐ ๐ก, ๐ฅ2, ๐ฆ2, ๐ง2 โค ๐( ๐ฅ1 โ ๐ฅ2 + ๐ฆ1 โ ๐ฆ2 + ๐ง1 โ ๐ง2 )
โฏ 1.5
And
๐ผ๐ ๐ก, ๐ฅ1, ๐ฆ1, ๐ง1, ๐ค1 โ ๐ผ๐ ๐ก, ๐ฅ2, ๐ฆ2, ๐ง2, ๐ค2 โค ๐ฟ( ๐ฅ1 โ ๐ฅ2 + ๐ฆ1 โ ๐ฆ2 +
๐ง1 โ ๐ง2 + ๐ค1 โ ๐ค2 ) โฏ 1.6
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๐ด๐ฅ1 ๐ก โ ๐ด๐ฅ2 ๐ก โค ๐บ ๐ฅ1 ๐ก โ ๐ฅ2 ๐ก
๐ต๐ฅ1 ๐ก โ ๐ต๐ฅ2 ๐ก โค ๐ป ๐ฅ1 ๐ก โ ๐ฅ2 ๐ก
โฏ 1.7
For all ๐ก โ ๐ 1
, ๐ฅ, ๐ฅ1, ๐ฅ2 โ ๐ท, ๐ฆ, ๐ฆ1, ๐ฆ2 โ ๐ท1, ๐ง, ๐ง1, ๐ง2 โ ๐ท2 and ๐ค, ๐ค1, ๐ค2 โ ๐ท3 where ๐, ๐, ๐พ, ๐, ๐ฟ and ๐บ, ๐ป are a
positive constants.
Consider the matrix
ฮ =
๐พ1
๐
3
๐พ1
๐๐๐พ2 2๐๐พ2
โฏ 1.8
Where
๐พ1 = ๐พ 1 + ๐บ + ๐ป + ๐ 1 + ๐บ + ๐ป , ๐พ2 = ๐ฟ[1 + ๐บ + ๐ป + ๐(1 + ๐บ + ๐ป)]
We define the non-empty sets as follows
๐ท๐ = ๐ท โ ๐
๐
2
(1 +
4๐
๐
)
๐ท1๐ = ๐ท1 โ ๐บ๐
๐
2
(1 +
4๐
๐
)
โฏ 1.9
and
๐ท2๐ = ๐ท2 โ ๐ป๐
๐
2
(1 +
4๐
๐
)
๐ท3๐ = ๐ท3 โ ๐๐
๐2
2
(1 +
4๐
๐
)
โฏ 1.10
Furthermore, we suppose that the greatest Eigen-value ๐ ๐๐๐ฅ of the matrix ฮ does not exceed unity, i.e.
ฮ = ๐พ1
๐
2
+ ๐๐พ2(2 + ๐พ1
๐
2
) < 1 โฏ 1.11
Lemma 2.1. Let ๐(๐ก) be a continuous (piecewise continuous) vector function in the interval 0 โค ๐ก โค ๐. Then
(๐ ๐ โ
1
๐
๐ก
0
๐(๐ )๐๐ )๐๐
๐
0
โค ๐ผ ๐ก max
๐กโ 0,๐
๐ ๐ก ,
Where ๐ผ ๐ก = 2๐ก(1 โ
๐ก
๐
) . (For the proof see [3]) .
Assume that
๐ด๐ฅ ๐ ๐ก, ๐ฅ0 = ๐ฆ ๐ ๐ก, ๐ฅ0 , ๐ต๐ฅ ๐ ๐ก, ๐ฅ0 = ๐ง ๐ ๐ก, ๐ฅ0 and
๐ค ๐ ๐ก, ๐ฅ0 = ๐(๐ , ๐ฅ ๐ ๐ , ๐ฅ0 , ๐ด๐ฅ ๐ ๐ , ๐ฅ0 , ๐ต๐ฅ ๐ ๐ , ๐ฅ0 )๐๐ ,
๐ก
๐กโ๐
๐ = 0,1,2, โฏ
II. APPROXIMATE SOLUTION
The investigation of a periodic approximate solution of the system (1.1) makes essential use of the
statements and estimates given below.
Theorem 2.1. If the system of integro-differential equations with impulsive action(1.1) satisfy the inequalities
(1.3) to (1.7) and the conditions (1.8), (1.10) has a periodic solution ๐ฅ = ๐ฅ ๐ก, ๐ฅ0 , passing through the point
0, ๐ฅ0 , ๐ฅ0 โ ๐ท๐ ,
๐ด๐ฅ0 โ ๐ท1๐ and ๐ต๐ฅ0 โ ๐ท2๐ , then the sequence of functions:
๐ฅ ๐ ๐ก, ๐ฅ0 = ๐ฅ0 + [๐(๐ , ๐ฅ ๐ ๐ , ๐ฅ0 , ๐ฆ ๐ ๐ , ๐ฅ0 , ๐ง ๐ ๐ , ๐ฅ0 , ๐ค ๐ ๐ , ๐ฅ0
๐ก
0
) โ
โ
1
๐
๐(๐ , ๐ฅ ๐ ๐ , ๐ฅ0 , ๐ฆ ๐ ๐ , ๐ฅ0 , ๐ง ๐ ๐ , ๐ฅ0 , ๐ค ๐ ๐ , ๐ฅ0 )
๐
0
๐๐ ]๐๐ +
+ ๐ผ๐
0<๐ก ๐<๐ก
(๐ฅ ๐ ๐ก๐, ๐ฅ0 , ๐ฆ ๐ ๐ก๐, ๐ฅ0 , ๐ง ๐ ๐ก๐, ๐ฅ0 , ๐ค ๐ ๐ก๐, ๐ฅ0 ) โ
โ
๐ก
๐
๐ผ๐(๐ฅ ๐ ๐ก๐, ๐ฅ0 , ๐ฆ ๐ ๐ก๐, ๐ฅ0 , ๐ง ๐ ๐ก๐, ๐ฅ0 , ๐ค ๐ ๐ก๐, ๐ฅ0 )
๐
๐=1
โฏ 2.1
With
๐ฅ0 ๐ก, ๐ฅ0 = ๐ฅ0 , ๐ = 0,1,2, โฏ
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is periodic in ๐ก of period ๐, and uniformly convergent as ๐ โ โ in
๐ก, ๐ฅ0 โ ๐ 1
ร ๐ท๐ = โโ,โ ร ๐ท๐ โฏ 2.2
To the vector function ๐ฅ0
(๐ก, ๐ฅ0) defined on the domain (1.2), which is periodic in ๐ก of period ๐ and satisfying
the system of integral equation
๐ฅ ๐ก, ๐ฅ0 = ๐ฅ0 + [๐(๐ , ๐ฅ ๐ , ๐ฅ0 , ๐ฆ ๐ , ๐ฅ0 , ๐ง ๐ , ๐ฅ0 , ๐ค ๐ , ๐ฅ0
๐ก
0
) โ
โ
1
๐
๐(๐ , ๐ฅ ๐ , ๐ฅ0 , ๐ฆ ๐ , ๐ฅ0 , ๐ง ๐ , ๐ฅ0 , ๐ค ๐ , ๐ฅ0 )
๐
0
๐๐ ]๐๐ +
+ ๐ผ๐
0<๐ก ๐<๐ก
(๐ฅ ๐ก๐, ๐ฅ0 , ๐ฆ ๐ก๐, ๐ฅ0 , ๐ง ๐ก๐, ๐ฅ0 , ๐ค ๐ก๐, ๐ฅ0 ) โ
โ
๐ก
๐
๐ผ๐(๐ฅ ๐ก๐, ๐ฅ0 , ๐ฆ ๐ก๐, ๐ฅ0 , ๐ง ๐ก๐, ๐ฅ0 , ๐ค ๐ก๐, ๐ฅ0 )
๐
๐=1
โฏ 2.3
Which a unique solution of the system (1.1) provided that
๐ฅ0
๐ก, ๐ฅ0 โ ๐ฅ0 โค
๐๐
2
(1 +
4๐
๐
) โฏ 2.4
and
๐ฅ0
๐ก, ๐ฅ0 โ ๐ฅ ๐ (๐ก, ๐ฅ0) โค ๐ ๐
1 โ ๐ โ1
๐
๐
2
(1 +
4๐
๐
) โฏ 2.5
for all ๐ โฅ 1, ๐ก โ ๐ 1
, where the eigen-value ๐ of the matrix ฮ is a positive fraction less than one.
Proof. Consider the sequence of functions ๐ฅ1 ๐ก, ๐ฅ0 , ๐ฅ2 ๐ก, ๐ฅ0 , โฏ, ๐ฅ ๐ ๐ก, ๐ฅ0 , โฏ,
defined by recurrence relation (2.1). Each of the functions of the sequence is periodic in ๐ก of period ๐.
Now, by Lemma 1.1, we have
๐ฅ ๐ ๐ก, ๐ฅ0 โ ๐ฅ0 = ๐ฅ0 + [๐(๐ , ๐ฅ ๐ ๐ , ๐ฅ0 , ๐ฆ ๐ ๐ , ๐ฅ0 , ๐ง ๐ ๐ , ๐ฅ0 , ๐ค ๐ ๐ , ๐ฅ0
๐ก
0
) โ
โ
1
๐
๐(๐ , ๐ฅ ๐ ๐ , ๐ฅ0 , ๐ฆ ๐ ๐ , ๐ฅ0 , ๐ง ๐ ๐ , ๐ฅ0 , ๐ค ๐ ๐ , ๐ฅ0 )
๐
0
๐๐ ]๐๐ +
+ ๐ผ๐
0<๐ก ๐<๐ก
(๐ฅ ๐ ๐ก๐, ๐ฅ0 , ๐ฆ ๐ ๐ก๐, ๐ฅ0 , ๐ง ๐ ๐ก๐, ๐ฅ0 , ๐ค ๐ ๐ก๐, ๐ฅ0 ) โ
โ
๐ก
๐
๐ผ๐(๐ฅ ๐ ๐ก๐, ๐ฅ0 , ๐ฆ ๐ ๐ก๐, ๐ฅ0 , ๐ง ๐ ๐ก๐, ๐ฅ0 , ๐ค ๐ ๐ก๐, ๐ฅ0 )
๐
๐=1
โ ๐ฅ0 โค
โค (1 โ
๐ก
๐
) ๐(๐ , ๐ฅ ๐ ๐ , ๐ฅ0 , ๐ฆ ๐ ๐ , ๐ฅ0 , ๐ง ๐ ๐ , ๐ฅ0 , ๐ค ๐ ๐ , ๐ฅ0 )
๐ก
0
๐๐ +
+
๐ก
๐
๐(๐ , ๐ฅ ๐ ๐ , ๐ฅ0 , ๐ฆ ๐ ๐ , ๐ฅ0 , ๐ง ๐ ๐ , ๐ฅ0 , ๐ค ๐ ๐ , ๐ฅ0 ) ๐๐
๐
๐ก
+
+ ๐ผ๐
0<๐ก ๐<๐ก
(๐ฅ ๐ ๐ก๐, ๐ฅ0 , ๐ฆ ๐ ๐ก๐, ๐ฅ0 , ๐ง ๐ ๐ก๐, ๐ฅ0 , ๐ค ๐ ๐ก๐, ๐ฅ0 ) +
+
๐ก
๐
๐ผ๐ ๐ฅ ๐ ๐ก๐, ๐ฅ0 , ๐ฆ ๐ ๐ก๐, ๐ฅ0 , ๐ง ๐ ๐ก๐, ๐ฅ0 , ๐ค ๐ ๐ก๐, ๐ฅ0 โค
๐
๐=1
โค ๐ผ ๐ก ๐ + 2๐๐
โค ๐
๐
2
(1 +
4๐
๐
) โฏ 2.6
From (1.7) and (2.4), we have
๐ด๐ฅ ๐ ๐ก, ๐ฅ0 โ ๐ด๐ฅ0 โค ๐๐
๐
2
(1 +
4๐
๐
)
๐ต๐ฅ ๐ ๐ก, ๐ฅ0 โ ๐ต๐ฅ0 โค ๐ป๐
๐
2
(1 +
4๐
๐
)
โฏ 2.7
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By mathematical induction, we have
๐ฅ ๐+1 ๐ก, ๐ฅ0 โ ๐ฅ ๐ (๐ก, ๐ฅ0) โค ๐ ๐ ๐ผ ๐ก + ๐ ๐ โค
๐
2
๐ ๐ + ๐ ๐ , โฏ 2.13
Where
๐ ๐+1 = ๐พ1
๐
2
๐ ๐ + ๐พ1 ๐ ๐ ,
๐ ๐+1 = ๐๐พ2 ๐๐ ๐ + 2๐๐พ2 ๐ ๐ , ๐0 = ๐ , ๐0 = 2๐๐ .
โฏ 2.14
๐ = 0,1,2, โฏ.
It is sufficient to show that all solutions of (2.14) approach zero as ๐ โ โ , i.e. it is necessary and sufficient that
the eigen values of the matrix ฮ are assumed to be within a unit circle.
It is well-known that the characteristic equation of the matrix ฮ is
๐พ1
๐
2
+ ๐๐พ2(2 + ๐พ1
๐
2
) < 1 โฏ 2.15
and this ensures that the sequence of functions (2.1) is convergent uniformly on the domain (2.2) as ๐ โ โ .
Let
lim
๐โโ
๐ฅ ๐ (๐ก, ๐ฅ0) = ๐ฅโ ๐ก, ๐ฅ0 โฏ 2.13
Since the sequence of functions (2.2) is periodic in ๐ก of period ๐, then the limiting is also periodic in ๐ก of period
๐.
Moreover, B lemma 1.1 and (2.15) and the following inequality
๐ฅ ๐+1 ๐ก, ๐ฅ0 โ ๐ฅ ๐ (๐ก, ๐ฅ0) โค ๐ฅ ๐+๐+1 ๐ก, ๐ฅ0 โ ๐ฅ ๐+๐(๐ก, ๐ฅ0) โค
๐โ1
๐=0
โค ๐ ๐+๐
๐โ1
๐=0
๐
๐
2
(1 +
4๐
๐
)
the inequalities (2.4) and (2.5) are satisfied for all ๐ โฅ 0 .
Finally, we have to show that ๐ฅ(๐ก, ๐ฅ0) is unique solution of (1.1). on the contrary, we suppose that there is at
least two different solutions ๐ฅ(๐ก, ๐ฅ0) and ๐(๐ก, ๐ฅ0) of (1.1).
From (2.3), the following inequality holds:
๐ฅ ๐ก, ๐ฅ0 โ ๐ ๐ก, ๐ฅ0 โค (1 โ
๐ก
๐
) ๐พ1
๐ก
0
๐ฅ ๐ , ๐ฅ0 โ ๐ ๐ , ๐ฅ0 ๐๐ +
+
๐ก
๐
๐พ1 ๐ฅ ๐ , ๐ฅ0 โ ๐ ๐ , ๐ฅ0
๐
๐ก
๐๐ + ๐พ2
0<๐ก ๐<๐ก
๐ฅ ๐ก๐, ๐ฅ0 โ ๐ ๐ก๐, ๐ฅ0 +
+
๐ก
๐
๐พ2
๐
๐=1
๐ฅ ๐ก๐, ๐ฅ0 โ ๐ ๐ก๐, ๐ฅ0 โฏ 2.16
Setting ๐ฅ ๐ก, ๐ฅ0 โ ๐ ๐ก, ๐ฅ0 = ๐ ๐ก , the inequality (2.16) can be written as:
๐(๐ก) โค (1 โ
๐ก
๐
) ๐พ1
๐ก
0
๐(๐ )๐๐ +
๐ก
๐
๐พ1 ๐(๐ )
๐
๐ก
๐๐ + ๐พ2
0<๐ก ๐<๐ก
๐(๐ก๐) +
๐ก
๐
๐พ2
๐
๐=1
๐(๐ก๐)
Let max ๐กโ[0,๐] ๐(๐ก) = ๐0 โฅ 0 . By iteration, we get:
๐ ๐ก โค ๐ ๐ ๐ผ ๐ก + ๐ ๐ โฏ 2.17
From (2.14), we have:
๐ ๐+1
๐ ๐+1
= ๐พ1
๐
2
๐พ1
๐๐๐พ2 2๐๐พ2
๐ ๐
๐ ๐
โฏ 2.18
which satisfies the initial conditions ๐0 = 0, ๐0 = ๐0 That is
๐ ๐
๐ ๐
=
๐พ1
๐
2
๐พ1
๐๐๐พ2 2๐๐พ2
๐
0
๐ ๐
โฏ 2.19
Hence it is clear that if the condition (2.15) is satisfied then ๐ ๐ โ 0 and ๐ ๐ โ 0 as ๐ โ โ.
From the relation (2.17) we get ๐(๐ก) โก 0 or ๐ฅ ๐ก, ๐ฅ0 = ๐ ๐ก, ๐ฅ0 , i.e. ๐ฅ(๐ก, ๐ฅ0) is a unique solution of (1.1). โ
III. EXISTENCE OF SOLUTION
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The problem of the existence of periodic solution of period ๐ of the system (1.1) is uniquely connected
with the existence of zeros of the function ฮ(๐ฅ0) which has the form
ฮ ๐ฅ0 =
1
๐
[ ๐(๐ก, ๐ฅ0
๐ก, ๐ฅ0 , ๐ฆ0
๐ก, ๐ฅ0 , ๐ง0
๐ก, ๐ฅ0 , ๐ค0
(๐ก, ๐ฅ0))
๐
0
๐๐ก +
+ ๐ผ๐
๐
๐=1
(๐ฅ0
๐ก๐, ๐ฅ0 , ๐ฆ0
๐ก๐, ๐ฅ0 , ๐ง0
๐ก๐, ๐ฅ0 , ๐ค0
(๐ก๐, ๐ฅ0))], โฏ 3.1
since this function is approximately determined from the sequence of functions
ฮm ๐ฅ0 =
1
๐
[ ๐(๐ก, ๐ฅ ๐ ๐ก, ๐ฅ0 , ๐ฆ ๐ ๐ก, ๐ฅ0 , ๐ง ๐ ๐ก, ๐ฅ0 , ๐ค ๐ (๐ก, ๐ฅ0))
๐
0
๐๐ก +
+ ๐ผ๐
๐
๐=1
(๐ฅ ๐ ๐ก๐, ๐ฅ0 , ๐ฆ ๐ ๐ก๐, ๐ฅ0 , ๐ง ๐ ๐ก๐, ๐ฅ0 , ๐ค ๐ (๐ก๐, ๐ฅ0))] โฏ 3.2
where ๐ฅ0
(๐ก, ๐ฅ0) is the limiting of the sequence of functions (2.1). Also
๐ฆ0
๐ก, ๐ฅ0 = ๐ด๐ฅ0
๐ก, ๐ฅ0 , ๐ง0
๐ก, ๐ฅ0 = ๐ต๐ฅ0
(๐ก, ๐ฅ0) and
๐ค0
๐ก, ๐ฅ0 = ๐(๐ , ๐ฅ0
๐ , ๐ฅ0 , ๐ด๐ฅ0
๐ , ๐ฅ0 , ๐ต๐ฅ0
(๐ , ๐ฅ0))
๐ก
๐กโ๐
๐๐ .
Now, we prove the following theorem taking in to ///// that the following inequality will be satisfied for all
๐ โฅ 1.
ฮ ๐ฅ0 โ ฮ ๐ (๐ฅ0) โค ๐ ๐
1 โ ๐ โ1
(๐พ1 +
๐
๐
๐พ2)
๐๐
2
(1 +
4๐
๐
) โฏ 3.3
Theorem 3.1. If the system of equations (1.1) satisfies the following conditions:
(๐) the sequence of functions (3.2) has an isolated singular point ๐ฅ0 = ๐ฅ0
, ฮ ๐ (๐ฅ0) โก 0, for all ๐ก โ ๐ 1
;
(๐๐) the index of this point is nonzero;
(๐๐๐) there exists a closed convex domain ๐ท4 belonging to the domain ๐ท๐ and possessing a unique singular point
๐ฅ0
such that on it is boundary ฮ ๐ท4
the following inequality holds
inf
xโฮ ๐ท4
ฮ ๐ (๐ฅ0
) โค ๐ ๐
1 โ ๐ โ1
(๐พ1 +
๐
๐
๐พ2)
๐๐
2
(1 +
4๐
๐
)
for all ๐ โฅ 1. Then the system (1.1) has a periodic solution ๐ฅ = ๐ฅ(๐ก, ๐ฅ0) for which ๐ฅ 0 โ ๐ท4.
Proof. By using the inequality (3.3) we can prove the theorem is a similar way to that of theorem 2.1.2 [2].
Remark 3.1. When ๐ ๐
= ๐ 1
, i.e. when ๐ฅ0
is a scalar, theorem 3.1 can be proved by the following.
Theorem 3.2. Let the functions ๐(๐ก, ๐ฅ, ๐ฆ, ๐ฆ, ๐ง, ๐ค) and ๐ผ๐(๐ฅ, ๐ฆ, ๐ง, ๐ค) of system (1.1) are defined on the interval
[๐, ๐] in ๐ 1
. Then the function (3.2) satisfies the inequalities:
min
๐+
๐๐
2
1+
4๐
๐
โค๐ฅ0โค๐โ
๐๐
2
1+
4๐
๐
ฮ ๐ ๐ฅ0
โค โ ๐ ๐
1 โ ๐ โ1
(๐พ1 +
๐
๐
๐พ2)
max
๐+
๐๐
2
1+
4๐
๐
โค๐ฅ0โค๐โ
๐๐
2
1+
4๐
๐
ฮ ๐ ๐ฅ0
โฅ ๐ ๐
1 โ ๐ โ1
(๐พ1 +
๐
๐
๐พ2)
โฏ 3.4
Then (1.1) has a periodic solution in ๐ก of period ๐ for which
๐ฅ0
0 โ [๐ +
๐๐
2
(1 +
4๐
๐
), ๐ โ
๐๐
2
(1 +
4๐
๐
)] .
Proof. Let ๐ฅ1 and ๐ฅ2 be any points of the interval
[๐ +
๐๐
2
(1 +
4๐
๐
), ๐ โ
๐๐
2
(1 +
4๐
๐
)] such that
ฮ ๐ ๐ฅ1 = min ฮ ๐ ๐ฅ0
๐+
๐๐
2
1+
4๐
๐
โค๐ฅ0โค๐โ
๐๐
2
1+
4๐
๐
,
ฮ ๐ ๐ฅ2 = max ฮ ๐ ๐ฅ0
๐+
๐๐
2
1+
4๐
๐
โค๐ฅ0โค๐โ
๐๐
2
1+
4๐
๐
.
โฏ 3.5
By using the inequalities (3.3) and (3.4) , we have
ฮ ๐ ๐ฅ1 = ฮ ๐ ๐ฅ1 + ฮ ๐ ๐ฅ1 โ ฮ ๐ ๐ฅ1 < 0 ,
ฮ ๐ ๐ฅ2 = ฮ ๐ ๐ฅ2 + (ฮ ๐
๐ฅ2 โ ฮ ๐ ๐ฅ2 ) > 0 .
3.6
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The continuity of ฮ(๐ฅ0
) and by using (3.6), there exists a point ๐ฅ0
, ๐ฅ0
โ ๐ฅ1, ๐ฅ2 , such that ฮ(๐ฅ0
) โก 0,
i.e. ๐ฅ = ๐ฅ(๐ก, ๐ฅ0) is a periodic solution in ๐ก of period ๐ for which ๐ฅ0
(0) โ [๐ +
๐๐
2
(1 +
4๐
๐
), ๐ โ
๐๐
2
(1 +
4๐
๐
)] . โ
Similar results can be obtained for other class of integro-differential equations of operators with impulsive
action.
In particular, the system of integro-differential equations which has the form
๐๐ฅ
๐๐ก
= ๐(๐ก, ๐ฅ, ๐ด๐ฅ, ๐ต๐ฅ, ๐ ๐ , ๐ฅ ๐ , ๐ด๐ฅ ๐ , ๐ต๐ฅ ๐ ๐๐
๐ ๐ก
๐ ๐ก
,
, ๐น ๐ก, ๐
๐ก
โโ
๐ ๐ , ๐ฅ ๐ , ๐ด๐ฅ ๐ , ๐ต๐ฅ ๐ ๐๐ , ๐ก โ ๐ก๐
ฮ๐ฅ = |๐ก=๐ก ๐
๐ผ๐(๐ฅ, ๐ด๐ฅ, ๐ต๐ฅ, ๐ ๐ , ๐ฅ ๐ , ๐ด๐ฅ ๐ , ๐ต๐ฅ ๐ ๐๐ ,
๐(๐ก ๐)
๐(๐ก ๐)
, ๐น ๐ก๐, ๐ ๐(๐ , ๐ฅ ๐ , ๐ด๐ฅ ๐ , ๐ต๐ฅ(๐ ))๐๐
๐ก ๐
โโ
)
โฏ 3.7
In this system (3.7), let the vector functions ๐ ๐ก, ๐ฅ, ๐ฆ, ๐ง, ๐ค , ๐(๐ก, ๐ฅ, ๐ฆ, ๐ง) and scalar functions ๐ ๐ก , ๐(๐ก)
are periodic in ๐ก of period ๐, defined and continuous on the domain.
๐ก, ๐ฅ, ๐ฆ, ๐ง, ๐ค, ๐ โ ๐ 1
ร ๐บ ร ๐บ1 ร ๐บ2 ร ๐บ3 ร ๐บ4 =
= โโ, โ ร ๐บ ร ๐บ1 ร ๐บ2 ร ๐บ3 ร ๐บ4 โฏ 3.8
Let ๐ผ๐ ๐ฅ, ๐ฆ, ๐ง, ๐ค, ๐ be a continuous vector function which are defined on the domain (3.8). A matrix ๐น ๐ก, ๐ is
defined and continuous in ๐ 1
ร ๐ 1
and satisfies the condition ๐น ๐ก + ๐, ๐ + ๐ = ๐น ๐ก, ๐ which ๐น ๐ก, ๐ โค
๐ฟ ๐โ๐พ ๐กโ๐
,
0 โค ๐ โค t โค T , where ๐ฟ, ๐พ are a positive constants. Suppose that the functions ๐(๐ก, ๐ฅ, ๐ฆ, ๐ง, ๐ค, ๐) and
๐ ๐ก, ๐ฅ, ๐ฆ, ๐ง , ๐ผ๐ ๐ฅ, ๐ฆ, ๐ง, ๐ค, ๐ are satisfying the following inequalities:
๐(๐ก, ๐ฅ, ๐ฆ, ๐ง, ๐ค, ๐) โค ๐ , ๐(๐ก, ๐ฅ, ๐ฆ, ๐ง) โค ๐ , โฏ 3.9
๐ ๐ก, ๐ฅ1, ๐ฆ1, ๐ง1, ๐ค1, ๐1 โ ๐ ๐ก, ๐ฅ2, ๐ฆ2, ๐ง2, ๐ค2, ๐2 โค ๐พโ
( ๐ฅ1 โ ๐ฅ2 + ๐ฆ1 โ ๐ฆ2 +
+ ๐ง1 โ ๐ง2 + ๐ค1 โ ๐ค2 + ๐1 โ ๐2 );
๐ ๐ก, ๐ฅ1, ๐ฆ1, ๐ง1 โ ๐ ๐ก, ๐ฅ2, ๐ฆ2, ๐ง2 โค ๐โ
๐ฅ1 โ ๐ฅ2 + ๐ฆ1 โ ๐ฆ2 + ๐ง1 โ ๐ง2 ;
โฏ 3.10
๐ผ๐ ๐ฅ1, ๐ฆ1, ๐ง1, ๐ค1, ๐1 โ ๐ผ๐ ๐ฅ2, ๐ฆ2, ๐ง2, ๐ค2, ๐2 โค ๐ฟโ
( ๐ฅ1 โ ๐ฅ2 + ๐ฆ1 โ ๐ฆ2 +
+ ๐ง1 โ ๐ง2 + ๐ค1 โ ๐ค2 + ๐1 โ ๐2 );
โฏ 3.11
and
๐ด๐ฅ1 ๐ก โ ๐ด๐ฅ2 ๐ก โค ๐บโ
๐ฅ1 ๐ก โ ๐ฅ2 ๐ก ,
๐ต๐ฅ1 ๐ก โ ๐ต๐ฅ2 ๐ก โค ๐ปโ
๐ฅ1 ๐ก โ ๐ฅ2 ๐ก .
โฏ 3.12
for all ๐ก โ ๐ 1
, ๐ฅ, ๐ฅ1, ๐ฅ2 โ ๐บ, ๐ฆ, ๐ฆ1, ๐ฆ2 โ ๐บ1, ๐ง, ๐ง1, ๐ง2 โ ๐บ2, ๐ค, ๐ค1, ๐ค2 โ ๐บ3,
๐, ๐1, ๐2 โ ๐บ4 , Provided that
๐ด๐ฅ ๐ก + ๐ = ๐ด๐ฅ ๐ก , ๐ต๐ฅ ๐ก + ๐ = ๐ต๐ฅ ๐ก , ๐ผ๐+๐ ๐ฅ, ๐ฆ, ๐ง, ๐ค, ๐ = ๐ผ๐(๐ฅ, ๐ฆ, ๐ง, ๐ค, ๐) and ๐ก๐+๐ = ๐ก๐ + ๐.
Define a non-empty sets as follows:
๐บ๐ = ๐บ โ
๐โ
๐
2
(1 +
4๐
๐
) , ๐บ1๐ = ๐บ1 โ ๐บโ
๐โ
๐
2
(1 +
4๐
๐
),
๐บ2๐ = ๐บ3 โ ๐ปโ
๐โ
๐
2
(1 +
4๐
๐
) , ๐บ4๐ = ๐บ4 โ ๐ฝ๐โ
๐โ
๐
2
(1 +
4๐
๐
) ,
๐บ5๐ = ๐บ5 โ
๐ฟ
๐พ
๐โ
๐โ
๐
2
(1 +
4๐
๐
) .
โฏ 3.13
Furthermore, the largest Eigen-value of ๐พ ๐๐๐ฅ of the matrix
ฮ =
๐พ1
๐
2
๐พ1
๐๐พ2 ๐ 2๐๐พ2
is less than unity, i.e.
8. Periodic Solutions for Nonlinear Systems of Integro-Differential Equationsโฆ
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1
2
[๐พ1
๐
3
+ 2๐๐พ2 + (
๐พ1 ๐
3
+ 2๐๐พ2)2 +
4๐๐พ1 ๐พ2 ๐
3
] < 1 , โฏ 3.14
where ๐พ1 = ๐พโ
[1 + ๐บโ
+ ๐ปโ
+
๐ฟ
๐พ
๐โ
+ ๐โ
(1 + ๐บโ
+ ๐ฝ)],
๐พ2 = ๐ฟโ
[1 + ๐บโ
+ ๐ปโ
+
๐ฟ
๐พ
๐โ
+ ๐โ
(1 + ๐บโ
+ ๐ฝ)] and ๐ฝ = max0โค๐กโค๐ ๐ ๐ก โ ๐(๐ก) .
Theorem 3.2. If the system of integro-differential equations with impulsive action (3.7) satisfying the above
assumptions and conditions has a periodic solution ๐ฅ = ๐ ๐ก, ๐ฅ0 , then there exists a unique solution which is the
limit function of a uniformly convergent sequence which has the form
๐ฅ ๐ ๐ก, ๐ฅ0 = ๐ฅ0 + [๐(๐ , ๐ฅ ๐ ๐ , ๐ฅ0 , ๐ฆ ๐ ๐ , ๐ฅ0 , ๐ง ๐ ๐ , ๐ฅ0 , ๐ค ๐ ๐ , ๐ฅ0 , ๐ ๐ (๐ , ๐ฅ0)
๐ก
0
) โ
โ
1
๐
๐(๐ , ๐ฅ ๐ ๐ , ๐ฅ0 , ๐ฆ ๐ ๐ , ๐ฅ0 , ๐ง ๐ ๐ , ๐ฅ0 , ๐ค ๐ ๐ , ๐ฅ0 , ๐ ๐ (๐ , ๐ฅ0))
๐
0
๐๐ ]๐๐ +
+ ๐ผ๐
0<๐ก ๐<๐ก
(๐ฅ ๐ ๐ก๐, ๐ฅ0 , ๐ฆ ๐ ๐ก๐, ๐ฅ0 , ๐ง ๐ ๐ก๐, ๐ฅ0 , ๐ค ๐ ๐ก๐, ๐ฅ0 , ๐ ๐ ๐ก๐, ๐ฅ0 ) โ
โ
๐ก
๐
๐ผ๐(๐ฅ ๐ ๐ก๐, ๐ฅ0 , ๐ฆ ๐ ๐ก๐, ๐ฅ0 , ๐ง ๐ ๐ก๐, ๐ฅ0 , ๐ค ๐ ๐ก๐, ๐ฅ0 , ๐ ๐ ๐ก๐, ๐ฅ0 )
๐
๐=1
โฏ 3.15
with
๐ฅ0 ๐ก, ๐ฅ0 = ๐ฅ0 , ๐ = 0,1,2, โฏ.
The proof is a similar to that of theorem 1.1.
If we consider the following mapping
ฮm ๐ฅ0 =
1
๐
[ ๐(๐ก, ๐ฅ ๐ ๐ก, ๐ฅ0 , ๐ฆ ๐ ๐ก, ๐ฅ0 , ๐ง ๐ ๐ก, ๐ฅ0 , ๐ค ๐ ๐ก, ๐ฅ0 , ๐ ๐ (๐ก, ๐ฅ0))
๐
0
๐๐ก +
+ ๐ผ๐
๐
๐=1 (๐ฅ ๐ ๐ก๐, ๐ฅ0 , ๐ฆ ๐ ๐ก๐, ๐ฅ0 , ๐ง ๐ ๐ก๐, ๐ฅ0 , ๐ค ๐ ๐ก๐, ๐ฅ0 , ๐ ๐ (๐ก๐, ๐ฅ0))].
โฏ 3.16
Then we can state a theorem similar to theorem 2.2, provided that ๐พ ๐๐๐ฅ =
1
2
[๐พ1
๐
3
+ 2๐๐พ2 +
(
๐พ1 ๐
3
+ 2๐๐พ2)2 +
4๐๐พ1 ๐พ2 ๐
3
] < 1 ,
Remark 3.2. It is clear that when we put ๐ผ๐ โก 0, we get a periodic solution for the systems (1.1) and (3.7)
without introducing impulsive action.
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