Periodic Solution for nonlinear system of differential
equations depending on the probability density function of
gamma distribution
Raad N. Butris
University Of Duhok, Faculty of Educational Science, School of Basic Education Mathematics Department
Abstract
.
The numerical-analytic method were introduced by Samoilenko has been used tostudy thePeriodic solution of a new nonlinear system of differential equations depending on the
probability density function of gamma distribution. Also these investigations lead us to
improving and extending the results of Samoilenko and extended his method .
Keywords . Numerical-analytic method existence, uniqueness, stability Periodic solution, probability density function of gamma dirstibution, nonlinear system of differential equations.
1. Introduction.
The theory of differential equations is one of the most fascinating and successful areas of
mathematics. Its results help us to prove important theorems and provide the inspiration for
many useful concepts in other areas of mathematics [1,6,7,10,12,13,14]. Many of the most
powerful techniques used in the application of mathematics to other sciences and engineering are
based on differential equations theory [4, 5, 6, 9, 1,17 ]. Because of its wide applicability, its
blend of geometric and analytic concepts, and the simplicity of many of its results differential
equations provides an excellent introduction to modern mathematics. Recent developments in
differential equations theory [7, 8, 13, 15,18]. Many author create and develop
numerical-analytic methods[2,3,4,8,9,10,11,12] and schemes to investigate periodic solution of integral
equations describing many applications in mathematical and engineering field.
Butris [3] has been used the numerical-analytic method to study the periodic solution for nonlinear system of differential equation depending on of gamma distribution which has the form
dx
dt = (𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼),𝑥𝑥) ⋯ (1.1)
where x∈D , and D is the closure of bounded domain and connected in Rn.
This study leads us to improving and extending Butris [3] results, to investigate the existence
and approximation of Periodic solution for nonlinear system of differential equations depending
on the probability density function of gamma distribution.
Consider the following system of differential equations depending on the probability
density function of gamma distribution which has the form
dx
dt = f(t,γ(t,α,β,µ), x) … (1.2)
where x∈ D , all real t and D is the closure of bounded domain and connected
in Rn. The vector function f(t,γ(t,α,β,µ), x) is defined on the domain:
(t,γ(t,α,β,µ), x)∈R1× [0, T] × D = (−∞,∞) × [0, T] × D, … (1.3)
is periodic in t of period T and continuous in the totality of variables and satisfies the inequalities:
|f(t,γ(t,α,β,µ), x)|≤ M| γ(t,α,β,µ)| , | γ(t,α,β,µ)|≤ Mγ, … (1.4)
|f(t,γ(t,α,β,µ), x1)−(t,γ(t,α,β,µ), x2)| ≤K| γ(t,α,β,µ) ||x1 −x2|
… (1.5)
for all t∈R1 and x, x1, x2 ∈D, where M = (M1, M2,⋯, Mn)is a positive constant vector and
Mγis a positive constant . The general formula for the probability density function of gamma
distribution is:-
γ(t,α,β,µ) = � t−α
β �
µ−1
exp�−t−αβ �
Γ( µ) , t≥ α , � … (1.6)
where T≤� t−α
β �
µ−1
exp�−t−αβ �
Γ(µ) ,α , β,µ are a positive constants.
We define the non-empty sets as follows:
Dγf = D−MMγT2 … . (1.7).
Furthermore, we suppose that the greatest Eigen value λmax of the matrix
Λ= Mγ KT2 does not exceed unity,
i.e.λmax(Λ) < 1. (1.8).
Lemma1.1. Let f(t) be a continuous vector function in the interval 0≤t≤ T, then
��(f(s)−T1 t
0
�f(s)ds)ds
T
0
� ≤ α(t) maxt∈[0,T]|f(t)|,
where α(t) = 2t(1− t
T) . (For the proof see [16 ]) .
By using lemma 1.1, we can state and prove the following lemma.
Lemma 1.2. Suppose that the function of the probability density γ(t,α,β,µ)is continuous
on the interval [0, T], then
�� γ(s,α,β,µ)−1T� γ(s,α,β,µ)ds
T
0
)ds
t
0
� ≤Mγα(t)
is hold for all values of α,β,µ.
Proof. Taking
�� γ(s,α,β,µ)−1T� γ(s,α,β,µ)ds
T
0
)ds
t
0
� ≤(1−T)t �|γ(s,α,β,µ)|ds
T
t
+Tt �|γ(s,α,β,µ)|
T
t
ds
=�1−Tt� �Mγ t
0
ds +Tt�Mγ T
t
ds
≤ Mγ��1−Tt�t +Tt(T−t)� =α(t)Mγ.
So that
�� γ(t,α,β,µ)−T1� γ(t,α,β,µ)ds
T
0
)ds
t
0
� ≤ α(t) Mγ ⋯(1.10)
for all t∈[0, T] and α(t)≤ T
2 .
2. Approximate of solution.
The investigation of approximate periodic solution of (1.2) is formulated by the following
theorem:
Theorem 2.1.If the system (1.2) satisfy the inequalities (1.4),(1.5) and the conditions
(1.6),(1.7) has a continuous periodic solution x = x(t,γ(t,α,β,µ), x0) , then the sequence of
functions
xm+1(t,γ(t,α), x0) = x0+�[f(s,γ(s,α,β,µ), xm(s,γ(s,α,β,µ), x0) t
0
−1T�(f(s,γ(s,α,β,µ), xm(s,γ(s,α,β,µ), x0)ds]ds T
0
⋯(2.1)
with
x0(t,γ(t,α,β,µ), x0) = x0 , m = 0,1,2,⋯
is uniformly convergent as m→ ∞ in the domain
(t,γ(t,α,β,µ), x0) ∈R1× [0, T] × Dγf (2.2)
to the limit function x0(t,γ(t,α), x0) which is defined on the domain (2.2), and satisfying the
system of integral equations
x(t,γ(t,α,β,µ), x0) = x0+�[f(s,γ(s,α,β,µ), x(s,γ(s,α,β,µ), x0) t
0
−1T�(f(s,γ(s,α,β,µ), x(s,γ(s,α,β,µ), x0)ds]ds T
0
… (2.3)
which is a continuous periodic solution of (1.2) provided that:
|𝑥𝑥0(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥
0)− 𝑥𝑥0|≤ 𝑀𝑀𝛾 𝛼𝛼(𝑡𝑡) … (2.4)
and
|𝑥𝑥0(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥
0)− 𝑥𝑥𝑚(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥0)| ≤ 𝛬𝑚(𝐸 − 𝛬)−1𝑀𝑀𝛾𝛼𝛼(𝑡𝑡) … (2.5)
for all 𝑚 ≥1 and 𝑡𝑡 ∈ 𝑅1, where 𝐸 is the identity matrix.
𝑷𝒓𝒐𝒐𝒇. By lemma 1.2 and by mathematical induction, we find that:
|𝑥𝑥𝑚(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥0)− 𝑥𝑥0|≤ 𝑀𝑀𝛾𝛼𝛼(𝑡𝑡) ⋯(2.8)
for all 𝑡𝑡 ∈ 𝑅1 and 𝑥𝑥0 ∈ 𝐷𝛾𝑓.
i.e. 𝑥𝑥𝑚(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥0)∈ 𝐷, for all 𝑡𝑡 ∈ 𝑅1 and 𝑥𝑥0 ∈ D𝛾𝑓.
We claim that the sequence of functions (2.1) is uniformly convergent on the
domain (2.2).
Suppose that the following inequality is true:
|𝑥𝑥𝑚(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥0)− 𝑥𝑥𝑚−1(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥0)|≤ 𝑀𝑀𝛾𝑚[𝐾𝑇2]𝑚−1𝛼𝛼(𝑡𝑡)
(2.10)
for all 𝑚 ≥1.
Now, we shall prove the following:
|𝑥𝑥𝑚+1(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥0)− 𝑥𝑥𝑚(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥0)|
≤ 𝐾[�1−𝑇� � 𝑀𝑡𝑡 𝛾 𝑡
0
|𝑥𝑥𝑚(𝑠,𝛾𝛾(𝑠,𝛼𝛼,𝛽,µ),𝑥𝑥0)− 𝑥𝑥𝑚−1(𝑠,𝛾𝛾(𝑠,𝛼𝛼,𝛽,µ),𝑥𝑥0)|𝑑𝑠
+𝑇 � 𝑀𝑡𝑡 𝛾 𝑇
𝑡
|𝑥𝑥𝑚(𝑠,𝛾𝛾(s,𝛼𝛼,𝛽,µ,𝑥𝑥0)− 𝑥𝑥𝑚−1(𝑠,𝛾𝛾(𝑠,𝛼𝛼,𝛽,µ,𝑥𝑥0)|𝑑𝑠]
≤ �1− 𝑡
𝑇� ∫ 𝑀𝛾 𝑡
0 𝑀𝑀𝛾𝑚�𝐾 𝑇 2�
𝑚−1
𝛼𝛼(𝑠)𝑑𝑠
+𝑇 � 𝑀𝑡𝑡 𝛾 𝑇
𝑡
𝑀 𝑀𝛾𝑚�𝐾𝑇2� 𝑚−1
𝛼𝛼(𝑠)𝑑𝑠
= 𝑀𝑀𝛾 �𝑀𝛾 𝐾𝑇2� 𝑚
𝛼𝛼(𝑡𝑡)
and hence
|𝑥𝑥𝑚+1(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥0)− 𝑥𝑥𝑚(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥0)|≤ 𝑀𝑀𝛾 �𝑀𝛾𝐾𝑇2� 𝑚
𝛼𝛼(𝑡𝑡) (2.11)
for all 𝑚 ≥0 .
From (2.11) we conclude that for any𝑘 ≥1, we have the inequality
|𝑥𝑥𝑚+𝑘(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥0)− 𝑥𝑥𝑚(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥0)| ≤ � 𝛬𝑚+𝑖𝑀𝑀𝛾 𝑘−1
𝑖=0
𝛼𝛼(𝑡𝑡)
such that
|𝑥𝑥𝑚+𝑘(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥0)−x𝑚(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥0)|
≤ �|𝑥𝑥𝑚+1+𝑖(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥0)− 𝑥𝑥𝑚+𝑖(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥0)| ∞
𝑖=0
≤ � 𝑀𝑀𝛾𝛼𝛼(𝑡𝑡) 𝛬𝑚+1+𝑖 ∞
𝑖=0
≤ 𝑀𝑀𝛾 𝛼𝛼(𝑡𝑡) 𝛬𝑚� 𝛬𝑖+1 ∞
𝑖=0
≤ 𝑀𝑀𝛾 𝛼𝛼(𝑡𝑡) 𝛬𝑚 (𝐸 − 𝛬)−1
So that
|𝑥𝑥𝑚+𝑘(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥0)− 𝑥𝑥𝑚(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥0)| ≤ 𝛬𝑚 (𝐸 − 𝛬)−1𝑀𝑀𝛾 𝛼𝛼(𝑡𝑡) (2.12)
for all 𝑘 ≥1.
From (2.12) and the condition (1.9), we find that:
𝑙𝑖𝑚 𝑚→∞𝛬
𝑚= 0. (2.13)
Relations (2.12) and (2.13) prove the uniform convergence of the sequence of functions (2.1)
on the domain (2.2).
Let
𝑙𝑖𝑚
𝑚→∞𝑥𝑥𝑚(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥0) =𝑥𝑥
0(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥
0) ⋯(2.14)
Since the sequence of functions (2.2) is a periodic and continuous in 𝑡𝑡,𝛾𝛾,𝑥𝑥 , then the
limiting function 𝑥𝑥0(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥0) is also periodic and continuous in 𝑡𝑡,𝛾𝛾,𝑥𝑥.
Moreover, by Lemma 1.2 and inequality (2.12) the inequalities (2.4) and (2.5) are holds.
3 Uniqueness of periodic solution.
We have to show that 𝑥𝑥(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥0) is a unique periodic solution of the
system (1.1).Assume that 𝑟(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥0) is another periodic solution of the system (1.1),
i.e.
𝑟(𝑡𝑡,𝛾𝛾(𝑠,𝛼𝛼,𝛽,µ),𝑥𝑥0) =𝑥𝑥0+�[𝑓(𝑠,𝛾𝛾(𝑠,𝛼𝛼,𝛽,µ),𝑟(𝑠,𝛾𝛾(𝑠,𝛼𝛼,𝛽,µ),𝑥𝑥0) 𝑡
0
−1
𝑇 � (𝑓(𝑠,𝛾𝛾(𝑠,𝛼𝛼,𝛽,µ),𝑟(𝑠,𝛾𝛾(𝑠,𝛼𝛼,𝛽,µ),𝑥𝑥0)𝑑𝑠]𝑑𝑠, 𝑇
0
(2.15)
Theorem2.𝑊𝑖𝑡𝑡ℎ the hypotheses and all conditions of the theorem1.1, the periodic
solution of 1.1is a unique continuous on the domain (1.2). Now, we prove that
𝑥𝑥(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥0) =𝑟(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥0) for all 𝑥𝑥0 ∈ 𝐷𝛾𝑓 and to do this, we need to drive the
following inequality, we need to drive the following inequality:
|𝑟(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥0)− 𝑥𝑥(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥0)|≤ 𝛬𝑚 (𝐸 − 𝛬)−1𝑀∗𝑀𝛾 𝛼𝛼(𝑡𝑡) (2.16)
𝑤ℎ𝑒𝑟𝑒𝑀∗= 𝑚𝑎𝑥𝑥
𝑥0∈𝐷𝛾𝑓�𝑓�𝑡𝑡,𝑟(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥0),𝑥𝑥(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥0)�|
Suppose that (2.16) is true for 𝑚= 𝑘, i.e.
|𝑟(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥0)− 𝑥𝑥(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥0)|≤ 𝛬𝑘 (𝐸 − 𝛬)−1𝑀∗𝑀𝛾 𝛼𝛼(𝑡𝑡)
For all 𝑡𝑡 ∈ 𝑅1 and 𝑥𝑥0 ∈ 𝐷𝛾𝑓.
Then
|𝑟(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥0)− 𝑥𝑥(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥0)|
≤ 𝐾[�1−𝑇� � 𝑀𝑡𝑡 𝛾 𝑡
0
|𝑟(𝑠,𝛾𝛾(𝑠,𝛼𝛼,𝛽,µ),𝑥𝑥0)− 𝑥𝑥(𝑠,𝛾𝛾(𝑠,𝛼𝛼,𝛽,µ),𝑥𝑥0)|𝑑𝑠
+𝑇 � 𝑀𝑡𝑡 𝛾 𝑇
𝑡 |𝑟(𝑠,𝛾𝛾(𝑠,𝛼𝛼,𝛽,µ),𝑥𝑥0)− 𝑥𝑥(𝑠,𝛾𝛾(𝑠,𝛼𝛼,𝛽,µ),𝑥𝑥0)|𝑑𝑠]
≤ 𝐾[�1− 𝑡𝑡
𝑇� � 𝑀𝛾 𝑡
0
𝛬𝑘(𝐸 − 𝛬)−1𝑀∗𝑀
𝛾𝛼𝛼(𝑠)𝑑𝑠
+𝑇𝑡∫ 𝑀𝑡𝑇 𝛼𝛬𝑘(𝐸 − 𝛬)−1𝑀∗𝑀𝛾𝛼𝛼(𝑠)𝑑𝑠] =𝛬𝑘+1(𝐸 − 𝛬)−1𝑀∗𝑀𝛾𝛼𝛼(𝑡𝑡).
By induction, inequality (2.16) is true for 𝑚= 0,1,2,⋯ ,
and thus from (2.14) and (2.16), we have:
𝑙𝑖𝑚
𝑚→∞|𝑟(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥0)− 𝑥𝑥𝑚(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥0)| = 0
and hence
𝑙𝑖𝑚
𝑚→∞𝑥𝑥𝑚(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥0) =𝑟(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥0)
By the relation (2.14), we get:
𝑥𝑥(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥0) =𝑟(𝑡𝑡,𝛾𝛾(𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ)𝑡𝑡,𝛼𝛼),𝑥𝑥0)
i.e. 𝑥𝑥(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥0) is a unique continuous solution of (1.1) on the domain (1.2). ∎
3. Existence of solution.
The problem of existence of a periodic solution of the system (1.1)
is uniquely connected with the existence of zeros of the function 𝛥(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥0) which has
the form:
𝛥(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥0) =
1𝑇∫ 𝑓(𝑠,𝛾𝛾(𝑠,𝛼𝛼,𝛽,µ),𝑥𝑥0(𝑠,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥 0)𝑑𝑠 𝑇
0 …………..⋯(3.1)
where 𝑥𝑥0(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼),𝑥𝑥0) is the limiting function of the sequence of functions (2.1).
The function (3.1) can find only approximately, say by computing the
following functions:
𝛥𝑚(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥0) =1𝑇∫ 𝑓0𝑇 (𝑠,𝛾𝛾(𝑠,𝛼𝛼,𝛽,µ),𝑥𝑥𝑚(𝑠,𝛾𝛾(𝑠,𝛼𝛼,𝛽,µ),𝑥𝑥0)𝑑𝑠,
m= 0,1,2,… ⋯(3.2)
Theorem3.1. Let all assumptions and conditions of theorem 2.1 were given. Then the
inequality:
|𝛥(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥0)−𝛥𝑚(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥0)|≤ 𝛬𝑚+1(𝐸 − 𝛬)−1𝑀𝑀𝛾 ⋯(3.3)
will be satisfied for all 𝑚 ≥0,𝑥𝑥0 ∈ 𝐷𝛾𝑓 .
Proof. By the relations (3.1) and (3.2), the estimate
|𝛥(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥0)−𝛥𝑚(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥0)|
≤𝐾𝑇 �|𝛾𝛾(𝑠,𝛼𝛼,𝛽,µ)|
𝑇
0
|𝑥𝑥0(𝑠,𝛾𝛾(𝑠,𝛼𝛼,𝛽,µ),𝑥𝑥
0)− 𝑥𝑥𝑚(𝑠,𝛾𝛾(𝑠,𝛼𝛼,𝛽,µ),𝑥𝑥0)|𝑑𝑠
≤𝐾𝑇 � 𝑀𝛼 𝑇
0
𝛬𝑚(𝐸 − 𝛬)−1𝑀𝑀
𝛾 𝛼𝛼(𝑠)𝑑𝑠
=𝛬𝑚+1(𝐸 − 𝛬)−1𝑀𝑀 𝛾
Thus the inequality (3.2) is hold for all 𝑚 ≥0 .
Next, we prove the following theorem taking into account at the inequality
(3.3) will be satisfied for all 𝑚 ≥0 .
Theorem 3.2. If the system (1.1) satisfies the following condition:
(i)The sequence of functions (3.2) has an isolated singular point 𝑥𝑥0 =𝑥𝑥0,
𝛥𝑚(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥0)≡ 0, for some 𝑡𝑡 ∈ 𝑅1.
(ii)The index of this point is nonzero;
(iii)There exists a closed convex domain 𝐷𝛾∗ belonging to domain 𝐷𝛾𝑓 and possessing a unique
singular point 𝑥𝑥0 such that on it is boundary 𝛤𝐷𝛾∗ the following inequality is holds
𝑖𝑛𝑓𝑥0∈𝛤𝐷𝛾∗‖𝛥𝑚(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥0)‖ ≥ �𝛬𝑚(𝐸 − 𝛬)−1𝑀𝑀𝛾� . . (3.4)
Where 𝑥𝑥0 ∈ 𝛤𝐷𝛾∗ for all 𝑚 ≥0 . Then the system (1.1) has a periodic solution
𝑥𝑥=𝑥𝑥(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥0) for which 𝑥𝑥(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥0) belongs to the domain 𝐷𝛾∗ .
Proof. By using the inequality (3.1) we can prove the theorem 3.1by the same proof of a
theorem7.1[13].
Remark 3.1[16]. When 𝑅𝑛 =𝑅1, i.e. when 𝑥𝑥0is a scalar, the existence of aperiodic
solution can be strengthens by giving up the requirement that the singular point shout be isolated,
thus we have
Theorem3.3. Let the system (1.1) is defined on the interval [𝑎,𝑏]. Suppose that for 𝑚 ≥0,
the function 𝛥𝑚(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥0) defined according to formula (3.2) satisfies the inequalities:
𝑚𝑖𝑛
𝑎+ℎ≤𝑥0≤𝑏−ℎ‖𝛥𝑚(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥0)‖ ≤ −𝜎𝑚 ;
𝑚𝑎𝑥𝑥
𝑎+ℎ≤𝑥0≤𝑏−ℎ‖𝛥𝑚(𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥0)‖ ≥ 𝜎𝑚 .
� ⋯(3.5)
Then the system (1.1) has a periodic solution 𝑥𝑥=𝑥𝑥(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼)𝑥𝑥0) for which
𝑥𝑥0 ∈[𝑎+ℎ,𝑏 − ℎ], where ℎ=�𝑀𝑀𝛾�𝑇2 and 𝜎𝑚 = �𝛬𝑚+1(𝐸 − 𝛬)−1𝑀𝑀𝛾� .
Proof. Let 𝑥𝑥1 and x2 be any two points on the interval [𝑎,𝑏] such that:
𝛥𝑚(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥1) =𝑎+ℎ≤𝑥𝑚𝑖𝑛
0≤𝑏−ℎ𝛥𝑚(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥0) ;
𝛥𝑚(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥2) =𝑎+ℎ≤𝑥𝑚𝑎𝑥𝑥
0≤𝑏−ℎ𝛥𝑚(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥0) .
� ⋯(3.6)
Taking into account inequalities (3.3) and (3.5), we have
𝛥(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥1)
= 𝛥𝑚(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥1) + [𝛥(𝑡𝑡,𝛾𝛾( 𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥1)− 𝛥𝑚(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥1)]
𝛥(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥2)
= 𝛥𝑚(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥2) + [𝛥(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥2)− 𝛥𝑚(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥2)]
…(3.7)
It follows from the inequalities (3.7) and the continuity of the function
𝛥(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥0), that there exist an isolated singular point 𝑥𝑥0,𝑥𝑥0 ∈ [𝑥𝑥1,𝑥𝑥2], such that 𝛥(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥0)≡ 0, this means that the system (1.1) has a periodic
continuous solution 𝑥𝑥=𝑥𝑥(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥0) for which 𝑥𝑥0 ∈[𝑎+ℎ,𝑏 − ℎ]. ∎.
4.Stability of solution.
In this section, we study the stability periodic solution of the system (1.2) by the following
theorem
Theorem4.1. If the function 𝛥(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥0) is defined by
𝛥:𝐷𝛾𝑓 → 𝑅𝑛,
𝛥(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥0) =1𝑇∫ 𝑓0𝑇 (𝑠,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥0(𝑡𝑡,𝛾𝛾(𝑠,𝛼𝛼,𝛽,µ),𝑥𝑥0)𝑑𝑠…… ⋯(3.8)
where𝑥𝑥0(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼),𝑥𝑥0) is a limit of the sequence of functions (2.1). Then the following
inequalities are holds
|𝛥(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥0)|≤ 𝑀𝑀𝛾 ⋯(3.8)
and
|𝛥(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥01)− 𝛥(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥02)|≤𝑇 𝛬2 (𝐸 − 𝛬)−1𝑀𝛾 ⋯(3.9)
for all 𝑥𝑥0,𝑥𝑥01, 𝑥𝑥02 ∈ 𝐷𝛾𝑓 .
Proof. From the properties function 𝑥𝑥0(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ,𝑥𝑥0) established theorem 2.1; it follows
that function 𝛥(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼),𝑥𝑥0) is continuous and bounded by 𝑀𝑀𝛼.
By using (3.7), we get:
|𝛥(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥01)− 𝛥(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥02)|
=�𝑇 � 𝑓1 (𝑠,𝛾𝛾(𝑠,𝛼𝛼,𝛽,µ),𝑥𝑥0(𝑠,𝛾𝛾(𝑠,𝛼𝛼,𝛽,µ),𝑥𝑥 01)𝑑𝑠 𝑇
0
−1
𝑇 �[𝑓(𝑠,𝛾𝛾(𝑠,𝛼𝛼,𝛽,µ),𝑥𝑥0(𝑡𝑡,𝛾𝛾(𝑠,𝛼𝛼,𝛽,µ),𝑥𝑥02)𝑑𝑠]𝑑𝑠 𝑇
0
≤𝐾𝑇 � 𝑀𝛾 𝑇
0
|𝑥𝑥0(𝑠,𝛾𝛾(𝑠,𝛼𝛼,𝛽,µ),𝑥𝑥
01)− 𝑥𝑥0(𝑠,𝛾𝛾(𝑠,𝛼𝛼,𝛽,µ),𝑥𝑥02)|𝑑𝑠
≤ 𝑀𝛾𝐾𝑇2 .𝑇2|𝑥𝑥0(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥01)− 𝑥𝑥0(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥02)|
=𝑇 𝛬2 |𝑥𝑥0(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥
01)− 𝑥𝑥0(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥02)|
and hence
|𝛥(0,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥01)− 𝛥(0,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥02)|
≤ 2𝑇 𝛬 |𝑥𝑥0(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥
01)− 𝑥𝑥0(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥02)|𝑀𝛾 ⋯(3.10)
where 𝑥𝑥01(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ,𝑥𝑥0) and 𝑥𝑥02(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ,𝑥𝑥0) are the solution of the integral equation:
𝑥𝑥(𝑡𝑡,𝛾𝛾(𝑡𝑡,𝛼𝛼,𝛽,µ),𝑥𝑥0𝑘) =𝑥𝑥0𝑘+� [𝑓(𝑠,𝛾𝛾(𝑠,𝛼𝛼,𝛽,µ),𝑥𝑥(𝑠,𝛾𝛾(𝑠,𝛼𝛼,𝛽,µ),𝑥𝑥0𝑘) 𝑡
0
−1𝑇 � (𝑓(𝑠,𝛾𝛾(𝑠,𝛼𝛼,𝛽,µ),𝑥𝑥(𝑠,𝛾𝛾(𝑠,𝛼𝛼,𝛽,µ),𝑥𝑥0𝑘)𝑑𝑠]𝑑𝑠 𝑇
0 ⋯(3.11)
with
x0k(t,γ(t,α,β,µ), x0) = x0k , k = 1,2.
From (3.11), we have:
|x0(t,γ(t,α,β,µ), x
01)−x0(t,γ(t,α,β,µ), x02)|≤ |x01−x02|
+K[�1−Tt� �Mγ t
0
|x0(s,γ(s,α,β,µ), x
01)−x0(s,γ(s,α,β,µ), x02)|ds
+Tt �Mγ T
t
|x0(s,γ(s,α,β,µ), x
01)−x0(s,γ(s,α,β,µ), x02)|ds]
≤|x01−x02| + MγKT2|x0(t,γ(t,α,β,µ), x01)−x0(t,γ(t,α,β,µ), x02)|α(t)
≤|x01−x02| +Λ|x0(t,γ(t,α,β,µ), x01)−x0(t,γ(t,α,β,µ), x02)|
Thus
|x0(t,γ(t,α,β,µ), x
01)−x0(t,γ(t,α,β,µ), x02)|≤ (E− Λ)−1|x01−x02|⋯(3.12)
Using the inequality (3.12) in (3.10), we get (3.9).
Remark 4.1[8]. The theorem 4.1 ensure the stability solution of the system (1.1) that is
when there is a slight change happen in the point x0,then a slight change will happen in the
function Δ(t,γ(t,α,β,µ), x0) .
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