On Stability of Nonlinear Differential System via Cone-Perturbing Liapunov Function Method ()
1. Introduction
Consider the non linear system of ordinary differential equations
(1.1)
and the perturbed system
(1.2)
Let Rn be Euclidean n-dimensional real space with any convenient norm
, and scalar product
. Let for some ![](//html.scirp.org/file/10-7402854x9.png)
![](//html.scirp.org/file/10-7402854x10.png)
where
denotes the space of continuous mappings
into
.
Consider the scalar differential equations with an initial condition
, (1.3)
(1.4)
and the perturbing equations
(1.5)
(1.6)
where
,
respectively.
Other mathematicians have been interested in properties of qualitative theory of nonlinear systems of differential equations. In last decade, in [1] , some different concepts of stability of system of ordinary differential Equations (1.1) are considered namely, say totally stability, practically stability of (1.1), and (1.2); and in [2] , methods of perturbing Liapunov function are used to discuss stability of (1.1). The authors in [3] discussed some stability of system of ordinary differential equations, and in [4] [5] the authors discussed totally and totally φ0-stability of system of ordinary differential Equations (1.1) using Liapunov function method that was played essential role for determine stability of system of differential equations. In [6] the authors discussed practically stability for system of functional differential equations.
In [7] , and [8] , the authors discussed new concept namely, φ0-equitable of the zero solution of system of ordinary differential equations using cone-valued Liapunov function method. In [4] , the author discussed and improved some concepts stability and discussed concept mix between totally stability from one side and φ0- stability on the other side.
In this paper, we will discuss and improve the concept of totally stability, practically stability of the system of ordinary differential Equations (1.1) with Liapunov function method, and comparison technique. Furthermore, we will discuss and improve the concept of totally φ0-stability, and practically φ0-stability of the system of ordinary differential Equations (1.1). These concepts are mix and lie somewhere between totally stability and practically stability from one side and φ0-stability on the other side. Our technique depends on cone-valued Liapunov function method, and comparison technique. Also we give some results of these concepts of the zero solution of differential equations.
The following definitions [8] will be needed in the sequal.
Definition 1.1. A proper subset
of
is called a cone if
![]()
where
and
denote the closure and interior of K respectively and
denotes the boundary of ![]()
Definition 1.2. The set
is called the adjoint cone if it satisfies the properties of the definition 3.1.
![]()
Definition 1.3. A function
is called quasimonotone relative to the cone K if
![]()
then there exists
such that
![]()
Definition 1.4. A function
is said to belong to the class
if
and
is strictly monotone increasing in r.
2. Totally Equistable
In this section we discuss the concept of totally equistable of the zero solution of (1.1) using perturbing Liapuniv functions method and Comparison principle method.
We define for
, the function
by
![]()
The following definition [1] will be needed in the sequal.
Definition 2.1. The zero solution of the system (1.1) is said to be
-totally equistable (stable with respect to permanent perturbations), if for every
there exist two positive numbers
and
such that for every solution of perturbed Equation (1.2), the inequality
![]()
holds, provided that
and
.
Definition 2.2. The zero solution of the Equation (1.3) is said to be
-totally equistable (stable with respect to permanent perturbations), if for every
, there exist two positive numbers
and
such that for every solution of perturbed Equation (1.5). The inequality
![]()
holds, provided that
and
.
Theorem 2.1. Suppose that there exist two functions
with![]()
and there exist two Liapunov functions
and
with![]()
where
for
and
denotes the complement of
satisfying the following con- ditions:
(H1)
is locally Lipschitzian in x.
![]()
(H2)
is locally Lipschitzian in x.
![]()
where
are increasing functions.
(H3)
![]()
(H4) If the zero solution of (1.3) is equistable, and the zero solution of (1.4) is totally equistable.
Then the zero solution of (1.1) is totally equistable.
Proof. Since the zero solution of the system (1.4) is totally equistable, given
, there exist two positive numbers
and
such that for every solution
of perturbed equation (1.6) the inequality
(2.1)
holds, provided that
and
.
Since the zero solution of (1.3) is equistable given
and
, there exists
such that
(2.2)
holds, provided that ![]()
From the condition (H2) we can find
such that
(2.3)
To show that the zero solution of (1.1) is
-totally equistable, it must show that for every
there exist two positive numbers
and
such that for every solution
of perturbed Equation (1.2). The inequality
![]()
holds, provided that
and
.
Suppose that this is false, then there exists a solution
of (1.2) with
such that
(2.4)
![]()
Let
and setting ![]()
Since
and
are Lipschitzian in x for constants
and
respectively.
Then
![]()
where
From the condition (H3) we obtain the differential inequality
![]()
for
Then we have
![]()
Let ![]()
Applying the comparison Theorem (1.4.1) of [1] , it yields
![]()
where
is the maximal solution of the perturbed Equation (1.6).
Define ![]()
To prove that
![]()
It must be show that
and
.
Choose
. From the condition (H1) and applying the comparison Theorem of [1] , it yields
![]()
where
is the maximal solution of (1.3).
From (2.2) at ![]()
(2.5)
From the condition (H2) and (2.4), at ![]()
(2.6)
From (2.3), we get
![]()
Since ![]()
From (2.1), we get
(2.7)
Then from the condition (H2), (2.4) and (2.7) we get ![]()
![]()
This is a contradiction, then it must be
![]()
holds, provided that
and
.
Therefore the zero solution of (1.1) is totally equistable.
3. Totally f0-Equistable
In this section we discuss the concept of Totally f0-equistable of the zero solution of (1.1) using cone valued perturbing Liapunov functions method and Comparison principle method.
The following definition [4] will be needed in the sequal.
Definition 3.1. The zero solution of the system (1.1) is said to be totally f0-equistable (f0-equistable with respect to permanent perturbations), if for every
,
and
, there exist two positive numbers
and
such that the inequality
![]()
holds, provided that
and
where
is the maximal solution of perturbed Equation (1.2).
Let for some ![]()
![]()
Theorem 3.1. Suppose that there exist two functions
with![]()
and let there exist two cone valued Liapunov functions
and
with
where
for
and
denotes the complement of
satisfying the following conditions:
(h1)
is locally Lipschitzian in
and
![]()
(h2)
is locally Lipschitzian in
and
![]()
where
are increasing functions.
(h3) ![]()
(h4) If the zero solution of (1.3) is f0-equistable, and the zero solution of (1.4) is totally f0-equistable. Then the zero solution of (1.1) is totally f0-equistable.
Proof. Since the zero solution of (1.4) is totally f0-equistable, given, given
there exist two positive numbers
and
such that the inequality
(3.1)
holds, provided that
and
. where
is the maximal solution of perturbed Equation (1.6).
Since the zero solution of the system (1.3) is f0-equistable, given
and
there exists
such that
(3.2)
holds, provided that
where
is the maximal solution of (1.3).
From the condition (h2) we can choose
such that
(3.3)
To show that the zero solution of (1.1) is T1-totally f0-equistable, it must be prove that for every
and
there exist two positive numbers
and
such that the inequality
![]()
holds, provided that
and
where
is the maximal solution of perturbed Equation (1.2).
Suppose that is false, then there exists a solution
of (1.2) with
such that
(3.4)
![]()
Let
and setting ![]()
Since
and
are Lipschitzian in x for constants
and
respectively.
Then
![]()
where
From the condition (h3) we obtain the differential inequality
![]()
for
Then we have
![]()
Let
. Applying the comparison Theorem of [1] , yields
![]()
Define ![]()
To prove that
![]()
It must be shown that
![]()
Choose
. From the condition (h1) and applying the comparison Theorem [1] , it yields
![]()
From (3.2) at ![]()
(3.5)
From the condition (h2) and (3.4), at ![]()
(3.6)
From (3.3), we get
![]()
Since ![]()
From (3.1), we get
(3.7)
Then from the condition (h2), (3.4) and (3.7) we get at ![]()
![]()
This is a contradiction, then
![]()
provided that
and
where
is the maximal solution of perturbed equation (1.2). Therefore the zero solution of (1.1) is totally f0-equistable.
4. Practically Equistable
In this section, we discuss the concept of practically equistable of the zero solution of (1.1) using perturbing Liapunov functions method and Comparison principle method.
The following definition [8] will be needed in the sequal.
Definition 4.1. Let
be given. The system (1.1) is said to be practically equistable if for
such that the inequality
(4.1)
holds, provided that
where
is any solution of (1.1).
In case of uniformly practically equistable, the inequality (4.1) holds for any
.
We define
.
Theorem 4.1. Suppose that there exist two functions
with![]()
and there exist two Liapunov functions
and
with![]()
where
and
denotes the complement of
satisfying the following conditions:
(I)
is locally Lipschitzian in x.
![]()
(II)
is locally Lipschitzian in x.
![]()
where
are increasing functions.
(III)
![]()
(IV) If the zero solution of (1.3) is equistable, and the zero solution of (1.4) is uniformly practically equistable.
Then the zero solution of (1.1) is practically equistable.
Proof. Since the zero solution of (1.4) is uniformly practically equistable, given
such that for every solution
of (1.4) the inequality
![]()
holds provided
.
Since the zero solution of the system (1.3) is equistable, given
and
there exist ![]()
such that for every solution
of (1.3)
![]()
holds provided that
.
From the condition (II) we can find
such that
![]()
To show that the zero solution of (1.1) practically equistable, it must be exist
such that for any solution
of (1.1) the inequality
![]()
holds, provided that
.
Suppose that this is false, then there exists a solution
of (1.1) with
such that
(4.5)
![]()
Let
and setting
![]()
From the condition (III) we obtain the differential inequality for ![]()
![]()
Let ![]()
Applying the comparison Theorem [8] , yields
![]()
where
is the maximal solution of (1.4).
To prove that
![]()
It must be show that
.
Choose
, from the condition (II) and applying the comparison Theorem of [1] , yields
![]()
where
is the maximal solution of (1.3).
From (4.3) at ![]()
(4.6)
From the condition (II) and (4.5), at ![]()
(4.7)
From (4.4), (4.6) and (4.7), we get
![]()
From (4.2), we get
(4.8)
Then from the condition (II), (4.5) and (4.8), we get at ![]()
![]()
This is a contradiction, then
![]()
provided that
.
Therefore the zero solution of (1.1) is practically equistable.
5. Practically f0-Equistable
In this section we discuss the concept of practically f0-equistable of the zero solution of (1.1) using cone valued perturbing Liapunov functions method and comparison principle method.
The following definitions [6] will be needed in the sequal.
Definition 5.1. Let
be given. The system (1.1) is said to be practically f0-equistable, if for
and
such that the inequality
(5.1)
holds, provided that
where
is the maximal solution of (1.1).
In case of uniformly practically f0-equistable, the inequality (5.1) holds for any
.
We define
![]()
Theorem 5.1. Suppose that there exist two functions
with![]()
and let there exist two cone valued Liapunov functions
and
with![]()
where
and
denotes the complement of
satisfying the following conditions:
(i)
is locally Lipschitzian in x relative to K.
![]()
(ii)
is locally Lipschitzian in x relative to K.
![]()
where
are increasing functions.
(iii) ![]()
(iv) If the zero solution of (1.3) is f0-equistable, and the zero solution of (1.4) is uniformly practically f0- equistable.
Then the zero solution of (1.1) is practically f0-equistable.
Proof. Since the zero solution of the system (1.4) is uniformly practically f0-equistable, given
for
such that the inequality
(5.2)
holds provided
, where
is the maximal solution of (1.4).
Since the zero solution of the system (1.3) is f0-equistable, given
and
there exist ![]()
such that the inequality
(5.3)
From the condition (ii), assume that
(5.4)
also we can choose
such that
(5.5)
To show that the zero solution of (1.1) is practically f0-equistable. It must be show that for
and
such that the inequality
![]()
holds, provided that
where
is the maximal solution of (1.1).
Suppose that is false, then there exists a solution
of (1.1) with
such that for
where ![]()
(5.6)
![]()
Let
and setting
![]()
From the condition (iii) we obtain the differential inequality
![]()
Let ![]()
Applying the comparison Theorem of [1] , yields
![]()
To prove that
![]()
It must be show that
![]()
Choose
From the condition (i) and applying the comparison Theorem of [1] , yield
![]()
From (5.3) at ![]()
(5.8)
From the condition (ii) and (5.6), at ![]()
(5.9)
From (5.5), (5.8) and (5.9), we get
![]()
From (5.2), we get
(5.10)
Then from the condition (ii), (5.4), (5.6) and (5.10), we get at ![]()
![]()
which leads to a contradiction, then it must be
![]()
holds, provided that
Therefore the zero solution of (1.1) is practically f0-equistable.
Acknowledgements
The authors would thank referees the manuscript for a valuable corrections of it.