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Totally equistable, totally
Φ
_{0}-equistable, practically equistable, and practically
Φ
_{0}-equistable of system of differential equations are studied. Cone valued perturbing Liapunov functions method and comparison methods are used. Some results of these properties are given.

Consider the non linear system of ordinary differential equations

and the perturbed system

Let R^{n} be Euclidean n-dimensional real space with any convenient norm

where

Consider the scalar differential equations with an initial condition

and the perturbing equations

where

Other mathematicians have been interested in properties of qualitative theory of nonlinear systems of differential equations. In last decade, in [_{0}-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 [

In [_{0}-equitable of the zero solution of system of ordinary differential equations using cone-valued Liapunov function method. In [_{0}- 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 φ_{0}-stability, and practically φ_{0}-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 φ_{0}-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 [

Definition 1.1. A proper subset

where

Definition 1.2. The set

Definition 1.3. A function

then there exists

Definition 1.4. A function

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 following definition [

Definition 2.1. The zero solution of the system (1.1) is said to be

holds, provided that

Definition 2.2. The zero solution of the Equation (1.3) is said to be

holds, provided that

Theorem 2.1. Suppose that there exist two functions

and there exist two Liapunov functions

where

(H_{1})

(H_{2})

where

(H_{3})

(H_{4}) 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

holds, provided that

Since the zero solution of (1.3) is equistable given

holds, provided that

From the condition (H_{2}) we can find

To show that the zero solution of (1.1) is

holds, provided that

Suppose that this is false, then there exists a solution

Let

Since

Then

where _{3}) we obtain the differential inequality

for

Let

Applying the comparison Theorem (1.4.1) of [

where

Define

To prove that

It must be show that

Choose_{1}) and applying the comparison Theorem of [

where

From (2.2) at

From the condition (H_{2}) and (2.4), at

From (2.3), we get

Since

From (2.1), we get

Then from the condition (H_{2}), (2.4) and (2.7) we get

This is a contradiction, then it must be

holds, provided that

Therefore the zero solution of (1.1) is totally equistable.

In this section we discuss the concept of Totally f_{0}-equistable of the zero solution of (1.1) using cone valued perturbing Liapunov functions method and Comparison principle method.

The following definition [

Definition 3.1. The zero solution of the system (1.1) is said to be totally f_{0}-equistable (f_{0}-equistable with respect to permanent perturbations), if for every

holds, provided that

Let for some

Theorem 3.1. Suppose that there exist two functions

and let there exist two cone valued Liapunov functions

where

(h_{1})

(h_{2})

where

(h_{4}) If the zero solution of (1.3) is f_{0}-equistable, and the zero solution of (1.4) is totally f_{0}-equistable. Then the zero solution of (1.1) is totally f_{0}-equistable.

Proof. Since the zero solution of (1.4) is totally f_{0}-equistable, given, given

holds, provided that

Since the zero solution of the system (1.3) is f_{0}-equistable, given

holds, provided that

From the condition (h_{2}) we can choose

To show that the zero solution of (1.1) is T_{1}-totally f_{0}-equistable, it must be prove that for every

holds, provided that

Suppose that is false, then there exists a solution

Let

Since

Then

where _{3}) we obtain the differential inequality

for

Let

Define

To prove that

It must be shown that

Choose_{1}) and applying the comparison Theorem [

From (3.2) at

From the condition (h_{2}) and (3.4), at

From (3.3), we get

Since

From (3.1), we get

Then from the condition (h_{2}), (3.4) and (3.7) we get at

This is a contradiction, then

provided that _{0}-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 [

Definition 4.1. Let

holds, provided that

In case of uniformly practically equistable, the inequality (4.1) holds for any

We define

Theorem 4.1. Suppose that there exist two functions

and there exist two Liapunov functions

where

(I)

(II)

where

(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

holds provided

Since the zero solution of the system (1.3) is equistable, given

such that for every solution

holds provided that

From the condition (II) we can find

To show that the zero solution of (1.1) practically equistable, it must be exist

holds, provided that

Suppose that this is false, then there exists a solution

Let

From the condition (III) we obtain the differential inequality for

Let

Applying the comparison Theorem [

where

To prove that

It must be show that

Choose

where

From (4.3) at

From the condition (II) and (4.5), at

From (4.4), (4.6) and (4.7), we get

From (4.2), we get

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.

In this section we discuss the concept of practically f_{0}-equistable of the zero solution of (1.1) using cone valued perturbing Liapunov functions method and comparison principle method.

The following definitions [

Definition 5.1. Let _{0}-equistable, if for

holds, provided that

In case of uniformly practically f_{0}-equistable, the inequality (5.1) holds for any

We define

Theorem 5.1. Suppose that there exist two functions

and let there exist two cone valued Liapunov functions

where

(i)

(ii)

where

(iv) If the zero solution of (1.3) is f_{0}-equistable, and the zero solution of (1.4) is uniformly practically f_{0}- equistable.

Then the zero solution of (1.1) is practically f_{0}-equistable.

Proof. Since the zero solution of the system (1.4) is uniformly practically f_{0}-equistable, given

holds provided

Since the zero solution of the system (1.3) is f_{0}-equistable, given

such that the inequality

From the condition (ii), assume that

also we can choose

To show that the zero solution of (1.1) is practically f_{0}-equistable. It must be show that for

holds, provided that

Suppose that is false, then there exists a solution

Let

From the condition (iii) we obtain the differential inequality

Let

Applying the comparison Theorem of [

To prove that

It must be show that

Choose

From (5.3) at

From the condition (ii) and (5.6), at

From (5.5), (5.8) and (5.9), we get

From (5.2), we get

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 _{0}-equistable.

The authors would thank referees the manuscript for a valuable corrections of it.

A. A.Soliman,W. F.Seyam, (2015) On Stability of Nonlinear Differential System via Cone-Perturbing Liapunov Function Method. Applied Mathematics,06,1769-1780. doi: 10.4236/am.2015.610157