On Stability of Nonlinear Differential System via Cone-Perturbing Liapunov Function Method

Totally equistable, totally φ0-equistable, practically equistable, and practically φ0-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.


Introduction
Consider the non linear system of ordinary differential equations (1.1) and the perturbed system (1.2) Let be Euclidean n -dimensional real space with any convenient norm , and scalar product . Let  The following definitions [1] will be needed in the sequal .

Definition 1.1
A proper subset is called a cone if where denotes the closure and interior of K respectively and denote 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

Definition 1.4
A function is said to belong to the class

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.

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

Totally equistable.
In this section we discuss the concept of Totally equistable of the zero solution of (1.1) using cone valued perturbing Liapunov functions method and Comparison principle method.
The following definition [3] will be needed in the sequal.

Definition 3.1
The zero solution of the system (1. Therefore the zero solution of (1.1) is totally equistable.

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 [5] 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 . where are increasing functions.
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 ( This is a contradiction ,then provided that . Therefore the zero solution of (1.1) is practically equistable.

practically
equistable In this section we discuss the concept of practically 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 . From the condition we obtain the differential inequality Applying the comparison Theorem of [7] , yields To prove that It must be show that

Choose
From the condition and applying the comparison Theorem [7 ] it yields From (5.3) at (5.8) From the condition 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 ,(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 equistable.