^{1}

^{*}

^{2}

In this paper, the notions of integral
Φ_{0}-stability of ordinary impulsive differential equations are introduced. The definition of integral
Φ_{0}-stability depends significantly on the fixed time impulses. Sufficient conditions for integral
Φ_{0}-stability are obtained by using comparison principle and piecewise continuous cone valued Lyapunov functions. A new comparison lemma, connecting the solutions of given impulsive differential system to the solution of a vector valued impulsive differential system is also established.

Impulsive differential equations have been developed in modeling impulsive problems in physics, population dynamics, ecology, biological systems, industrial robotics, optimal control, bio-technology and so forth. In view of the vast applications, the fundamental and qualitative properties i.e. stability, boundedness etc. of such equations are studied extensively in past decades. Several types of stability have been defined and established in literature by academicians for impulsive ordinary differential equations. Various techniques such as scalar valued piecewise continuous Lyapunov functions, vector valued piecewise continuous Lyapunov functions, Rajumikhin method, comparison principle etc. have been employed to establish stability results.

To the best of our knowledge, the concept of integral stability and

where,

The paper is organized as follows:

In Section 2, some preliminaries notes and definitions are given. In Section 3, a new comparison lemma, connecting the solutions of given impulsive ordinary differential system to the solution of a vector valued impulsive differential system is worked out. This lemma plays an important role in establishing the main results of the paper. Sufficient conditions for integral

Let

For any

Let

Together with system (1), let us consider, its perturbed IDS:

where,

Let

Let us define the following:

Definition 1. A proper subset K of

Definition 2. The set

The set

Definition 3. A function

Consider the following sets:

Definition 4. A function

1.

2.

3. For each

And for

Now referring [

Definition 5. Let

Definition 6. Let

Throughout in the paper it was assumed that

Let us consider the following comparison impulsive differential systems (referring [

and

along with its perturbed system

where

Definition 7. The zero solution of (1) is said to be

Definition 8. The zero solution of (1) is said to be integrally stable, if for every

Definition 9. The trivial solution of (1) is said to be integrally

and, for every

Lemma 1: Consider the comparison system (3) and assume that

(i)

(ii)

(iii)

Let

Proof: Let

Define

where M is the Lipschitz constant in

Therefore we have

Also

Then by theorem (1.4.3) in [

Theorem 1: Let us assume the following:

1. Let

2. There exist

(i)

(ii) For

where

(iii)

3. For any number

(iv)

(v) For

holds for any

where

(vi)

where

4. The system (3) and (4) have solutions, for any initial point

5. For any initial point

Let the zero solution of (3) be

Proof: Since

where

Let

As

As the zero solution of (3) is

where

As

Again in view of the fact that the perturbations in (5), depend only on t and system (4) is

holds provided that

Since

Select

Let

If possible let this be false. Therefore there exists a point

Case 1: Let

In this case first we note that

For if

Now let us consider the interval

Subcase 1.1: Let there exists

If

where

As

Now

Now from inequality (13) and condition (iv) of theorem, we get

Let us define the function

Now, for

Again for

For the impulsive differential system (5) which is the perturbed system of (4), set the perturbations on RHS of (5) as

Therefore (19) and (20) can be written as

and

If we consider the comparison system (5) with maximal solution

where H is the interval of existence of maximal solution

Now by using the inequality (7) for

Let us choose a point

Now let us define a continuous function

and the sequence of numbers.

We see that if (7) holds then from (22), for every

let

From inequalities (17) and (18) we see,

and hence from (11), we get

Now from the choice of

which yields

Subcase 1.2: Let there exist a point

Choose

Case 2: If

Let us select

Now by adopting the procedure as in case 1, we get the inequalities (21) and (25). Then by using these inequalities along with the conditions (iv) and (vi) of the statement of theorem, we have

and that again is a contradiction .Therefore inequality (14) is valid.

Thus in all the cases, validity of (14) proves that system (1) is integrally

Results in [

AnjuSood,Sanjay K.Srivastava, (2015) Integral Φ_{0}-Stability of Impulsive Differential Equations. Open Journal of Applied Sciences,05,651-660. doi: 10.4236/ojapps.2015.510064