1. Introduction
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
-stability were introduced for ordinary differential equations by Lakshmikantham in 1969 [1] and by Akpan in 1992 [2] respectively. Later, these stabilities were developed in [3] and [4] by Akpan, Soliman and Abdalla but for ordinary differential equations. In 2010, Integral stability was established for impulsive functional differential equations by Hristova. Motivated by these works, in this paper, we introduce and establish integral
-stability for impulsive ordinary differential equations:
(1)
where,
,
, ![](//html.scirp.org/file/10-2310492x15.png)
,
,
and
are a sequence of instantaneous impulse operators and have been used to depict abrupt changes such as shocks, harvesting, natural disasters etc. and K is a cone defined in Section 2.
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
-stability are obtained by employing comparison principle and piecewise continuous cone valued Lyapunov functions.
2. Preliminaries
Let
denote the n-dimensional Euclidean space with any convenient norm
and the scalar product
,
,
,
.
For any
,
, we will write
iff
for all ![]()
Let
be the solution of system (1), having discontinuities of the first type (left continuous) at the moments when they meet the hyper planes
.
Together with system (1), let us consider, its perturbed IDS:
(2)
where,
,
.
Let
,
so that the trivial solution of (1) and (2) exists.
Let us define the following:
Definition 1. A proper subset K of
is called a cone if (i)
(ii)
(iii)
(iv)
(v)
, where
and
are interior and closure of
respectively.
denotes the boundary of
.
Definition 2. The set
is called the adjoint cone if it satisfies the properties (i)-(v) of definition 1.
The set
iff
for some
.
Definition 3. A function
is said to be quasi monotone relative to the cone
if for each
and
imply that there exists
such that
and
.
Consider the following sets:
![]()
![]()
![]()
![]()
![]()
.
Definition 4. A function
is said to belong to class
if:
1.
is a continuous function in
;
2.
is Lipschitz continuous relative to cone K, in its second argument;
3. For each
,
and
exist.
And for
we define derivative of the function
along the trajectory of the system (1) by
.
Now referring [5] , let us define the following:
Definition 5. Let
The function
is said to be
-weakly decrescent, if there exists a
and a function
such that the inequality
implies that
.
Definition 6. Let
The function
is said to be
-strongly decrescent, if there exists a
and a function
such that the inequality
implies that
.
Throughout in the paper it was assumed that
.
Let us consider the following comparison impulsive differential systems (referring [3] for Ordinary differential systems)
(3)
and
(4)
along with its perturbed system
(5)
where
is quasi monotone non decreasing in its second argument and
is quasi monotone non decreasing satisfying
, ![]()
,
,
,
,
,
and
are to be chosen later such that
.
Definition 7. The zero solution of (1) is said to be
-stable, if for every
and for any
there exists a positive function
, which is continuous in
for each
such that the inequality
implies that
,
where
and
is the maximal solution of (1) relative to the cone k.
Definition 8. The zero solution of (1) is said to be integrally stable, if for every
and for any
there exists a positive function
, which is continuous in
for each
such that for any solution
of perturbed system (2) , the inequality
holds provided that
and for every
, the perturbations
and
of RHS of (2) satisfy
.
Definition 9. The trivial solution of (1) is said to be integrally
-stable, if for every
and for any
there exists a positive function
, which is continuous in
for each
such that for any solution
of perturbed system (2) and for
, the inequality
holds provided that
(6)
and, for every
, the perturbations
and
of RHS of (2) satisfy
. (7)
3. Main Results
Lemma 1: Consider the comparison system (3) and assume that
(i)
where
is quasi monotone non decreasing in its second argument;
(ii)
such that
and satisfies
![]()
(iii)
such that
for ![]()
Let
be the maximal solution of (3) existing on J. Then for any solution
of (1) existing on J, we have
provided that
.
Proof: Let
be the solution of (1) existing for
such that
.
Define
for
such that
. Then for small
, we have
![]()
,
where M is the Lipschitz constant in
.
Therefore we have
![]()
Also
and
.
Then by theorem (1.4.3) in [6] , we observe the desired inequality
for all.
Theorem 1: Let us assume the following:
1. Let
and ![]()
2. There exist
such that
(i)
is
-weakly decrescent
(ii) For
the inequality
holds for all ![]()
where
monotone non decreasing in its second argument
(iii)
for all
where
is monotone non decreasing, satisfying ![]()
3. For any number
there exists
such that
(iv)
for
where ![]()
(v) For
the inequality
![]()
holds for any
![]()
where
is monotone non decreasing in its second argument.
(vi)
for
![]()
where
, ![]()
4. The system (3) and (4) have solutions, for any initial point
.
5. For any initial point
, the system (1) has solution.
Let the zero solution of (3) be
-stable, and scalar IDE (4) is integrally
-stable, then the system (1) will be integrally
-stable.
Proof: Since
is
-weakly decrescent, therefore there exists a
and a function
such that the inequality
implies that
(8)
where
.
Let
be a fixed time. Choose a number
such that
.
As
, there exist Lipschitz constants
and
of
and
respectively. Let
.
As the zero solution of (3) is
-stable, therefore for every
and for any
there exists a positive function
for each
such that the inequality
implies that
, (9)
where
is the maximal solution of (3)
As
, there exists
and hence
such that
. (10)
Again in view of the fact that the perturbations in (5), depend only on t and system (4) is
-integrally stable, there exists a function
, continuous in
for each
(take in particular
) such that for every solution
of perturbed system (5), the inequality
(11)
holds provided that
and for every
, the perturbation terms
and
satisfy
. (12)
Since
,
let us choose
such that
and
where
is a function satisfying
.
Select
,
such that the inequalities
and
hold (13)
Let
be the solution of (2). Now we will prove that if the inequalities (6) and (7) are satisfied then
(14)
If possible let this be false. Therefore there exists a point
such that
(15)
Case 1: Let
for any
. Then the solution
is continuous at
. Therefore ![]()
In this case first we note that
.
For if
, then by the choice of
we get
which is a contradiction to (15).
Now let us consider the interval ![]()
Subcase 1.1: Let there exists
such that
and
![]()
If
is the maximal solution of (3) with
, then in view of the assumptions (ii) and (iii) of theorem, using lemma 1, we obtain
(16)
where
is a solution of (1), starting at
.
As
is chosen therefore we have
and
by using (8) and then (10), we get
![]()
Now
by virtue of (9) gives:
(17)
Now from inequality (13) and condition (iv) of theorem, we get
(18)
Let us define the function
,
by ![]()
Now, for
and
,
, in view of (v) of theorem and lipschitz condition on
and
, we have
(19)
Again for
such that
, by using condition (vi) of theorem and Lipschitz conditions on
and
, we get
(20)
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
, through the point
where
, using (19), (20) and lemma 1, we get
![]()
where H is the interval of existence of maximal solution
(21)
Now by using the inequality (7) for
in the interval
and from the choice of
,
(22)
Let us choose a point
such that
.
Now let us define a continuous function
given by
![]()
and the sequence of numbers.
![]()
We see that if (7) holds then from (22), for every ![]()
(23)
let
be the maximal solution of (5), through the point
where the perturbations terms are defined by
and
. Note that here we have
.
From inequalities (17) and (18) we see,
i.e.
(24)
and hence from (11), we get
for. (25)
Now from the choice of
, inequalities (21), (25) and condition (iv) of statement of theorem, we get
![]()
which yields
, a contradiction and therefore the inequality (14) is valid for
.
Subcase 1.2: Let there exist a point
for some
such that
and
.
Choose
satisfying
with
Now if we take
in place of
and repeat the proof of subcase 1.1 we arrive at contradiction that assures the validity of (14).
Case 2: If
for some
then from (15),
and
.
Let us select
such that ![]()
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
-stable.
4. Conclusion
Results in [1] [4] [7] have been exploited and extended to establish the new type of stability i.e. integral
-stability for the impulsive differential systems. Sufficient conditions are obtained by employing comparison principle and piecewise continuous cone valued Lyapunov functions.