Oscillation of Higher Order Linear Impulsive Dynamic Equations on Time Scales ()
1. Introduction
Since Stefen Hilger formed the definition of derivatives and integrals on time scales, several authors has expounded on various aspects of the new theory, see the papers by Agarwal et al. [1] and the references cited therein.
A book on the subject of time scale, i.e., measure chain, by Bohner and Peterson [2] summarize and organizes much of time scale calculus on time scale and references given therein.
A time scale 
is an arbitrary closed subset reals, and the cases when this tie sale is equal to the reals or to the integers represent the classical theories of differential and of difference equations.
In recent years there has been much research activity concerning the oscillation and non-oscillation of solution of some differential equations on time scales,we refer the reader to the few papers [3-7].
In [4], the authors considered the second order dynamic equation
and some sufficient conditions for oscillation of all solution on unbounded time scales are given. But, the oscillation criteria are not considered the impulsive influence. It is rarely about the oscillation of higher order impulsive dynamic equations on time scales.
In this paper we shall consider the following linear higher order impulsive dynamic equation
(1)
where n is even, 
, 
is positive real-valued rd-continuous functions defined on the time scales and
(H1): 
Throughout the remainder of the paper, we assume that, for each 
the points of impulses 
are right dense (rd for short). In order to define the solutions of the problem (1), we introduce the following space
Definition 1. A function 
is said to be a solution of (1), if it satisfies
a.e. on
, and for each 
satisfies the impulsive condition 
and the initial conditions
,
.
Before doing so, let us first recall that a solution of (1) is a nontrivial real function 
satisfying Equation (1) for
. A solution 
of (1) is said to be oscillatory if it is neither eventually positive nor eventually negative, otherwise it is non-oscillatory. Equation (1) is said to be oscillatory if all its solutions are oscillatory. Our attention is restricted to those solutions of (1) which exist on some half line 
and satisfy 
for any 
2. Preliminaries
A time scale 
is an arbitrary non-empty closed subset of the real numbers
. Since we are interested in oscillatory behavior, we suppose that the time scale under consideration is not bounded above, i.e., it is a time scale interval of the form
. On any time scale we define the forward and backward jump operators by
(2)
A point t is said to be left-dense if
, right-dense if 
left scattered if
, and right-scattered if 
The graininess 
of the time scale is define by 
The set 
is derived from 
as follow: If 
has a left-scattered maximum m, then 
otherwise, 
For a function 
(the range 
of may actually be replaced by any Banach space), the (delta) derivative is defined by
(3)
A function 
is said to be re-continuous at each right-dense point and if there exists a finite left limit in all left-dense points, and f is said to be differentiable if its derivative exists, the derivative and the shift operator 
are related by the formula
(4)
We will make use of the following product and quotient rules for the derivative of the product fg and the quotient f/g of two differentiable functions f and g
(5)
(6)
The integration by parts formula reads
(7)
Remark 1. We note that if
, then
, 
, 
and (1) becomes the higher order differential equation
If 
then
, 
,
and (1) becomes the higher order difference equation
If
, 
then 
and (1) becomes the higher order difference equation
If 
then
, 
and (1) becomes the higher order difference equation
If 
then 
and 
and (1) becomes the higher order difference equation
3. Main Results
In the following, we will prove some lemmas, which will be useful for establishing oscillation criteria .
Lemma 1. Let 
and
. Then
implies, for all 
(8)
See
.
Lemma 2. Assume that 
and
(9)
then for 
(10)
Proof. Let
, use Lemma 1, we obtain
Hence (10) is true for
. Now assume that
(10) holds for 
for some integer
. Then for
, it follows from (9) and Lemma 1, we get
Using (9), we obtain from (10)
which on simplification gives the estimate (10) for
, by induction, we get (10) holds for
.
Lemma 3. Let 
be a solution of (1), and conditions (H1) are satisfied. Suppose that there exists an 
and some
, such that
, 
for
. Then, there exists some 
such that 
for
.
Proof. Without loss of generality, let
. Assume that for any
,
. By
,
, 
, we have that 
is monotonically nondecreasing on
. For
, we have
Integrating the above inequality, we have
(11)
Similar to (11),
(12)
By 
and (11), (12), we have
Applying induction we have, for any natural number m,
(13)
By condition (H1) and 
for all sufficiently large m, we have
. i.e., there exists a natural number N, when
, we have
. By
again, we have
, for
. When
, we have
, where
. The proof of Lemma 3 is completed.
Lemma 4. Let 
be a solution of (1) and conditions (H1) are satisfied. Suppose that there exists an 
and some 
such that
, 
for
. 
is not always equal to 0 in 
for
. Then we have 
for all sufficiently large t.
Proof. Without loss of generality, let
. We claim that 
for any
.
If it is not true, then there exists some 
such that
. Since
, 
is monotonically non-increasing in 
for
. And because 
is not always equal to 0 in
, there exists some 
such that 
is not always equal to 0 in
. Without loss of generality,we can assume
, that is, 
is not always equal to 0 in
.So we have
For
, we have
By induction, for
, we have 
. So we have
By Lemma 3, for all sufficiently large t, we have
. Similarly, we can conclude, by using Lemma 3 repeatedly, that for all sufficiently large t,
. This is a contradiction with
! Hence, we have 
for any
. So we have 
for all sufficiently large t. The proof of Lemma 4 is completed.
Lemma 5. Let 
be a solution of (1) and conditions (H1) are satisfied. Suppose 
and 
for
. Then there exist some 
and 
such that for
,
(14)
Proof. Let
, for 
by (1) and 
is nonnegative and is not always equal to 0 in any
,
is not always equal to 0 in
, by Lemma 4. So we have 
for all sufficiently large t. Without loss of generality, let
,
. So 
is monotonically non-decreasing in
.
If for any
, 
, then
.
If there exists some
, 
, by 
is monotonically nondecreasing and
, then 
for
. So there exists some
, when
, then one of the following statements holds:
(A1) 
(B1) 
when (A1) holds, by Lemma 3, then
, for all sufficiently large t. By Lemma 3 over and over again, at last, for all sufficiently large t, we have
When (B1) holds, by Lemma 4, then
, for all sufficiently large t. By deducing further, there exists some
, when
, then one of the following statements holds:
(A2) 
(B2) 
discuss the above over and over,eventually, there exists some 
and
, when
, we have
The proof of Lemma 5 is completed.
Remark 2. If 
is an eventually negative solution of (1),we have conclusions similar to Lemma 4 and Lemma 5.
Theorem 1. If conditions (H1) hold, and
(15)
then every solution of (1) is oscillatory.
Proof. Let 
be a non-oscillatory solution of (1). Without loss of generality, let 
By Lemma 5 and (1), there exists 
when
, we have
(16)
Let
. when
, 
is monotonically non-increasing in 
and 
is monotonically increasing in
.
By (1), we have
(17)
Integrating (17) from 
to 
we have
(18)
by the above equation and 
is monotonically increasing, we have
then
(19)
similar to (19), we have
(20)
By (19), (20) and 
being monotonically increasing,
similarly ,we have
then
By induction we have,for any natural number
,
(21)
By (15), (21) and
, for all sufficiently lager m, we have
This contradicts
, for
. Hence, every solution of (1) is oscillatory. The proof of theorem 1 is completed.
Corollary 1. Assume the conditions (H1) holds, and there exists a positive integer 
such that 
for
. If
, then every solution of (1) is oscillatory.
Proof. Without loss of generality, let
. By
, we get
, therefore
Let
, 
,we get that (15) of Theorem 1 holds. By Theorem 1, we know that every solution of (1) is oscillatory.
Corollary 2. Assume the condition (H1) holds and there exist a positive integer 
and some positive integer
, such that
, for
. Furthermore, assume that
, then every solution of (1) is oscillatory.
Proof. By
, we have
Let
, 
, we get that (15) of Theorem 1 holds. By Theorem 1, we know that every solution of (1) is oscillatory.
4. Example
Example. Consider
(22)
where n is even, 
, 
, 
, 
,
. For condition (H1)
when 
From the above, the condition (H1) holds.
Let
,
By Corollary 2, we know that every solution of (22) is oscillatory.
NOTES