ABSTRACT

In this paper, we will establish some oscillation criteria for the higher order linear dynamic equation on time scale in term of the coefficients and the graininess function. We illustrate our results with an example.

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.

REFERENCES

NOTES

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