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In this paper, we study oscillatory properties of solutions for the nonlinear impulsive hyperbolic equations with several delays. We establish sufficient conditions for oscillation of all solutions.

The theory of partial functional differential equations can be applied to many fields, such as biology, population growth, engineering, control theory, physics and chemistry, see the monograph [

The theory of impulsive partial differential systems makes its beginning with the paper [

To the best of our knowledge, there is little work reported on the oscillation of second order impulsive partial functional differential equation with delays. Motivated by this observation, in this paper we study the oscillation of nonlinear forced impulsive hyperbolic partial differential equation with several delays of the form

with the boundary conditions

and the initial condition

Here

Euclidean N-space

In the sequal, we assume that the following conditions are fulfilled:

(H1)

piece wise continuous in t with discontinuities of first kind only at

(H2)

(H3)

(H4)

Let us construct the sequence

By a solution of problem (1), (2) ((1),(3)) with initial condition (4), we mean that any function

1. If

2. If

3. If

4. If

or

Here the number

We introduce the notations:

The solution

This paper is organized as follows: Section 2, deals with the oscillatory properties of solutions for the problem (1) and (2). In Section 3, we discuss the oscillatory properties of solutions for the problem (1) and (3). Section 4 presents some examples to illustrate the main results.

To prove the main result, we need the following lemmas.

Lemma 2.1. Suppose that

and

Lemma 2.2. Let

are satisfies the impulsive differential inequality

where

has an eventually positive solution.

Proof. Let

For

By Green’s formula, and the boundary condition we have

where

Also from condition (H2), and Jenson’s inequality we can easily obtain

Thus,

where

For

According to

Hence, we obtain that

This completes the proof.

Lemma 2.3. Let

have no eventually positive solution, then each nonzero solution of the problem (1)-(2) is oscillatory in the domain G.

Proof. Let

From Lemma 2.2, it follows that the function

If

is a positive solution of the following impulsive hyperbolic equation

and satisfies

where

For

According to

Thus, it follows that the function

Now, if we set

Lemma 2.4. Let

has no eventually positive solution, then each nonzero solution of the problem (1), satisfying the boundary condition

is oscillatory in the domain G.

Proof. Let

From Lemma 2.2, it follows that the function

If

and satisfies

For

According to

Thus it follows that the function

Lemma 2.5. Assume that

(A1) the sequence

(A2)

(A3) for

where

Then

Proof. The proof of the lemma can be found in [

Lemma 2.6. Let

Assume that there exists

hold, then

Proof. The proof of the lemma can be found in [

We begin with the following theorem.

Theorem 2.1. If condition (14), and the following condition

hold, where

then every solution of the problem (1), (2) oscillates in G.

Proof. Let

From Lemma 2.4, we know that

For

Then we have

Substitute (16)-(18) into (11) and then we obtain,

Hence we have

or

From above inequality and condition

From (12)-(13), we obtain

and

Let

Then according to Lemma 2.5, we have

Since

Next we consider the problem (1) and (3). To prove our main result we need the following lemmas.

Lemma 3.1. Suppose that

and

Proof. The proof of the lemma can be found in [

Lemma 3.2. Let

are satisfies the impulsive differential inequality

where

has the eventually positive solution

Proof. Let

For

Lemma 3.1 and then integrating (1) with respect to x over

By Green’s formula, and the boundary condition we have

where

From condition (H2), we can easily obtain

The proof is similar to that of Lemma 2.1 and therefore the details are omitted.

Lemma 3.3. Let

have no eventually positive solution, then each nonzero solution of the problem (1), (3) is oscillatory in the domain G.

Proof. The proof is similar to Lemma 2.3, and hence the details are omitted.

Futhermore, if we set

Lemma 3.4. Let

has no eventually positive solution, then each nonzero solution of the problem (1), satisfying the boundary condition

is oscillatory in the domain G.

Proof. The proof is similar to Lemma 2.4, and hence the details are omitted.

Using the above lemmas, we prove the following oscillation result.

Theorem 3.1. If condition (14) and the following condition

hold, where

then every solution of the problem (1), (3) oscillates in G.

Proof. Let

From Lemma 3.4, we know that

For

Then we have

We substitute (29)-(31) into (25) and can obtain the following inequality,

then we have

From (26)-(27), we can obtain

It follows that

Let

Then according to Lemma 2.5, we have

Since

Theorem 3.2. If condition (14) and the following condition

hold for some

Proof. The proof is obvious and hence the details are omitted.

In this section, we present some examples to illustrate the main results.

Example 4.1. Consider the impulsive differential equation

and the boundary condition

Here

Moreover

so (14) holds. We take

thus

Hence (28) holds. Therefore all conditions of Theorem 3.1 are satisfied. Hence every solution of the problem (33), (34) oscillates in

Example 4.2. Consider the impulsive differential equation

and the boundary condition

Here

of the problem (35), (36) oscillates in

The authors thank Prof. E. Thandapani for his support to complete the paper. Also the authors express their sincere thanks to the referee for valuable suggestions.

VadivelSadhasivam,JayapalKavitha,ThangarajRaja, (2015) Forced Oscillation of Nonlinear Impulsive Hyperbolic Partial Differential Equation with Several Delays. Journal of Applied Mathematics and Physics,03,1491-1505. doi: 10.4236/jamp.2015.311175