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In this paper, we obtained some sufficient conditions for the oscillation of all solutions of the second order neutral differential equation of the form where , and . Examples are provided to illustrate the main results.

In this paper, we are concerned with the oscillatory behavior of solutions of the second order nonlinear neutral differential equation of the form

where

(C_{1})

(C_{2})

(C_{3})

(C_{4})

By a solution of Equation (1), we mean a continuous function x defined on an interval

tions satisfying condition

tions. As usual, a solution of Equation (1) is called oscillatory if it is neither eventually positive nor eventually negative; otherwise we call it nonosicllatory.

From the literature, it is known that second order neutral functional differential equations have applications in problems dealing with vibrating masses attached to an elastic bar and in some variational problems. For further applications and questions regarding existence and uniqueness of solutions of neutral functional differential equations, see [

In recent years, there has been an increasing interest in establishing conditions for the oscillation or nonoscilla- tion of solution of neutral functional differential equations, see [

In [

Ye and Xu [

In [

In [

Equation (1) when

Equation (1). In Section 2, we use Riccati transformation technique to obtain some sufficient conditions for the oscillation of all solutions of Equation (1). Examples are provided in Section 3 to illustrate the main results.

In this section, we obtain some new oscillation criteria for the Equation (1). We begin with the following theorem.

Theorem 2.1 If

and

where

is oscillatory.

Proof. Suppose that

First assume that

or

Integrating (4) from

a contradiction to (2.1).

If

Clearly

Dividing the last inequality by

Letting

Therefore,

From (5), we have

Next, we introduce another function

Clearly

Similarly, we introduce another function

Clearly

Dividing the last inequality by

Letting

Differentiating (5), we obtain

Differentiating (8), we have

Differentiating (10), we have

Inview of (12), (13) and (14), we can obtain

From (4) and (15), we obtain

Multiplying (16) by

From the above inequality, we obtain

Thus, it follows that

By (7), (9) and (11), we obtain that

which contradicts (3). The proof is now complete.

Corollary 2.1. Assume that

Proof. The proof follows from Theorem 2.1.

Theorem 2.2. Assume that

then every solution of Equation (1) is oscillatory.

Proof. Let

It follows from (C_{2}) and (7) that

Inview of (9), we have

From (11), we obtain

Therefore from (18), we obtain

which is a contradiction with (17). The proof is now complete.

Corollary 2.2. Assume that

Proof. The proof follows from Theorem 2.2.

To prove our next theorem, we need a class of function

Following [

respect to

Define the operator

for

then, it is easy to see that

Theorem 2.3. Assume that

and

where

Proof. Let

First assume that

Then

Since

Then

Since

Then

Since

Now applying the operator

From the last inequality, we obtain

or

Taking the sup limit in the last inequality, we obtain a contradiction with (22).

Next consider the case

From the last inequality, we obtain

or

Taking the sup limit in the last inequality, we obtain a contradiction with (23). The proof is now completed.

Remark 2.1. With different choices of functions

For example, if we take

From Theorem 2.3, we obtain the following oscillation criteria for Equation (1).

Corollary 2.3. Assume that

and

where

In this section, we provide three examples to illustrate the main results.

Example 3.1. Consider the neutral differential equation

Here

is easy to see that all conditions of Theorem 2.1 are satisfied and hence every solution of Equation (31) is oscillatory.

Example 3.2. Consider the neutral differential equation

Here

We conclude this paper with the following remark.

Remark 3.1. The results presented in [

equations

plement and generalize some of the known results in the literature.

RamalingamArul,Venkatachalam SubramaniyamShobha, (2015) Oscillation of Second Order Nonlinear Neutral Differential Equations with Mixed Neutral Term. Journal of Applied Mathematics and Physics,03,1080-1089. doi: 10.4236/jamp.2015.39134