1. Intrduction

By comparison with some first difference equations whose oscillatory characters are known and by means of a Riccati transformation technique, we obtain several new sufficient conditions for the oscillation of all solutions of the nonlinear neutral difference equation of the form

(1.1)

Where is a fixed integer, denotes the forward difference operator defined by and . Throughout this paper, we will assume the following hypotheses:

(A₁)

(A₂) are quotient of odd positive integers.

(A₃) .

(A₄)

(A₅) such that

In addition, we will make use of the following conditions:

is a nonnegative real valued function, a constant.

We set . By a solution of equation (1.1) we mean a nontrivial sequence defined on , which satisfies equation (1.1) for all . A solution of equation (1.1) is said to be oscillatory if it is neither eventually positive nor eventually negative and nonoscillatory otherwise. Equation (1.1) is called oscillatory if all its solutions are oscillatory. In recent years, there has been an increasing interest in the study of the problem of determining the oscillation and non-oscillation of solutionsof difference equations of the form (1.1) and its special cases. In[1], Graef et al. proved several theorems provided sufficient conditions foroscillation of all solutions of the third order difference equation of the form

(1.2)

depend on condition

(1.3)

In[2], via comparison with first order oscillatory difference equations, Agarwal et al. proved several theorems provided sufficient conditions for oscillation of all solutions of the third order difference equation of the form

(1.4)

depend on condition

(1.5)

In[3], by a Riccati transformation technique, Schmeidel studied the oscillatory and asymptotic behavior of solutions of the third order difference equation

(1.6)

using the condition

(1.7)

In[4], via comparison with first order oscillatory difference equations, Grace et al. discussed the oscillatory behavior of the solutions of the difference equation of the form

(1.8)

under the condition

(1.9)

In[5], by a Riccatitransformation technique, Selvaraj et al. established some sufficient conditionsforoscillation of all solutions of the third order non-linear difference equation of the form

(1.10)

using the condition

(1.11)

In[6], Saker et al. studied the oscillatory behavior of solutions of the equation

(1.12)

using the condition (1.7) and

(1.13)

In[7], Thandapani et al. considered the third order difference equation of the form

(1.14)

under the condition

(1.15)

In[8], using condition (1.11) Selvaraj et al. considered nonlinear third-order difference equations of the form

(1.16)

and they study the oscillatory behavior of solutions of equation (1.16). In[9], Saker investigated the third-order difference equation (1.12) using the condition (1.13) and the authorobtained some Hille and Nehari type criteria for the oscillation of equation (1.12). For further results concerning the oscillatory and asymptotic behavior of thirdorder difference equation we refer to the books[10, 11, 12] and the references cited therein. Our results improve and extend some known results in the literature. The paper is organized as follows. In Section 2, we will state and prove the main oscillation theorems, in Section 3, and Section 4, we will give some remarks and provide some examples to illustrate the main results.

2. Main Results

In this section, we establish some new oscillation criteria for the equation (1.1) under the following conditions

(2.1)

(2.2)

(2.3)

In the following results, we shall use the following notations:

We assume that there exists a double sequence and such that

We begin with some useful lemmas, which will be used in obtaining our main results. The proof of the following Lemmas are similar to that of (Lemma 2.1, 2.2 and 2.3 respectively in[13]) and hence the details are omitted.

Lemma 2.1. Let be an eventually positive solution of the equation (1.1) and suppose that

satisfies

Then there exists such that

(2.4)

Lemma 2.2. Assume that (2.1) holds. Let be an eventually positive solution of equation (1.1). Then for sufficiently large , there are only two possible cases:

Lemma 2.3. Assume that (2.1) holds. Let be an eventually positive solution of the equation (1.1) and suppose that Case (II) of Lemma 2.2 holds. If

(2.5)

then .

Theorem 2.1. Let (2.1) and (2.5) hold. If the first order delay equation

(2.6)

is oscillatory then equation (1.1) is oscillatory or .

Proof. To the contrary assume that (1.1) has a nonoscillatory solution. Then, without loss of generality, there is a such that . From the proof of Lemma 2.2 there are two possible cases. Assume that (I) holds. From Lemma (2.1), we have

where . Summing the above inequality from , we obtain

(2.7)

There exists a with , such that

(2.8)

Since and , then

(2.9)

From (2.8), we obtain

From equation (1.1), and the last inequality , we obtain,

Summing the last inequality from to , we get

The sequence is obviously strictly decreasing. Hence, by the discrete analog of Theorem 1 in[14] we conclude that there exists a positive solution of equation (2.6) which tends to zero. This contradicts that (2.6) is oscillatory. If the (II) holds, we are then back to the proof of Lemma 2.3 to show that . The proof is complete.

Theorem 2.2. Assume that (2.2), (2.5) and (2.6) hold. If

(2.10)

then every solution of equation (1.1) oscillatory or

Proof. Let is eventually positive solution of (1.1) for . Say . Based on condition (2.2), there exist three possible cases (I), (II) (as those of Lemma 2.2), and

Assume that (I) holds. Then we are back to the proof of Theorem 2.1 to get contradiction to (2.6). Assume that (II) holds. Then we are back to the proof of Lemma 2.3 to show that . Assume that (III) holds. Then, we have

There exists a with , such that

From equation (1.1), (2.9), and the last inequality, we obtain,

(2.11)

where . It is clear that and . It follows that

Summing the last inequality from n to , we obtain

where . There exists a with , such that

Summing (2.11) from to and using the above inequality, we find

In view of , we see that

where Summing the above inequality from , we obtain

which contradicts the condition (2.10). The proof is complete.

Theorem 2.3. Assume that (2.3), (2.5), (2.6) and (2.10) hold. If

(2.12)

then every solution of equation (1.1) oscillatory or

Proof. Let is eventually positive solution of (1.1) for . Say . By (2.3), there exist four possible cases: (I), (II), (III) (as those of Theorem 2.2) and

Assume that (I) holds. Then we are back to the proof of Theorem 2.1 to get contradiction to (2.6). Assume that (II) holds. Then we are back to the proof of Lemma 2.3 to show that . Assume that (III) holds. Then we are back to the proof of Theorem 2.2 to get contradiction to (2.10). Assume that (IV) holds. We one can choose with , such that

where . Thus equation (1.1), (2.9) and yield

where . Summing the above inequality from , we find

Again summing the above inequality from , we find

where . Finally, summing the above inequality from we have

.

From condition (2.12), we get then, , which contradicts the fact that is a positive solution of (1.1). The proof is complete.

Theorem 2.4. Let , (2.1) and (2.5) hold. Further, assume that thereexists a positive nondecreasing sequence , such that

(2.13)

Then every solution of equation (1.1) oscillatory or

Proof. To the contrary assume that (1.1) has a nonoscillatory solution. Then, without loss of generality, there is a such that From the proof of Lemma 2.2 there are two possible cases. Assume that (I) holds. Define the sequence by

(2.14)

Then . From (2.14), we have

Now, by using the inequality

(2.15)

then, we have

Thus

(2.16)

From , there exists with for all such that

(2.17)

Since we get

(2.18)

From (2.16), (2.17) and (2.18), we obtain

From (2.9), we obtain

Which yields after summing by parts

Using the inequality , we have

(2.19)

Then,

Hence,

Hence,

which is contrary to (2.13). Assume that (II) holds. Then we are back to the proof of Lemma 2.3 to show that . This completes the proof of Theorem 2.4.

Corollary 2.1. Assume that all the assumptions of Theorem 2.4 hold, except the condition (2.13) is replaced by

Then equation(1.1) is oscillatory or

When , we obtain the following result

Corollary 2.2. Assume that all the assumptions of Theorem 2.4 hold, except the condition (2.13) is replaced by

(2.20)

Then equation (1.1) is oscillatory or

Theorem 2.5. Assume that , (2.2), (2.5), and (2.13) hold. If

(2.21)

then every solution of equation (1.1) oscillatory or .

Proof. Let is eventually positive solution of (1.1) for . Say Based on condition (2.2), there exist three possible cases. Assume that (I) holds. Then we are back to the proof of Theorem 2.1 to get contradiction to (2.13). Assume that (II) holds. Then we are back to the proof of Lemma 2.3 to show that . Assume that (III) holds. Then, there exists such that . Then, we have

Summing the above inequality from , we obtain

(2.22)

Hence there exists a such that

From equation (1.1), , (2.9) and the last inequality, we obtain

(2.23)

where . It is clear that and . It follows that

(2.24)

Since as we can choose such that and thus

Thus

Substituting back in (2.24), we have

(2.25)

where . Summing this inequality from , we see that

where . Summing again from , we have

or equivalently

Summing from we have

By condition (2.21), we have then, as which contradicts the fact that . The proof is complete.

Theorem 2.6. Assume that , (2.3), (2.5), (2.13), and (2.21) hold. If

(2.26)

then every solution of equation (1.1) oscillatory or .

Proof. Let is eventually positive solution of (1.1) for . Based on condition (2.3), there exist four possible cases. Assume that (I) holds. Then we are back to the proof of Theorem 2.4 to get contradiction to (2.13). Assume that (II) holds. Then we are back to the proof of Lemma 2.3 to show that . Assume that (III) holds. Then we are back to the proof of Theorem 2.5 to get contradiction to (2.21). Assume that (IV) holds. Since is non-increasing sequence there exists a negative constant such that

Dividing by and summing the last inequality from we obtain

Summing the last inequality from we obtain

From condition (2.26), we get . Since then, as , which contradicts the fact that is a positive solution of (1.1). The proof is complete.

Theorem 2.7. Let , (2.1) and (2.5) hold. Further, assume that thereexists a positive nondecreasing sequence , such that

(2.27)

Then every solution of equation (1.1) oscillatory or .

Proof. To the contrary assume that (1.1) has a nonoscillatory solution. Then, without loss of generality, there is a such that From the proof of Lemma 2.2 there are two possible cases. Assume that (I) holds. Define the sequence by

(2.28)

Then . From (2.28) and we have

(2.29)

From Lemma 2.1, there exists with for all such that

(2.30)

From (2.18) and (2.29), we have

(2.31)

From (1.1), (2.28), and (2.31), we have

(2.32)

Therefore, we have

which yields after summing by parts

Using the inequality we have

Then,

Hence,

Hence,

which is contrary to (2.27). Assume that (II)holds. Then we are back to the proof of Lemma 2.3 to show that . This completes the proof of Theorem 2.7.

From Theorem 2.7, if we get the following result

Corollary 2.3. Assume that all the assumptions of Theorem 2.7 hold, except the condition (2.27) is replaced by

Then equation(1.1) is oscillatory or .

Theorem 2.8. Assume that (2.2), (2.5), (2.27) and (2.10) hold. Then every solution of equation (1.1) oscillatory or .

Proof. The proof is similar to that of Theorem 2.2, Theorem 2.7 and hence the details are omitted.

Theorem 2.9. Assume that (2.3), (2.5), (2.27), (2.10) and (2.12) hold. Then every solution of equation (1.1) oscillatory or

Proof. The proof is similar to that of Theorem 2.3, Theorem 2.7 and hence the details are omitted.

3. Conclusions

In this paper, we established some new sufficient conditions which insure that every solution of this equation either oscillates or converges to zero. Our results improved and expanded some known results, see e.g. the following results :

Remark 3.1. If Then Theorem 2.3 reduced to a special case Theorem 2.1 in[4].

Remark 3.2. If Then Theorem 2.4 extended and improved Theorem 6 in[6].

Remark 3.3. . Then Theorem 2.4 reduced to a special case of Theorem 1 in[5].

Remark 3.4. If Then Theorem 2.4 extended and improved Theorem 3 in[3].

Remark 3.5. If Then Theorem 2.4 reduced to a special case of Theorem 1 in[1].

Remark 3.6. If Then Theorem 2.4 extended and improved Theorem 2.7 and Theorem 2.8 in[7].

Remark 3.7. If Then Corollary 2.2 reduced to a special case of Theorem 2 in[1].

Remark 3.8. If . Then Theorem 2.7 extended and improved Theorem 2.5 and Theorem 2.6 in[7].

Remark 3.9. If . Then Theorem 2.9 extended and improved Theorem 15 in[6].

Remark 3.10. If . Then we extended and improved Theorems in[2].

Remark 3.11. If Then we reduced to Theorems in[8].

Remark 3.12. If Then we reduced to Theorems in[15].

4. Examples

In this section we will show the applications of our oscillation criteria by three examples. We will see that the equations in the example is oscillatory or tend to zero based on the results in Section 2.

Example 4.1. Consider the difference equation

(4.1)

If we take then all conditions of theorem 2.4 are also satisfied. Hence every solution of (4.1) is oscillatory or satisfies .

Example 4.2. Consider the third order nonlinear neutral difference equation

(4.2)

If we take then, Corollary 2.2 asserts that every solution of (4.2) is oscillatory or tend to zero.

Example 4.3. Consider the third order difference equation

By choosing . Then, by Theorem 2.7, we conclude that every solution of (4.3) either oscillates or tend to zero.

ACKNOWLEDGMENTS

The authors would like to thank the anonymous referees very much for valuable suggestions, corrections and comments, which results in a great improvement in the original manuscript.