Pure and Applied Mathematics Journal
Volume 5, Issue 1, February 2016, Pages: 9-14

Oscillations of Neutral Difference Equations of Second Order with Positive and Negative Coefficients

Hussain Ali Mohamad, Hala Majid Mohi

Dept. of Mathematics, College of Science for Women, University of Baghdad, Baghdad, Iraq

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(H. A. Mohamad)
(H. M. Mohi)

To cite this article:

Hussain Ali Mohamad, Hala Majid Mohi. Oscillations of Neutral Difference Equations of Second Order with Positive and Negative Coefficients. Pure and Applied Mathematics Journal. Vol. 5, No. 1, 2016, pp. 9-14. doi: 10.11648/j.pamj.20160501.12


Abstract: In this paper some necessary and sufficient conditions are obtained to guarantee the oscillation for bounded and all solutions of second order nonlinear neutral delay difference equations. In Theorem 5 and Theorem 8, We have studied the oscillation criteria as well as the asymptotic behavior, where was established some sufficient conditions to ensure that every solution are either oscillates or  as  Examples are given to illustrate the obtained results.

Keywords: Oscillation, Neutral Difference Equations, Second Order Difference Equations


1. Introduction

In this paper the oscillation for bounded and all solutions of second order neutral delay difference equation with positive and negative coefficients:

(1)

will be studied, where  is the forward difference operator,  are nonnegative infinite sequences of real numbers and  are infinite sequences of real numbers.  is function . The purpose of this research is to obtain new sufficient conditions for the oscillation of all solutions of equation (1). The following assumptions are used:

 There exists a sequence  such that  and

;

 .

2. Main Result

The next results provide sufficient conditions for the oscillation of all bounded solutions of Eq. (1). For a simplicity set

(2)

Let the sequence  be defined as

(3)

and the sequence  be defined as

(4)

The following theorem based on Theorem 7.6.1, [3] pp. 184:

Theorem 1. ([3], pp. 184)

Assume that  is a nonnegative sequence of real numbers and let  be a positive integer. Suppose that

Then

i.     The difference inequality

cannot have eventually positive solutions.

ii.   The difference inequality

cannot have eventually negative solutions.

Theorem 2. ([12], pp.10) Let  then

 for all if and only if .

 for all if and only if .

Theorem 3. Suppose that  is bounded  let , (), and  hold, in addition to

(5)

Then every bounded solution of equation (1) oscillates.

Proof. Assume for the sake of contradiction that {} be positive bounded solution of eq. (1) for , then from equations (1), (2) and (3) we obtain

(6)

Hence, ,  are monotone sequences. We claim that  for , otherwise,, , thus,  and  as . From (3) we get

then  as , which is a contradiction. Hence our claim is established. We have two cases for :

Case 1: ; Case 2:

Case 1: , then there exists  such that  for . Since  is bounded, let , so there exists a subsequence  of  such that ,  . From (3) we get

Since  is arbitrary, by Theorem 2, it follows that for sufficiently large  we get

As , it follows that  which is a contradiction.

Case 2: . By taking the summation of both sides of (6) from to , , we get

(7)

From (3) we get

Since  is arbitrary, it follows that

(8)

Substituting (8) in (7) to obtain

By theorem 1-ii and in virtue of (5), it follows that the last inequality cannot have eventually negative solution, which is a contradiction.

Example 4. Consider the difference equations

(E1)

where m =1, k=3, l= 2,  , ,

  

  

  

  

By Theorem 3, it follows that every bounded solution of  oscillates, for instance  is such a solution.

Theorem 5. Suppose that , (,  hold, in addition to

(9)

(10)

Then every solution {} of equation (1) either oscillates or  as

Proof. For the sake of contradiction, assume that {} be an eventually positive solution of eq. (1), then from equations (1), (2) and (3) it follows that (6) hold, that is

Hence ,  are monotone sequences. If  for , thus  and  as . From (3) we obtain

which implies that  as .

If  for , we have two cases to consider for :

Case 1:  Case 2:

Case 1: . Then  where .

If , From (3) we get

which implies that , otherwise if  is bounded it follows from the last inequality , which is a contradiction.

If , then there exists such that , for . If , then from (3) we get

 is arbitrary, so for sufficiently large  we get

By taking the summation of both sides of (6) from to , it follows that

In virtue of (9) the last inequality implies that  Leads to a contradiction.

Case 2: . In this case  is bounded, we claim that  is bounded, otherwise there exists a subsequence  of  such that   and , from (3) we get

which implies that , a contradiction.

By taking the summation of both sides of (6) from to, , it follows that

(11)

From (3) we get

Since  is arbitrary, it follows that for sufficiently large :

Substituting the last inequality in (11) we obtain

By theorem 1-ii and in virtue of (10), it follows that the last inequality cannot have eventually negative solution, which is a contradiction.

In the next theorem we will use the sequence  already defined in (4).

Theorem 6. Suppose that  , (, and () hold, in addition to

(12)

Then every bounded solution of equation (1) oscillates.

Proof. For the sake of contradiction, assume that {} be an eventually positive bounded solution of eq. (1), then from equations (1), (2) and (4) we obtain

(13)

Hence, ,  are monotone sequences, we claim that  for , otherwise for , hence  and  as . Let , then , where  are positive constants. From (4) we obtain

which implies that  as , which is a contradiction. Our claim has been established, then it remains to consider two possible cases for the existence of nonoscillatory solution of eq. (1) for :

Case 1:  Case 2:

Case 1: . Then there existssuch that, , for . Since  is bounded, let   so there exists a sequence  such that, . From (4) we get

Since  is arbitrary, then by theorem 2.2, it follows for sufficiently large  that:

as , we get from the last inequality  which is a contradiction.

Case 2: . By taking the summation of both sides of (13) from to,  it follows that

(14)

From (4) we get

Since  is arbitrary, it follows that

Substituting the last inequality in (14) we obtain

By Theorem 1-i, and in virtue of (12), it follows that the last inequality cannot have eventually positive solution, which is a contradiction.

Example 7. Consider the difference equation

(E2)

where k=1, m=1, l = 2,  

,

  

  

  

  

By theorem 5, every bounded solution of  oscillates, for instance  is such a solution.

Theorem 8. Suppose that , ( hold, in addition to (12) and

(15)

Then every solution {} of equation (1) either oscillates or  as

proof. For the sake of contradiction, assume that {} be an eventually positive solution of eq. (1), then from equations (1), (2) and (4) it follows that (13) hold, that is

Hence, ,  are monotone sequences. If  for , thus  and  as . From (4) we obtain

which implies that  as .

If  for  we have two cases to consider for :

Case 1:  Case 2:

Case 1: . Then  where .

If , From (4) we get

which implies that

If , then there exists such that , for . If . From (4) we get

By taking summation to both sides of (13) from to , it follows that

In virtue of (12) the last inequality implies that  Leads to a contradiction.

Case 2: . By taking the summation of both sides of (13) from to, , it follows that

(16)

From (4) we get

Since  is arbitrary, it follows that

Substituting the last inequality in (16) we obtain

By Theorem 1-i and in virtue of (12), it follows that the last inequality cannot have eventually positive solution, which is a contradiction.

3. Conclusion

1.   In this paper we used two series  and , and obtained necessary and sufficient conditions for every solution of neutral difference equation of second order with positive and negative coefficients, to be oscillates or tends to infinity as  

2.   In condition  we can use  where  is constant and the results remain true.

3.   The conditions  and  can be improved, and established new conditions.


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