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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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.
References