Abstract and Applied Analysis

Volume 2014 (2014), Article ID 392097, 9 pages

http://dx.doi.org/10.1155/2014/392097

Research Article

Oscillations of Difference Equations with Several Oscillating Coefficients

L. Berezansky,¹ G. E. Chatzarakis,² A. Domoshnitsky,³ and I. P. Stavroulakis⁴

¹Department of Mathematics, Ben-Gurion University of Negev, 84105 Beer-Sheva, Israel

²Department of Electrical and Electronic Engineering Educators, School of Pedagogical and Technological Education (ASPETE), N. Heraklion, 14121 Athens, Greece

³Department of Mathematics and Computer Sciences, Ariel University of Samaria, 40700 Ariel, Israel

⁴Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece

Received 18 April 2014; Accepted 10 May 2014; Published 1 June 2014

Academic Editor: Tongxing Li

Copyright © 2014 L. Berezansky et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We study the oscillatory behavior of the solutions of the difference equation where , are real sequences with oscillating terms, , are general retarded (advanced) arguments, and denotes the forward (backward) difference operator . Examples illustrating the results are also given.

1. Introduction

In the present paper, we study the oscillatory behavior of the solutions of the difference equation

where , , are real sequences with oscillating terms, and , are sequences of integers such that

and the (dual) advanced difference equation

where , , are real sequences with oscillating terms and , , are sequences of integers such that

Here, and . Also, as usual, denotes the forward difference operator and denotes the backward difference operator .

Strong interest in is motivated by the fact that it represents a discrete analogue of the differential equation (see [1] and the references cited therein)

where, for every , is an oscillating continuous real-valued function in the interval , and is a continuous real-valued function on such that

while, represents a discrete analogue of the advanced differential equation (see [1] and the references cited therein)

where, for every , is an oscillating continuous real-valued function in the interval and is a continuous real-valued function on such that

By a solution of , we mean a sequence of real numbers which satisfies for all . Here,

It is clear that, for each choice of real numbers , there exists a unique solution of which satisfies the initial conditions , , and .

By a solution of the advanced difference equation , we mean a sequence of real numbers which satisfies for all .

A solution of [] is called oscillatory, if the terms of the sequence are neither eventually positive nor eventually negative. Otherwise, the solution is said to be nonoscillatory.

In the last few decades, the oscillatory behavior of all solutions of difference equations has been extensively studied when the coefficients are nonnegative. See, for example, [2–20] and the references cited therein. However, for the general case when are allowed to oscillate, it is difficult to study the oscillation of [], since the difference of any nonoscillatory solution of [] is always oscillatory. Thus, a small number of papers are dealing with this case. See, for example, [1, 21–32] and the references cited therein.

For (3) and (5) with oscillating coefficients, Fukagai and Kusano [1] established the following theorems.

Theorem 1 (see [1, Theorem ]). Assume (4) and that there is a continuous nondecreasing function such that for , . Suppose moreover that there is a sequence such that , the intervals are disjoint, and

If there is a constant such that

then all solutions of (3) oscillate.

Theorem 2 (see [1, Theorem ii]). Assume (6) and that there is a continuous nondecreasing function such that for , . Suppose moreover that there is a sequence such that , the intervals are disjoint, and

If there is a constant such that

then all solutions of (5) oscillate.

For and with oscillating coefficients, recently, Bohner et al. [21, 23] established the following theorems.

Theorem 3 (see [23, Theorem 2.4]). Assume (1) and that the sequences are increasing for all . Suppose also that for each there exists a sequence such that and

where

If, moreover,

where , then all solutions of oscillate.

Theorem 4 (see [23, Theorem 3.4]). Assume (2) and that the sequences are increasing for all . Suppose also that for each there exists a sequence such that and

where

If, moreover,

where , then all solutions of oscillate.

Theorem 5 (see [21, Theorem 2.1]). Assume (1) and that the sequences are increasing for all . Suppose also that for each there exists a sequence such that ,

If, moreover,

then all solutions of oscillate.

Theorem 6 (see [21, Theorem 3.1]). Assume (2) and that the sequences are increasing for all . Suppose also that for each there exists a sequence such that ,

If, moreover,

then all solutions of oscillate.

In the present paper, the authors study further [] and derive new sufficient oscillation conditions when neither (14) [(17)] nor (20) [(23)] is satisfied (cf. [6–8] and the references cited therein in the case of the equations [] with nonnegative coefficients , ). Examples illustrating the results are also given.

2. Retarded Equations

In this section, we present new sufficient conditions for the oscillation of all solutions of when the conditions (14) and (20) are not satisfied, under the assumption that the sequences are increasing for all . To that end, the following lemma provides a useful tool.

Lemma 7. Assume that (1) holds, the sequences are increasing for all and is a nonoscillatory solution of . Suppose also that for each there exists a sequence , such that , and (12) where is defined by (13). Set

where .

If , then

Proof. Since the solution of is nonoscillatory, it is either eventually positive or eventually negative. As is also a solution of , we may restrict ourselves only to the case where eventually.

By (12), it is obvious that there exists such that

Also, by (24) we have

where is an arbitrary real number with .

In view of (26) and (27), gives

for every . This guarantees that the sequence is decreasing on .

Assume that , where is defined by (24). From inequality (28), it is clear that there exists such that

This is because in the case where , there exists such that (30) is satisfied, while in the case where , then , and, therefore,

That is, in both cases (30) is satisfied.

Now, we will show that . Indeed, in the case where , since , it is obvious that . In the case where , then . Assume, for the sake of contradiction, that . Hence, and then

which contradicts (28). Thus, in both cases, we have . Therefore

Summing up from to , and using the fact that the function is decreasing and the function (as defined by (13)) is increasing, we have

or

which, in view of (30), gives

Summing up from to , and using the same arguments, we have

or

which, in view of (34), gives

Combining inequalities (37) and (40), we obtain

or

Thus

In view of (43), inequality (42) gives

which, in view of (40) becomes

Thus

or

Hence,

which, for arbitrarily small values of , implies (25).

The proof of the lemma is complete.

Theorem 8. Assume that (1) holds, the sequences are increasing for all and is defined by (13). Suppose also that for each there exists a sequence such that , (12) and define by (24), where .

If , and

then all solutions of oscillate.

Proof. Assume, for the sake of contradiction, that is an eventually positive solution of . Then there exists such that

Therefore, by we have

for every . This guarantees that the sequence is decreasing on .

Summing up from to , and using the fact that the function is decreasing and the function (as defined by (13)) is increasing, we obtain

Consequently,

which gives

Assume that and (49) holds. Then by Lemma 7, inequality (25) is fulfilled, and so (54) leads to

which contradicts condition (49).

The proof of the theorem is complete.

3. Advanced Equations

Oscillation of all solutions of is described by the theorem below. Note that the proof is an easy modification of the proof of Theorem 8 and hence is omitted.

Theorem 9. Assume (2) holds, the sequences are increasing for all and is defined by (16). Suppose also that for each there exists a sequence such that , (15) and

where .

If and

then all solutions of oscillate.

Remark 10. When , then the conditions (49) and (57) reduce to the conditions (14) and (17), respectively. However the improvement is clear when . The lower bound in (49) and (57) is 0.946475699. That is, when , our conditions (49) and (57) essentially improve (14) and (17).

4. Examples

The significance of the results is illustrated in the following examples.

Example 1. Consider the retarded difference equation

where , , and are oscillating coefficients, as shown in Figure 1.

In view of (13), it is obvious that . Observe that for

we have for every , where

For

we have for every , where

and, for

we have for every , where

Therefore,

Observe that

Now,

Observe that

that is, condition (49) of Theorem 8 is satisfied and, therefore, all solutions of equation (58) oscillate.

On the other hand,

Observe that for every , for every , where

and for every , where

Therefore,

Also,

Therefore none of the conditions (14) and (20) is satisfied.

Figure 1

Example 2. Consider the advanced difference equation

where and are oscillating coefficients, as shown in Figure 2.

Figure 2

In view of (16), it is obvious that . Observe that for

we have for every , where

Also, for

we have for every , where

Therefore,

Observe that

Now

Also

Observe that

that is, condition (57) of Theorem 9 is satisfied and, therefore, all solutions of equation (74) oscillate.

On the other hand,

Observe that for every and for every , where

Therefore,

Also,

Therefore none of the conditions (17) and (23) is satisfied.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors would like to thank the referees for the constructive remarks which improved the presentation of the paper.

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