Discrete Dynamics in Nature and Society
Volume 2018 (2018), Article ID 9416319, 13 pages
https://doi.org/10.1155/2018/9416319

Research Article

Oscillation Conditions for Difference Equations with a Monotone or Nonmonotone Argument

G. M. Moremedi¹ and I. P. Stavroulakis1,2

¹Department of Mathematical Sciences, University of South Africa, Pretoria 0003, South Africa
²Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece

Correspondence should be addressed to I. P. Stavroulakis; ipstav@uoi.gr

Received 5 September 2017; Accepted 5 April 2018; Published 21 June 2018

Academic Editor: Manuel De la Sen

Copyright © 2018 G. M. Moremedi and I. P. Stavroulakis. 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.

Consider the first-order delay difference equation with a constant argument and the delay difference equation with a variable argument where is a sequence of nonnegative real numbers, is a positive integer, and is a sequence of integers such that for all and . A survey on the oscillation of all solutions to these equations is presented. Examples illustrating the results are given.

1. Introduction

In the last few decades the oscillation theory of delay differential equations has been extensively developed. The oscillation theory of discrete analogues of delay differential equations has also attracted growing attention in the recent few years. The reader is referred to [1–36] and the references cited therein. In particular, the problem of establishing sufficient conditions for the oscillation of all solutions of the delay difference equation with a constant argument

and the delay difference equation with a variable argument

where is a sequence of nonnegative real numbers, is a positive integer, , and is a sequence of integers such that for all and , has been the subject of many recent investigations. Strong interest in (1) and is motivated by the fact that they represent the discrete analogues of the delay differential equations (see [22, 37, 38] and the references cited therein):

Set . Clearly, is a finite positive integer.

By a solution of (1) [] we mean a sequence which is defined for and which satisfies (1) [] for .

A solution is said to be oscillatory if the terms of the solution are not eventually positive or eventually negative. Otherwise, the solution is called nonoscillatory.

For convenience, we will assume that inequalities about values of sequences are satisfied eventually for all large .

For the general theory of these equations, the reader is referred to [1–3, 22, 39].

Besides the purely mathematical problem, the interest in the behavior of the solutions to difference equations with retarded arguments is justified by the fact that the mathematical modeling of many real-world problems leads to difference equations where the unknown function depends on the past history rather than only the present state. This interest grows stronger as difference equations naturally arise from discretization of differential equations. As a consequence, many researchers have been concerned with the study of qualitative behavior of solutions to difference equations, in particular, the study of oscillation of solutions.

In 1969 and in 1974 Pielou (see [22, p. 194]) considered the delay difference equation

as the discrete analogue of the delay logistic equation

where and are the growth rate and the carrying capacity of the population, respectively. Pielou’s interest in (3) was in showing that “the tendency to oscillate is a property of the populations themselves and is independent of the extrinsic factors.” That is, population sizes oscillate “even though the environment remains constant.” According to Pielou, “oscillations can be set up in a population if its growth rate is governed by a density dependent mechanism and if there is a delay in the response of the growth rate to density changes. When this happens the size of the population alternately overshoots and undershoots its equilibrium level.”

The blowfly (Lucilia cuprina) studied in 1954 by Nicholson (see [22, p. 194]) is an example of a laboratory population which behaves in the manner described above.

It is noteworthy that a first-order linear difference equation of the form without retarded argument does not possess oscillatory solutions when . A small delay may change this situation, as one can see below, even in the case of equations with constant delays and constant coefficients, and this certainly adds interest in the investigation of the oscillatory character of the above equations (1) and .

A great part of the existing literature on the oscillation of concerns the case where the argument is nondecreasing, while only a small number of papers are dealing with the general case of arguments being not necessarily monotone. See, for example, [4–6, 11, 12, 30] and the references cited therein. The consideration of nonmonotone arguments may lead to better approximation of the natural phenomena described by difference equations because quite often there appear natural disturbances (e.g., noise in communication systems) that affect all the parameters of the equation and therefore the “fair” (from a mathematical point of view) monotone arguments are, in fact, nonmonotone almost always. In view of this, for the case of an interesting question is whether we can state oscillation criteria considering the argument to be not necessarily monotone.

In this paper, we present a survey on the oscillation of solutions to (1) and in both cases that the argument is monotone or nonmonotone and examples which illustrate the significance of the results.

2. Oscillation Criteria for (1)

In 1981, Domshlak [15] studied this problem in the case where Then, in 1989, Erbe and Zhang [21] established the following oscillation criteria for (1).

Theorem 1 (see [21]). Assume that

or

or

Then all solutions of (1) oscillate.

In the same year 1989 Ladas et al. [25] proved the following theorem.

Theorem 2 (see [25]). Assume that

Then all solutions of (1) oscillate.

Therefore they improved the condition by replacing the of by the arithmetic mean of the terms in .

Note that this condition is sharp in the sense that the fraction on the right hand side cannot be improved, since when is a constant, say , then this condition reduces to

which is a necessary and sufficient condition for the oscillation of all solutions to (1). Moreover, concerning the constant in and it should be emphasized that, as it is shown in [21], if

then (1) has a nonoscillatory solution.

In 1990, Ladas [24] conjectured that (1) has a nonoscillatory solution if

holds eventually. However this conjecture is not correct and a counterexample was given in 1994 by Yu et al. [36]. Moreover, in 1999 Tang and Yu [33], using a different technique, showed that if

then (1) has a nonoscillatory solution. Therefore, as a corollary (see [33] Corollary 2]), Tang and Yu presented an affirmative answer to the so-called “corrected Ladas conjecture”; that is, if

then (1) has a nonoscillatory solution.

Very recently Karpuz [23] studied this problem and derived the following conditions. If

then every solution of (1) oscillates, while if there exists such that

then (1) has a nonoscillatory solution.

From the above conditions, using the arithmetic-geometric mean, it follows that if

then (1) has a nonoscillatory solution. That is, Karpuz replaced condition by , which is a weaker condition.

It is interesting to establish sufficient conditions for the oscillation of all solutions of (1) when both and are not satisfied. (For (2) and this question has been investigated by many authors; see, e.g., [30] and the references cited therein.)

In 1993, Yu et al. [35] and Lalli and Zhang [26], trying to improve , established the following (false) sufficient oscillation conditions for (1)

respectively.

Unfortunately, the above conditions and are not correct. This is due to the fact that they are based on the following (false) discrete version of Koplatadze-Chanturia Lemma. (See [14, 18].)

Lemma A (false). Assume that is an eventually positive solution of (1) and that

Then

As one can see, the erroneous proof of Lemma A is based on the following (false) statement. (See [14, 18].)

Statement A (False). If (9) holds, then for any large , there exists a positive integer such that and

It is obvious that all the oscillation results which have made use of Lemma A or Statement A are incorrect. For details on this problem see the paper by Cheng and Zhang [14].

Here it should be pointed out that the following statement (see [25, 28]) is correct and it should not be confused with Statement A.

Statement 3 (see [25, 28]). If

then for any large , there exists a positive integer with such that

In 1995, Stavroulakis [28], based on this correct Statement 3, proved the following theorem.

Theorem 4 (see [28]). Assume that

Then all solutions of (1) oscillate.

In 1999 Domshlak [18] and in 2000 Cheng and Zhang [14] established the following lemmas, respectively, which may be looked upon as (exact) discrete versions of Koplatadze-Chanturia Lemma.

Lemma 5 (see [18]). Assume that is an eventually positive solution of (1) and that condition (12) holds. Then

Lemma 6 (see [14]). Assume that is an eventually positive solution of (1) and that condition (12) holds. Then

Based on these lemmas, the following theorem was established in 2004 by Stavroulakis [29].

Theorem 7 (see [29]). Assume that

Then either one of the conditions

or

implies that all solutions of (1) oscillate.

Remark 8 (see [29]). From the above theorem it is now clear that

is the correct oscillation condition by which the (false) condition should be replaced.

Remark 9 (see [29]). Observe the following:

(i)When ,

(since, from the above-mentioned conditions, it makes sense to investigate the case when ) and therefore condition implies .

(ii)When ,

while

Therefore, in this case conditions and are independent.

(iii)When ,

and therefore condition implies .

(iv)When , condition or implies .(v)When , condition may hold but condition may not hold.

We illustrate these by the following examples.

Example 10 (see [29]). Consider the equation

where

Here and it is easy to see that

Thus condition is satisfied and therefore all solutions oscillate. Observe, however, that condition is not satisfied.
If, on the other hand, in the above equation

then it is easy to see that

In this case condition is satisfied and therefore all solutions oscillate. Observe, however, that condition is not satisfied.

Example 11 (see [29]). Consider the equation

where

Here and it is easy to see that

We see that condition is satisfied and therefore all solutions oscillate. Observe, however, that

that is, condition is not satisfied.

In 1995, Chen and Yu [13], following the above-mentioned direction, derived a condition which, formulated in terms of and , says that all solutions of (1) oscillate if and

In 1998, Domshlak [17] studied the oscillation of all solutions and the existence of nonoscillatory solution of (1) with -periodic positive coefficients , . It is very important that, in the following cases where , and , the results obtained are stated in terms of necessary and sufficient conditions and it is very easy to check them.

In 2000, Tang and Yu [34] improved condition to the condition

where is the greater root of the algebraic equation

In 2001, Shen and Stavroulakis [27], using new techniques, improved the previous results as follows.

Theorem 12 (see [27]). Assume that and that there exists an integer such that

where and are the greater real roots of the equations

respectively. Then all solutions of (1) oscillate.

Notice that when , (see [27]), and so condition reduces to

where . Therefore, from Theorem 12, we have the following corollary.

Corollary 13 (see [27]). Assume that and that holds. Then all solutions of the equation

oscillate.

A condition derived from and which can be easily verified is given in the next corollary.

Corollary 14 (see [27]). Assume that and that

Then all solutions of (34) oscillate.

Remark 15 (see [27]). Observe that when , condition reduces to

which cannot be improved in the sense that the lower bound cannot be replaced by a smaller number. Indeed, by condition (Theorem 2.3 in [21]), we see that (34) has a nonoscillatory solution if

Note, however, that even in the critical state where

(34) can be either oscillatory or nonoscillatory. For example, if , then (34) will be oscillatory in case and nonoscillatory in case (the Kneser-like theorem, [16]).

Example 16 (see [27]). Consider the equation

where is a constant. It is easy to see that

Therefore, by Corollary 14, all solutions oscillate. However, none of the conditions – is satisfied.

The following corollary concerns the case when .

Corollary 17 (see [27]). Assume that and that

where are as in Theorem 12. Then all solutions of (1) oscillate.

Following this historical (and chronological) review we also mention that in the (critical) case where

the oscillation of (1) has been studied in 1994 by Domshlak [16] and in 1998 by Tang [31] (see also Tang and Yu [32]). In a case when is asymptotically close to one of the periodic critical states, unimprovable results about oscillation properties of the equation

were obtained by Domshlak in 1999 [19] and in 2000 [20].

3. Oscillation Criteria for

In this section we study the delay difference equation with variable argument where is a sequence of nonnegative real numbers and is a sequence of integers such that for all and .

In 2008, Chatzarakis et al. [7] investigated for the first time the oscillatory behavior of equation in the case of a general nonmonotone delay argument and derived the following theorem.

Theorem 18 (see [7]). Assume that

If

then all solutions of equation oscillate.

Remark 19 (see [7]). Clearly, the sequence of integers is nondecreasing and for all

Also in the same year Chatzarakis et al. [8] derived the following theorem.

Theorem 20 (see [8]). Assume that

Then all solutions of oscillate.

Remark 21 (see [8]). Note that condition (43) is not a limitation since, by , if is a nondecreasing function, all solutions of oscillate.

Remark 22 (see [8]). Condition is optimal for under the assumption that , since in this case the set of natural numbers increases infinitely in the interval for .

In [8] an example is also presented to show that the condition is optimal, in the sense that it cannot be replaced by the nonstrong inequality.

As it has been mentioned above, it is an interesting problem to find new sufficient conditions for the oscillation of all solutions of the delay difference equation , in the case where neither nor is satisfied.

In 2008 Chatzarakis et al. [7] and in 2008 and 2009 Chatzarakis et al. [9, 10] derived the following conditions.

Theorem 23 (see [7, 9, 10]). (I) Assume that . Then either one of the conditions:

or

implies that all solutions of oscillate.
(II) If and, in addition, for all large and

or if and, in addition, for all large and

then all solutions of are oscillatory.

Remark 24. Observe the following:
(i) When , it is easy to verify that

and therefore condition is weaker than conditions (46), (45), and (44).
(ii) When , it is easy to show that

and therefore, in this case, inequality (47) improves inequality and, especially, when , the lower bound in is 0.8929094 while that in (47) is 0.7573593.

In 2011, Braverman and Karpuz [6] studied also in the case of nonmonotone delays. More precisely the following were derived in [6].

Theorem 25 (see [6]). There is no constant such that the inequalities

imply oscillation of .

Remark 26 (see [6]). Obviously, there is no constant such that

implies oscillation of .

Remark 27. At this point it should be emphasized that conditions and imply that all solutions of oscillate without the assumption that is monotone. Note that in instead of the sequence , defined by (42), is considered, which is nondecreasing and for all

Theorem 28 (see [6]). If

then every solution of oscillates.

Using the upper bound of the ratio for possible nonoscillatory solutions of , presented in [7–10], the above result was essentially improved in 2014 by Stavroulakis [30].

Theorem 29 (see [30]). Assume that

where

Then all solutions of oscillate.

Remark 30. Observe that, as , condition (54) reduces to (52). However the improvement is clear as . Actually, when , the value of the lower bound in (54) is equal to ≈0.863457014, while when and , the lower bound in (54) is 0.7573593. That is, in all cases condition (54) of Theorem 29 essentially improves condition (52) of Theorem 28.

Example 31 (see [30] cf. [6]). Consider the equation

where

Observe that for this equation, and it is easy to see that

and so

Thus,

and therefore none of the known oscillation conditions , , (44), (45), , and (52) is satisfied. However,

that is, the conditions of Theorem 29 are satisfied and therefore all solutions to (56) oscillate.
In 2015, Braverman et al. [5] established the following iterative oscillation conditions. If for some

or

where

and , then all solutions of oscillate.
In 2017 Asteris and Chatzarakis [4] and Chatzarakis and Shaikhet [12] proved that if for some

or

where

with and , then all solutions of oscillate.

Very recently Chatzarakis et al. [11] established the following conditions which essentially improve all the related conditions in the literature.

Theorem 32 (see [11]). (i) If there exists an such that for sufficiently large , then all solutions of are oscillatory.
(ii) If for some we have , for sufficiently large , and

where

then all solutions of are oscillatory.

Theorem 33 (see [11]). Assume that and for some

or

where is defined by (69). Then all solutions of are oscillatory.

The following example illustrates that the conditions of Theorems 32 and 33 essentially improve known results in the literature, yet they indicate a type of independence among some of them. The calculations were made by the use of MATLAB software.

Example 34 (see [11]). Consider the retarded difference equation

with , and (see Figure 1(a))

Clearly, is nonmonotone. For the function defined by (42), we have (see, also, Figure 1(b)) that

The left-hand side in (62) attains its maximum at , , for every . Specifically,

or

The computation immediately implies that if , then

For example, for , we obtain

That is, conditions (62) and (63) are not satisfied for .

(a)

(b)

Figure 1: The graphs of and .

Similarly, if , then

For example, for and by using an algorithm on MATLAB software, we obtain

and therefore

Thus

That is, conditions (65) and (66) are not satisfied for .

On the contrary, for every , we have

For example, for and by using an algorithm on MATLAB software, we obtain

and therefore

Thus

that is, condition (68) of Theorem 32 is satisfied for . Therefore, all solutions of (72) are oscillatory.

Observe that, for , we obtain

That is, conditions , , , (52)(62) (for ) and (55)(63) (for ) are not satisfied.

Also, it is clear that condition is not applicable for (72).

In addition, by using an algorithm on MATLAB software, we obtain

which means that condition (71) is not satisfied.

The improvement of condition (68) over other conditions will be clear by comparing the values on the left-side of these conditions. For example, comparison with conditions , (52)(62) (for ) and (65) (for ) gives percentage improvement of 76.7%, 40.21%, and 20.15%, respectively. Also, while conditions (62), (63), (65), and (66) do not lead to oscillation from the first iteration, condition (68) is satisfied from the first iteration, showing that, in our example, (68) is much faster than (62), (63), (65), and (66).

4. Conclusion

In the present paper we are concerned with the oscillatory behavior of linear delay difference equations with constant or time-varying argument. The problem is interesting since the corresponding first-order linear difference equation without delay does not possess oscillatory solutions when and also by the fact that the mathematical modeling of many real-world problems lead to difference equations with retarded arguments. We present the most interesting sufficient oscillation conditions in the case of monotone or nonmonotone arguments and give examples which illustrate the applicability and the significance of the results.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

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