Oscillation Conditions for Difference Equations with a Monotone or Nonmonotone Argument

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. Consider the first-order delay difference equation with a constant argumentΔx(n)+p(n)x(n−k) = 0, n = 0, 1, 2, . . . , and the delay difference equation with a variable argument Δx(n) + p(n)x(τ(n)) = 0, n = 0, 1, 2, . . . , where p(n) is a sequence of nonnegative real numbers, k is a positive integer, Δx(n) = x(n + 1) − x(n), and τ(n) is a sequence of integers such that τ(n) ≤ n − 1 for all n ≥ 0 and limn→∞τ(n) = ∞. A survey on the oscillation of all solutions to these equations is presented. Examples illustrating the results are given.

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 (1)  without retarded (() ≡ ) argument does not possess oscillatory solutions when () < 1.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 (1)  .
A great part of the existing literature on the oscillation of (1)  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 (1)  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 (1)  in both cases that the argument () is monotone or nonmonotone and examples which illustrate the significance of the results.
In the same year 1989 Ladas et al. [25] proved the following theorem.
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   /( + 1) +1 in (C 2 ) and (C 4 ) 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 +1 for all large , (N 2 ) 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  0 ≥ 1 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 (N 2 ) by (N 3 ), which is a weaker condition.It is interesting to establish sufficient conditions for the oscillation of all solutions of (1) when both (C 3 ) and (C 4 ) are not satisfied.(For (2) and (2)  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 (C 3 ), established the following (false) sufficient oscillation conditions for (1) respectively.Unfortunately, the above conditions (F 1 ) and (F 2 ) 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.
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.
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.
Theorem 7 (see [29]).Assume that Then either one of the conditions 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 (F 1 ) should be replaced.
(ii) When  = 3, while Therefore, in this case conditions (C 6 ) and (C 7 ) are independent.
We illustrate these by the following examples.
Example 10 (see [29]).Consider the equation where Here  = 3 and it is easy to see that , Thus condition (C 7 ) is satisfied and therefore all solutions oscillate.Observe, however, that condition (C 6 ) is not satisfied.
If, on the other hand, in the above equation then it is easy to see that , In this case condition (C 6 ) is satisfied and therefore all solutions oscillate.Observe, however, that condition (C 7 ) is not satisfied.
Example 11 (see [29]).Consider the equation where Here  = 16 and it is easy to see that that is, condition (C 3 ) is not satisfied.
In 2000, Tang and Yu [34] improved condition (C 8 ) to the condition where  2 is the greater root of the algebraic equation In 2001, Shen and Stavroulakis [27], using new techniques, improved the previous results as follows.
Corollary 13 (see [27]).Assume that 0 ≤  ≤ 1/4 and that (C 11 ) holds.Then all solutions of the equation A condition derived from (C 11 ) and which can be easily verified is given in the next corollary.
Remark 15 (see [27]).Observe that when  = 1/4, condition which cannot be improved in the sense that the lower bound Therefore, by Corollary 14, all solutions oscillate.However, none of the conditions (C 1 )-(C 9 ) is satisfied.
The following corollary concerns the case when  > 1.
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].
In 2008, Chatzarakis et al. [7] investigated for the first time the oscillatory behavior of equation (1)  in the case of a general nonmonotone delay argument () and derived the following theorem.
Also in the same year Chatzarakis et al. [8] derived the following theorem.
In [8] an example is also presented to show that the condition (C  2 ) 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 (1)  , in the case where neither (C  1 ) nor (C  2 ) 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 0 <  ≤ 1/.Then either one of the conditions: implies that all solutions of (1)  oscillate.
In 2011, Braverman and Karpuz [6] studied (1)  also in the case of nonmonotone delays.More precisely the following were derived in [6].
Remark 27.At this point it should be emphasized that conditions (C  1 ) and (C  2 ) imply that all solutions of (1)  oscillate without the assumption that () is monotone.Note that in (C  1 ) instead of () the sequence (), defined by (42), is considered, which is nondecreasing and () ≤ () for all  ≥ 0.
Very recently Chatzarakis et al. [11] established the following conditions which essentially improve all the related conditions in the literature.
(ii) If for some ℓ ∈ N we have  ℓ () < 1, for sufficiently large , and where then all solutions of (1)  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.

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 () < 1 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.