Oscillation Conditions for Difference Equations with a Monotone or Nonmonotone Argument
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.
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.
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 .
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 equationas the discrete analogue of the delay logistic equationwhere 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 the same year 1989 Ladas et al.  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 in and it should be emphasized that, as it is shown in , if then (1) has a nonoscillatory solution.
In 1990, Ladas  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. . Moreover, in 1999 Tang and Yu , using a different technique, showed that ifthen (1) has a nonoscillatory solution. Therefore, as a corollary (see  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.
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.,  and the references cited therein.)
Lemma A (false). Assume that is an eventually positive solution of (1) and that Then
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 .
Based on these lemmas, the following theorem was established in 2004 by Stavroulakis .
Remark 8 (see ). 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 ). 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 ). 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 ). 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 1998, Domshlak  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  improved condition to the condition where is the greater root of the algebraic equation
In 2001, Shen and Stavroulakis , using new techniques, improved the previous results as follows.
Corollary 13 (see ). Assume that and that holds. Then all solutions of the equation oscillate.
Remark 15 (see ). 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 ), 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, ).
The following corollary concerns the case when .
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  and in 1998 by Tang  (see also Tang and Yu ). 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  and in 2000 .
In 2008, Chatzarakis et al.  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 ). Assume thatIfthen all solutions of equation oscillate.
Remark 19 (see ). Clearly, the sequence of integers is nondecreasing and for all
Also in the same year Chatzarakis et al.  derived the following theorem.
Theorem 20 (see ). Assume thatThen all solutions of oscillate.
Remark 22 (see ). Condition is optimal for under the assumption that , since in this case the set of natural numbers increases infinitely in the interval for .
In  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.
Theorem 23 (see [7, 9, 10]). (I) Assume that . Then either one of the conditions:orimplies that all solutions of oscillate.
(II) If and, in addition, for all large andor if and, in addition, for all large andthen all solutions of are oscillatory.
Remark 24. Observe the following:
(i) When , it is easy to verify thatand 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.
Theorem 25 (see ). There is no constant such that the inequalitiesimply oscillation of .
Remark 26 (see ). Obviously, there is no constant such thatimplies 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 ). Ifthen every solution of oscillates.
Theorem 29 (see ). Assume thatwhereThen 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  cf. ). Consider the equationwhere Observe that for this equation, and it is easy to see that and soThus,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.  established the following iterative oscillation conditions. If for some orwhereand , then all solutions of oscillate.
In 2017 Asteris and Chatzarakis  and Chatzarakis and Shaikhet  proved that if for some orwherewith and , then all solutions of oscillate.
Very recently Chatzarakis et al.  established the following conditions which essentially improve all the related conditions in the literature.
Theorem 32 (see ). (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 , andwherethen 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 ). 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)) thatThe left-hand side in (62) attains its maximum at , , for every . Specifically,orThe computation immediately implies that if , thenFor example, for , we obtain