# Oscillation Conditions for Difference Equations with a Monotone or Nonmonotone Argument

Authors: G. M. Moremedi; I. P. Stavroulakis
Journal: Discrete Dynamics in Nature and Society
 (2018)

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## Abstract

Consider the first-order delay difference equation with a constant argumentΔxn+pnxn-k=0,n=0,1,2,…, and the delay difference equation with a variable argument Δxn+pnxτn=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.


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## Body

## 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 (1)Δxn+pnxn-k=0,n=0,1,2,…,and the delay difference equation with a variable argument 1′Δxn+pnxτn=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)=∞, has been the subject of many recent investigations. Strong interest in (1) and 1′ 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): (2)x′t+ptxt-τ=0,pt≥0,τ>0,2′x′t+ptxτt=0,pt≥0,τt<t.Setw=-minn≥0τ(n). Clearly, w is a finite positive integer.By asolution of (1) [1′] we mean a sequence x(n) which is defined for n≥-k[n≥-w] and which satisfies (1) [1′] for n≥0.A solutionx(n) is said to beoscillatory if the terms x(n) of the solution are not eventually positive or eventually negative. Otherwise, the solution is callednonoscillatory.For convenience, we will assume that inequalities about values of sequences are satisfied eventually for all largen.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(3)Nt+1=αNt1+βNt-k,α∈1,∞,β∈0,∞,k∈Nas the discrete analogue of the delay logistic equation3′N′t=rNt1-Nt-τK,where r and K 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 form1′ without retarded (τ(n)≡n) argumentdoes not possess oscillatory solutions when pn<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 of1′ concerns the case where the argument τn 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 τ(n) 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 τ(n) 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 k=1. Then, in 1989, Erbe and Zhang [21] established the following oscillation criteria for (1).Theorem 1 (see [21]).
Assume thatC1β≔liminfn→∞⁡pn>0,limsupn→∞⁡pn>1-βorC2liminfn→∞⁡pn>kkk+1k+1orC3A≔limsupn→∞⁡∑i=n-knpi>1.Then all solutions of (1) oscillate.In the same year 1989 Ladas et al. [25] proved the following theorem.Theorem 2 (see [25]).
Assume thatC4liminfn→∞⁡1k∑i=n-kn-1pi>kkk+1k+1.Then all solutions of (1) oscillate.Therefore they improved the conditionC2 by replacing the p(n) of C2 by the arithmetic mean of the terms p(n-k),…,p(n-1) in C4.Note that this condition is sharp in the sense that the fraction on the right hand side cannot be improved, since whenp(n) is a constant, say p(n)=p, then this condition reduces to (4)p>kkk+1k+1,which is anecessary and sufficient condition for the oscillation of all solutions to (1). Moreover, concerning the constant kk/k+1k+1 in C2 and C4 it should be emphasized that, as it is shown in [21], if N1sup⁡pn<kkk+1k+1,then (1) has a nonoscillatory solution.In 1990, Ladas [24] conjectured that (1) has a nonoscillatory solution if (5)1k∑i=n-kn-1pi≤kkk+1k+1holds 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(6)∑i=n-knpi≤1efor  all  largen,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 N2∑i=n-knpi≤kk+1k+1for  all  largen,then (1) has a nonoscillatory solution.Very recently Karpuz [23] studied this problem and derived the following conditions. If(7)liminfn→∞⁡infλ≥1⁡1λ∏i=n-kn1+λpi>1,then every solution of (1) oscillates, while if there exists λ0≥1 such that (8)1λ0∏i=n-kn1+λ0pi≤1for  all  largen,then (1) has a nonoscillatory solution.From the above conditions, using the arithmetic-geometric mean, it follows that ifN3∑i=n-knpi≤kk+1kfor  all  largen,then (1) has a nonoscillatory solution. That is, Karpuz replaced condition N2 by N3, which is a weaker condition.It is interesting to establish sufficient conditions for the oscillation of all solutions of (1) when both C3 and C4 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 C3, established the following (false) sufficient oscillation conditions for (1) F10<α≔liminfn→∞⁡∑i=n-kn-1pi≤kk+1k+1,A>1-α24,F2∑i=n-knpi≥d>0for  largen,A>1-d481-d34+1-d32-1,respectively.Unfortunately, the above conditionsF1 and F2 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 thatx(n) is an eventually positive solution of (1) and that (9)∑i=n-knpi≥M>0for  largen.Then(10)xn>M24xn-kfor  largen.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 N, there exists a positive integer n such that n-k≤N≤n and (11)∑i=n-kNpi≥M2,∑i=Nnpi≥M2.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(12)∑i=n-kn-1pi≥M>0for  largen,then for any large n, there exists a positive integer n∗ with n-k≤n∗≤n such that (13)∑i=n-kn∗pi≥M2,∑i=n∗npi≥M2.In 1995, Stavroulakis [28], based on this correct Statement 3, proved the following theorem.Theorem 4 (see [28]).
Assume that(14)0<α≤kk+1k+1,C5limsupn→∞⁡pn>1-α24.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 thatx(n) is an eventually positive solution of (1) and that condition (12) holds. Then (15)xn>M24xn-kfor  largen.Lemma 6 (see [14]).
Assume thatx(n) is an eventually positive solution of (1) and that condition (12) holds. Then (16)xn>Mkxn-kfor  largen.Based on these lemmas, the following theorem was established in 2004 by Stavroulakis [29].Theorem 7 (see [29]).
Assume that(17)0<α≤kk+1k+1.Then either one of the conditions C6limsupn→∞⁡∑i=n-kn-1pi>1-α24or C7limsupn→∞⁡∑i=n-kn-1pi>1-αkimplies that all solutions of (1) oscillate.Remark 8 (see [29]).
From the above theorem it is now clear that(18)0<α≔liminfn→∞⁡∑i=n-kn-1pi≤kk+1k+1,limsupn→∞⁡∑i=n-kn-1pi>1-α24is the correct oscillation condition by which the (false) condition F1 should be replaced.Remark 9 (see [29]).
Observe the following:(i)
Whenk=1,2,(19)αk>α24,(since, from the above-mentioned conditions, it makes sense to investigate the case when α<k/(k+1)k+1) and therefore condition C6 implies C7.(ii)
Whenk=3, (20)α3>α24whenα>14while (21)α3<α24whenα<14.Therefore, in this case conditions C6 and C7 are independent.(iii)
Whenk≥4,(22)αk<α24,and therefore condition C7 implies C6.(iv)
Whenk<12, condition C6 or C7 implies C3.(v)
Whenk≥12, condition C6 may hold but condition C3 may not hold.We illustrate these by the following examples.Example 10 (see [29]).
Consider the equation(23)xn+1-xn+pnxn-3=0,n=0,1,2,…,where (24)p2n=110,p2n+1=110+6495sin2⁡nπ2,n=0,1,2,….Here k=3 and it is easy to see that (25)α=liminfn→∞⁡∑i=n-3n-1pi=310<344,limsupn→∞⁡∑i=n-3n-1pi=310+6495>1-α3.Thus condition C7 is satisfied and therefore all solutions oscillate. Observe, however, that condition C6 is not satisfied.
If, on the other hand, in the above equation(26)p2n=8100,p2n+1=8100+7461000sin2⁡nπ2,n=0,1,2,…,then it is easy to see that (27)α=liminfn→∞⁡∑i=n-3n-1pi=24100<344,limsupn→∞⁡∑i=n-3n-1pi=24100+7461000>1-α24.In this case condition C6 is satisfied and therefore all solutions oscillate. Observe, however, that condition C7 is not satisfied.Example 11 (see [29]).
Consider the equation(28)xn+1-xn+pnxn-16=0,n=0,1,2,…,where (29)p17n=p17n+1=⋯=p17n+15=2100,p17n+16=2100+6551000,n=0,1,2,….Here k=16 and it is easy to see that (30)α=liminfn→∞⁡∑i=n-16n-1pi=32100<161717,limsupn→∞⁡∑i=n-16n-1pi=32100+6551000=0.975>1-α24.We see that condition C6 is satisfied and therefore all solutions oscillate. Observe, however, that (31)A=limsupn→∞⁡∑i=n-16npi=34100+6551000=0.995<1;that is, condition C3 is not satisfied.In 1995, Chen and Yu [13], following the above-mentioned direction, derived a condition which, formulated in terms of α and A, says that all solutions of (1) oscillate if 0<α≤kk+1/(k+1)k+1 andC8A>1-1-α-1-2α-α22.In 1998, Domshlak [17] studied the oscillation of all solutions and the existence of nonoscillatory solution of (1) with r-periodic positive coefficients p(n), p(n+r)=p(n). It is very important that, in the following cases where {r=k},{r=k+1},{r=2},{k=1,r=3}, and {k=1,r=4}, 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 C8 to the condition C9A>λ2k1-kln⁡λ2-1-α-1-α-α22,where λ2 is the greater root of the algebraic equation(32)kλk1-λ=α.In 2001, Shen and Stavroulakis [27], using new techniques, improved the previous results as follows.Theorem 12 (see [27]).
Assume that0≤α≤kk+1/(k+1)k+1 and that there exists an integer l≥1 such that C10limsupn→∞⁡∑i=1kpn-i+d¯α-k∏i=1k∑j=1kpn-i+j+∑m=0l-1dαk-m+1k∑i=1k∏j=0m+1pn-kj+i>1,where d¯(α) and d(α/k) are the greater real roots of the equations (33)dk+1-dk+αk=0,dk+1-dk+αk=0,respectively. Then all solutions of (1) oscillate.Notice that whenk=1, d(α)=d¯(α)=(1+1-4α)/2 (see [27]), and so condition C10 reduces to C11limsupn→∞⁡Cpn+pn-1+∑m=0l-1Cm+1∏j=0m+1pn-j-1>1,where C=2/(1+1-4α),α=liminfn→∞pn. Therefore, from Theorem 12, we have the following corollary.Corollary 13 (see [27]).
Assume that0≤α≤1/4 and that C11 holds. Then all solutions of the equation (34)xn+1-xn+pnxn-1=0oscillate.A condition derived fromC11 and which can be easily verified is given in the next corollary.Corollary 14 (see [27]).
Assume that0≤α≤1/4 and that C12limsupn→∞⁡pn>1+1-4α22.Then all solutions of (34) oscillate.Remark 15 (see [27]).
Observe that whenα=1/4, condition C12 reduces to (35)limsupn→∞⁡pn>14which cannot be improved in the sense that the lower bound 1/4 cannot be replaced by a smaller number. Indeed, by condition N1 (Theorem 2.3 in [21]), we see that (34) has a nonoscillatory solution if (36)sup⁡pn<14.Note, however, that even in the critical state where (37)limn→∞⁡pn=14,(34) can be either oscillatory or nonoscillatory. For example, if p(n)=1/4+c/n2, then (34) will be oscillatory in case c>1/4 and nonoscillatory in case c<1/4 (the Kneser-like theorem, [16]).Example 16 (see [27]).
Consider the equation(38)xn-1-xn+14+asin4⁡nπ8xn-1=0,where a>0 is a constant. It is easy to see that (39)liminfn→∞⁡pn=liminfn→∞⁡14+asin4⁡nπ8=14,limsupn→∞⁡pn=limsupn→∞⁡14+asin4⁡nπ8=14+a.Therefore, by Corollary 14, all solutions oscillate. However, none of the conditions C1–C9 is satisfied.The following corollary concerns the case whenk>1.Corollary 17 (see [27]).
Assume that0≤α≤kk+1/(k+1)k+1 and that C13limsupn→∞⁡∑i=n-kn-1pi>1-d¯α-kαk-kdα/k-kβ21-dα/k-kβ,where d¯(α),d(α/k) 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(40)1k∑i=n-kn-1pi≥kkk+1k+1,limn→∞⁡1k∑i=n-kn-1pi=kkk+1k+1,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 p(n) is asymptotically close to one of the periodic critical states, unimprovable results about oscillation properties of the equation (41)xn+1-xn+pnxn-1=0were obtained by Domshlak in 1999 [19] and in 2000 [20].


## 3. Oscillation Criteria for1′

In this section we study the delay difference equation with variable argument1′ where Δx(n)=x(n+1)-x(n),p(n) is a sequence of nonnegative real numbers and τ(n) is a sequence of integers such that τ(n)≤n-1 for all n≥0 and limn→∞τ(n)=∞.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 τ(n) and derived the following theorem.Theorem 18 (see [7]).
Assume that(42)σn≔max0≤s≤n⁡τs,n≥0.IfC1′limsupn→∞⁡∑j=σnnpj>1,then all solutions of equation 1′ oscillate.Remark 19 (see [7]).
Clearly, the sequence of integersσ(n) is nondecreasing and τ(n)≤σ(n) for all n≥0.Also in the same year Chatzarakis et al. [8] derived the following theorem.Theorem 20 (see [8]).
Assume that(43)limsupn→∞⁡∑i=τnn-1pi<+∞,C2′α≔liminfn→∞⁡∑i=τnn-1pi>1e.Then all solutions of 1′ oscillate.Remark 21 (see [8]).
Note that condition (43) is not a limitation since, by C1′, if τ is a nondecreasing function, all solutions of 1′ oscillate.Remark 22 (see [8]).
ConditionC2′ is optimal for 1′ under the assumption that limn→+∞⁡(n-τ(n))=∞, since in this case the set of natural numbers increases infinitely in the interval [τ(n),n-1] for n→∞.In [8] an example is also presented to show that the condition C2′ 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 equation1′, in the case where neither C1′ nor C2′ 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 that0<α≤1/e. Then either one of the conditions:(44)limsupn→∞⁡∑j=σnnpj>1-1-1-α2,(45)limsupn→∞⁡∑j=σnnpj>1-121-α-1-2α,orC5′limsupn→∞⁡∑j=σnnpj>1-121-α-1-2α-α2,implies that all solutions of 1′ oscillate.
(II) If0<α≤1/e and, in addition, p(n)≥1-1-α for all large n and(46)limsupn→∞⁡∑j=σnnpj>1-α1-1-α1-αor if 0<α≤6-42 and, in addition, p(n)≥α/2 for all large n and(47)limsupn→∞⁡∑j=σnnpj>1-142-3α-4-12α+α2,then all solutions of 1′ are oscillatory.Remark 24.
Observe the following:
(i) When0<α≤1/e, it is easy to verify that(48)1-α-1-2α-α22>α1-1-α1-α>1-α-1-2α2>1-1-α2and therefore condition C5′ is weaker than conditions (46), (45), and (44).
(ii) When0<α≤6-42, it is easy to show that (49)142-3α-4-12α+α2>121-α-1-2α-α2and therefore, in this case, inequality (47) improves inequality C5′ and, especially, when α=6-42≃0.3431457, the lower bound in C5′ is 0.8929094 while that in (47) is 0.7573593.In 2011, Braverman and Karpuz [6] studied 1′ also in the case of nonmonotone delays. More precisely the following were derived in [6].Theorem 25 (see [6]).
There is no constantΛ>0 such that the inequalities(50)limsupn→∞⁡n-τnpn>Λ,limsupn→∞⁡∑i=τnn-1pi>Λimply oscillation of 1′.Remark 26 (see [6]).
Obviously, there is no constantΛ>0 such that(51)limsupn→∞⁡∑i=τnnpi>Λimplies oscillation of 1′.Remark 27.
At this point it should be emphasized that conditionsC1′ and C2′ imply that all solutions of 1′ oscillate without the assumption that τ(n) is monotone. Note that in C1′ instead of τ(n) the sequence σ(n), defined by (42), is considered, which is nondecreasing and τ(n)≤σ(n) for all n≥0.Theorem 28 (see [6]).
If(52)limsupn→∞⁡∑j=σnnpj∏i=τjσn-111-pi>1,then every solution of 1′ oscillates.Using the upper bound of the ratiox(τ(n))/x(n) for possible nonoscillatory solutions x(n) of 1′, presented in [7–10], the above result was essentially improved in 2014 by Stavroulakis [30].Theorem 29 (see [30]).
Assume that(53)α=liminfn→∞⁡∑i=τnn-1pi,(54)limsupn→∞⁡∑j=σnnpj∏i=τjσn-111-pi>1-cα,where(55)cα=121-α-1-2α-α2if0<α≤1/e,142-3α-4-12α+α2if0<α≤6-42,pn≥α2.Then all solutions of 1′ oscillate.Remark 30.
Observe that, asα→0, condition (54) reduces to (52). However the improvement is clear as α→1/e. Actually, when α=1/e, the value of the lower bound in (54) is equal to ≈0.863457014, while when p(n)≥α/2 and α=6-42≃0.3431457, 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(56)Δxn+pnxτn=0,n=0,1,2,…,where (57)pn=10-4,nis  even1e,nis  odd,τn=n-1,nis  evenn-2,nis  odd.Observe that for this equation, σ(n)=τ(n) and it is easy to see that (58)∑i=τnn-1pi=1e,nis  even1e+10-4,nis  odd,∑i=τnnpi=1e+10-4,nis  even2e+10-4,nis  odd∑j=σnnpj∏i=τjσn-111-pi≃0.58,nis  even0.95,nis  oddand so(59)limsupn→∞⁡∑i=τnnpi=2e+10-4≃0.7358,limsupn→∞⁡∑i=τnn-1pi=1e+10-4≃0.3679,liminfn→∞⁡∑i=τnn-1pi=1e,limsupn→∞⁡∑j=σnnpj∏i=τjσn-111-pi≃0.95.Thus,(60)0.7358<1α=1e,0.7358<1-1-1-α2≃0.9579,0.7358<1-121-α-1-2α≃0.9409,0.7358<1-121-α-1-2α-α2≃0.8634,0.95<1and therefore none of the known oscillation conditions C1′, C2′, (44), (45), C5′, and (52) is satisfied. However,(61)limsupn→∞⁡∑j=σnnpj∏i=τjσn-111-pi≃0.95>1-121-α-1-2α-α2≃0.8634;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 r∈N(62)limsupn→∞⁡∑j=σnnpjar-1σn,τj>1,or(63)limsupn→∞⁡∑j=σnnpjar-1σn,τj>1-1-α-1-2α-α22,where(64)a1n,k=∏i=kn-11-pi,ar+1n,k=∏i=kn-11-piar-1i,τiand α=liminfn→∞⁡∑j=τnn-1pj, then all solutions of 1′ oscillate.
In 2017 Asteris and Chatzarakis [4] and Chatzarakis and Shaikhet [12] proved that if for some l∈N(65)limsupn→∞⁡∑i=σnnpi∏j=τiσn-111-plj>1,or(66)limsupn→∞⁡∑i=σnnpi∏j=τiσn-111-plj>1-1-α-1-2α-α22,where(67)pln=pn1+∑i=τnn-1pi∏j=τiσn-111-pl-1jwith p0(n)=p(n) and α=liminfn→∞⁡∑j=τ(n)n-1p(j), then all solutions of 1′ 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 anl≥1 such that Pl(n)≥1 for sufficiently large n, then all solutions of 1′ are oscillatory.
(ii) If for somel∈N we have Pl(n)<1, for sufficiently large n, and(68)limsupn→∞⁡∑i=σnnpi∏j=τiσn-111-Plj>1,where(69)Pln≔pn1+∑i=τnn-1piexp⁡∑j=τin-1pj∏m=τjj-111-Pl-1m,then all solutions of 1′ are oscillatory.Theorem 33 (see [11]).
Assume that0<α≤1/e and for some l∈N(70)limsupn→∞⁡∑i=σnnpi∏j=τiσn-111-Plj>1-1-α-1-2α-α22,or(71)limsupn→∞⁡∑i=σnnpi∏j=τin11-Plj>21-α-1-2α-α2,where Pl(n) is defined by (69). Then all solutions of 1′ are oscillatory.The following example illustrates that the conditions of Theorems32 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(72)Δxn+pxτn=0,n∈N0,with 0<p≤1/e, and (see Figure 1(a))(73)τn=n-1,ifn=3μn-2,ifn=3μ+1n-4,ifn=3μ+2,μ∈N0.Clearly, τ is nonmonotone. For the function σ defined by (42), we have (see, also, Figure 1(b)) that(74)σn=max0≤s≤nτs=n-1,ifn=3μn-2,ifn=3μ+1n-3,ifn=3μ+2,μ∈N0.The left-hand side in (62) attains its maximum at n=3μ+2, μ∈N0, for every r∈N. Specifically,(75)∑j=3μ-13μ+2pja2-13μ-1,τj=p1a23μ-1,τ3μ-1+p1a23μ-1,τ3μ+p1a23μ-1,τ3μ+1+p1a23μ-1,τ3μ+2=p1a23μ-1,3μ-5+p1a23μ-1,3μ-1+p1a23μ-1,3μ-1+p1a23μ-1,3μ-2=p1∏i=3μ-53μ-21-pa1-1i,τi+p·1+p·1+p1∏i=3μ-23μ-21-pa1-1i,τior(76)∑j=3μ-13μ+2pja2-13μ-1,τj=2p+p1-p9+p1-p21-p2-p1-p4-p1-2p1-p2-p21-p4-p1-2p.The computation immediately implies that if p∈0.145,0.1564, then(77)limsupn→∞⁡∑j=σnnpja2-1σn,τj<1.For example, for p=29/200=0.145, we obtain(78)α=liminfn→∞⁡∑j=τnn-1pj=liminfμ→∞⁡∑j=3μ-13μ-1pj=29200=0.145<1e,limsupn→∞⁡∑j=σnnpja2-1σn,τj≃0.8437652<1,0.8437652<1-1-α-1-2α-α22≃0.9875.That is, conditions (62) and (63) are not satisfied for r=2.Figure 1
The graphs ofτ(n) and σ(n).
(a) (b)Similarly, ifp∈0.145,0.1564, then(79)limsupn→∞⁡∑i=σnnpi∏j=τiσn-111-p1j<1.For example, for p=29/200=0.145 and by using an algorithm on MATLAB software, we obtain (80)Φ1n=∑i=σnnpi∏j=τiσn-111-p1j=∑i=σnnpi∏j=τiσn-111-pj1+∑k=τjj-1pk∏m=τkσj-11/1-pmand therefore(81)Φ13μ+2=∑i=3μ-13μ+229200∏j=τi3μ-211-29/2001+∑k=τjj-129/200∏m=τkσj-11/1-29/200≃0.8529.Thus(82)limsupn→∞⁡Φ1n≃0.8529<1,0.8529<1-1-α-1-2α-α22≃0.9875.That is, conditions (65) and (66) are not satisfied for l=1.On the contrary, for everyp∈0.145,0.1564, we have(83)limsupn→∞⁡∑i=σnnpi∏j=τiσn-111-P1j>1.For example, for p=29/200=0.145 and by using an algorithm on MATLAB software, we obtain(84)F1n=∑i=σnnpi∏j=τiσn-111-P1j=∑i=σnnpi×∏j=τiσn-111-pj1+∑k=τjj-1pkexp⁡∑w=τkj-1pw∏m=τww-11/1-pmand therefore(85)F13μ+2=∑i=3μ-13μ+229200×∏j=τi3μ-211-29/2001+∑k=τjj-129/200exp⁡∑w=τkj-129/200∏m=τww-11/1-29/200≃1.0248.Thus(86)limsupn→∞⁡F1n=limsupn→∞⁡∑i=hnnpi∏j=τiσn-111-P1j≃1.0248>1;that is, condition (68) of Theorem 32 is satisfied for l=1. Therefore, all solutions of (72) are oscillatory.Observe that, forp=29/200=0.145, we obtain(87)limsupn→∞⁡∑j=σnnpj=limsupμ→∞⁡∑j=3μ-13μ+2pj=4·29200=0.58<1,α=0.145<1e,0.58<1-1-α-1-2α-α22≃0.9875,limsupn→∞⁡∑j=σnnpj∏i=τjσn-111-pi=limsupμ→∞⁡∑j=3μ-13μ+229200∏i=τj3μ-211-29/200=29200·limsupμ→∞⁡∏i=τ3μ-13μ-211-29/200+∏i=τ3μ3μ-211-29/200+∏i=τ3μ+13μ-211-29/200+∏i=τ3μ+23μ-211-29/200=29200·limsupμ→∞⁡∏i=3μ-53μ-211-29/200+∏i=3μ-13μ-211-29/200+∏i=3μ-13μ-211-29/200+∏i=3μ-23μ-211-29/200=29200·limsupμ→∞⁡11-29/2004+1+1+11-29/200≃0.7309<1,0.7309<1-1-α-1-2α-α22≃0.9875.That is, conditions C1′, C2′, C5′, (52)≡(62) (for r=1) and (55)≡(63) (for r=1) are not satisfied.Also, it is clear that conditionC4 is not applicable for (72).In addition, by using an algorithm on MATLAB software, we obtain(88)limsupn→∞⁡∑i=σnnpi∏j=τin11-P1j≃4.6956<21-α-1-2α-α2≃80.1448,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 C1′, (52)≡(62) (for r=1) and (65) (for l=1) 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 delaydoes not possess oscillatory solutions when p(n)<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.



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