# Modified Oscillation Results for Advanced Difference Equations of Second-Order

Authors: G. E. Chatzarakis; N. Indrajith; S. L. Panetsos; E. Thandapani
Journal: Journal of Applied Mathematics
 (2022)

---

## Abstract

In this paper, we present a new method to establish the oscillation of advanced second-order difference equations of the formΔηℓΔυℓ+ρℓυσℓ=0, using the ordinary difference equation ΔηℓΔυℓ+qℓυℓ+1=0. The obtained results are new and improve the existing criteria. We provide examples to illustrate the main results.


---

## Body

## 1. Introduction

This paper is concerned with the second-order advanced difference equation(1)ΔηℓΔυℓ+ρℓυσℓ=0,ℓ≥ℓ0,where ℓ∈ℕℓ0=ℓ0,ℓ0+1,ℓ0+2,⋯,, ℓ0 is a nonnegative integer, and(C1) ηℓ, ρℓ, and qℓ are positive real sequences for ℓ≥ℓ0(C2) σℓ is a monotone increasing sequence of integers with σℓ≥ℓ+1 for ℓ≥ℓ0(C3) Ωℓ=∑s=ℓ0ℓ−11/ηs⟶∞ as ℓ⟶∞By a solution ofE, we mean a nontrivial sequence υℓ that satisfies E for all ℓ≥ℓ0. A solution υℓ of E is said to be oscillatory if it is neither eventually negative nor eventually positive. Otherwise, it is said to be nonoscillatory. Equation (1) is called oscillatory if all its solutions are oscillatory.Oscillation phenomena take part in different models described by various differential equations, partial differential equations, and dynamic equations on time scales; see, for instance, the papers [1–6] for more details. In particular, we refer the reader to the papers [4, 6] for models from mathematical biology and physics where oscillation and/or delay actions may be formulated by means of cross-diffusion terms. In recent years, there have appeared several criteria on the oscillation of (1) for the retarded case, that is, σℓ≤ℓ−1 and limℓ⟶∞σℓ=∞, using either comparison methods or/and Riccati transformation technique. On the majority, these studies use the comparison methods, which is considered to be the most powerful tool in the oscillation theory of difference equations (see, for example, [7–17] and the references cited therein). Another particular method appearing in several studies is also the summation averaging method (see, for example, [1, 18–21] and the references cited therein).From the literature, it is well known that not many oscillation results are available by using comparison methods. In [22, 23], the authors obtained oscillation of the advanced difference equation (1) from that of the ordinary difference equation
(2)ΔηℓΔυℓ+ρℓυℓ+1=0,without explicitly using the information about the advanced argument σℓ. Very recently in [22], the authors studied the oscillation of E by assuming that ηℓ≥1 for all ℓ≥ℓ0.In this paper, we present a new method which produces the oscillation of (1) without the restriction ηℓ≥1. Thus, our results generalize and complement to those reported in [14, 15, 22, 23].


## 2. Main Results

Without loss of generality, in studying the nonoscillatory solutions of (1), we can restrict our attention only to positive solutions.Lemma 1.
Letυℓ be a positive solution of (1). Then, ηℓΔυℓ>0 and ΔηℓΔυℓ<0, eventually.Proof.
The proof can be found in Lemma1, [22], and the details are omitted.Next we present an oscillation criterion for equation (2) which will be used to prove our main results.Lemma 2.
Assume that(3)limℓ⟶∞∑s=ℓ0ℓρsis convergent and υℓ is a positive solution of E1. Then, there is an integer ℓ1∈ℕℓ0 such that
(4)ωℓ≥∑s=ℓ∞ρs+∑s=ℓ∞ωs+1ωsηs,forℓ∈ℕℓ1,where ωℓ=ηℓΔυℓ/υℓ for ℓ≥ℓ1.Proof.
From Lemma1, there is an ℓ1∈ℕℓ0 such that ωℓ>0 for all ℓ≥ℓ1. Taking into account the fact that ηℓΔυℓ is positive and decreasing, we have
(5)Δωℓ=ΔηℓΔυℓυℓ+1−ηℓΔυℓυℓυℓ+1Δυℓ=−ρℓ−ωℓηℓΔυℓηℓυℓ+1≤−ρℓ−ωℓωℓ+1ηℓ.Summing up the last inequality fromℓ to j and then taking j⟶∞, we obtain
(6)ωℓ≥∑s=ℓ∞ρs+∑s=ℓ∞ωsωs+1ηs,for all ℓ∈ℕℓ1.Now let us define(7)Qℓ,0=∑s=ℓ∞ρs,j=0,ℓ≥ℓ1,(8)Qℓ,j=∑s=ℓ∞Qs+1;j−1Qs;j−1ηs+Qℓ;0,j≥1,ℓ≥ℓ1.Theorem 3.
Let condition (3) hold. Then, (2) is oscillatory provided one of the following two conditions holds:
(H1) There exists an integer j∈ℕ such that Qℓ;0,⋯,Qℓ;j−1 defined by (8) satisfying
(9)∑s=ℓ1∞Qs+1,j−1Qs,j−1ηs=∞.
(H2) There exists an integer ℓ≥ℓ1 such that
(10)limj⟶∞supQℓ;j=∞.Proof.
Assume, for the sake of contradiction, that equation (2) is nonoscillatory. Let (H1) hold, and let υℓ be a positive solution of (2) for all ℓ∈ℕℓ1. By Lemma 2, we have for j=0(11)ωℓ≥∑s=ℓ∞ρs+∑s=ℓ∞ωsωs+1ηs≥Qℓ,0,ℓ∈ℕℓ1.
Thus,(12)∑s=ℓ1∞Qs+1,0Qs,0ηs≤∑s=ℓ1∞ωs+1ωsηs≤ωℓ1
by (11), which contradicts (9). Similarly for j≥1 from (8) and (4), we have
(13)ωℓ≥Qℓ,jforj=1,2,⋯and hence,
(14)∑s=ℓ1∞Qs+1,jQs,jηs≤∑s=ℓ1∞ωs+1ωsηs≤ωℓ1<∞,which is again a contradiction.
Next, suppose that (H2) hold. Clearly, in view of ωℓ≥Qℓ;j for j=1,2,⋯, we get from (4) that limj⟶∞Qℓ1;j≤ωℓ1<∞, which is a contradiction. The proof of the theorem is complete.Theorem 4.
Assume that(15)Ωℓ∑s=ℓ∞ρs≥β>14,
eventually. Then,E is oscillatory.Proof.
From (15) and (8) for j=0, we have Qℓ;0≥β/Ωℓ. From (8) for j=1, we get
(16)Qℓ;1=∑s=ℓ∞Qs+1,0Qs,0ηs+Qℓ,0≥β2∑s=ℓ∞1Ωs+1Ωsηs+βΩℓ=β2∑s=ℓ∞Δ−1Ωs+βΩℓ=β+β2Ωℓ.
Hence,(17)Qℓ;1≥β1Ωℓ,
whereβ1=β+β2.
In general,(18)Qℓ;j=∑s=ℓ∞Qs+1,j−1Qs,j−1ηs+Qℓ;0≥βjΩℓ,
whereβj=β+βj−12 and j=1,2,⋯. Clearly, β=β0<β1<β2<⋯. If βj converges to some positive number α, then
(19)α=β0+α2.
But there is no real positive solution for such an equationβ0>1/4. Thus, limj⟶∞βj=∞. Then, we have limj⟶∞Qℓ;j=∞. Thus, from Theorem 3, it follows that (2) is oscillatory. Now using Theorem 3.5 of [23], we see that (1) is oscillatory. The proof of the theorem is complete.Next assume that the opposite condition of (15), namely,
(20)Ωℓ∑s=ℓ∞ρs≥βbutβ≤14,holds.Theorem 5.
Letυℓ be a positive solution of (1) and
(21)Ωℓ∑s=ℓ∞ρs≥β>0,
eventually. Then, there is an integerL such that for ℓ≥L,
(22)υℓΩβℓ,
is monotonically nondecreasing.Proof.
The proof is similar to that of Theorem3 of [22], and hence, the details are omitted.Next we state a comparison result, containing an advanced argument.Theorem 6.
Let (21) hold. If the difference equation
(23)ΔηℓΔυℓ+ΩσℓΩℓ+1βρℓυℓ+1=0
is oscillatory, then (1) is oscillatory.Proof.
The proof is similar to that of Theorem4 in [22] and hence omitted.The above theorem ensures that any oscillation criterion established for (23) leads to an oscillation criterion for (1).Theorem 7.
Let (21) hold. Assume that there is a constant β1 such that
(24)Ωℓ∑s=ℓ∞ΩσsΩs+1βρs≥β1>14,
eventually. Then, (1) is oscillatory.Proof.
The condition (24) guarantees that (23) oscillates, which in turn implies that (1) is oscillatory. This completes the proof.Next, we provide an example, illustrating this result.Example 1.
Consider the second-order advanced Euler type difference equation(25)Δ1ℓ+1Δυℓ+bℓℓ+1ℓ+2υλℓ=0,ℓ≥1,
withb>0 and λ≥2 is an integer.NowΩℓ≃ℓ2/2. With β=b/4 and by Theorem 7, equation (25) is oscillatory provided that
(26)bλb/4>1.For example, ifb=1/2, then it is required that λ≥18.Note that the result in [22] cannot be applicable to (25) since ηℓ=1/ℓ+1<1.If condition (24) fails to hold β1≤1/4, then we can derive a new oscillation criterion using the constant β1.Theorem 8.
Let (21) hold. Assume that υℓ is a positive solution of (1) and
(27)Ωℓ∑s=ℓ∞ΩσsΩs+1βρs≥β1>0,
eventually. Then,(28)υℓΩβ1ℓ
is monotonically nondecreasing.Proof.
Use [22], Theorem 2.6, to complete the proof.Theorem 9.
Let (21) and (24) hold. If the difference equation
(29)ΔηℓΔυℓ+ΩσℓΩℓ+1β1ρℓυℓ+1=0
is oscillatory, then so is (1).Theorem 10.
Let (21) and (24) hold. If there exists a constant β2 such that
(30)Ωℓ∑s=ℓ∞ΩσsΩs+1β1ρs≥β2>14
eventually, then (1) is oscillatory.The proofs of Theorems9 and 10 follow from Theorems 6 and 7, and hence, the details are omitted.Example 2.
Consider the difference equation (25). For this equation β1=bλ2b. By Theorem 10, we see that (25) is oscillatory provided that
(31)bλ2β1>14.
Sinceβ1>b, Theorem 10 improves Theorem 7.For convenience, let us use the additional condition that there is a positive constantδ such that
(32)ΩσℓΩℓ+1≥δ>1,eventually. In view of (21), conditions (24) and (30) can be written in simpler form as
(33)β1=δββ>14,(34)β2=δβ1β>14,respectively. Repeating the above process, we have the increasing sequenceβjj=0∞ defined as β0=β(35)βj+1=δβjβ.Now as in [22], Theorem 2.9, one can generalize the oscillation criteria obtained in Theorems 7 and 10.Theorem 11.
Let (21) and (32) hold. If there exists a positive integer m such that βj≤1/4 for j=0,1,⋯,m−1 and βm>1/4, then (1) is oscillatory.Example 3.
Consider the second-order advanced difference equation(36)Δ1ℓ+1Δυℓ+0.5ℓℓ+1ℓ+2υ5ℓ=0,ℓ≥1.For this equationβ0=1/8 and δ=25. Through direct calculations, we get
(37)β1=0.18692,β2=0.22815.Thus, Theorems7 and 10 fail for equation (36). But
(38)β3=0.26053>0.25,and hence, Theorem 11 guarantees the oscillation of (36).


## 3. Conclusion

In this paper, we have derived a new comparison method to obtain the oscillation of second-order advanced difference equation which removed the restriction imposed on the coefficientηℓ such that ηℓ≥1 as in [22]. Thus, the oscillation criteria obtained in this paper improved and complemented to the existing results. It is an interesting problem to extend the results of this paper to equation (1) when it is in noncanonical form.



---

*Source: 3407776-2022-06-30.xml*