# Oscillation of Fourth-Order Nonlinear Homogeneous Neutral Difference Equation

Authors: G. Sumitha; R. Kodeeswaran; S. Noeiaghdam; S. Balamuralitharan; V. Govindan
Journal: International Journal of Differential Equations
 (2022)

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## Abstract

In this paper, we establish the solution of the fourth-order nonlinear homogeneous neutral functional difference equation. Moreover, we study the new oscillation criteria have been established which generalize some of the existing results of the fourth-order nonlinear homogeneous neutral functional difference equation in the literature. Likewise, a few models are given to represent the significance of the primary outcomes.


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## Body

## 1. Introduction

The investigation of the conduct of solutions of functional difference equations is a significant space of examination and is quickly developing because of the advancement of time scales and the time-scale analytics (see, e.g., [1, 2]). Most papers on higher-order nonlinear neutral difference equations manage the presence of positive solutions and the asymptotic conduct of solutions. We refer the per user to a portion of the works [3–9] and the references referred to in that.In [10], Migda investigated the asymptotic properties of nonoscillatory solutions of neutral difference equation of the form(1)Δmxϱ+p∘ϱxϱ−ξ+fϱ,xςϱ=hϱ.The papers [9, 10] are tantamount. Be that as it may, more accentuation might be given to [9], which manages the oscillatory, nonoscillatory, and asymptotic characters. Some oscillation criteria have been set up by applying the discrete Taylor series [11]. The inspiration of the current work has come from two bearings [9, 12] and the second is expected to [13].In 2008, Tripathy [9] investigated oscillatory and asymptotic behaviour of solutions of a class of fourth-order nonlinear neutral difference equations of the form(2)Δ2rnΔ2yn+pnyn−m+qnGyn−k=0.Also,(3)Δ2rnΔ2yn+pnyn−m+qnGyn−k=fn.Under the condition∑n=0∞n/rn<∞, for different ranges of pn and qn≥0. Also, no super linearity or sublinearity conditions are imposed on G∈CR,R. It is interesting to observe that the nature of the function rn influences the behaviour of solutions of (2) or (3).In 2013, Tripathy [12] discussed oscillatory behaviour of solutions of a class of fourth-order neutral functional differences (2) and (3) under the conditions ∑n=0∞n/rn=∞ and rn>0,qn>0.Parhi and Tripathy [13] studied oscillation of a class of nonlinear neutral difference equations of higher order and the behaviour of its solutions is studied separately.In this present work, we study the oscillation behaviour of the fourth-order nonlinear homogeneous neutral functional difference equation.


## 2. Oscillation Behaviour of Neutral Difference Equation

In this section, we establish the solution of the fourth-order homogeneous neutral functional difference equation of the form(4)Δ2r∘ϱΔ2y∘ϱ+p∘ϱy∘ϱ−ξ+q∘ϱGy∘ϱ−ς=0,where(1)
Δ is a difference operator(2)
Δy∘ϱ=y∘ϱ+1−y∘ϱ(3)
r∘ϱ,p∘ϱ,q∘ϱ,andy∘ϱ are real valued functions defined on Nϱ0=ϱ0,ϱ0+1,…,ϱ≥ϱ0≥0(4)
G∈Cℛ,ℛ is nondecreasing(5)
ξ,ς>0 are constants with(5)∑ϱ=0∞ϱr∘ϱ<∞.A solutiony∘ϱ of (4) is oscillatory if y∘ϱy∘ϱ+1≠0 for every integer ϱ>N>0. Otherwise, it is nonoscillatory. The NFDE (4) is oscillatory, if all its solutions are oscillatory.For the oscillation of (4), we define the operators(6)ℒ1u∘ϱ=Δℒ0u∘ϱ=Δu∘ϱℒ2u∘ϱ=r∘ϱΔℒ1u∘ϱℒ3u∘ϱ=Δℒ2u∘ϱℒ4u∘ϱ=Δℒ3u∘ϱ.Also, the notations(7)Dk∘,ε=∑t=εk∘−2k∘−ϵ−1ϵ−εr∘ϵEk∘,ε=∑t=εk∘−1k∘−ϵ−1ϵ+1−εr∘ϵℱk∘,ε=∑t=εk∘−1k∘−ϵ−1r∘ϵℳk∘,ε=∑t=εk∘−1ϵ+1−εϵ−εr∘ϵ..Lemma 1.
Let (5) hold andu∘be a real valued function withℒ4u∘ϱ≤0for largeϱ. Ifu∘ϱ>0, then one of these (8) to (9) holds and ifu∘ϱ<0, then one of these (9) to (13) holds, where(8)ℒ1u∘ϱ>0,ℒ2u∘ϱ>0,andℒ3u∘ϱ>0,(9)ℒ1u∘ϱ>0,ℒ2u∘ϱ<0,andℒ3u∘ϱ>0,(10)ℒ1u∘ϱ>0,ℒ2u∘ϱ<0,andℒ3u∘ϱ<0,(11)ℒ1u∘ϱ<0,ℒ2u∘ϱ>0,andℒ3u∘ϱ>0,(12)ℒ1u∘ϱ<0,ℒ2u∘ϱ<0,andℒ3u∘ϱ>0,(13)ℒ1u∘ϱ<0,ℒ2u∘ϱ<0,andℒ3u∘ϱ<0.Lemma 2.
Letu∘ϱ>0and Lemma 1hold. Then,C1ℛϱ≤u∘ϱ≤ϱC2forC1>0&C2>0, where(14)ℛϱ=∑s=ϱ∞s−ϱr∘s,for largeϱ.Theorem 1.
Letς≥2ξ0≤p∘ϱ≤d<∞. If (5) and(15)Gu∘u∘≥φ>0,u∘≠0∈ℛ,(16)Gu∘v∘≥Gu∘Gv∘,G−u∘=−Gu∘,u∘,v∘>0∈ℛ,(17)Gu∘+Gv∘≥λGu∘+v∘,λ>0,(18)Qϱ=minq∘ϱ,q∘ϱ−ξ,ϱ≥ξ,(19)limsupk∘⟶∞∑j∘=k∘−ξk∘Qj∘GDj∘−ς,k∘−ς>1+Gdλφ,(20)limsupk∘⟶∞∑j∘=k∘−ξk∘Qj∘GΕj∘−ς,k∘ς>1+Gdλφ,(21)limsupε⟶∞∑j∘=ε−ς−2ε−ς−1Qj∘Gℳj∘−ς,k∘−ς>1+Gdλφ,(22)limsupε⟶∞∑k∘=j∘+ξ−ς−2j∘+ξ−ςQj∘Gℳj∘−ς,k∘−ς>1+Gdλφ,hold, then (4) is oscillatory.Proof.
Lety∘ϱ be a nonoscillatory solution of (4) and y∘ϱ>0 for ϱ≥ϱ0>μ.
If we set(23)zϱ=y∘ϱ+p∘ϱy∘ϱ−ξ,then (4) becomes(24)ℒ4zϱ=−q∘ϱGy∘ϱ−ς<0.
Hence, we findϱ1≥ϱ0 such that ℒ1zϱ,1=1,2,3 are eventually of one sign on ϱ1,∞.
Consider the potential cases 2.3 to 2.6 of Lemma1.
Case 2.5. For k∘−1≥ε≥ϱ1, it follows from the Taylor series that(25)−zk∘=−zε−k∘−εΔzk∘+∑ϵ=εk∘−1ϵ+1−εΔ2zϵ≤∑ϵ=εk∘−1ϵ+1−εΔ2zϵ.
Also,(26)ℒ2zϵ−ℒ2zε=∑s=εϵ−1ℒ3zs≤ϵ−εℒ3zε,which implies(27)ℒ2zϵ≤ϵ−εℒ3zε.
That is,(28)Δ2zϵ≤ϵ−εℒ3zεr∘ϵ.
Consequently,(29)zk∘≥−∑ϵ=εk∘−ϵϵ+1−εϵ−εr∘ϵℒ3zε=−ℒ3zεℳk∘,ε.
Forj∘−ς≥k∘−ς+2≥ϱ1+2, the above disparity can be composed as(30)zj∘−ς≥−ℒ3zε−ςℳj∘−ς,k∘−ς.
Using (4), it is not difficult to check that(31)ℒ4zϱ+q∘ϱGy∘ϱ−ς+Gdℒ4zϱ−ξ+Gdq∘ϱ−ξGy∘ϱ−ς−ξ=0.
Expected to (16), (17) and (31) give(32)0≥ℒ4zj∘+Gdℒ4zj∘−ξ+λQj∘Gzj∘−ς≥ℒ4zj∘+Gdℒ4zj∘−ξ+λQj∘Gℳj∘−ς,k∘−ςG−ℒ3zε−ς.
That is,(33)λ∑j∘=ε−ς−2ε−ς−1Qj∘Gℳj∘−ς,k∘−ςG−ℒ3zε−ς≤−∑j∘=ε−ς−2ε−ς−1ℒ4zj∘+Gdℒ4zj∘−ξ≤−ℒ3zε−ς−Gdℒ3zε−ς−ξ≤−1+Gdℒ3zε−ς.
As a result,(34)∑j∘=ε−ς−2ε−ς−1Qj∘Gℳj∘−ς,k∘−ς≤1+Gdλ−ℒ3zε−ςG−ℒ3zε−ς≤1+Gdλφ,which is logical inconsistency to (21) due to (15).
Case 2.6. For k∘−1≥ε≥ϱ1, it follows from (25) that(35)zε=zk∘−k∘−εΔzk∘+∑ϵ=εk∘−1ϵ+1−εΔ2zϵ≥∑ϵ=εk∘−1ϵ+1−εΔ2zϵ.
Since(36)ℒ2zϵ−ℒ2zε=∑s=εϵ−1ℒ3zs≥ϵ−εℒ3zϵ−1,we have(37)ℒ2zϵ≥ϵ−εℒ3zϵ−1.
That is,(38)Δ2zϵ≤ϵ−εℒ3zεr∘ϵ.
Consequently,(39)zε≥∑ϵ=εk∘−ϵϵ+1−εϵ−εr∘ϵℒ3zϵ−1=ℒ3zk∘−2ℳk∘,ε.
Forj∘−ς≥k∘−ς+2≥ϱ1+2, the above disparity can be composed as(40)zk∘−ς≥ℒ3zj∘−ς−2ℳj∘−ς,k∘−ς.
Applying (16) and (17) in (31) yields that(41)0≥ℒ4zk∘+Gdℒ4zk∘−ξ+λQk∘Gzk∘−ς.
Expected to (40), the above equation becomes(42)λ∑k∘=j∘+ξ−ς−2j∘+ξ−ςQj∘Gℳj∘−ς,k∘−ςG−ℒ3zj∘−ς−2≤−∑k∘=j∘+ξ−ς−2j∘+ξ−ςℒ4zk∘+Gdℒ4zk∘−ξ≤ℒ3zj∘+ξ+ς−2+Gdℒ3zj∘−ς−2≤1+Gdℒ3zj∘−ς−2.
Therefore,(43)∑k∘=j∘+ξ−ς−2j∘+ξ−ςQk∘Gℳj∘−ς,k∘−ς≤1+Gdλℒ3zj∘−ς−2Gℒ3zj∘−ς−2≤1+Gdλφ,,which is logical inconsistency to (22) due to (15).
Cases 2.3 and 2.4 can be managed also as the above cases.
Finally,y∘ϱ<0 for ϱ≥ϱ0. Using (16) and xϱ=−y∘ϱ in (4), we obtain xϱ>0 and(44)Δ2r∘ϱΔ2xϱ+p∘ϱxϱ−ξ+q∘ϱGxϱ−ς=0.
Proceeding as above, we see that every solution of (44) oscillates.
This completes the proof of theorem.Theorem 2.
Let−1≤p∘ϱ≤0andς≥2ξ. If (5) to (16) and(45)limsupk∘⟶∞∑j∘=k∘−ξk∘q∘j∘GDj∘−ς,k∘−ς>1φ,(46)limsupk∘⟶∞∑j∘=k∘−ξk∘q∘j∘GΕj∘−ς,k∘−ς>1φ,(47)limsupε⟶∞∑j∘=ε−ς−2ε−ς−1q∘j∘Gℳj∘−ς,k∘−ς>1φ,(48)limsupj⟶∞∑k∘=j∘+ξ−ς−2j∘+ξ−ςq∘j∘Gℳj∘−ς,k∘−ς>1φ,(49)limsupk∘⟶∞∑j∘=k∘+ξ−ςk∘+ξ−ςq∘j∘Gℱk∘,+ξ−ς,j∘+ξ−ς>1φ,(50)limsupl⟶∞∑j∘=ε+ξ−ςε+ξ−ς+1q∘j∘Gℳk∘,+ξ−ς,j∘+ξ−ς>1φ,(51)limsupk∘⟶∞∑j∘=k∘+ξ−ς−2k∘q∘j∘Gℳj∘+ξ−ς,k∘+ξ−ς>1φ,hold, then every solution of (4) oscillates.Proof.
Let the opposition thaty∘ϱ is a nonoscillatory solution of (4) and y∘ϱ>0 for ϱ≥ϱ0>μ.
The casey∘ϱ<0 for ϱ≥ϱ0>μ can be dealt with similarly.
Settingzϱ by (23), we get (24).
Consequently, we findϱ1>ϱ0 such that zϱ and ℒizϱ,i=1,2,3 are eventually of one sign on ϱ1,∞.
Letzϱ>0. Then, zϱ≤y∘ϱ for ϱ≥ϱ2>ϱ1 and (24) becomes(52)ℒ2zϱ+q∘ϱGzϱ−ς≤0.
Applying Lemma1 to (52) and Theorem 1, we get contradictions to (45) to (48) expected to ς>2ξ>ξ.
Next,zϱ<0 for ϱ≥ϱ1 and zϱ≥−y∘ϱ−ξ for ϱ≥ϱ2≥ implies y∘ϱ−ς≥−zϱ+ξ−ς.
By Lemma1, the cases of 2.4 to 2.8 hold for.
Case 2.4. Since(53)−ℒ2zϵ≥ℒ2zk∘−1−ℒ2zϵ=∑s=lk∘−2ℒ3zs≥k∘−ϵ−1ℒ3zk∘−2.,it follows that(54)−Δ2zϵ≥k∘−ϵ−1r∘ϵℒ3zk∘−2.
Fork≥ϵ+2>ϱ1, summing the above inequality from ϵ=ε to k∘−1, we have(55)Δzε≥ℒ3zk∘−2∑ϵ=εk∘−1k∘−ϵ−1r∘ϵ.
That is,(56)zε+1−zε≥ℒ3zk∘−2∑ϵ=εk∘−1k∘−ϵ−1r∘ϵ,which implies(57)−zε≥ℒ3zk∘−2∑ϵ=εk∘−1k∘−ϵ−1r∘ϵ=ℒ3zk∘−2ℱk∘,ε≥ℒ3zk∘ℱk∘,ε,for k≥ϵ+2>ϱ2.
Therefore, fork∘+ξ−ς≥j∘+ξ−ς≥ϵ+ξ−ς+2>ϱ2,(58)−zj∘+ξ−ς≥ℒ3zk∘+ξ−ςℱk∘,+ξ−ς,j∘+ξ−ς.
Since (4) can be viewed as(59)ℒ4zj∘+q∘j∘G−zj∘+ξ−ς≤0,using (58), (59), and (16) yields(60)ℒ4zj∘+q∘j∘Gℒ3zk∘+ξ−ςGℱk∘,+ξ−ς,j∘+ξ−ς≤0.
Summing the last inequality formj∘=k∘+ξ−ς to k∘+ς−ξ, we get(61)Gℒ3zk∘+ξ−ς∑j∘=k∘+ξ−ςk∘+ς−ξq∘j∘Gℱk∘,+ξ−ς,j∘+ξ−ς≤ℒ3zk∘+ξ−ς.
Hence,(62)limsupk∘⟶∞∑j∘=k∘+ξ−ςk∘+ς−ξq∘j∘Gℱk∘,+ξ−ς,j∘+ξ−ς≤1φ,which gives the contradiction to (49).
Case 2.5. From (25), we have(63)−zε=−zk∘+k∘−εΔzk∘−∑ϵ=εk∘−1ϵ+1−εΔ2zϵ≥−∑ϵ=εk∘−1ϵ+1−εΔ2zϵ,for k∘−1≥ε≥ϱ1 and(64)ℒ2zϵ−ℒ2zε=∑s=εϵ−1ℒ3zs≤ϵ−εℒ3zε,which implies that(65)ℒ2zϵ≤ϵ−εℒ3zε.
That is,(66)−Δ2zϵ≥ϵ−εr∘ϵℒ2zε.
Consequently,(67)−zε≥−∑ϵ=εk∘−1ϵ+1−εϵ−εr∘ϵℒ3zε=−ℒ3zεℳk∘,ε.
Also hence,(68)−zj∘+ξ−ς≥−ℒ3zj∘+ξ−ςℳk∘,+ξ−ς,j∘+ξ−ς≥−ℒ3zj∘+ξ−ς+2ℳk∘,+ξ−ς,j∘+ξ−ς,holds for k∘+ξ−ς≥j∘+ξ−ς≥ϵ+ξ−ς+2>ϱ2, using (16) and (68) in (59) and summing from ϵ+ξ−ς to ϵ+ξ−ς+1, we obtain(69)G−ℒ3zϵ+ξ−ς+2∑j∘=ϵ+ξ−ςϵ+ξ−ς+1q∘j∘Gℳk∘,+ξ−ς,j∘+ξ−ς≤−ℒ3zϵ+ξ−ς+2.
Therefore,(70)∑j∘=ϵ+ξ−ςϵ+ξ−ς+1q∘j∘Gℳk∘,+ξ−ς,j∘+ξ−ς≤−ℒ3zϵ+ξ−ς+2G−ℒ3zϵ+ξ−ς+2≤1φ,which is logical inconsistency to (50).
Case 2.6. We use (25) and it follows that(71)−zk∘=−zε+k∘−εΔzk∘+∑ϵ=εk∘−1ϵ+1−εΔ2zϵ≥∑ϵ=εk∘−1ϵ+1−εΔ2zϵ,for k∘−1≥ε≥ϱ1. Since(72)ℒ2zϵ−ℒ2zε=∑s=εϵ−1ℒ3zs≤ϵ−εℒ3zϵ−1,we have(73)Δ2zϵ≥ℒ3zϵ−1ϵ−εr∘ϵ,which implies that(74)−zk∘≥∑ϵ=εk∘−1ϵ+1−εℒ3zϵ−1ϵ−εr∘ϵ=ℒ3zk∘−2ℳk∘,ε.
Hence, forj∘+ξ−ς≥k∘+ξ−ς+2>ϱ1+2, it follows that(75)−zj∘+ξ−ς≥ℒ3zj∘+ξ−ς−2ℳj∘+ξ−ς,k∘+ξ−ς.
Consequently, (59) becomes(76)∑j∘=ϵ+ξ−ς−2k∘q∘j∘Gℳj∘+ξ−ς,k∘+ξ−ςGℒ3zj∘+ξ−ς−2≤ℒ3zk∘+ξ−ς−2.
As a result,(77)∑j∘=ϵ+ξ−ς−2k∘q∘j∘Gℳj∘+ξ−ς,k∘+ξ−ς≤ℒ3zk∘+ξ−ς−2Gℒ3zj∘+ξ−ς−2≤1φ,which is logical inconsistency to (51).
In both cases 2.7 and 2.8, we see thatlimn⟶∞zϱ=−∞.
On the other hand,zϱ<0 for ϱ≥ϱ1 implies y∘ϱ≤y∘ϱ−ξ for ϱ≥ϱ1.
That is,(78)y∘ϱ≤y∘ϱ−ξ≤y∘ϱ−2ξ≤⋯≤y∘ϱ1.
Hence,y∘ϱ is bounded and ⇒zϱ is bounded, which is a contradiction.
This completes the proof of the theorem.Example 1.
Consider the NFDE,(79)Δ2ϱeϱΔ2y∘ϱ+p∘ϱy∘ϱ−1+q∘ϱGy∘ϱ−3=0,ϱ>3.
Here,(80)p∘ϱ=e−2+e−ϱ;q∘ϱ=e2−12e+12e+ϱe+ϱeϱ−e+12ϱ+1;,r∘ϱ=ϱeϱandgu∘=4ue2=φu.
Clearly, all conditions of Theorem1 are satisfied.
Hence, (79) is oscillatory and y∘ϱ=−1ϱ is one of the oscillatory solutions of (79).


## 3. Conclusion

In this paper, we inferred new properties of the nonoscillatory solutions and using these outcomes, some new adequate are introduced for the concentrated on NFDE to have the purported property oscillatory. Our outcomes improve and supplement many known outcomes for NFDEs as well as for ordinary functional difference equations also. At last, we give two models that show the meaning of the fundamental results.



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