# Oscillation of Third-Order Nonlinear Generalized Difference Equation with Multiple Neutral Terms

Authors: M. Sumathy; P. Venkata Mohan Reddy; M. Maria Susai Manuel; M. Syed Ali
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In this paper, the authors discuss the oscillatory behaviour of a third order generalized difference equation with multiple neutral terms. We have also applied Riccati transformation and Philo type technique to derive new oscillation criteria for the difference equation in question. Suitable examples are provided to validate our main results.


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## Body

## 1. Introduction

The theory of difference equations has grown immensely over the past few decades. In the field of probability theory, statistical analysis, combinatorial analysis, electrical networks, and sociology, difference equations has emerged as mathematical models describing real life challenges.In the recent past, the study of oscillation and nonoscillation for second order nonlinear difference equations has garnered a great deal of attention [1, 2]. The latest research also emphasizes the different kinds of difference equations, including ordinary, linear, nonlinear, superlinear, quasilinear, sublinear, delay, and neutral delay difference equations. Interestingly, one can refer the oscillatory behavior for sublinear neutral delay second and third order difference equations in [3, 4]. An investigation of the oscillation of the second-order quasilinear neutral delay difference equations can be seen in [5, 6]. The study of the Oscillation of second order half-linear difference equations has also been given in [7]. Importantly, the oscillation criteria for higher order neutral equations can be seen in [8].The literature regarding the present study is referred in [9, 10]. In a revealing manner, tracking a maneuvering target, and fault diagnosis of wind turbine gearbox is performed in [11] by using the properties of Riccati difference equation. In [12], for the study of the Mittag-Leffer stability analysis of fractional discrete time neural networks, a class of semilinear fractional difference equations are used.A few applications of specific kinds of nonlinear third order delay difference equations are prominent in the study of Mathematical Biology, Economics, and many other fields of Mathematics that involve discrete models [13–16]. Moreover, oscillatory solution of third order delay difference equations are used to remove speckle noise in the field of image processing which can be seen in [17]. The suitable smoothing filter for the edge mask computed using the third order difference equation is examined in [18].Our main focus in this paper is on the oscillatory behavior of the third-order difference equations. In the earlier research, plethora of methods about the oscillatory property of third order difference equation were presented. For instance, in [19] the third order difference equation under consideration is as follows:(1)ΔaξΔbξΔxξ+qξfxξ−m+1=hξ,ξ≥ξ0,where aξ,bξ,qξ are positive real sequences, m is positive integer, f∈CR,R with ufu>0 for u≠0 and ∑ξ=ξ0∞1/aξ=∑ξ=ξ0∞1/bξ=∞. Using the Riccati transformation technique, sufficient conditions for the existence of oscillatory solutions are provided. In [20], the authors considered the following equation:(2)ΔcξΔdξΔyξ+pξyξ−k+qξfxξ−m=eξ,ξ≥ξ0−m.Withcξ,dξ as positive real sequences such that ∑ξ=ξ0∞1/cξ=∑ξ=ξ0∞1/dξ=∞. Some sufficient conditions are established for the oscillation of solutions of equation (2) by using Riccati transformation technique. In [21], the authors considered the nonlinear delay difference equation as follows:(3)ΔaξΔbξΔxξα1α2+qξfxσξ=0,ξ≥ξ0,where aξ,bξ,qξ are positive real sequences, σξ is sequence of integers such that ∑ξ=ξ0∞1/aξ1/α2=1/bξ1/α1=∞. By reducing the order of the equation, the main results are obtained, as well as certain sufficient conditions are provided for the oscillation of solutions of equation (3) by adopting Riccati transformation.Furthermore, in [4], the oscillation property of the generalized third order sublinear neutral delay difference equation of the form is as follows:(4)Δlα2ξΔlα1ξΔlzξ+qξxβk−σl=0,is discussed using Riccati type transformations.For further study on third order difference equations, one can refer [22] also. Motivated by the above literature available on the oscillation criteria for different class of difference equations involving the conventional difference operator Δ, we wish to generalize the results for the more generalized difference equation involving the generalized difference operator Δl. Hence, in this paper we consider the third-order non-linear generalized difference equations with multiple neutral terms of the form(5)Δla1ξΔla2ξΔlxξγ1γ2+∑j=1ncjξyγ3gjξ=0,0<ξ0<ξ,where xξ=yξ+bξyf¯ξ, γi>0 for i=1,2,3 is a ratio of odd positive integers. Here, Δl is the generalized difference operator defined by Δlxξ=xξ+l−xξ≡zξ, ξ∈Nlξ0=ξ0,ξ0+l,ξ0+2l,⋯, ξ0∈0,∞, l∈0,∞ and its inverse is defined by the following equation:(6)xξ=xξ0+j+∑r=0ξ−ξ0−j−l/lzk0+j+rl,k∈Nlj.For the validity of our discussion, we consider the following conditions on equation (5)(i)
aiξ is a positive sequence and ∑s=ξ0∞ai−1/γ3−is=∞, for i=1,2.(ii)
bξ is a real sequence with 1<bξ.(iii)
ciξ>0.(iv)
f¯ξ is strictly increasing with ξ>f¯ξ, and limξ⟶∞f¯ξ=∞.(v)
gjξ with limξ⟶∞gjξ=∞ where j=1,2,…,n.(vi)
miξ=ξ−ξi−j−l/l, ξi¯=ξi+j, and j=ξ−ξ0−ξ−ξ0/ll.We focus only on the solutionsyξ of equation (5), defined for some ξ0<ξ1 satisfying supyξ:ξ>K>0 for all K>ξ. We assume that a proper solution exists for equation (5). A proper solution yξ of equation (5) is said to be oscillatory if yξyξ+l<0 for large ξ and is nonoscillatory otherwise. Also, if all the solutions of equation (5) are oscillatory, then the equation itself is oscillatory.Forthwith, in this paper we establish necessary conditions for the oscillation of solutions of third-order nonlinear generalized difference equations with multiple neutral terms. Many well-known oscillation criteria that have appeared in the prevailing research have been extended and improved by the results presented here.


## 2. Main Results

The following notations and lemmas are useful in order to prove our main results.(7)ϕ1ξ=1bf¯−1ξ1−1bf¯−1f¯−1ξ,(8)ϕ2ξ=1bf¯−1ξ1−R3f¯−1f¯−1ξ,ξ2bf¯−1f¯−1ξR3f¯−1ξ,ξ2,(9)Φ1ξ=∑j=1ncjξϕ1γ3giξ,Φ2ξ=∑j=1ncjξϕ2γ3giξ,R1ξ,ξ1=∑r=0m1ξ1a11/γ2ξ1¯+rl,R2ξ,ξ1=R1ξ,ξ1a2ξ1/γ1forξ≥ξ1,R3ξ,ξ2=∑r1=0m2ξR2ξ2¯+r1l,ξ1forξ≥ξ2>ξ1.Lemma 1.
Ifyξ is an eventually positive solution of equation (5), then xξ satisfies either C1. xξ>0, Δlxξ>0, a2ξΔlxξγ1>0 and a1ξa2ξΔlxξγ1γ2≤0, or C2. xξ>0, Δlxξ<0, a2ξΔlxξγ1>0, and a1ξa2ξΔlxξγ1γ2≤0.Proof.
Letyξ be a positive solution of equation (5) for every ξ≥ξ0. By defining xξ, with xξ≥yξ>0 for ξ≥ξ1∈Nlξ0, from equation (5), we have the following equation:(10)Δla1ξΔla2ξΔlxξγ1γ2=−∑j=1ncjξyγ3gjξ<0.
This impliesa1ξΔla2ξΔlxξγ1γ2 is decreasing on ξ1,∞, which is either positive or negative. Furthermore, we must have to prove that a1ξΔla2ξΔlxξγ1γ2>0 for ξ≥ξ1≥ξ0. Otherwise, we have a constant L1γ2>0 in such a way that(11)Δla2ξΔlxξγ1<−L1a11/γ2ξ<0,forξ≥ξ1.
Hence, by equation (6)(12)a2ξΔlxξγ1≤a2ξ1¯Δlxξ1¯γ1−L1∑r=0m1ξ1a11/γ2ξ1¯+rl.
Lettingξ⟶∞, then using condition (i), we have limξ⟶∞a2ξΔlxξγ1=−∞. Subsequently, there exists a ξ2≥ξ1 also a constant L2γ1>0, so that(13)a2ξΔlxξγ1<−L2γ1,forξ≥ξ2.
Dividing the above inequality bya2ξ and by applying summation from ξ2 to ξ−l, we get the following equation:(14)xξ<xξ2¯−L2∑t=0m2ξ1a21/γ2ξ2¯+tl.
Lettingξ⟶∞ and using condition (i), we have xξ⟶−∞. Thus, xξ<0 eventually which is contradictory to the fact xξ>0, which gives Δla2ξΔlxξγ1 is positive, that is Δla2ξΔlxξγ1γ2>0 holds.
It can be known fromΔla2ξΔlxξγ1>0, is a monotonically increasing in the interval ξ2,∞. Therefore Δlxξ is ultimately either positive or negative. Hence, we obtain either C1 or C2 for xξ, which completes the proof.Lemma 2.
Let equation (7) holds, and let yξ be an eventually positive solution of equation (5) with xξ satisfying condition C2 of Lemma 1. If(15)∑r=0∞1a21/γ1ξ5¯+rl∑s=rm4ξ1a11/γ2ξ4¯+sl∑t=sm3ξΦ1ξ3¯+tl1/γ21/γ1=∞,
Then,yξ of equation (5) converges to zero when ξ⟶∞.Proof.
Letyξ be an eventually positive solution of equation (5). Then, there exits ξ∈Nlξ0 such that, for ξ1≤ξ, 0<yξ, 0<yf¯ξ, 0<yg1ξ for i=1,2,…,n. From the definition of xξ, we obtain the following equation:(16)yξ=1bf¯−1ξxf¯−1ξ−yf¯−1ξ=xf¯−1ξbf¯−1ξ−xf¯−1f¯−1ξ−yf¯−1f¯−1ξbf¯−1ξbf¯−1f¯−1ξ≥xf¯−1ξbf¯−1ξ−xf¯−1f¯−1ξbf¯−1ξxf¯−1f¯−1ξ.
Throughf¯ξ<ξ, from (iv) and the fact thatxξ is decreasing, we obtain the following equation:(17)xf¯−1f¯−1ξ≤xf¯−1ξ.
Using this in equation (16), we get the following equation:(18)yξ≥ϕ1ξxf¯−1ξ,(19)ygiξ≥ϕ1giξxf¯−1giξ,i=1,2,…,n,for ξ2≤ξ. Using equation (19) in equation (5) we get the following equation:(20)Δla1ξΔla2ξΔlxξγ1γ2+∑j=1ncjξϕ1γ3giξxγ3f¯−1giξ≤0,for ξ2≤ξ. From (iv)-(v), xξ is decreasing, and from equation (20) we obtain the following equation:(21)Δla1ξΔla2ξΔlxξγ1γ2+xγ3f¯−1ξ∑j=1ncjξϕ1γ3giξ≤0,forξ2≤ξ.
Sincexξ>0 and Δlxξ<0, a constant M exists, such that(22)limξ⟶∞xξ=M<∞,where M≥0. If M>0, then there exists ξ2≤ξ3, such that ξ2<f¯−1f¯−1ξ and(23)xξ=M,forξ3≤ξ.
Summing twice equation (21) from ξ3≤ξ4≤ξ to ∞, we get the following equation:(24)−Δlzξ≥Mγ3/γ1γ2a21/γ1ξ4¯∑s=0∞1a11/γ2ξ4¯+sl∑t=sm3ξΦ1ξ3¯+tl1/γ21/γ1.
Summing the above inequality fromξ5 to ∞, we obtain the following equation:(25)zξ≥Mγ3/γ1γ2∑r=0∞1a21/γ1ξ5¯+rl∑s=rm4ξ1a11/γ2ξ4¯+sl∑t=sm3ξΦ1ξ3¯+tl1/γ21/γ1,which is contradictory to equation (15), and hence it is true that M=0. Hence, limξ⟶∞xξ=0. Since 0<yξ≤xξ for ξ1≤ξ<∞, it is clear that limξ⟶∞yξ=0.Theorem 1.
Letgjξ≤f¯ξ for j=1 to n, Φ1ξ>0, Φ2ξ>0 and equation (15) hold. If there exists functions hξ and Hξ∈CNlξ0,R such that a2ξHξ∈CNlξ0,R and(26)∑s=0∞Aξ3¯+sl−MB1+γ1γ2ξ3¯+slCγ1γ2ξ3¯+sl=∞,forξ2<ξ3≤ξ,where(27)M=γ1γ2γ1γ21+γ1γ21+γ1γ2,(28)ψξ=N1,ifγ1γ2≤γ3,N2R3ξ,ξ2γ3/γ1γ2−1,ifγ3<γ1γ2,(29)Aξ=−hξΔla1ξHξ+hξ∑j=1ncjξϕ2gjξγ3R3f¯−1gjξ,ξ2R3ξ,ξ2γ3+γ3γ1γ2hξR3ξ,ξ2ψξa1ξ+lHξ+l1/γ1γ2+1,(30)Bξ=Δlhξhξ+l+γ3+γ3γ1γ2hξR3ξ,ξ2ψξa1ξ+lHξ+l1/γ1γ2hξ+l,(31)Cξ=γ3hξR3ξ,ξ2ψξhξ+l1/γ1γ2+1,for ξ2<ξ, then every solution of equation (5) is either oscillatory or approaches zero as ξ tends to ∞.Proof.
Let equation (5) have a nonoscillatory solution yξ on Nlξ0. Let us assume that there exists ξ1∈Nlξ0 such that, for ξ≥ξ1, yfξ>0, and ygjξ>0, equations (7) and (8) hold and xξ satisfies either C1 or C2 for j=1 to n. Equation (16) follows from the proof of Lemma 2 with the assumption that C1 holds. Since a1ξΔla2ξΔlxξγ1γ2 is decreasing, we observe that(32)a2ξΔlxξγ1=a2ξ1¯Δlzξ1¯γ1+∑r=0m1ξΔla2ξ1¯+rlΔlxξ1¯+rlγ1≥∑r=0m1ξa11/γ2ξ1¯+rlΔla2ξ1¯+rlΔlxξ1¯+rlγ1a11/γ2ξ1¯+rl≥a11/γ2ξ+lΔla2ξ+lΔlxξ+lγ1R1ξ,ξ1.
From the above equation, withξ≥ξ2>ξ1, we have the following equation:(33)Δla2ξΔlxξγ1R1ξ,ξ1=a1−1/γ2ξa11/γ2ξR1ξ,ξ1Δla2ξΔlxξγ1−a2ξΔlxξγ1R1ξ,ξ1R1ξ+l,ξ1≤0.
Thus,a2ξΔlxξγ1R1ξ,ξ1−1 is decreasing for ξ≥ξ2 and further, this fact yields as follows:(34)xξ=xξ2¯+∑r1=0m2ξΔlxξ2¯+r1lR2ξ2¯+r1l,ξ1R2ξ2¯+r1l,ξ1≥R3ξ,ξ2R2ξ,ξ1Δlxξ.
From the above equation, withξ≥ξ3>ξ2, we obtain the following equation:(35)ΔlxξR3ξ,ξ2=R3ξ,ξ2Δlxξ−xξR2ξ,ξ1R3ξ,ξ2R3ξ+l,ξ2≤0.which implies that xξR3ξ,ξ2−1 is decreasing for ξ≥ξ3.
Next, in view of the fact thatxξR3ξ,ξ2−1 is decreasing for ξ≥ξ3 and ξ>f¯ξ or f¯−1ξ≤f¯−1f¯−1ξ, yields(36)xf¯−1f¯−1ξ≤R3f¯−1f¯−1ξ,ξ2xf¯−1ξR3f¯−1ξ,ξ2.
Using equation (34) in (15), we obtain the following equation:(37)yξ≥1bf¯−1ξ1−R3f¯−1f¯−1ξ,ξ2bf¯−1f¯−1ξR3f¯−1ξ,ξ2xf¯−1ξ=ϕ2ξxf¯−1ξ.
This implies that(38)ygjξ≥ϕ2gjξxf¯−1gjξ,
Forj=1 to n and for all ξ≥ξ3. Using equation (38) in equation (5) we get the following equation:(39)Δla1ξΔla2ξΔlxξγ1γ2+∑j=1ncjξϕ2gjξγ3xγ3f¯−1gjξ≤0,
Define the Riccati transformation(40)wξ=hξa1ξΔla2ξΔlxξγ1γ2xγ3ξ+a1ξHξforξ1≤ξ.
Now,(41)Δlwξ≤Δlhξhξ+lwξ+l+hξΔla1ξHξ−hξ∑j=1ncjξϕ2gjξγ3xγ3f¯−1gjξxγ3ξ−γ3hξa1ξ+lΔla2ξ+lΔlxξ+lγ1γ2Δlxξxξ+lγ3+1.
Using the factxξR3ξ,ξ2−1 is nonincreasing for ξ3≤, and f¯ξ≥gjξ implies f¯−1gjξ≤ξ, which gives the following equation:(42)xf¯−1gjξxξ≥R3f¯−1gjξ,ξ2R3ξ,ξ2,j=1,2,…,n,for ξ4≤ξ. Substituting equations (32), (34), (40) and (42) into equation (41), we get the following equation:(43)Δlwξ≤Δlhξhξ+lwξ+l+hξΔla1ξHξ−hξ∑j=1ncjξϕ2gjξγ3R3f¯−1gjξ,ξ3R3ξ,ξ3γ3−γ3hξR2ξ,ξ1xξ+lγ3/γ1γ2−1wξ+lhξ+l−a1ξ+lHξ+l1/γ1γ2+1.
Next, to computexξγ3/γ1γ2−1, consider the following cases.Case 1.
Assumeγ1γ2≤γ3. Without loss of generality we assume that there exists a constant n1 and Δlxξ>0, where(44)n1≔xξ1≤xξforξ2≤ξ,which implies that(45)N1≔n1γ3/γ1γ2−1≤xγ3/γ1γ2−1ξforξ2≤ξ.Case 2.
Assumeγ1γ2>γ3. From equation (35), there exists a constant n2, such that(46)xξR3ξ,ξ2≤xξR3ξ3,ξ2≔n2forξ3≤ξ.
Hence,(47)xξγ3/γ1γ2−1≥n2γ3/γ1γ2−1R3ξ,ξ2γ3/γ1γ2−1≔N2R3ξ,ξ2γ3/γ1γ2−1forξ3≤ξ.
Combining equation (43) with equations (45) and (47), we get the following equation:(48)Δlwξ≤Δlhξhξ+lwξ+l+hξΔla1ξHξ,−hξ∑j=1ncjξϕ2gjξγ3R3f¯−1gjξ,ξ2R3ξ,ξ2γ3−γ3hξR3ξ,ξ2ψξwξ+lhξ+l−a1ξ+lHξ+l1/γ1γ2+1.
By the inequality(49)A1+1/β−A−B1+1/β≤B1/β1+1βA−1βBfor1≤β.
We obtain the following equation:(50)wξ+lhξ+l−a1ξ+lHξ+l1/γ1γ2+1≥wξ+lhξ+l1/γ1γ2+1+1γ1γ2a1ξ+lHξ+l1/γ1γ2+1−1+1γ1γ2a1ξ+lHξ+l1/γ1γ2+1hξ+lwξ+l.
Substituting equation (50) into equation (48), we obtain the following equation:(51)Δlwξ≤Δlhξhξ+lwξ+l+hξΔla1ξHξ−hξ∑j=1ncjξϕ2gjξγ3R3f¯−1gjξ,ξ2R3ξ,ξ2γ3−γ3hξR3ξ,ξ2ψξwξ+lhξ+l1/γ1γ2+1−γ3γ1γ2hξR3ξ,ξ2ψξa1ξ+lHξ+l1/γ1γ2+1+γ3hξR3ξ,ξ2ψξ1+1γ1γ2a1ξ+lHξ+l1/γ1γ2hξ+lwξ+l,(52)Δlwξ≤hξΔla1ξHξ−hξ∑j=1ncjξϕ2gjξγ3R3f¯−1gjξ,ξ2R3ξ,ξ2γ3−γ3γ1γ2hξR3ξ,ξ2ψξa1ξ+lHξ+l1/γ1γ2+1+Δlhξhξ+l+γ3+γ3γ1γ2hξR3ξ,ξ2ψξa1ξ+lHξ+l1/γ1γ2hξ+lwξ+l−γ3hξR3ξ,ξ2ψξhξ+l1/γ1γ2+1wξ+l1/γ1γ2+1,
Using equation (29), (30), and (31), the above equation can be expressed as follows:(53)Δlwξ=−Aξ+Bξwξ+l−Cξwξ+l1/γ1γ2+1.
By applying the inequality(54)Xw−Yw1+1/β≤ββ1+β1+βX1+βYβ,
We obtain the following equation:(55)Δlwξ≤−Aξ+γ1γ2γ1γ21+γ1γ21+γ1γ2B1+γ1γ2ξCγ1γ2ξ.
Summing the above inequality fromξ3 to ξ yields(56)∑s=0m3ξAξ3¯+sl−MB1+γ1γ2ξ3¯+slCγ1γ2ξ3¯+sl≤wξ3¯−wξ<wξ3¯,which is contradictory to equation (66). With the assumption C2, and by Lemma 2, yξ converges to zero as ξ⟶∞. The proof is now completed.
Now, we shall establish oscillation criteria and the convergence of solutions to equation (5) based on Philos type technique. Let us define the functions d, D:Nl×Nl⟶R, such that(i)
Dξ,ξ=0 with ξ≥ξ0≥0;(ii)
Dξ,ξ1>0 with ξ>ξ1≥ξ0;(iii)
Δlξ1Dξ,ξ1=Dξ,ξ1+l−Dξ,ξ1≤0 for ξ>ξ1≥ξ0, and(57)Δlξ1Dξ,ξ1+Bξ1Dξ,ξ1=−dξ,ξ1.Theorem 2.
Letgjξ≤fξ for j=1 to n, Φ1ξ>0, Φ2ξ>0, and equation (15) hold. If there exists functions hξ and Hξ∈CNlξ0,R such that a2ξHξ∈CNlξ0,R and(58)limsupξ⟶∞1Dξ,ξ1∑s=0m1ξDξ,ξ1¯+slAξ1¯+sl−Mdξ,ξ1¯+sl1+γ1γ2Cγ1γ2ξ¯+slDγ1γ2ξ,ξ¯+sl=∞,for all ξ0≤ξ1≤ξ, where hξ, Aξ, Bξ and Cξ are defined as in Theorem 1, then as ξ tends to ∞, every solution of equation (5) is either oscillatory or tends to zero.Proof.
Let (5) has a nonoscillatory solution yξ on Nlξ0. Without loss of generality we assume that there exists ξ1∈Nlξ0 such that, for ξ≥ξ1, yξ, yf¯ξ>0 and ygjξ>0, equations (7) and (8) holds and xξ satisfies either C1 or C2 for j=1 to n. Suppose that C1 is true. Equation (52) is obtained by using the same arguments as in the proof of Theorem 1. Given equation (40), the inequality (53) assumes the following form:(59)Aξ≤−Δlwξ+Bξwξ+l−Cξwξ+l1/γ1γ2+1.
Multiplying equation (59) by Dξ,s and summing the resulting inequality from ξ1 to ξ−l for every ξ1≤ξ, we obtain the following equation:(60)∑s=0m1ξDξ,ξ1¯+slAξ1¯+sl≤−∑s=0m1ξDξ,ξ1¯+slΔlwξ1¯+sl+∑s=0m1ξBξ1¯+slDξ,ξ1¯+slwξ1¯+sl+l−∑s=0m1ξCξ1¯+slDξ,ξ1¯+slwξ1¯+sl+l1/γ1γ2+1.
Applying summation by parts, we get the following equation:(61)∑s=0m1ξDξ,ξ1¯+slAξ1¯+sl≤Dξ,ξ1wξ1−∑s=0m1ξCξ¯+slDξ,ξ¯+slwξ¯+sl+l1/γ1γ2+1+∑s=0m1ξΔlξ1Dξ,ξ1¯+sl+Bξ1¯+slDξ,ξ1¯+slwξ1¯+sl+l,≤Dξ,ξ1wξ1−∑s=0m1ξCξ¯+slDξ,ξ¯+slwξ¯+sl+l1/γ1γ2+1+∑s=0m1ξdξ,ξ1¯+slwξ1¯+sl+l,≤Dξ,ξ1wξ1+∑s=0m1ξdξ,ξ1¯+slwξ1¯+sl+l−Cξ¯+slDξ,ξ¯+slwξ¯+sl+l1/γ1γ2+1.
Using equation (54), the above inequality becomes as follows:(62)∑s=0m1ξDξ,ξ1¯+slAξ1¯+sl≤Dξ,ξ1wξ1+∑s=0m1ξMdξ,ξ1¯+sl1+γ1γ2Cγ1γ2ξ¯+slDγ1γ2ξ,ξ¯+sl,and this implies that(63)1Dξ,ξ1∑s=0m1ξDξ,ξ1¯+slAξ1¯+sl−Mdξ,ξ1¯+sl1+γ1γ2Cγ1γ2ξ¯+slDγ1γ2ξ,ξ¯+sl≤wξ1,which stands contrary to equation (58). If we assume C2, then by Lemma 2, we get the desired result.Corollary 1.
Assume that all the conditions of Theorem2 are met and if equation (58) is replaced by the following conditions(64)limsupξ⟶∞1Dξ,ξ1∑s=0m1ξDξ,ξ1¯+slAξ1¯+sl=∞,limsupξ⟶∞1Dξ,ξ1∑s=0m1ξMdξ,ξ1¯+sl1+γ1γ2Cγ1γ2ξ¯+slDγ1γ2ξ,ξ¯+sl<∞,then also, every solution of equation (5) either tends to zero as ξ⟶∞ or oscillatory.Remark 1.
One may chooseDξ,ξ∗ in a proper manner to develop various oscillation criteria for equation (5). Some alternatives are as follows:(65)Dξ,ξ∗=ξ−ξ∗lλ,λ≥1,ξ0≤ξ∗≤ξ,Dξ,ξ∗=logξ+1ξ∗+1lλ,λ≥1,ξ0≤ξ∗≤ξ,where nlλ=nn−ln−2l⋯n−λ−1l.
Subsequently, the oscillation criteria for equation (5) are established in the case gjξ≥fξ.Theorem 3.
Letgjξ≥fξ for j=1 to n and Φ1ξ>0, Φ2ξ>0, and equation (15) hold. If there exists functions hξ and Hξ∈CNlξ0,R such that a2ξHξ∈CNlξ0,R and(66)∑s=0∞A∗ξ3¯+sl−MB1+γ1γ2ξ3¯+slCγ1γ2ξ3¯+sl=∞,forξ2<ξ3≤ξ.where(67)A∗ξ=hξΔla1ξHξ−hξΦ2ξ+γ3γ1γ2hξR3ξ,ξ2ψξa1ξ+lHξ+l1/γ1γ2+1,for all ξ0≤ξ1≤ξ, where hξ,Aξ, Bξ, and Cξ are defined in Theorem 1, then every solution to equation (5) is either oscillatory or tends to zero, as ξ approaches ∞.Proof.
Let equation (5) have a nonoscillatory solution yξ on Nlξ0. Without loss of generality we assume that there exists ξ1∈Nlξ0 such that, for ξ≥ξ1, yξ, yf¯ξ>0 and ygjξ>0, equation (7) and (8) holds and xξ satisfies either C1 or C2 for j=1 to n. Suppose that C1 is valid. Using the procedure as followed in the proof of Theorem 1, the process is smooth in arriving equation (52). By considering the fact that f¯ξ≤gjξ, we have the following equation:(68)ξ≤f¯−1gjξ,j=1ton.
Consequently, given thatxξ is increasing, we get the following inequality:(69)xf¯−1gjξxξ≥1,j=1ton.
Using equation (69) in equation (48), we obtain the following equation:(70)Δlwξ≤Δlhξhξ+lwξ+l+hξΔla1ξHξ−hξ∑j=1ncjξϕ2gjξγ3−γ3hξR3ξ,ξ2ψξwξ+lhξ+l−a1ξ+lHξ+l1/γ1γ2+1≤−A∗ξ+Bξwξ+l−Cξwξ+l1/γ1γ2+1,where Bξ and Cξ are presented as in Theorem 1. By adopting the same procedure as in Theorem 1, we can complete the proof.Corollary 2.
Letgjξ≥f¯ξ for j=1 to n and Φ1ξ>0, Φ2ξ>0, and equation (15) holds. If there exist functions hξ and Hξ∈CNlξ0,R such that a2ξHξ∈CNlξ0,R and(71)limsupξ⟶∞1Dξ,ξ1∑s=0m1ξDξ,ξ1¯+slA∗ξ1¯+sl−Mdξ,ξ1¯+sl1+γ1γ2Cγ1γ2ξ¯+slDγ1γ2ξ,ξ¯+sl=∞,for all ξ0≤ξ1≤ξ, where hξ, Bξ,and Cξ are defined in Theorem 1, A∗ξ is defined in Theorem 3, then every solution of equation (5) is either oscillatory or approaches zero as ξ tends to ∞.


## 3. Examples

Example 1.
Assume the third-order nonlinear generalized difference equation with multiple neutral terms(72)ΔlΔlΔlyξ+20yξ−2l1/33+∑j=1j=4cjξygjξ=0,ξ≥4l.
Herea1ξ=a2ξ=1, bξ=20, γ1=1/3, γ2=3, γ3=1, cjξ=42jj+3>0, f¯ξ=ξ−2l<ξ,and gjξ=ξ−jl, which implies limξ⟶∞f¯ξ=limξ⟶∞f¯ξ−2l=∞ and limξ⟶∞g1ξ=limξ⟶∞ξ=∞. Then, we obtain the following equation:(73)R1ξ,ξ1=ξ−ξ1l⇒R1ξ,2l=ξ−2ll,R2ξ,ξ1=ξ−ξ1l3⇒R2ξ,2l=ξ−2ll3,R3ξ,ξ2=ξ−ξ22ξ−ξ2−l24l4⇒R3ξ,2l=ξ2−5ξl+6l224l4,ϕ1ξ=19400≥0,ϕ2ξ=1201−ξ+l2ξ+2l220ξ2ξ−l2≥0,Φ1ξ=119.7≥0,Φ2ξ=∑j=142110j2+3j1−ξ−jl+l2ξ−jl+2l220ξ−jl2ξ−jl−l2≥0,where M=1/4 and ϕξ=N1. If we choose hξ=ξ and Hξ=ξ, it is easy to verify all the conditions of Theorem 1. Hence, all solutions of equation (72) are oscillatory. Infact yξ=−13ξ/l is one such oscillatory solution.Example 2.
Assume the third-order non-linear generalized difference equation with multiple neutral terms(74)ΔlΔlξ−2lΔlyξ+2yξ−2l31/5+∑j=1263/5l2ξ−5−2jl1/5y3/5ξ−4−2jl=0.
Herea1ξ=1, a2ξ=ξ−2l, bξ=2, γ1=3, γ2=1/5, γ3=3/5, c1ξ=63/52ξ−3l1/5/l, c2ξ=63/52ξ−l1/5/l, f¯ξ=ξ−2l<ξ, g1ξ=ξ−2l, and g2ξ=ξ, which implies limξ⟶∞f¯ξ=limξ⟶∞f¯ξ−2l=∞, limξ⟶∞g1ξ=limξ⟶∞ξ−2l=∞ and limξ⟶∞g2ξ=limξ⟶∞ξ=∞. Then, we obtain R1ξ,ξ1=R1ξ,2l=ξ−2l/l, R2ξ,ξ1=R2ξ,2l=1, R3ξ,ξ2=R3ξ,2l=ξ−2l/l,(75)R3f¯−1ξ,2l=R3ξ+2l,2l=ξl,R3f¯−1f¯−1ξ,2l=R3ξ+4l,2l=ξ+2ll,ϕ1ξ=14≥0,ϕ2ξ=ξ−2l4ξ≥0,Φ1ξ=643/5×1l2ξ−3l1/5+2ξ−l1/5≥0,Φ2ξ=6ξ−4l4ξ−2l3/52ξ−3l1/5l+6ξ−2l4ξ3/52ξ−l1/5l≥0.where M=1/4 and ψξ=N1. If we choose hξ=ξ and ξ=ξ, it is observed that all the conditions of Theorem 1 are satisfied. Hence, all solutions of equation (74) are oscillatory. We observed that yξ=−15ξ/l is one such oscillatory solution of equation (74).Example 3.
Consider the third-order nonlinear generalized difference equation with multiple neutral terms(76)ΔlΔlξ−2lΔlyξ+3yξ−2l3+∑i=15ciξy3giξ=0.
Herea1ξ=1, a2ξ=ξ−2l, bξ=3, γ1=1, γ2=3, γ3=3, c1ξ=27l3+441/2lξ−l2−189/2l2ξ−l, c2ξ=−152l3+132l2ξ−768lξ2, c3ξ=250l3+lξ+l2, c4ξ=−152l3−132l2ξ+2l−768lξ+2l2, c5ξ=27l3+441/2lξ+3l2+189/2l2ξ+3l, f¯ξ=ξ−2l<ξ, g1ξ=ξ−l, g2ξ=ξ, g3ξ=ξ+l, g4ξ=ξ+2l, and g5ξ=ξ+3l with hξ=Hξ=ξ, then it is easy to verify that all the conditions of Theorem 1 are satisfied. Infact yξ=l/ξ is one such solution of equation (74) which converges to zero as ξ⟶∞.


## 4. Conclusion

As part of this study, we established newly enhanced oscillation criteria for the third order nonlinear generalized difference equation with multiple neutral terms, which is new in the literature. Implementation of Riccati transformation and Philos type technique enhances the quality of our findings. Our findings are well supported by the presentation of few examples.


## 5. Suggestion for Future Research

By definingxξ=yξ+∑i=1biξyf¯iξ, the research can be extended for various class of generalized difference equation.



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*Source: 8103094-2022-12-29.xml*