# Oscillation Results for a Class of Nonlinear Fractional Order Difference Equations with Damping Term

Authors: A. George Maria Selvam; Jehad Alzabut; Mary Jacintha; Abdullah Özbekler
Journal: Journal of Function Spaces
 (2020)

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## Abstract

The paper studies the oscillation of a class of nonlinear fractional order difference equations with damping term of the formΔψλzηλ+pλzηλ+qλF∑s=λ0λ−1+μλ−s−1−μys=0, where zλ=aλ+bλΔμyλ, Δμ stands for the fractional difference operator in Riemann-Liouville settings and of order μ, 0<μ≤1, and η≥1 is a quotient of odd positive integers and λ∈ℕλ0+1−μ. New oscillation results are established by the help of certain inequalities, features of fractional operators, and the generalized Riccati technique. We verify the theoretical outcomes by presenting two numerical examples.


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## Body

## 1. Introduction and Background

The objective of this paper is to provide oscillation theorems for the equation(1)Δψλzηλ+pλzηλ+qλF∑s=λ0λ−1+μλ−s−1−μys=0,where λ∈ℕλ0+1−μ, η>0 is a quotient of odd positive integers, Δμ is the fractional difference operator in the sense of Riemann-Liouville (RL) and of order μ, 0<μ≤1, and
(2)zλ=aλ+bλΔμyλ.Theμth fractional sum for μ>0, (see [1]) is defined by
(3)Δa−μfλ=1Γμ∑s=aλ−μλ−s−1μ−1fs,λ∈ℕa+μ,where the fractional sum Δ−μ is defined from ℕa to ℕa+μ, fs is defined for s≡amod1 and Δa−μfλ is defined for λ≡a+μmod1. The falling function is
(4)λμ=Γλ+1Γλ+1−μ.where Γ is the Gamma function, given by
(5)Γt≔∫0∞st−1e−sds,for t∈ℝ+≔0,∞.Letμ and N be positive integers such that N=μ, namely, μ∈N−1,N. Set ν=N−μ. Then, μth fractional difference (see [2]) is defined as
(6)Δμfλ=ΔN−νfλ=ΔNΔ−νfλ.The following conditions are assumed to hold throughout this work:(i)
(H1)ψλ∈Ct0,∞,pλ∈Ct0,∞, and qλ∈Ct0,∞ are positive sequences with ψλ>pλ(ii)
(H2)(7)Δaλbλ≠0for λ∈λ0,∞ and
(8)limλ→∞∑k=λ0λ−1akbk<∞(iii)
(H3)aλ is a nonnegative sequence on λ0,∞ for some λ0>0. There exists a n0>0 such that bλ≥n0 for λ∈λ0,∞(iv)
(H4)F is a monotone decreasing function satisfying ΨFΨ>0 such that
(9)FΨΨη≥L>0forΨ≠0A nontrivial solution of Eq. (1) is called oscillatory if it has arbitrarily large zeros. Otherwise, it is called nonoscillatory. Eq. (1) is called oscillatory if all its solutions are oscillatory.In the literature, Wang et al. [3] extended some oscillation results from integer-order differential equation to the fractional-order differential equation
(10)Daαxt+qtfxt=0;t∈a,+∞,a>0,where Daα denotes the standard RL differential operator of order α with 0<α≤1,qt is a positive real-valued function, f·:0,∞⟶0,∞ is a continuous function satisfying
(11)fuuIα−2≥K>0,and I2−α denotes RL integral operator. Xiang et al. [4] investigated conditions for oscillation of fractional-order differential equation
(12)atwγt−bλf∫t∞s−t−αxsds=0;t≥t0>0,where wt=pt+qtD−αxt, α∈0,1 is a fractional-order, γ>0 is a quotient of odd positive integers, and D−αx is the RL right-sided fractional derivative of order α of x defined by
(13)D−αxt≔−1Γ1−αddt∫t∞s−t−αxsds,for t∈ℝ+.In this paper and motivated by the above work, we intend to carry forward the oscillation results from fractional differential Eq. (12) to the fractional difference Eq. (1). Moreover, we consider Eq. (1) under a damping term.Fractional difference calculus is evolving as a powerful tool for studying problems in many fields such as biology, mechanics, control systems, ecology, electrical networks, electrochemical of corrosion, chemical physics, optics, and signal processing, economics and so forth (see [5–12]) and the references therein. Basic definitions and properties of fractional difference calculus were presented by Goodrich and Peterson [1]. In addition, there are other research works dealing with fractional difference equations which have helped to build up some of the basic theory in this area (see [2, 13–15]).Fractional difference equations are applied to model physical processes which vary with time and space and its nonlocal property enables to model systems with memory effect. The study of the qualitative analysis of these equations has gained momentum in recent years and thus numerous publications have been reviled [16–20].Of late, the investigation of the oscillation of solutions for fractional order difference equations has accelerated with several articles (see [21–29]). In what follows, we state some lemmas and preliminaries that will contribute in proving our main results.Definition 1 (see [2]).
For anyμ≥0, the falling factorial is known as
(14)sμ=Γs+1Γs+1−μ,where Γ is as in (5).Lemma 2 (see [23]).
Letyλ be a solution of (1) and
(15)Ψλ=∑s=λ0λ−1+μλ−s−1−μys,then
(16)ΔΨλ=Γ1−μΔμyλ.Lemma 3 (see [30]).
The product and quotient rules of the difference operatorΔ are defined to be
(17)Δxλyλ=Δxλyλ+1+xλΔyλ,and
(18)Δxλyλ=Δxλyλ−xλΔyλyλyλ+1,where Δxλ=xλ+1−xλ.Lemma 4 (see [31]).
Letη≥1 be a quotient of two positive odd integers. If
(19)Ψλ+1>Ψλ>0Ψλ+1<Ψλ<0,then
(20)ΔΨηλ≥ΔΨληΔΨηλ≤ΔΨλη.Lemma 5 (see [32]).
Letξ,ζ∈ℝwithζ>0. Then, the inequality
(21)ξX−ζX2≤ξ24ζ,holds for all X∈ℝ.


## 2. Main Results

Herein, new oscillation theorems for Eq. (1) are established by using mathematical inequalities, the properties of RL sum and difference operators, and the generalized Riccati technique.Define the sequence(22)uλ=∏s=λ0λ−1ψsψs−ps.Thenuλ>0,
(23)uλ+1=ψλψλ−pλuλ,and hence
(24)Δuλ=uλ+1−uλ=pλψλ−pλuλ.Theorem 6.
Assume thatH1−H4 hold. If
(25)limλ→∞∑v=λ0λ−11uvψv1/η=∞,and
(26)limλ→∞∑v=λ2λ−1Luvqv−Δuv2ψv+1Nη4uv+1=∞,then Eq. (1) is oscillatory.Proof.
Suppose thatyλ is a nonoscillatory solution of Eq. (1). Without loss of generality, we may assume that yλ is an eventually positive solution of Eq. (1). Then, there exits λ1∈λ0,∞ such that yλ>0, Ψλ>0 for λ∈λ1,∞, where Ψλ is defined in Lemma 2. Considering the assumption H4 and using Eq. (1), we get
(27)Δψλzηλ+pλzηλ=−qλFΨλ,or
(28)Δψλzηλ+pλzηλ≤−LqλΨηλ,where zλ is defined in (2). Hence, we proceed from (22) and (17) to
(29)Δuλψλzηλ=uλ+1Δψλzηλ+Δuλψλzηλ=uλ+1Δψλzηλ+uλpλψλ−pλψλzηλ=uλ+1Δψλzηλ+uλ+1ψλ−pλψλpλψλ−pλψλzηλ=uλ+1Δψλzηλ+pλzηλ≤−Luλ+1qλΨηλ<0.
Then,uλψλzηλ is strictly decreasing on λ1,∞ and is eventually of constant sign. Since uλ>0, ψλ>0, and η>0 is a quotient of odd positive integers, we observe that zλ is eventually of constant sign.
First, we show that(30)zλ>0forλ∈λ1,∞.
If not, then there exitsλ2≥λ1 such that zλ2<0, and we obtain
(31)uλψλzηλ<uλ2ψλ2zηλ2=d<0,which implies
(32)zηλ<duλψλ<0,for λ∈λ2,∞ that is
(33)zλ<duλψλ1/η<0.
So we arrive at thatzλ<0 on λ2,∞. Hence for λ∈λ2,∞, we get
(34)aλbλ+Δμyλ<zλbλ<1bλduλψλ1/η<1n0duλψλ1/η.
From (16), we have
(35)aλbλ+ΔΨλΓ1−μ=aλbλ+Δαyλ<1n0duλψλ1/η,and summing from λ2 to λ−1, we obtain
(36)∑v=λ2λ−1ΔΨv<Γ1−μ∑v=λ2λ−11n0duvψv1η−avbv,and hence we have
(37)Ψλ<Ψλ2+Γ1−μ∑v=λ2λ−11n0duvψv1/η−avbv.
Now, by lettingλ⟶∞, we get
(38)limλ→∞Ψλ<limλ→∞Ψλ2+Γ1−μ∑v=λ2λ−11n0duvψv1/η−avbv<Ψλ2+Γ1−μ∑v=λ2∞1n0duvψv1/η−avbv=−∞,which contradicts that Ψλ>0, λ∈λ1,∞. Therefore zλ>0 for λ∈λ2,∞.
Now, since(39)zλbλ=aλbλ+Δμyλ=aλbλ+ΔΨλΓ1−μ,and we have
(40)ΔΨλΓ1−μ=zλ−aλbλ<zλ−aλn0<zλn0,from (16), we obtain
(41)ΔΨλ<zλn0.
Define the generalized Riccati function(42)wλ=uλψλzηλΨηλ;λ∈λ1,∞.
It is clear thatwλ>0. Using (17) and (18) for λ∈λ0,∞, we have
(43)Δwλ=Δuλψλ+1zηλ+1Ψηλ+1+uλΔψλzηλΨηλ=Δuλwλ+1uλ+1+uλΔψλzηλΨηλ−ψλzηλΔΨηλΨηλΨηλ+1=Δuλwλ+1uλ+1+uλΔψλzηλΨηλ+1−uλψλzηλΨηλΨηλ+1ΔΨηλ=Δuλwλ+1uλ+1−uλpλzηλ+qλfΨλΨηλ+1−uλψλzηλΨηλΨηλ+1ΔΨηλ=Δuλwλ+1uλ+1−uλpλzηλΨηλ+1−uλqλfΨλΨηλ+1−uλψλzηλΨηλΨηλ+1ΔΨηλ.
Since(44)uλψλzηλΨηλΨηλ+1>0,and ΔΨλ>0, then Ψλ+1−Ψλ>0; it follows from H4 that
(45)fΨλΨηλ+1>fΨλ+1Ψηλ+1≥L,and ΔΨηλ>ΔΨλη by Lemma 4.
Using the fact thatuλψλzηλ is strictly decreasing, we arrive at
(46)uλψλzηλΨηλΨηλ+1ΔΨηλ≥uλψλzηλΨηλΨηλ+1ΔΨλη.
Also we have(47)ΔΨλη<zλn0η,from (41).
Now substituting (45), (46), and (47) in (43), we get
(48)Δwλ<Δuλwλ+1uλ+1−Luλqλ−uλψλzηλΨηλΨηλ+1zηλn0η<Δuλwλ+1uλ+1−Luλqλ−uλ+1ψλ+1zηλ+1Ψηλ+1Ψηλ+1zηλ+1n0η<Δuλwλ+1uλ+1−Luλqλ−uλ+1ψλ+1zηλ+12uλ+1ψλ+1Ψηλ+12n0η<Δuλwλ+1uλ+1−Luλqλ−uλ+1ψλ+1zηλ+1Ψηλ+12×1uλ+1ψλ+1n0η<Δuλwλ+1uλ+1−Luλqλ−1uλ+1ψλ+1n0ηw2λ+1.
Using (48) and (21) with X=wλ+1, a=Δuλ/uλ+1, and
(49)b=1uλ+1ψλ+1n0η,we obtain
(50)Δwλ<−Luλqλ+Δuλ2ψλ+1uλ+1n0η4uλ+12=−Luλqλ+Δuλ2ψλ+1n0η4uλ+1,and hence
(51)Luλqλ−Δuλ2ψλ+1n0η4uλ+1<−Δwλ.
Summing up (51) from λ2 to λ−1, we have
(52)∑v=λ2λ−1Luvqv−Δuv2ψv+1n0η4uv+1<−∑v=λ2λ−1Δwv=−wλ−wλ2=wλ2−wλ,and hence
(53)∑v=λ2λ−1Luvqv−Δuv2ψv+1n0η4uv+1<wλ2.
Lettingλ⟶∞,
(54)limλ→∞∑v=λ2λ−1Luvqv−Δuv2ψv+1n0η4uv+1<wλ2<∞,which contradicts with (26).Theorem 7.
Assume that (25)and (26) hold and there exists a positive sequence Ht,s such that
(1)
Hλ,λ=0 for λ≥λ0; Hλ,v>0 for λ>v≥λ0(2)
Δ2Hλ,v=Hλ,v+1−Hλ,v<0 for λ>v≥λ0
If(55)limλ→∞1Hλ,λ0∑v=λ0λ−1LuvqvHλ,v−h+2λ,vuv+1ψv+1n0η4Hλ,v=∞,then Eq. (1) is oscillatory, where uλ is defined in Theorem 6 and
(56)h+λ,v=Δ2Hλ,v+ΔuvHλ,vuv+1.Proof.
Suppose thatyλ is a nonoscillatory solution of (1). Without loss of generality, we may assume that yλ is an eventually positive solution of (1). Then, there exists λ1∈λ0,∞ such that yλ>0, Ψλ>0 for λ∈λ1,∞. Proceeding as in proof of Theorem 6 we arrive at (48). Multiplying (48) by Hλ,v and summing up from λ2 to λ−1, we obtain
(57)∑v=λ2λ−1ΔwvHλ,v<∑v=λ2λ−1Δuvwv+1Hλ,vuv+1−∑v=λ2λ−1LuvqvHλ,v−∑v=λ2λ−1Hλ,vw2v+1uv+1ψv+1n0η.
Using summation by parts formula, we get(58)−∑v=λ2λ−1ΔwvHλ,v=−Hλ,vwλv=λ2v=λ−1+∑v=λ2λ−1wλ+1Δ2Hλ,v=Hλ,λ2wλ2+∑v=λ2λ−1wv+1Δ2Hλ,v.
Hence, from (57)
(59)∑v=λ2λ−1LuvqvHλ,v<Hλ,λ2wλ2+∑v=λ2λ−1wv+1Δ2Hλ,v+∑v=λ2λ−1Δuvwv+1Hλ,vuv+1−∑v=λ2λ−1Hλ,vw2v+1uv+1ψv+1n0η<∑v=λ2λ−1Δ2Hλ,v+ΔuvHλ,vuv+1wv+1−∑v=λ2λ−1Hλ,vw2v+1uv+1ψv+1n0η+Hλ,λ2wλ2<Hλ,λ2wλ2+∑v=λ2λ−1h+λ,vwv+1−Hλ,vw2v+1uv+1ψv+1n0η.
Now by using (21) with X=wv+1, ξ=h+λ,v and
(60)ζ=Hλ,vuv+1ψv+1n0η,we obtain
(61)∑v=λ2λ−1LuvqvHλ,v−h+2λ,vuv+1ψv+1n0η4Hλ,v<Hλ,λ2wλ2<Hλ,λ0wλ2,for λ>λ2>λ1>λ0. Then
(62)∑v=λ0λ−1LuvqvHλ,v−h+2λ,vuv+1ψv+1n0η4Hλ,v=∑v=λ0λ2−1LuvqvHλ,v−h+2λ,vuv+1ψv+1n0η4Hλ,v+∑v=λ2λ−1LuvqvHλ,v−h+2λ,vuv+1ψv+1n0η4Hλ,v<∑v=λ0λ2−1LuvqvHλ,v−h+2λ,vuv+1ψv+1n0η4Hλ,v+Hλ,λ0wλ2<Hλ,λ0∑v=λ0λ2−1Luvqv+Hλ,λ0wλ2,which yields
(63)1Hλ,λ0∑v=λ0λ−1LuvqvHλ,v−h+2λ,vuv+1ψv+1n0η4Hλ,v<∑v=λ0λ2−1Luvqv+wλ2.
Taking limit asλ⟶∞, we get
(64)limλ→∞1Hλ,λ0∑v=λ0λ−1LuvqvHλ,v−h+2λ,vuv+1ψv+1n0η4Hλ,v<∑v=λ0λ2−1Luvqv+wλ2<∞,which contradicts with (55).
Next, we consider the condition(65)limλ→∞∑v=λ0λ−11uvψv1/η<∞,which implies that (25) does not hold. Under this condition, we have the following result.Theorem 8.
Assume thatH1−H4, (26), and (65) hold. If
(66)limλ→∞∑s=λ2λ−11usψs∑v=s∞uv+1qv1/η−asbs=∞,then every solution yλ is oscillatory or satisfies limλ→∞Ψλ=0, where Ψλ is defined in Lemma 2.Proof.
Assume thatyλ is a nonoscillatory solution of (1). Without loss of generality, assume that yλ is eventually a positive solution of (1). Proceeding like in the proof of Theorem 6, we get that (28) holds. Then, there are two signs of zλ. When zλ>0 is eventually positive, we conclude from the proof of Theorem 6 that equation (1) is oscillatory.
Next, assume thatzλ is eventually negative, then there exits λ2>λ0 such that zλ<0 for λ≥λ2. Since zλ=aλ+bλΔμyλ, we have
(67)aλ+bλΔΨλΓ1−μ=zλ<0,and hence
(68)ΔΨλ<−Γ1−μaλbλ.
On the other hand, SinceH2 holds, we get
(69)limλ→∞aλbλ=0.
Now, taking the limit of the both sides of (68) as λ tends to ∞, we get
(70)limλ→∞ΔΨλ≤0.
SinceΨλ>0 for λ∈λ1,∞, we have
(71)limλ→∞Ψλ=β≥0.
Claim thatβ=0. If not, then Ψλ≥β for λ∈λ2,∞. Now we have
(72)Δuλψλzηλ≤−Luλ+1qλΨηλ,≤−Luλ+1qλβη,by (29). Summing up from λ to ∞, we have
(73)∑v=λ∞Δuvψvzηv=limv→∞uvψvzηv−uλψλzηλ≤−βηL∑v=λ∞uv+1qv,which yields
(74)uλψλzηλ≥limv→∞uvψvzηv+βηL∑v=λ∞uv+1qv≥βηL∑v=λ∞uv+1qv,and hence
(75)aλbλ+ΔΨλΓ1−μ=zλ>L1/ηβ1uλψλ∑v=λ∞uv+1qv1/η.
The last inequality above implies that(76)ΔΨλ>Γ1−μL1/ηβ1uλψλ∑v=λ∞uv+1qv1/η−aλbλ.
Summing up fromλ2 to λ−1, we get
(77)Ψλ>Ψλ2+Γ1−μL1/ηβ∑s=λ2λ−11usψs∑v=s∞uv+1qv1/η−asbs.
Taking the limit of the both sides of the above inequality asλ tends to ∞, we end up with
(78)limλ→∞Ψλ=∞,which contradicts to Ψλ>0 for λ∈λ1,∞. Therefore, we obtain β=0, that is
(79)limλ→∞Ψλ=0.


## 3. Applications

Example 9.
Consider the equation(80)Δλ2xηλ+lxηλ+1λ2∑s=λ0λ−1+μλ−s−1−μys−3=0,where μ∈0,1,
(81)xλ=e−λ+1λΔμyλ,λ∈ℕ2=2,3,4,⋯,and that η>0 is a quotient of odd positive integers in ℕ2.
Comparing with Eq. (1), we get ψλ=λ2, aλ=e−λ, bλ=1/λ, pλ=λ, Fy=y−3, qλ=1/λ2, η=3, λ0=2, n0=4, L=1/3, and that
(82)fyyη=1y6>ε=L>0fory≠0,where ε is a certain positive number. It is clear that assumptions H1−H4 hold.
Further from (22), we have
(83)uλ=∏v=λ0λ−1ψvψv−pv=∏v=2λ−1v2v2−v=λ−1;Δuλ=uλ+1−uλ=1;Δaλbλ=Δλe−λ=λ+1e−λ+11−e+e−λ≠0;limλ→∞∑v=λ0λ−1avbv=limλ→∞∑v=1λ−1ve−v<∞;limλ→∞∑v=λ0λ−11uvψv1/η=limλ→∞∑v=2λ−11v2v−11/3>∑v=2∞1v=∞;limλ→∞∑v=λ2λ−1Luvqv−Δuv2ψv+1n0η4uv+1=limλ→∞∑v=2λ−1v−13v2−16v+12v=∞.
Thus, conditions (25) and (26) are satisfied. Therefore, all solutions of (80) are oscillatory by Theorem 6.Example 10.
Consider the equation(84)Δ1λzηλ+1λ2zηλ+1λ−1∑s=λ0λ−1+μλ−s−1−μys−1=0,where μ∈0,1, zλ=λ2+λΔμyλ, λ∈ℕ2, and that η>0 is a quotient of odd positive integers in ℕ2. Comparing with (1), we have ψλ=1/λ, aλ=λ2, bλ=λ, pλ=1/λ2, Fy=y−1, qλ=1/λ−1, η=1, λ0=2, n0=2, L=1/2, and that
(85)fyyη=1y2>ε=L>0fory≠0,where ε is a certain positive number. It is clear that assumptions H1−H4 hold.
Further from (22), we have
(86)uλ=∏v=λ0λ−1ψvψv−pv=∏v=2λ−1vv−1=λ−1;Δuλ=uλ+1−uλ=1;Δaλbλ=Δλ2λ=Δλ≠0;limλ→∞∑v=λ0λ−1avbv=limλ→∞∑v=1λ−1v2v=limλ→∞∑v=λ0λ−1v<∞;limλ→∞∑v=λ0λ−11uvψv1/η=limλ→∞∑v=2λ−1vv−1=∞.
We define the double sequenceHλ,v as follows:
(1)
Hλ,v=2λ−v2>0 for λ>v>2(2)
Hλ,2=λ−22>0 for λ>s=2(3)
Δ2Hλ,v=Hλ,v+1−Hλ,v=2λ−v+12−2λ−v2(4)
Δ2Hλ,v=2v−4λ+1<0 for λ>v≥2
and(87)h+λ,v=Δ2Hλ,v+ΔuvHλ,vuv+1=3v2−8lv+v+4λ2v.
Then(88)1Hλ,λ0∑v=λ2λ−1LuvqvHλ,v−h+2λ,vuv+1ψv+1n0η4Hλ,v=1λ−22∑v=λ2λ−12λ−v22−3v2−8lv+v+4λ222v+12λ−v2,for λ2>λ1, and hence
(89)limλ→∞1Hλ,λ0∑v=λ2λ−1LuvqvHλ,v−h+2λ,vuv+1ψv+1n0η4Hλ,v=limλ→∞1λ−22∑v=λ2λ−12λ−v22−3v2−8lv+v+4λ222v+12λ−v2=∞.
Therefore, condition (55) is satisfied. We deduce that all solutions of Eq. (84) are oscillatory by Theorem 7.


## 4. A Concluding Remark

In this paper, we obtained new oscillation theorems for a class of fractional difference equation. The main outcomes are proved via the means of mathematical inequalities, properties of fractional operators, and generalized Riccati technique. We claim that the concluded results have merit and considered as an extension for the corresponding fractional differential equations. Particular examples that are consistent to the main results are demonstrated at the end of the paper.The consideration of Eq. (1) with forcing term of the form
(90)Δψλzηλ+pλzηλ+qλF∑s=λ0λ−1+μλ−s−1−μys=gλ,could be further investigated. We leave this for future consideration.



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