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

The paper studies the oscillation of a class of nonlinear fractional order difference equations with damping term of the form Δ1⁄2ψðλÞzηðλÞ + pðλÞzηðλÞ + qðλÞFð∑λ−1+μ s=λ0 ðλ − s − 1Þ ð−μÞyðsÞÞ = 0, where zðλÞ = aðλÞ + bðλÞΔμyðλÞ, Δ 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.


Introduction and Background
The objective of this paper is to provide oscillation theorems for the equation where λ ∈ ℕ λ 0 +1−μ , η > 0 is a quotient of odd positive integers, Δ μ is the fractional difference operator in the sense of Riemann-Liouville (RL) and of order μ, 0 < μ ≤ 1, and The μ th fractional sum for μ > 0, (see [1]) is defined by where the fractional sum Δ −μ is defined from ℕ a to ℕ a+μ , f ðsÞ is defined for s ≡ a mod ð1Þ and Δ −μ a f ðλÞ is defined for λ ≡ ða + μÞ mod ð1Þ. The falling function is where Γ is the Gamma function, given by for t ∈ ℝ + ≔ ð0,∞Þ.
In the literature, Wang et al. [3] extended some oscillation results from integer-order differential equation to the fractional-order differential equation where D α a denotes the standard RL differential operator of order α with 0 < α ≤ 1, qðtÞ is a positive real-valued function, f ð·Þ: ½0,∞Þ ⟶ ½0,∞Þ is a continuous function satisfying and I 2−α denotes RL integral operator. Xiang et al. [4] investigated conditions for oscillation of fractional-order differential equation where wðtÞ = pðtÞ + qðtÞðD α − xÞðtÞ, α ∈ ð0, 1Þ is a fractional-order, γ > 0 is a quotient of odd positive integers, and ðD α − xÞ is the RL right-sided fractional derivative of order α of x defined by 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][6][7][8][9][10][11][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][14][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][17][18][19][20].
Of late, the investigation of the oscillation of solutions for fractional order difference equations has accelerated with several articles (see [21][22][23][24][25][26][27][28][29]). In what follows, we state some lemmas and preliminaries that will contribute in proving our main results.

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 Then uðλÞ > 0, and hence and then Eq. (1) is oscillatory.
Proof. Suppose that yðλÞ is a nonoscillatory solution of Eq.

Journal of Function Spaces
Proof. Assume that yðλÞ is a nonoscillatory solution of (1). Without loss of generality, assume that yðλÞ 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 z ðλÞ. When zðλÞ > 0 is eventually positive, we conclude from the proof of Theorem 6 that equation (1) is oscillatory. Next, assume that zðλÞ is eventually negative, then there exits λ 2 > λ 0 such that zðλÞ < 0 for λ ≥ λ 2 . Since z ðλÞ = aðλÞ + bðλÞΔ μ yðλÞ, we have and hence On the other hand, Since ðH 2 Þ holds, we get Now, taking the limit of the both sides of (68) as λ tends to ∞, we get Since ΨðλÞ > 0 for λ ∈ ½λ 1 ,∞Þ, we have Claim that β = 0. If not, then ΨðλÞ ≥ β for λ ∈ ½λ 2 ,∞Þ. Now we have by (29). Summing up from λ to ∞, we have which yields and hence The last inequality above implies that Summing up from λ 2 to λ − 1, we get Taking the limit of the both sides of the above inequality as λ tends to ∞, we end up with which contradicts to ΨðλÞ > 0 for λ ∈ ½λ 1 ,∞Þ. Therefore, we obtain β = 0, that is