# Oscillation of Nonlinear Fractional Dynamic Equations with Forcing Term

Authors: S. Manikandan; V. Muthulakshmi; S. Harikrishnan; Porpattama Hammachukiattikul
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, interval oscillation criteria for the nonlinear damped dynamic equations with forcing terms on time scales within conformable fractional derivatives are established. Our approach is determined from the implementation of generalized Riccati transformation, some properties of conformable time-scale fractional calculus, and certain mathematical inequalities. Also, we extend the study of oscillation to conformable fractional Euler-type dynamic equation. Examples are presented to emphasize the validity of the main theorems\enleadertwodots.


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## Body

## 1. Introduction

The consideration of dynamic equations on time scales has attracted many researchers because of its wide applications in the field of science and engineering. The theory of time scales was presented by Hilger [1] to unify the discrete and continuous analysis. It not only unifies the continuous and discrete cases but also gives new areas in between such as q-calculus [2]. The qualitative analysis of solutions of dynamic equations on different time scales has received considerable notice. In particular, the investigation of the oscillation of solutions to dynamic equations [3–5], dynamic equations with damping term [6–8], and dynamic equations on various time scales [9–11] has gained extensive attention. Fractional calculus is a generalization of integration and differentiation to any order. Recently, it has been realized that the fractional calculus has numerous applications in engineering, signal processing, economics and finance, probability and statistics, neural networks, and thermodynamics; see for illustrations [12–16] and the citations therein. In recent times, the importance has been given to fractional order calculus rather than integer order due to its applications in engineering such as neural networks, electrical and mechanical engineering, and population dynamics. The fractional dynamic equations on time scales have been studied by only few authors [17–19].In [19], Feng and Meng established the asymptotic and oscillatory behavior of the following dynamic equation of fractional order on time scales using the generalized Riccati transformation technique:(1)atrtxαtαγα+ptrtxαtαγ+qtfxt=0,t∈t0,∞T.Alzabut et al. [17] considered the following nonlinear damped dynamic equation with a conformable fractional derivative:(2)rtxαtγα+ptxαtγ+qtfxt=0,t∈t0,∞T.Here, the authors established the oscillation of the above equation when the nonlinear functionf is increasing and nonincreasing. Besides, the results are carried out in light of the following two cases: ∫t0∞Δαs/rs=∞ and ∫t0∞Δαs/rs<∞. Motivated by the above discussion, in this work, we established the oscillation results of nonlinear conformable fractional dynamic equations with a forced term.In the recent papers [18, 20], the conformable time-scale fractional calculus has been introduced. Applications of the obtained results demonstrate that the newly defined calculus will be applied to investigate oscillation for both fractional differential and fractional difference equations at the same time. Therefore, the determination of oscillation of solutions of conformable fractional dynamic equations has become a promising topic for researchers. To the best of our observation, papers [17, 19] are the only research that has studied the oscillation of conformable fractional dynamic equations.


## 2. Problem Formulation and Preliminaries

In this article, we are concerned with a class of nonlinear conformable fractional dynamic equations with damped and forced terms on time scales of the kind:(3)φηyα¯ηα¯+aηyα¯η+bηfyη=hη,η∈η0,∞T^,where T^ denotes an arbitrary time scale, η0,∞T^=η0,∞∩T^, ⋅α¯ is the conformable fractional dynamic operator of order α¯0<α¯≤1, φη,aη,bη∈Crdη0,∞T^,0,∞, and f∈Cℝ,ℝ such that yfy>0 and the function hη=≤0,α¯1,β¯1,≥0,α¯2,β¯2, where α¯i,β¯i⊂η0,∞T^ is in Crdη0,∞T^,0,∞. We constitute new interval criteria for oscillation of the solutions of equation (3) when the nonlinear function f is increasing and nonincreasing and extend the results to the Euler-type fractional dynamic equations.By a solution of (3), we insist a nontrivial function yη∈ℝ fulfilling (3) for η≥η0. If a solution of (3) is neither eventually positive (EP) nor eventually negative (EN), then it is called oscillatory. Or else, it is said to be nonoscillatory. If all solutions of (3) are oscillatory, then (3) is called oscillatory.Before we proceed to the main results, we present essential preliminaries on conformable time-scale fractional calculus that will be used to justify further discussion. Terms and definitions are adopted from the papers [2, 18].Definition 1 (see [2]).
On any time scaleT^, ση=infυ∈T^,υ>η and ρη=supυ∈T^,υ<η are defined as the forward and backward jump operators, respectively.
A pointη∈T^ is known as right-dense if ση=η, left-dense if ρη=η, right-scattered if ση>η, and left scattered if ρη<η.
The graininess functionμη of the time scale is given by μη=ση−η.
The setT^k=T^\ρsupT^,T^ifsupT^<∞T^ifT^=∞.Definition 2 (see [2]).
A real-valued functionf defined on T^ is known as rd-continuous if at all left-dense points, a finite left limit of f exists, and if it is continuous at each right-dense point.Definition 3 (see [2]).
A functionζ:T^⟶ℝ is known as regressive if 1+μηζη≠0forη∈T^k. ℛ is the collection that consists of all rd-continuous regressive functions f:T^⟶ℝ. We define ℛ+:=g∈ℛ:1+μηgη>0,η∈T^.Definition 4 (see [2]).
Ifζ∈ℛ, then the function defined by(4)eζη,υ=exp∫υηΘμυζυΔυ,forη∈T^,υ∈T^k,is called exponential, where Θhz=log1+hz/h,h≠0,z,h=0.
Also,eζη,υ is a nonzero real function and eζση;η0=1+μηζηeζη,η0.Definition 5 (see [18]).
Forη∈T^k, α¯∈0,1 and g:T^⟶ℝ, the conformable fractional derivative of order α¯ for g at η is gα¯η (if it exists) so that, for every positive ϵ, there is a δ-neighborhood N fulfilling(5)gση−gνη1−α¯−gα¯ηση−ν≤ϵση−ν,for allν∈N.Definition 6 (see [18]).
IfGα¯η=gη,η∈T^k, then G is said to be an α¯-order antiderivative of g and the integral of g given by(6)∫abgηΔα¯η=∫abgηηα¯−1Δη=Gb−Ga,wherea,b∈T^,is named as Cauchy α¯-fractional.Theorem 1 (see [18]).
By the definition ofα¯- order conformable fractional derivative, fα¯η=η1−α¯fΔη. Also, f is decreasing (increasing) for η>0, if fα¯η<0>0 for η>0.Theorem 2 (see [19]).
Letζ˜η=ηα¯−1ζη,α¯∈0,1. If ζ˜∈ℛ, then, for the IVP xα¯η=ζηxη and xη0=1onT^, the unique solution is the exponential function eζ˜η,η0 for fixed η0∈T^.Theorem 3 (see [18]).
Letg,f be real-valued α¯ differentiable functions defined on T^ at a point υ in T^k. Then,(i)
gfα¯υ=gα¯υfυ+gσυfα¯υ=gυfα¯υ+gα¯υfσυ(ii)
g/fα¯υ=gα¯υfυ−gυfα¯υ/fυfσυ,providedfυfσυ≠0(iii)
gσυ=gυ+μυυ1−α¯gα¯υTheorem 4 (see [18]).
Assumeα¯∈0,1; g,f:T^⟶ℝ be rd-continuous. Then,(7)∫abgηfα¯ηΔα¯η=gηfηab−∫abgα¯ηfσηΔα¯η,for a,b∈T^.Definition 7.
AssumeD=η,υ|η≥υ≥η0. Then, class G is a collection of functions G∈CrdD,ℝ so that Gη,η=0 for η≥η0; Gη,υ>0 for η>υ≥η0 and G has continuous α¯-partial fractional derivatives Gα¯ηη,υ and Gα¯υη,υ with respect to η and υ, respectively, such that Gα¯ηη,υ=h1η,υGη,υ and Gα¯υη,υ=−h2η,υGη,υ, where h1,h2∈LlocD,0,∞T^.For simplicity, we use the notion as follows:(8)Φη=−ηα¯−1aηφη.


## 3. Main Results

This part supplies the main theorems of the work. We will present the results in two folds based on the monotonicity off and extend the results for fractional Euler-type dynamic equation.Lemma 1.
Suppose thatΦη∈ℛ+ and ∫η0∞Δα¯υ/φυ=∞ hold.(i)
If (3) has an EP solution y, then there is a sufficiently large η∗∈η0,∞T^ so that(9)φηyα¯ηeΦηη,η0α¯<0,yα¯η>0on the intervalη∗,∞T^.(ii)
If (3) has an EN solution y, then there is a sufficiently large η∗∈η0,∞T^ so that(10)φηyα¯ηeΦηη,η0α¯>0,yα¯η<0on the intervalη∗,∞T^.Proof.
(i)
Sincey is EP, there is an η1 such that yη>0 on η1,∞T^. Now, we have(11)φηyα¯ηeΦηη,η0α¯=eΦηη,η0φηyα¯ηα¯−φηyα¯ηeΦηη,η0α¯eΦηη,η0eΦηση,η0=eΦηη,η0φηyα¯ηα¯+aηyα¯ηeΦηη,η0eΦηη,η0eΦηση,η0=−bηfyη+hηeΦηση,η0.By the assumptions, we takeα¯1,β¯1≥η1 such that hη≤0 on α¯1,β¯1T^ with α¯1<β¯1 and yη>0. Then, on the interval α¯1,β¯1T^, we have(12)φηyα¯ηeΦηη,η0α¯=−bηfyη+hηeΦηση,η0<0.SinceΦ∈ℛ+, we acquire that eΦηση,η0>0 and φηyα¯η/eΦηη,η0 is decreasing on η1,∞T^. Therefore, by the assumption that φ is positive and Φ∈ℛ+, we get that yα¯η is either negative or positive.Assume thatyα¯<0 on η2,∞T^ for sufficiently large η2>η1. Thus,(13)yη−yη2=∫η2ηφυyα¯υφυΔα¯υ≤φη2yα¯η2∫η2ηΔα¯υφυ.Lettingη⟶∞, we get limη⟶∞yη=−∞. This leads to a contradiction that yη is EP. Therefore, yα¯η is EP.(ii)
Sincey is EN, there is a sufficiently large η1 so that yη>0 on η0,∞T^. Now, we have(14)φηyα¯ηeΦηη,η0α¯=eΦηη,η0φηyα¯ηα¯−φηyα¯ηeΦηη,η0α¯eΦηη,η0eΦηση,η0=eΦηη,η0φηyα¯ηα¯+aηyα¯ηeΦηη,η0eΦηη,η0eΦηση,η0=−bηfyη+hηeΦηση,η0.
By the assumptions, we can chooseα¯2,β¯2≥η1 such that hη≥0 on α¯2,β¯2T^ with α¯2<β¯2 and yη<0. Then, on α¯2,β¯2T^, we attain(15)φηyα¯ηeΦηη,η0α¯=−bηfyη+hηeΦηση,η0>0.
SinceΦ∈ℛ+, we get that eΦηση,η0>0 and φηyα¯η/eΦηη,η0 is increasing on η1,∞T^. Therefore, by the assumption of φη>0 and Φη∈ℛ+, we get that yα¯η is either negative or positive.
Assume thatyα¯η>0 on η2,∞T^ for sufficiently large η2>η1. Therefore,(16)yη−yη2=∫η2ηφυyα¯υφυΔα¯υ≥φη2yα¯η2∫η2ηΔα¯υφυ.
Lettingη⟶∞, we get limη⟶∞yη=∞, and this leads to a contradiction that yη is EN. Therefore, yα¯ is EN.


### 3.1. Oscillation Criteria for (3) When f Is Not Necessarily Increasing

To establish oscillation criteria, we make use of the following assumption:(H1)fy/y≥K∀y≠0 and for some K>0.Theorem 5.
LetΦη∈ℛ+. If ∫η0∞Δα¯υ/φυ=∞ and H1 holds. If there exists ci∈α¯i,β¯i,i=1,2, a function Θη>0 on T,∞T^ and G∈G for sufficiently large T such that(17)1Gci,α¯i∫α¯iciGυ,α¯iKbυΘυeΦυσυ,η0−φυzα¯υGυ,α¯i+h1υ,α¯iΘσυ24Gυ,α¯iΘυeΦυυ,η0Δα¯υ+1Gβ¯i,ci∫ciβ¯iGβ¯i,υKbυΘυeΦυσυ,η0−φυzα¯υGβ¯i,υ−h2β¯i,υΘσυ24Gβ¯i,υΘυeΦυυ,η0Δα¯υ>0,then (3) is oscillatory.Proof.
Assume that (3) has a nonoscillatory solution y. Then, y is either EN or EP for η≥η1≥η0. Define a generalized Riccati function as follows:(18)wη=Θηφηyα¯ηyηeΦηη,η0.
By Lemma1, clearly wη≥0 and(19)wα¯η=ΘyηeΦηη,η0φηyα¯ηα¯+aηyα¯ηeΦηη,η0eΦηη,η0eΦηση,η0+yηzα¯η−Θηyα¯ηyσηyηφσηyα¯σηeΦηση,η0Θηyη−bηfyη+hηeΦηση,η0+zα¯ηwΘση−Θηφσηyα¯ηyα¯σηyσηyηeΦηση,η0≤−KbηΘηeΦηση,η0+ΘηhηyηeΦηση,η0+zα¯ηwΘση−Θηyα¯ηφσηyα¯σηyηyσηeΦηση,η0.
By assumptions, we takeα¯i,β¯i≥η1 for i=1,2 so that hη≤0 on α¯1,β¯1 with α¯1<β¯1 and yη is EP, or hη≥0 on α¯2,β¯2 with α¯2<β¯2 and yη is EN. On the intervals α¯1,β¯1 and α¯2,β¯2, the above inequality implies that(20)wα¯η≤−KbηΘηeΦηση,η0+zα¯ηwΘση−ΘηeΦηη,η0φηwΘση2.
By multiplyingGη,υ and taking α¯- fractional integral for (20) from ci to η, we have(21)∫ciηKbυΘυeΦυσυ,η0Gη,υΔα¯υ≤−∫ciηwα¯υGη,υΔα¯υ+∫ciηzα¯ηwΘσηGη,υΔα¯υ−∫ciηΘυeΦυυ,η0φυwΘση2Gη,υΔα¯υ≤Gη,ciwci+∫ciηwσυGα¯η,υΔα¯υ+∫ciηzα¯υwσυGη,υΘσυΔα¯υ−∫ciηΘυeΦυυ,η0φυwΘσυ2Gη,υΔα¯υ≤Gη,ciwci+∫ciηGη,υzα¯υ−h2η,υΘσυGη,υwσυΘσυΔα¯υ−∫ciηGη,υΘυeΦυυ,η0φυwσυΘσυ2Δα¯υ≤Gη,ciwcid+∫ciηGη,υφυzα¯υGη,υ−Θσυh2η,υ24Gη,υΘυeΦυυ,η0Δα¯υ.
Lettingη⟶β¯i−, we get(22)∫ciβ¯iGη,υKbηΘηeΦηση,η0−φυzα¯υGη,υ−Θσυh2η,υ24Gη,υΘυeΦυυ,η0Δα¯υ≤Gβ¯i,ciwci.
By multiplyingGυ,η and taking integration of α¯-order from η to ci for (20), we have(23)∫ηciGυ,ηKbυΘυeΦυσυ,η0Δα¯υ≤−∫ηciGυ,ηwα¯υΔα¯υ+∫ηciGυ,ηzα¯υwσυΘσυΔα¯υ−∫ηciGυ,ηΘυeΦυυ,η0φυwσυΘσυ2Δα¯υ≤−Gci,ηwci+∫ηciGα¯υ,ηwσυΔα¯υ+∫ηciGυ,ηzα¯υwσυΘσυΔα¯υ−∫ηciGυ,ηΘυeΦυυ,η0φυwσυΘσυ2Δα¯υ≤−Gci,ηwci+∫ηciGυ,ηzα¯υ+h1υ,ηΘσυGη,υwσυΘσυΔα¯υ−∫ηciGυ,ηΘυeΦυυ,η0φυwσυΘσυ2Δα¯υ≤−Gci,ηwci+∫ηciGυ,ηφυzα¯υGυ,η+h1υ,ηΘσυ24Gυ,ηΘυeΦυυ,η0Δα¯υ.
Lettingη⟶α¯i+, we get(24)∫α¯iciGυ,ηKbηΘηeΦηση,η0−φυzα¯υGυ,η+h1η,υΘσυ24Gυ,ηΘυeΦυυ,η0Δα¯υ≤−Gci,α¯iwci.
Dividing (33) by Gβ¯i,ci and (24) by Gci,α¯i, we attain(25)1Gβ¯i,ci∫ciβ¯iGβ¯i,υKbυΘυeΦυσυ,η0−φυzα¯υGβ¯i,υ−h2β¯i,υΘσυ24Gβ¯i,υΘυeΦυυ,η0Δα¯υ≤wci,1Gci,α¯i∫α¯iciGυ,α¯iKbυΘυeΦυσυ,η0−φυzα¯υGυ,α¯i+h1υ,α¯iΘσυ24Gυ,α¯iΘυeΦυυ,η0Δα¯υ≤−wci.
Adding the above two inequalities, we get a contradiction to hypothesis.Remark 1.
Assume that the condition of Theorem5 holds for every T≥0 and function Gυ,η∈G for sufficiently large T. If there exist some ci∈α¯i,β¯i,i=1,2 so that(26)∫α¯iciGυ,α¯iKbυΘυeΦυσυ,η0−φυzα¯υGυ,α¯i+h1υ,α¯iΘσυ24Gυ,α¯iΘυeΦυυ,η0Δα¯υ>0,∫ciβ¯iGβ¯i,υKbυΘυeΦυσυ,η0−φυzα¯υGβ¯i,υ−h2β¯i,υΘσυ24Gβ¯i,υΘυeΦυυ,η0Δα¯υ>0,for i=1,2, then (3) is oscillatory.LetG=Gη−υ∈G. Then, h1η−υ=h2η−υ, and we use hη−υ for them. By using this Gη−υ in Theorem 5, we have the corollary as follows.Corollary 1.
Suppose that all the assumptions of Theorem5 hold and for every T≥η0, there is a T≤α¯1<2c1−α¯1≤α¯2<2c2−α¯2 so that hη=≤0,η∈α¯1,2c1−α¯1,≥0,η∈α¯2,2c2−α¯2. If there is a G:=Gη−υ∈G such that(27)1Gci−α¯i∫α¯iciGυ−α¯i2KbυΘυeΦυσυ,η0−φυzα¯υ2Gυ−α¯i+h2υ−α¯iΘ2συ2Gυ−α¯iΘυeΦυυ,η0Δα¯υ>0,then (3) is oscillatory.


### 3.2. Oscillation Criteria for (3) When f Is Increasing

To establish oscillation criteria, we make use of the following assumption:(H2)
f′ exists and f′y≥M for all y≠0 and for some M>0Theorem 6.
LetΦη∈ℛ+. If H2,∫η0∞Δα¯υ/φυ=∞ holds. If there exists ci∈α¯i,β¯i,i=1,2, a function Θη>0 on T,∞T^ and G∈G for sufficiently large T such that(28)frac1Gci,α¯i∫α¯iciGυ,α¯ibυΘυeΦυσυ,η0−φηzα¯υGυ,α¯i+h1υ,α¯iΘσυ24MGυ,α¯iΘυeΦυυ,η0Δα¯υ+1Gβ¯i,ci∫ciβ¯iGβ¯i,υbυΘυeΦυσυ,η0−φηzα¯υGβ¯i,υ−h2β¯i,υΘσυ24MGβ¯i,υΘυeΦυυ,η0Δα¯υ>0,then (3) is oscillatory.Proof.
Assume that (3) has a nonoscillatory solution y. Define a generalized Riccati function as follows:(29)wη=Θηφηyα¯ηfyηeΦηη,η0.
Clearly,wη≥0 and(30)wα¯η=ΘηfyηeΦηη,η0φηyα¯ηα¯+aηeΦηη,η0yα¯ηeΦηση,η0eΦηη,η0+fyηzα¯η−Θηfyηα¯fyσηfyηφσηyα¯σηeΦηση,η0=Θηfyη−bηfyη+hηeΦηση,η0+zα¯ηwΘση−Θηφσηfyηα¯yα¯σηfyσηfyηeΦηση,η0=−bηΘηeΦηση,η0+hηΘηfyηeΦηση,η0+zα¯ηwΘση−Θηφσηfyηα¯yα¯σηfyσηfyηeΦηση,η0.
By the hypothesis, we takeα¯i,β¯i≥η1 for i=1,2 such that hη≤0 on α¯1,β¯1 with α¯1<β¯1 and yη is EP, or hη≥0 on α¯2,β¯2 with α¯2<β¯2 and yη is EN. On the intervals α¯1,β¯1 and α¯2,β¯2, we have(31)wα¯η≤−bηΘηeΦηση,η0+zα¯ηwΘση−Θηf′yηyα¯ηφσηyα¯σηfyηfyσηeΦηση,η0≤−bηΘηeΦηση,η0+zα¯ηwΘση−MΘηeΦηη,η0φηwz2ση.
By multiplyingGη,υ and taking α¯- fractional integral for (31) from ci to η, we have(32)∫ciηbυΘυeΦυσυ,η0Gη,υΔα¯υ≤−∫ciηwα¯υGη,υΔα¯υ+∫ciηzα¯υwσυΘσυGη,υΔα¯υ−∫ciηMΘυeΦυυ,η0φυwσυΘσυ2Gη,υΔα¯υ≤Gη,ciwci+∫ciηwσυGα¯η,υΔα¯υ+∫ciηzα¯υwσυGη,υΘσυΔα¯υ−∫ciηMΘυeΦυυ,η0φυwσυΘσυ2Gη,υΔα¯υ≤Gη,ciwci+∫ciηGη,υzα¯υ−h2η,υΘσυGη,υwσυΘσυΔα¯υ−∫ciηGη,υMΘυeΦυυ,η0φυwσυΘσυ2Δα¯υ≤Gη,ciwci+∫ciηφυzα¯υGη,υ−h2η,υΘσυ24MGη,υΘυeΦυυ,η0Gη,υΔα¯υ.
Lettingη⟶β¯i−, we get(33)∫ciβ¯iGβ¯i,υbυΘυeΦυσυ,η0−φυzα¯υGβ¯i,υ−h2β¯i,υΘσυ24MGβ¯i,υΘυeΦυυ,η0Δα¯υ≤Gβ¯i,ciwci.
By multiplyingGυ,η and taking α¯- fractional integral for (31) from η to ci, we get(34)∫ηciGυ,ηbυΘυeΦυσυ,η0Δα¯υ≤−∫ηciGυ,ηwα¯υΔα¯υ+∫ηciGυ,ηzα¯υwσυΘσυΔα¯υ−∫ηciGυ,ηMΘυeΦυυ,η0φυwσυΘσυ2Δα¯υ≤−Gci,ηwci+∫ηciGα¯υ,ηwσυΔα¯υ+∫ηciGυ,ηzα¯υwσυΘσυΔα¯υ−∫ηciGυ,ηMΘυeΦυυ,η0φυwσυΘσυ2Δα¯υ≤−Gci,ηwci+∫ηciGυ,ηzα¯υ+h1υ,ηΘσυGη,υwσυΘσυΔα¯υ−∫ηciGυ,ηMΘυeΦυυ,η0φυwσυΘσυ2Δα¯υ≤−Gci,ηwci+∫ηciGυ,ηφυzα¯υGυ,η+h1υ,ηΘσυ24MGυ,ηΘυeΦυυ,η0Δα¯υ.
Lettingη⟶α¯i+, we get(35)∫α¯iciGυ,α¯ibυΘυeΦυσυ,η0−φυzα¯υGυ,α¯i−h1υ,α¯iΘσυ24MGυ,α¯iΘυeΦυυ,η0Δα¯υ≤−Gci,α¯iwci.
Divide (22) by Gβ¯i,ci and (35) by Gci,α¯i, we obtain(36)1Gβ¯i,ci∫ciβ¯iGβ¯i,υbυΘυeΦυσυ,η0−φυzα¯υGβ¯i,υ−h2β¯i,υΘσυ24MGβ¯i,υΘυeΦυυ,η0Δα¯υ≤wci,1Gci,α¯i∫α¯iciGυ,α¯ibυΘυeΦυσυ,η0−φυzα¯υGυ,α¯i−h1υ,α¯iΘσυ24MGυ,α¯iΘυeΦυυ,η0Δα¯υ≤−wci.
Adding the above two inequalities, imply a contradiction to hypothesis.


### 3.3. Oscillation Criteria for Fractional Euler Equation

We consider fractional Euler-type dynamic equation:(37)yα¯ηα¯+aηα¯yα¯η+bη2α¯yη=hη,η∈η0,∞T^.Theorem 7.
Assume that−a/η∈ℛ+. If there exists ci∈α¯i,β¯i,i=1,2, a function Θη>0 on T,∞T^ and G∈G, for sufficiently large T such that(38)1Gci,α¯i∫α¯iciGυ,α¯ibυΘυυ2α¯eΦυσυ,η0−zα¯υGυ,α¯i−h1υ,α¯iΘσυ24Gυ,α¯iΘυeΦυυ,η0Δα¯υ+1Gβ¯i,ci∫ciβ¯iGβ¯i,υbυΘυυ2α¯eΦυσυ,η0−φηzα¯υGβ¯i,υ−h2β¯i,υΘσυ24Gβ¯i,υΘυeΦυυ,η0Δα¯υ>0,then (37) is oscillatory.Proof.
The proof is the same as that of Theorem6.Remark 2.
Suppose the conditions in Theorem5 are replaced by the assumption given as follows:(39)hη=≥0,α¯1,β¯1,≤0,α¯2,β¯2,then Theorem 5 is accomplished.Remark 3.
In this article, we introduced the interval oscillation criteria for conformable fractional dynamic equations with Riccati transformation. So far, there are many results concerning the fractional model with oscillation criteria (see [19]). However, to the best of our cognizance, there are no results on the oscillation of fractional dynamic equations with forcing term.


## 3.1. Oscillation Criteria for (3) When f Is Not Necessarily Increasing

To establish oscillation criteria, we make use of the following assumption:(H1)fy/y≥K∀y≠0 and for some K>0.Theorem 5.
LetΦη∈ℛ+. If ∫η0∞Δα¯υ/φυ=∞ and H1 holds. If there exists ci∈α¯i,β¯i,i=1,2, a function Θη>0 on T,∞T^ and G∈G for sufficiently large T such that(17)1Gci,α¯i∫α¯iciGυ,α¯iKbυΘυeΦυσυ,η0−φυzα¯υGυ,α¯i+h1υ,α¯iΘσυ24Gυ,α¯iΘυeΦυυ,η0Δα¯υ+1Gβ¯i,ci∫ciβ¯iGβ¯i,υKbυΘυeΦυσυ,η0−φυzα¯υGβ¯i,υ−h2β¯i,υΘσυ24Gβ¯i,υΘυeΦυυ,η0Δα¯υ>0,then (3) is oscillatory.Proof.
Assume that (3) has a nonoscillatory solution y. Then, y is either EN or EP for η≥η1≥η0. Define a generalized Riccati function as follows:(18)wη=Θηφηyα¯ηyηeΦηη,η0.
By Lemma1, clearly wη≥0 and(19)wα¯η=ΘyηeΦηη,η0φηyα¯ηα¯+aηyα¯ηeΦηη,η0eΦηη,η0eΦηση,η0+yηzα¯η−Θηyα¯ηyσηyηφσηyα¯σηeΦηση,η0Θηyη−bηfyη+hηeΦηση,η0+zα¯ηwΘση−Θηφσηyα¯ηyα¯σηyσηyηeΦηση,η0≤−KbηΘηeΦηση,η0+ΘηhηyηeΦηση,η0+zα¯ηwΘση−Θηyα¯ηφσηyα¯σηyηyσηeΦηση,η0.
By assumptions, we takeα¯i,β¯i≥η1 for i=1,2 so that hη≤0 on α¯1,β¯1 with α¯1<β¯1 and yη is EP, or hη≥0 on α¯2,β¯2 with α¯2<β¯2 and yη is EN. On the intervals α¯1,β¯1 and α¯2,β¯2, the above inequality implies that(20)wα¯η≤−KbηΘηeΦηση,η0+zα¯ηwΘση−ΘηeΦηη,η0φηwΘση2.
By multiplyingGη,υ and taking α¯- fractional integral for (20) from ci to η, we have(21)∫ciηKbυΘυeΦυσυ,η0Gη,υΔα¯υ≤−∫ciηwα¯υGη,υΔα¯υ+∫ciηzα¯ηwΘσηGη,υΔα¯υ−∫ciηΘυeΦυυ,η0φυwΘση2Gη,υΔα¯υ≤Gη,ciwci+∫ciηwσυGα¯η,υΔα¯υ+∫ciηzα¯υwσυGη,υΘσυΔα¯υ−∫ciηΘυeΦυυ,η0φυwΘσυ2Gη,υΔα¯υ≤Gη,ciwci+∫ciηGη,υzα¯υ−h2η,υΘσυGη,υwσυΘσυΔα¯υ−∫ciηGη,υΘυeΦυυ,η0φυwσυΘσυ2Δα¯υ≤Gη,ciwcid+∫ciηGη,υφυzα¯υGη,υ−Θσυh2η,υ24Gη,υΘυeΦυυ,η0Δα¯υ.
Lettingη⟶β¯i−, we get(22)∫ciβ¯iGη,υKbηΘηeΦηση,η0−φυzα¯υGη,υ−Θσυh2η,υ24Gη,υΘυeΦυυ,η0Δα¯υ≤Gβ¯i,ciwci.
By multiplyingGυ,η and taking integration of α¯-order from η to ci for (20), we have(23)∫ηciGυ,ηKbυΘυeΦυσυ,η0Δα¯υ≤−∫ηciGυ,ηwα¯υΔα¯υ+∫ηciGυ,ηzα¯υwσυΘσυΔα¯υ−∫ηciGυ,ηΘυeΦυυ,η0φυwσυΘσυ2Δα¯υ≤−Gci,ηwci+∫ηciGα¯υ,ηwσυΔα¯υ+∫ηciGυ,ηzα¯υwσυΘσυΔα¯υ−∫ηciGυ,ηΘυeΦυυ,η0φυwσυΘσυ2Δα¯υ≤−Gci,ηwci+∫ηciGυ,ηzα¯υ+h1υ,ηΘσυGη,υwσυΘσυΔα¯υ−∫ηciGυ,ηΘυeΦυυ,η0φυwσυΘσυ2Δα¯υ≤−Gci,ηwci+∫ηciGυ,ηφυzα¯υGυ,η+h1υ,ηΘσυ24Gυ,ηΘυeΦυυ,η0Δα¯υ.
Lettingη⟶α¯i+, we get(24)∫α¯iciGυ,ηKbηΘηeΦηση,η0−φυzα¯υGυ,η+h1η,υΘσυ24Gυ,ηΘυeΦυυ,η0Δα¯υ≤−Gci,α¯iwci.
Dividing (33) by Gβ¯i,ci and (24) by Gci,α¯i, we attain(25)1Gβ¯i,ci∫ciβ¯iGβ¯i,υKbυΘυeΦυσυ,η0−φυzα¯υGβ¯i,υ−h2β¯i,υΘσυ24Gβ¯i,υΘυeΦυυ,η0Δα¯υ≤wci,1Gci,α¯i∫α¯iciGυ,α¯iKbυΘυeΦυσυ,η0−φυzα¯υGυ,α¯i+h1υ,α¯iΘσυ24Gυ,α¯iΘυeΦυυ,η0Δα¯υ≤−wci.
Adding the above two inequalities, we get a contradiction to hypothesis.Remark 1.
Assume that the condition of Theorem5 holds for every T≥0 and function Gυ,η∈G for sufficiently large T. If there exist some ci∈α¯i,β¯i,i=1,2 so that(26)∫α¯iciGυ,α¯iKbυΘυeΦυσυ,η0−φυzα¯υGυ,α¯i+h1υ,α¯iΘσυ24Gυ,α¯iΘυeΦυυ,η0Δα¯υ>0,∫ciβ¯iGβ¯i,υKbυΘυeΦυσυ,η0−φυzα¯υGβ¯i,υ−h2β¯i,υΘσυ24Gβ¯i,υΘυeΦυυ,η0Δα¯υ>0,for i=1,2, then (3) is oscillatory.LetG=Gη−υ∈G. Then, h1η−υ=h2η−υ, and we use hη−υ for them. By using this Gη−υ in Theorem 5, we have the corollary as follows.Corollary 1.
Suppose that all the assumptions of Theorem5 hold and for every T≥η0, there is a T≤α¯1<2c1−α¯1≤α¯2<2c2−α¯2 so that hη=≤0,η∈α¯1,2c1−α¯1,≥0,η∈α¯2,2c2−α¯2. If there is a G:=Gη−υ∈G such that(27)1Gci−α¯i∫α¯iciGυ−α¯i2KbυΘυeΦυσυ,η0−φυzα¯υ2Gυ−α¯i+h2υ−α¯iΘ2συ2Gυ−α¯iΘυeΦυυ,η0Δα¯υ>0,then (3) is oscillatory.


## 3.2. Oscillation Criteria for (3) When f Is Increasing

To establish oscillation criteria, we make use of the following assumption:(H2)
f′ exists and f′y≥M for all y≠0 and for some M>0Theorem 6.
LetΦη∈ℛ+. If H2,∫η0∞Δα¯υ/φυ=∞ holds. If there exists ci∈α¯i,β¯i,i=1,2, a function Θη>0 on T,∞T^ and G∈G for sufficiently large T such that(28)frac1Gci,α¯i∫α¯iciGυ,α¯ibυΘυeΦυσυ,η0−φηzα¯υGυ,α¯i+h1υ,α¯iΘσυ24MGυ,α¯iΘυeΦυυ,η0Δα¯υ+1Gβ¯i,ci∫ciβ¯iGβ¯i,υbυΘυeΦυσυ,η0−φηzα¯υGβ¯i,υ−h2β¯i,υΘσυ24MGβ¯i,υΘυeΦυυ,η0Δα¯υ>0,then (3) is oscillatory.Proof.
Assume that (3) has a nonoscillatory solution y. Define a generalized Riccati function as follows:(29)wη=Θηφηyα¯ηfyηeΦηη,η0.
Clearly,wη≥0 and(30)wα¯η=ΘηfyηeΦηη,η0φηyα¯ηα¯+aηeΦηη,η0yα¯ηeΦηση,η0eΦηη,η0+fyηzα¯η−Θηfyηα¯fyσηfyηφσηyα¯σηeΦηση,η0=Θηfyη−bηfyη+hηeΦηση,η0+zα¯ηwΘση−Θηφσηfyηα¯yα¯σηfyσηfyηeΦηση,η0=−bηΘηeΦηση,η0+hηΘηfyηeΦηση,η0+zα¯ηwΘση−Θηφσηfyηα¯yα¯σηfyσηfyηeΦηση,η0.
By the hypothesis, we takeα¯i,β¯i≥η1 for i=1,2 such that hη≤0 on α¯1,β¯1 with α¯1<β¯1 and yη is EP, or hη≥0 on α¯2,β¯2 with α¯2<β¯2 and yη is EN. On the intervals α¯1,β¯1 and α¯2,β¯2, we have(31)wα¯η≤−bηΘηeΦηση,η0+zα¯ηwΘση−Θηf′yηyα¯ηφσηyα¯σηfyηfyσηeΦηση,η0≤−bηΘηeΦηση,η0+zα¯ηwΘση−MΘηeΦηη,η0φηwz2ση.
By multiplyingGη,υ and taking α¯- fractional integral for (31) from ci to η, we have(32)∫ciηbυΘυeΦυσυ,η0Gη,υΔα¯υ≤−∫ciηwα¯υGη,υΔα¯υ+∫ciηzα¯υwσυΘσυGη,υΔα¯υ−∫ciηMΘυeΦυυ,η0φυwσυΘσυ2Gη,υΔα¯υ≤Gη,ciwci+∫ciηwσυGα¯η,υΔα¯υ+∫ciηzα¯υwσυGη,υΘσυΔα¯υ−∫ciηMΘυeΦυυ,η0φυwσυΘσυ2Gη,υΔα¯υ≤Gη,ciwci+∫ciηGη,υzα¯υ−h2η,υΘσυGη,υwσυΘσυΔα¯υ−∫ciηGη,υMΘυeΦυυ,η0φυwσυΘσυ2Δα¯υ≤Gη,ciwci+∫ciηφυzα¯υGη,υ−h2η,υΘσυ24MGη,υΘυeΦυυ,η0Gη,υΔα¯υ.
Lettingη⟶β¯i−, we get(33)∫ciβ¯iGβ¯i,υbυΘυeΦυσυ,η0−φυzα¯υGβ¯i,υ−h2β¯i,υΘσυ24MGβ¯i,υΘυeΦυυ,η0Δα¯υ≤Gβ¯i,ciwci.
By multiplyingGυ,η and taking α¯- fractional integral for (31) from η to ci, we get(34)∫ηciGυ,ηbυΘυeΦυσυ,η0Δα¯υ≤−∫ηciGυ,ηwα¯υΔα¯υ+∫ηciGυ,ηzα¯υwσυΘσυΔα¯υ−∫ηciGυ,ηMΘυeΦυυ,η0φυwσυΘσυ2Δα¯υ≤−Gci,ηwci+∫ηciGα¯υ,ηwσυΔα¯υ+∫ηciGυ,ηzα¯υwσυΘσυΔα¯υ−∫ηciGυ,ηMΘυeΦυυ,η0φυwσυΘσυ2Δα¯υ≤−Gci,ηwci+∫ηciGυ,ηzα¯υ+h1υ,ηΘσυGη,υwσυΘσυΔα¯υ−∫ηciGυ,ηMΘυeΦυυ,η0φυwσυΘσυ2Δα¯υ≤−Gci,ηwci+∫ηciGυ,ηφυzα¯υGυ,η+h1υ,ηΘσυ24MGυ,ηΘυeΦυυ,η0Δα¯υ.
Lettingη⟶α¯i+, we get(35)∫α¯iciGυ,α¯ibυΘυeΦυσυ,η0−φυzα¯υGυ,α¯i−h1υ,α¯iΘσυ24MGυ,α¯iΘυeΦυυ,η0Δα¯υ≤−Gci,α¯iwci.
Divide (22) by Gβ¯i,ci and (35) by Gci,α¯i, we obtain(36)1Gβ¯i,ci∫ciβ¯iGβ¯i,υbυΘυeΦυσυ,η0−φυzα¯υGβ¯i,υ−h2β¯i,υΘσυ24MGβ¯i,υΘυeΦυυ,η0Δα¯υ≤wci,1Gci,α¯i∫α¯iciGυ,α¯ibυΘυeΦυσυ,η0−φυzα¯υGυ,α¯i−h1υ,α¯iΘσυ24MGυ,α¯iΘυeΦυυ,η0Δα¯υ≤−wci.
Adding the above two inequalities, imply a contradiction to hypothesis.


## 3.3. Oscillation Criteria for Fractional Euler Equation

We consider fractional Euler-type dynamic equation:(37)yα¯ηα¯+aηα¯yα¯η+bη2α¯yη=hη,η∈η0,∞T^.Theorem 7.
Assume that−a/η∈ℛ+. If there exists ci∈α¯i,β¯i,i=1,2, a function Θη>0 on T,∞T^ and G∈G, for sufficiently large T such that(38)1Gci,α¯i∫α¯iciGυ,α¯ibυΘυυ2α¯eΦυσυ,η0−zα¯υGυ,α¯i−h1υ,α¯iΘσυ24Gυ,α¯iΘυeΦυυ,η0Δα¯υ+1Gβ¯i,ci∫ciβ¯iGβ¯i,υbυΘυυ2α¯eΦυσυ,η0−φηzα¯υGβ¯i,υ−h2β¯i,υΘσυ24Gβ¯i,υΘυeΦυυ,η0Δα¯υ>0,then (37) is oscillatory.Proof.
The proof is the same as that of Theorem6.Remark 2.
Suppose the conditions in Theorem5 are replaced by the assumption given as follows:(39)hη=≥0,α¯1,β¯1,≤0,α¯2,β¯2,then Theorem 5 is accomplished.Remark 3.
In this article, we introduced the interval oscillation criteria for conformable fractional dynamic equations with Riccati transformation. So far, there are many results concerning the fractional model with oscillation criteria (see [19]). However, to the best of our cognizance, there are no results on the oscillation of fractional dynamic equations with forcing term.


## 4. Examples

In this part, in view of the conditions acquired in Section3, we introduce some examples to represent the adequacy of the suggested nonlinear fractional model and the merits of our methodology.Example 1.
We take the nonlinear conformable fractional damped dynamic equation:(40)η−1y1/2η1/2+η−3y1/2η+η2yη1+eyη=sinη,η∈1,∞.
Here,T^=ℝ; α¯=1/2; φη=η−1; aη=η−3; bη=η2; fyη=yηγ1+eyη; η0=1. Clearly, fy/y=1+eyη≥1=K, ση=η, μη=0, Φη=−η−5/2,(41)∫η0∞Δα¯υφυ=∫η0∞υ−1/2υ−1dυ=∫η0∞υ1/2dυ=∞,1≥eΦηη,η0=exp∫η0ηΦυdυ≥1+∫1ηΦυdυ=1−∫1ηυ−5/2dυ=1+23η−3/2−1≥13.
Letα¯1=nπ, β¯1=α¯2=n+1π, β¯2=n+2π, ci=α¯i+β¯i/2,i=1,2;n=1,2,…, Θη=η5/2, and Gη,υ=η−υ. Then, we get(42)1Gci,α¯i∫α¯iciGυ,α¯iKbυΘυeΦυσυ,η0−φυzα¯υGυ,α¯i+h1υ,α¯iΘσυ24Gυ,α¯iΘυeΦυυ,η0Δα¯υ+1Gβ¯i,ci∫ciβ¯iGβ¯i,υKbυΘυeΦυσυ,η0−φυzα¯υGβ¯i,υ+h2β¯i,υΘσυ24Gβ¯i,υΘυeΦυυ,η0Δα¯υ≥2π∫α¯iciυ−α¯iυ9/2−3s−15/2υ2υ−α¯i+υ−α¯iυ5/224υ−α¯iυ5/2υ1/2dυ+2π∫ciβ¯iβ¯i−υυ9/2−3s−15/2υ2β¯i−υ+β¯i−υυ5/224β¯i−υυ5/2υ1/2dυ>0.
Hence, by Theorem5, (40) is oscillatory.Example 2.
We take the nonlinear conformable fractional damped dynamic equation:(43)Δ1/2η−1/2Δ1/2yη+η−5/2Δ1/2yη+η1/2yη=cos2t.
Here,η∈2,∞ℤ; T^=ℤ; α¯=1/2; φη=η−1/2; aη=η−5/2; bη=η1/2; η0=2; fyη=yη.
Therefore, we getf′y=1=M, ση=η+1, μη=1, Φη=−η−5/2, 1+μηΦη=1−η−5/2≥1−1/2>0 which implies Φη∈ℛ+. Also,(44)∫η0∞Δα¯υφυ=∫2∞υα¯−1Δυφυ=∑υ=2∞υα¯−1υ−1/2=∑υ=2∞1=∞,1>eΦηη,η0=exp∫η0ηΦυΔυ≥1+∫2ηΦυΔυ=1−∑υ=2η−1υ−5/2≥1−∫1η−1υ−5/2dυ=1+23η−1−3/2−1>13.
Letα¯1=nπ+π/4,β¯1=α¯2=n+1π+π/4,β¯2=n+2π+π/4,ci=α¯i+β¯i/2,i=1,2;n=1,2,… and Θη=η2. Then, all the constraints of Theorem 6 are satisfied with Gη,υ=η−υ, and (40) is oscillatory.Example 3.
We take the following conformable fractional Euler-type dynamic equations(45)y1/2η1/2+aη−1/2y1/2η+bη−1yη=cosη,where η∈1,∞ℤ, T^=ℝ, α¯=1/2, and η0=1. Then, μη=0 and e−a/υη,η0=exp∫η0η−a/υdυ=η−a since ση=η. If we let α¯1=nπ+π/2, β¯1=α¯2=n+1π+π/2, β¯2=n+2π+π/2, ci=α¯i+β¯i/2,i=1,2,n=1,2,… and Θη=η2, then all the constraints of Theorem 7 are fulfilled with Gη,υ=η−υ for suitable choice of a and b. Hence, we conclude that equation (45) is oscillatory.


## 5. Conclusion

In this article, we established the interval oscillation criteria for conformable fractional damped dynamic equations with forcing term. The obtained results are improved in the sense which provides sufficient criteria for the forced oscillation of the considered equation with various conditions. Finally, numerical examples are also presented to validate the theoretical results of this study, and in addition, the developed methodology has been tested on the forced oscillation of a fractional Euler-type dynamic equation with an example.It would be interesting to extend the results proposed in this article to more complex systems, including fractional dynamic equations with time delay. This will be examined in our future works.



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