# The Oscillatory of Linear Conformable Fractional Differential Equations of Kamenev Type

Authors: Hui Liu; Run Xu
Journal: Discrete Dynamics in Nature and Society
 (2020)

---

## Abstract

In this paper, the oscillatory of the Kamenev-type linear conformable fractional differential equations in the form ofptyα+1tα+yα+1t+qtyt=0 is studied, where t≥t0 and 0<α≤1. By employing a generalized Riccati transformation technique and integral average method, we obtain some oscillation criteria for the equation. We also give some examples to illustrate the significance of our results.


---

## Body

## 1. Introduction

The fractional derivative originated in the 17th century and formed a relatively complete form in the 19th century. In the existing literature, we can find many definitions of the fractional integral and fractional differential given by these scholars, such as Riemann–Liouville, Caputo, Weyl, Hadamard, and Chen. In the past decades, fractional differential equations have been widely used in viscoelasticity, electrical networks, signal processing, systems identification, and other natural phenomena and physical problems, fluid flow, rheology, and many other fields. There are many literature studies that have investigated the existence, uniqueness, and stability of fractional differential equations and a series of conclusions about the qualitative of solutions, see [1–9] and their references.In 2014, a new definition, which is called the conformable derivative, was proposed in the literature [10], and the properties and calculations of conformable derivatives have been studied in the literature [11–14]. It almost satisfied all the properties of the normal derivative: the derivative of constant is zero, the rules for the product and quotient of two functions, and the chain rule. Because it combines the best properties, many research studies have increased their research on it in recent years. Although there is much about the oscillations of fractional differential equations (see [15–28] and the references therein), there is little about the conformable fractional differential equations of Kamenev type.In 1978, Kamenev [7] studied the oscillation of the following:(1)ptyαtα+qtyt=0,t≥t0,where p∈Ct0,∞,0,∞,q∈Ct0,∞,R,0<α≤1 and q might change signs. And the above equation is oscillation under the condition(2)lim sup1tm∫t0tt−smqsds=∞,m>1.In 2019, Shao and Zhaowen [14] established new oscillation criteria of Kamenev type for linear conformable of the above equation, where p∈Ct0,∞,0,∞,q∈Ct0,∞,R,0<α≤1 and q might change signs.Inspired by the above, by extending the order of the equation containing conformable fractional derivative and increasing the number of terms, we make it have new application value.In this paper, we discuss the oscillatory behavior about linear conformable fractional differential equations of Kamenev type of the following:(3)ptyα+1tα+yα+1t+qtyt=0,t≥t0,where p∈Ct0,∞,0,∞,q∈Ct0,∞,R,0<α≤1.In this paper, we always assume that the following conditions are in force:(C1): there exist a constant k∈N+ such that yαt/yα+1t≥k, t≥t0.(C2): Ht,s is a continuous function and satisfies the following:(1)
Ht,t=0,Ht,s>0 for t>s≥t0.(2)
H has a continuous and nonpositive partial derivative on D0 with respect to the second variable, where D0=t,s:t>s≥t0.(3)
Leth:D0⟶R+ be a continuous function and satisfy(4)Hst,s=−ht,sHt,s.


## 2. Main Results

First, we present the definition and properties of conformable fractional derivatives and integral.(D1): the left conformable fractional derivative starting from t0 of a function f: t0,∞⟶R of order 0<α≤1 is defined by(5)Tt0αft=fαt=limϵ⟶0ft+ϵt−t01−α−ftϵ,wheret0=0, we write Tα. If Tt0αft exists on t0,t1, then(6)Tt0αft0=limt⟶t0+fαt.Iff is differentiable, then(7)Tt0αft=fαt=t−t01−αf′t.(D2): if functions f,g∈C0,∞,R are α-differential for t0>0,α∈0,1 and continuous at t0.(1)
For all real constantsaandb, we can get Tt0αaf+bg=aTt0αf+bTt0αg(2)
Tt0αfg=fTt0αg+gTt0αf(3)
For allp, Tt0αtp=ptp−α(4)
Tt0αf/g=gTt0αf−fTt0αg/g2, for g≠0(5)
Whenc is a constant, we have Tt0αc=0(D3): let f,g:t0,t1⟶R be two functions, and fg is differentiable. Then,(8)∫t0t1fsgαsdt0αs=fsgs|t0t1−∫t0t1gsfαsdt0αs.Theorem 1.
Suppose that (C1) and (C2) hold, and there exists a function rt∈Ct0,∞,R+ such that(9)lim supt⟶∞1Ht,t0∫t0+1tHt,srsqs−164kz2sHt,sdt0αs=∞,then equation (3) is oscillatory, where M+s=max0,rαsps−rs,(10)Zs=rsps−3/2M+sHt,ss−t01−αht,s.Proof.
On the contrary, we suppose that equation (3) has a nonoscillation solution yt≠0, t≥t0. Defining V(t) as a generalized Riccati substitution,(11)Vt=rtptyα+1tyt.
From (C1), (3), and (D2), we obtain(12)Vαt=rαtptyα+1tyt+rtptyα+1tytα=rαtrtVt+rtptyα+1tαyt−yαtptyα+1ty2t=rαtrtVt−rtqt−rtyα+1tyt−rtptyαtyα+1ty2t=rαtrtVt−rtqt−Vtpt−V2trtpt⋅yαtyα+1t≤rαtpt−rtrtptVt−krtptV2t−rtqt≤M+trtptVt−krtptV2t−rtqt,that is,(13)Vαt≤M+trtptVt−krtptV2t−rtqt.
Multiply both sides of (13) by Ht,s, and integrate from T to t0T≥t0≥t0. In accordance with (D1), (C2), (13), and the integration by parts, we get(14)∫TtHt,srsqsdt0αs≤−∫TtHt,sVαsdt0αs+∫TtHt,sM+srspsVsdt0αs−∫TtHt,skrspsV2sdt0αs=Ht,svT−∫TtvsHαt,sdt0αs+∫TtHt,sM+srspsVsdt0αs−∫TtHt,skrspsV2sdt0αs=Ht,sVT−∫Tts−t01−αht,sHt,sVsdt0αs+∫TtHt,sM+srspsVsdt0αs−∫TtHt,skrspsV2sdt0αs=Ht,TVT−∫TtHt,skrspsV2s+s−t01−αht,sHt,s−Ht,sM+srspsVsdt0αs.
Letλ=2,A=s−t01−αht,s⋅Ht,s1/4 and B=1/2M+s/rsps⋅Ht,s3/4/s−t01−αht,s.
From(15)Aλ−λABλ−1≥λ−1Bλ,λ>1,we can have(16)∫TtHt,srsqsdt0αs≤Ht,TVT−∫TtHt,skrspsV2s+14M+2srsps2Ht,s⋅Ht,ss−t01−αht,sVsdt0αs=Ht,TVT−∫TtkHt,srspsVs+18kZs2dt0αs+∫Tt116kZ2Sdt0αs≤Ht,TVT+∫Tt164kZ2Sdt0αs,where Zs=rsps−3/2M+sHt,s/s−t01−αht,s.
Fort≥T≥t0, we have(17)∫TtHt,srsqs−164kZ2Sdt0αs≤Ht,TVT≤Ht,TVT≤Ht,t0VT.
LettingT=t0+1, we get(18)∫t0+1tHt,srsqs−164kZ2Sdt0αs≤Ht,t0Vt0+1.
Dividing both sides of the above inequality byHt,t0 and taking the upper limit in both sides as t⟶∞, we have the contradict with (9). The proof is complete.
Lettingm be an integer with m>1, then Ht,t=0,Ht,s>0. We have ht,s=mt−sm−2/2, so equation (9) is transformed to the following.Corollary 1.
If there exists a functionrt∈Ct0,∞,R+ and(19)limsupt⟶∞1tm∫t0+1tt−smrsqs−164kJ2sHt,sdt0αs=∞,then equation (3) is oscillatory, where Js=rsps−3/2M+st−sm+2/2/s−t01−αm.Before we introduce Theorem2, we present the properties of the conformable fractional integral.(D4): the left conformable fractional integral of order 0<α≤1 starting at t0 is defined by(20)It0αft=∫t0ts−t0α−1fsds=∫t0tfsdt0αs.We call the conformable fractional integral of a given functionf is α-integrable if it exists.(D5): assume that f:α,∞⟶R is continuous and 0<α≤1. For all t>t0, we can get(21)It0αTt0αft=∫t0ts−t0α−1s−t01−αf′sds=ft−ft0,Tt0αIt0αft=ft.(D6) (Cauchy–Schwarz inequality with conformable fractional derivative). Let f,g:t0,∞⟶R be two functions such that f2 and g2 are α-integrable. Then,(22)∫t0tfsgαsdt0αs2≤∫t0tfs2dt0αs⋅∫t0tgs2dt0αs.Theorem 2.
LetHt,s be defined as in (C2) and (C1) also hold. Suppose that(23)0<infs≥t0liminft⟶∞Ht,sHt,t0≤∞,(24)limsupt⟶∞1Ht,t0∫t0tZ2s16kdt0αs<∞,hold. For every T≥t0, if there exists functions S∈Ct0,∞,R, rt∈Ct0,∞,R+ such that(25)limsupt⟶∞1Ht,s∫TtHt,srsqs−164kZsdt0αs≥ST,then equation (3) is oscillatory provided(26)∫t0+1∞kS+2srspsdt0αs=∞,where rt∈Ct0,∞, Zs=rsps−3/2M+sHt,s/s−t01−αht,s, and S+t=maxSt,0.Proof.
On the contrary, for allt≥t0, we suppose that equation (3) has a nonoscillation solution yt≠0. Define V(t) as a generalized Riccati substitution:(27)Vt=rtptyα+1tyt.
Proceeding as the proof of Theorem1, for t≥T≥t0, from (16), we can get(28)∫TtHt,srsqs−164kZsdt0αs≤Ht,TVT−∫TtkHt,srspsVs+18kZs2dt0αs.
Dividing both sides of the above inequality byHt,T, we obtain(29)1Ht,T∫TtHt,srsqs−164kZsdt0αs≤VT−1Ht,T∫TtkHt,srspsVs+18kZs2dt0αs.
Immediately, it shows the following:(30)limsupt⟶∞1Ht,s∫TtHt,srsqs−164kZsdt0αs≤VT−liminft⟶∞1Ht,s∫TtkHt,srspsVs+18kZs2dt0αs.
So, for allT≥t0, from (25), we have(31)VT≥ST+liminft⟶∞1Ht,s∫TtkHt,srspsVs+18kZs2dt0αs.
It means that, for everyT≥t0,(32)VT≥ST,liminft⟶∞1Ht,s∫TtkHt,srspsVs+18kZs2dt0αs<∞.
Then, for everyT0>t0, we obtain(33)∞>liminft⟶∞1Ht,T0∫T0tkHt,srspsVs+18kZs2dt0αs≥liminft⟶∞1Ht,T0∫T0tkHt,srspsV2sdt0αs+1Ht,T0∫T0t14Ht,srspsZsVsdt0αs.
Fort≥T0>t0, we let(34)Et=1Ht,T0∫T0tkHt,srspsV2sdt0αs,Ft=1Ht,T0∫T0t14Ht,srspsZsVsdt0αs.
From (3), we have(35)liminft⟶∞Ft+Et<∞.
Next, we claim that(36)∫T0∞k⋅V2srspsdt0αs<∞.
From (23), we can find a constant β that will allow(37)infs≥t0liminft⟶∞Ht,sHt,t0>β>0.
If (36) is not true, then for every τ>0, we can find T1>T0 such that(38)∫T0tk⋅V2srspsdt0αs≥2τβ,t>T1.
From (D3), we obtain the following:(39)Et=1Ht,T0∫T0tHt,s∫T0sk⋅V2ξrξpξdt0αξαdt0αs=−1Ht,T0∫T0t∫T0sk⋅V2ξrξpξdt0αξHt,ssαdt0αs≥−1Ht,T0∫T1t∫T0sk⋅V2ξrξpξdt0αξHt,ssαdt0αs≥−2τβ1Ht,T0∫T1tHt,ssαdt0αs≥2τβHt,T1Ht,T0,t≥T1>T0.
Thus, (4) and (7) give Ht,ssα≤0, for t≥T1>T0, and we have Ht,T0≥Ht,T1>0.
However,(40)liminft⟶∞Ht,T1Ht,T0>β,T1>T0.
Noticing thatT0>t0 can be arbitrarily close to t0 and H is continuous, we choose T2≥T1 such that, for t≥T2,(41)Ht,T1Ht,T0≥β2.
So, from (39) and (41), we have(42)Et≥τ,t≥T2,and τ>0 is arbitrary, and we obtain(43)limt⟶∞Et=∞.
In the intervalT0,∞, we select a sequence ti, i=1,2,…, which makes limi⟶∞ti=∞, and(44)limi⟶∞Eti+Fti=liminft⟶∞Et+Ft.
From (35), we can always find a constant M which lets(45)Eti+Fti≤M,i=1,2,….
Through (43) and (45), we gain(46)limi⟶∞Eti=∞,(47)limi⟶∞Fti=−∞.
The above also show that(48)limi⟶∞F2tiEti=∞.
Using (46), divide both sides of (45) by Eti, and because i is sufficiently large, we receive(49)1+FtiEti≤MEti<12.
So, we can get(50)FtiEti<−12.
By (D6), and for every positive integer i, we gain(51)F2ti=1H2ti,T0∫T0ti14Hti,srspsZsVsdt0αs2≤1Hti,T0∫T0ti116kZ2sdt0αs⋅1Hti,T0∫T0tikHti,srspsV2sdt0αs=1Hti,T0∫T0tiZ2s16kdt0αsEti.
That is,(52)F2tiEti≤1Hti,T0∫T0tiZ2s16kdt0αs.
Using (37), we obtain(53)liminft⟶∞Ht,T0Ht,t0>β.
Then, there existsT0′≥T0, for t≥T0′ such that(54)Ht,T0Ht,t0≥β.
So, we have for sufficiently largei,(55)Hti,T0Hti,t0≥β.
For any largei, (52) is going to be equal to(56)F2tiEti≤1β1Hti,t0∫T0tiZ2s16kdt0αs≤1β1Hti,t0∫t0tiZ2s16kdt0αs.
Sincelimi⟶∞F2ti/Eti=∞, that implies(57)limsupt⟶∞1Ht,t0∫t0tZ2s16kdt0αs=∞,which contradicts with (24). This gives (36), that is,(58)∫t0+1∞k⋅V2srspsdt0αs<∞,and by using VT≥ST>0, we can gain(59)∫t0+1∞k⋅S+2srspsdt0αs≤∫t0+1∞k⋅V2srspsdt0αs<∞,lim,which contradicts with (26). The proof is complete.Corollary 2.
If there exists a functionrt∈Ct0,∞,R+ and a function S∈Ct0,∞,R, Ht,s=t−sm,m>1. So, (24) and (25) are replaced by(60)limsupt⟶∞1tm∫t0tJ2s16kdt0αs<∞,(61)limsupt⟶∞1tm∫Ttt−smrsqs−164kJsdt0αs≥ST,where Js=rsps−3/2M+st−sm+2/2/s−t01−αm; then, equation (3) is oscillatory.


## 3. Example

Consider the oscillation for the conformable fractional differential equations(62)yα+1tα+yα+1t+yt=0.t≥t0,0<α≤1.Here,yαt is the conformable fractional derivative. Compared with (1.1), here ps=1, and let rs=ps=1, that is, Js=0. Using Corollary 1, we can verify that(63)limsupt⟶∞1tm∫t0+1tt−smrsqs−164kJ2sHt,sdt0αs=1tm∫t0+1tt−smdt0αs=1tm∫t0+1tt−sms−toα−1ds≥1tm∫t0+1tt−sms−T1α−1ds.Lettings−T1=ξ, we have(64)∫t0+1tt−sms−T1α−1ds=∫0t−T1ξα−1t−ξ+T1mdξ=∫0t−T1ξα−1t−T1−ξmdξ=∫0t−T1ξα−11−ξt−T1mt−T1mdξ=t−T1m+1t−T1α−1∫0t−T1ξt−T1α−11−ξt−T1mdξt−T1.Next, lettingv=ξ/t−T1, then the above one(65)=t−T1m+1t−T1α−1∫0t−T1ξt−T1α−11−ξt−T1mdξt−T1=t−T1m+α∫01vα−11−vmdv.That is,(66)∫t0+1tt−sms−T1α−1ds=t−T1m+αBm+1,α,where B⋅,⋅ is the classical beta function defined by(67)Bm,n=∫011−sm−1sn−1ds.So, we have1/tm∫t0+1tt−smdt0αs=1/tmt−T1m+αBm+1,α=∞, which shows (19) holds. From Corollary 1, we have equation (3) is oscillatory.


## 4. Conclusion

In the paper, we concerned with the oscillation of the Kamenev type for linear conformable fractional differential equations by using generalized Riccati transformation, Cauchy–Schwarz inequality for the conformable fractional derivative, and average inequality technique. Because conformable derivative combines the best qualities of fractional derivatives, it is now well developed. We will continue to refine this part of the theory and continue to study the oscillation of fractional differential equations.



---

*Source: 3857592-2020-04-09.xml*