# Kamenev Type Oscillatory Criteria for Linear Conformable Fractional Differential Equations

Authors: Jing Shao; Zhaowen Zheng
Journal: Discrete Dynamics in Nature and Society
 (2019)

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## Abstract

Using integral average method and properties of conformable fractional derivative, new Kamenev type oscillation criteria are given firstly for conformable fractional differential equations, which improve known results in oscillation theory. Examples are also given to illustrate the effectiveness of the main results.


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## Body

## 1. Introduction

Fractional differential equations are used in various fields such as physics, mathematics, biology, biomedical sciences, and finance as well as other disciplines (see [1, 2]). In the last few decades, a lot of attention was paid to finding the more suitable definitions of fractional derivatives. There are many definitions in the existent literature, such as the Riemann-Liouville, Caputo, Riesz, Riesz-Caputo, Weyl, Grunwald-Letnikov, Hadamard, and Chen derivatives, etc. In 2014, R. Khalil et al. [3] have suggested a new fractional derivative, which is called the conformable derivative. This new definition satisfies almost all the requirements of standard derivative, for example, the conformable derivative of a constant is zero, and the chain rule (see [3–7] and references therein).In this paper, we consider the linear conformable fractional differential equation(1)ptyαtα+qtyt=0,t≥t0,where p∈C([t0,∞),(0,∞)), q∈C([t0,∞),R), 0<α≤1, and q might change signs.A nontrivial solution of (1) is called oscillatory if it has arbitrarily large zeros; otherwise, it is said to be nonoscillatory. Equation (1) is said to be oscillatory if all its solutions are oscillatory.Whenα=1, we have the following second-order differential equation:(2)pty′t′+qtyt=0.There are a lot of papers involving the oscillation for (2) and other linear, nonlinear, damped, and forced differential equations or Hamiltonian systems (see [8–11]) since the foundation work of Wintner [11] (see also for [12–25]). Especially, if p(t)≡1, we obtain the second-order linear Hill equation(3)y′′t+qtyt=0.In 1978, Kamenev [8] established a new oscillation criterion of differential equation (1), using integral average method, which has the result of Wintner as a particular case. The obtained result in [8] states that the condition (4)limsupt→∞1tm∫t0tt-smqsds=∞,for some integer m>1, is sufficient for the oscillation of (3).In 1989, Philos [26] improved the Kamenev type criterion to obtain the following results for (3).Theorem 1.
LetH:D≡{(t,s):t≥s≥t0}→R be a continuous function which satisfies H(t,t)=0 for t≥t0, H(t,s)>0 for t>s≥t0; H has a continuous and nonpositive partial derivative on D0≡{(t,s):t>s≥t0} with respect to the second variable. Moreover, let h:D0→R be a continuous function with (5)-∂H∂st,s=ht,sHt,sfor all (t,s)∈D0. Then (3) is oscillatory if (6)limsupt→∞1Ht,t0∫t0tHt,sqs-14h2t,sds=∞.Theorem 2.
LetH and h be defined as in Theorem 1. Suppose that(7)0<infs≥t0limt→∞infHt,sHt,t0≤∞and (8)limsupt→∞1Ht,t0∫t0th2t,sds<∞.Moreover, there exists a continuous function A:[t0,∞)→R such that, for every T≥t0, (9)limsupt→∞1Ht,T∫TtHt,sqs-14h2t,sds≥AT,where A+(t)=max{A(t),0}. Then (3) is oscillatory provided(10)∫t0∞A+2tdt=∞.Theorem 3.
LetH and h be defined as in Theorem 1. Suppose that (7) and (11)liminft→∞1Ht,t0∫t0tHt,sqsds<∞hold. Moreover, there exists a continuous function A:[t0,∞)→R such that (12)liminft→∞1Ht,T∫TtHt,sqs-14h2t,sds≥AT,for every T≥t0. Then (10) implies the oscillation of (3).As a new research field, the oscillation of fractional differential equations has been widely studied by many authors (e.g., see [27–35] and the references quoted therein). Among these oscillation criteria, most of them are major in the fractional differential equations with forcing terms. The Kamenev type oscillation criteria of fractional differential equations without forcing terms are obtained [27] under the frame of Caputo derivative. However, to our best knowledge, the study of oscillatory behavior of linear conformable fractional differential equation has not been seen in literature. The purpose of this paper is to obtain new Kamenev type oscillatory criteria for (1) using integral average method based on the properties of conformable fractional derivatives and integral. By investigating some new properties of this derivative, the classical oscillatory problem of (2) can be extended to (1). These oscillation criteria improve the results mentioned above. Examples are also given to illustrate the effectiveness of our main results.


## 2. Main Results

Firstly, we give some definitions and properties of conformable fractional derivatives and integral, which are important in the proofs of the main results.Definition 4.
The left conformable fractional derivative starting fromt0 of a function f:[t0,∞)→R of order α with 0<α≤1 is defined by (13)Tt0αft=fαt=limε→0ft+εt-t01-α-ftε,when α=1, this derivative of f(t) coincides with f′(t). If (Tt0αf)(t) exists on (t0,t1) then (14)Tt0αft0=limt→t0+fαt.Definition 5.
Letα∈(0,1]. Then the left conformable fractional integral of order α starting at t0 is defined by (15)It0αft=∫t0ts-t0α-1fsds≔∫t0tfsdt0αs.If the conformable fractional integral of a given function f exists, we call that f is α-integrable.Lemma 6 (see [4]).
Ifα∈(0,1] and f∈C1([t0,∞),R), then, for all t>t0, we have (16)It0αTt0αft=ft-ft0and (17)Tt0αIt0αft=ft.Lemma 7 (see [3]).
(1)Tt0α(af+bg)=aTt0α(f)+bTt0α(g) for all real constant a, b.
(2)Tt0α(fg)=fTt0α(g)+gTt0α(f).
(3)Tt0α(tp)=ptp-α, for all p.
(4)Tt0α(f/g)=(gTt0αf-fTt0αg)/g2.
(5)Tt0α(c)=0, where c is a constant.Lemma 8 (see [6]).
Letf,g:[t0,t1]→R be two functions such that fg is differentiable. Then (18)∫t0t1fsgαsdt0αs=fsgst0t1-∫t0t1gsfαsdt0αs.Lemma 9 (Cauchy-Schwarz inequality with conformable fractional derivative).
Letf,g:[t0,∞)→R be two functions such that f2 and g2 are α-integrable. Then (19)∫t0tfsgsdt0αs2≤∫t0tfs2dt0αs∫t0tgs2dt0αs.Theorem 10.
LetH(t,s) and h(t,s) be defined as that in Theorem 1. Then (1) is oscillatory if(20)
l
i
m
s
u
p
t
→
∞
1
H
t
,
t
0
∫
t
0
+
1
t
H
t
,
s
q
s
-
1
4
s
-
t
0
2
-
2
α
p
s
h
2
t
,
s
d
t
0
α
s
=
∞
.Proof.
Suppose on the contrary that there exists a nontrivial solutiony(t) of (1) which is not oscillatory. Without loss of generality, we may suppose that y(t)≠0 for all t≥t0. Define V(t)=p(t)y(α)(t)/y(t) for t≥t0; by the properties of conformable fractional derivative, we get the Riccati equation (21)Vαt+qt+1ptV2t=0,t≥t0.Thus, for every t, T with t≥T≥t0, we have(22)∫TtHt,sqsdt0αs=-∫TtHt,sVαsdt0αs-∫TtHt,sV2spsdt0αs.Using Lemma 8, for t≥T≥t0, we get(23)∫TtHt,sqsdt0αs=-∫TtHt,sVαsdt0αs-∫TtHt,sV2spsdt0αs=Ht,TVT+∫TtHt,ss-t01-αht,sVsdt0αs-∫TtHt,sV2spsdt0αs=Ht,TVT+14∫Tts-t02-2αpsh2t,sdt0αs-∫TtHt,sVsps+s-t01-α2psht,s2dt0αs≤Ht,TVT+14∫Tts-t02-2αpsh2t,sdt0αs.Thus, for every t≥T≥t0,(24)∫TtHt,sqs-14s-t02-2αpsh2t,sdt0αs≤Ht,TVT≤Ht,TVT≤Ht,t0VT.This inequality holds for T=t0+1, which means(25)∫t0+1tHt,sqs-14s-t02-2αpsh2t,sdt0αs≤Ht,t0Vt0+1.
Dividing both sides of (25) by H(t,t0), and taking the upper limit in both sides as t→∞, the right hand side is always bounded, which contradicts hypothesis (20). This completes the proof of Theorem 10.Remark 11.
The singularity of integral att0 has no affection on the oscillation of conformable fractional differential equations since oscillation is a qualitative property at infinite. The results are still true if the lower bound of the integration in (20) is replaced by any other point larger than t0.Let us consider the functionH defined by (26)Ht,s=t-sm,t≥s≥t0,where m is an integer with m>1. Then H(t,t)=0, H(t,s)>0 for t>s≥t0, and by direct calculation, we have (27)ht,s=mt-sm-2/2,t>s≥t0.Under a modification of the hypotheses of Theorem10, we can obtain the following result.Corollary 12.
If there exists an integerm>1 such that (28)
l
i
m
s
u
p
t
→
∞
1
t
m
∫
t
0
+
1
t
t
-
s
m
q
s
-
m
2
4
t
-
s
m
-
2
s
-
t
0
2
-
2
α
d
t
0
α
s
=
∞
,then (1) is oscillatory.Ifp(t)≡1 in (1), we can get (29)limt→∞1tm∫t0+1tm24t-sm-2s-t02-2αdt0αs<∞,and then we further obtain the following.Corollary 13.
Ifp(t)≡1 in (1) and there exists an integer m>1 such that (30)limsupt→∞1tm∫t0+1tt-smqsdt0αs=∞,then (1) is oscillatory.Theorem 14.
LetH and h be defined as in Theorem 10. Moreover, suppose that(31)0<infs≥t0liminft→∞Ht,sHt,t0≤∞and(32)limsupt→∞1Ht,t0∫t0tpss-t021-αh2t,sdt0αs<∞hold. If there exists a function A∈C([t0,∞),R) such that(33)
l
i
m
s
u
p
t
→
∞
1
H
t
,
T
∫
T
t
H
t
,
s
q
s
-
1
4
p
s
s
-
t
0
2
1
-
α
h
2
t
,
s
d
t
0
α
s
≥
A
Tfor every T≥t0, then (1) is oscillatory provided(34)∫t0+1∞A+2tptdt0αt=∞.Proof.
Suppose that (1) possesses a nonoscillatory solution y(t); without loss of generality, we may also suppose that y(t)≠0 for t≥t0. Define V(t)=p(t)y(α)(t)/y(t); as in the proof of Theorem 10, we get(35)∫TtHt,sqs-14s-t02-2αpsh2t,sdt0αs=Ht,TVT-∫TtHt,sVsps+s-t01-α2psht,s2dt0αsfor all t, T with t≥T≥t0. Then we have (36)1Ht,T∫TtHt,sqs-14s-t02-2αpsh2t,sdt0αs=VT-1Ht,T∫TtHt,sVsps+s-t01-α2psht,s2dt0αs.Hence,(37)limsupt→∞1Ht,T∫TtHt,sqs-14s-t02-2αpsh2t,sdt0αs=VT-liminft→∞1Ht,T∫TtHt,sVsps+s-t01-α2psht,s2dt0αs.Thus, by condition (33), we obtain for T≥t0(38)VT≥AT+liminft→∞1Ht,T∫TtHt,sVsps+s-t01-α2psht,s2dt0αs.This implies that(39)VT≥ATfor every T≥t0, and (40)liminft→∞1Ht,T∫TtHt,sVsps+s-t01-α2psht,s2dt0αs<∞,T≥t0.Then for every T0>t0, we get (41)∞>liminft→∞1Ht,T0∫T0tHt,sVsps+s-t01-α2psht,s2dt0αs≥liminft→∞1Ht,T0∫T0tHt,spsV2sdt0αs+1Ht,T0∫T0tHt,ss-t01-αht,sVsdt0αs,i.e., we have(42)liminft→∞Pt+Qt<∞,where(43)Pt=1Ht,T0∫T0tHt,sV2spsdt0αs,t≥T0>t0and(44)Qt=1Ht,T0∫T0tHt,ss-t01-αht,sVsdt0αs,t≥T0>t0.Now we claim that(45)∫T0∞V2tptdt0αt<∞.By condition (31), there exists a constant γ with(46)infs≥t0limt→∞infHt,sHt,t0>γ>0.If (45) is not true, then for each λ>0, there exists a T1>T0 such that(47)∫T0tV2spsdt0αs≥2λγ,t≥T1.Then for all t≥T1, we have (48)Pt=1Ht,T0∫T0tHt,s∫T0sV2ξpξdt0αξαdt0αs=-1Ht,T0∫T0t∫T0sV2ξpξdt0αξHt,ssαdt0αs≥-1Ht,T0∫T1t∫T0sV2ξpξdt0αξHt,ssαdt0αs≥-2λγ1Ht,T0∫T1tHt,ssαdt0αs=2λγHt,T1Ht,T0.However, (49)liminft→∞Ht,T1Ht,t0>γ,using the continuity of H and noticing that T0>t0 can be arbitrarily close to t0, we can choose a T1′≥T1 such that (50)Ht,T1Ht,T0≥γ2for every t≥T1′. Hence, we have P(t)≥λ, for all t≥T1′. Since λ>0 is arbitrary, we have(51)limt→∞Pt=∞.
Next, let a sequenceti,i=1,2,⋯, in the interval [T0,∞), be selected such that limi→∞ti=∞, and (52)limi→∞Pti+Qti=liminft→∞Pt+Qt.By (42), there exists a constant C0 such that(53)Pti+Qti≤C0,i=1,2,⋯.Using (51) and (53), we have(54)limi→∞Pti=∞and hence(55)limi→∞Qti=-∞.By (53) and (54), we obtain (56)1+QtiPti≤MPti<12,provided that i is sufficiently large. Therefore, (57)QtiPti<-12,for all large i. Then (55) ensures that(58)limi→∞Q2tiPti=∞.Using Lemma 9 and Cauchy-Schwarz inequality, we have, for any positive integer i, (59)Q2ti=1H2ti,T0∫T0tihti,sHti,ss-t01-αVsdt0αs2≤1Hti,T0∫T0tipss-t021-αh2ti,sdt0αs·1Hti,T0∫T0tiHti,spsV2sdt0αs≤1Hti,T0∫T0tipss-t021-αh2ti,sdt0αsPti,and consequently (60)Q2tiPti≤1Hti,T0∫T0tipss-t021-αh2ti,sdt0αsfor all large i. By (46), we get (61)liminft→∞Ht,T0Ht,t0>γ,and then there exists a T0′′≥T0, such that (62)Ht,T0Ht,t0≥γ,for every t≥T0′′. Hence, (63)Hti,T0Hti,t0≥γ,for sufficiently large i. Therefore, (64)Q2tiPti≤1γ1Hti,t0∫T0tipss-t021-αh2ti,sdt0αs≤1γ1Hti,t0∫t0tipss-t021-αh2ti,sdt0αsfor all large i. Using (58), we have(65)limi→∞1Hti,t0∫t0tipss-t021-αh2ti,sdt0αs=∞.This implies that (66)limsupt→∞1Ht,t0∫t0tpss-t021-αh2t,sdt0αs=∞,which contradicts condition (32). This gives (45). Since T0>t0 can be selected arbitrarily, we have (67)∫t0+1∞V2spsdt0αs<∞.Using (39), we have (68)∫t0+1∞A+2spsdt0αs≤∫t0+1∞V2spsdt0αs<∞,which contradicts condition (34). This completes the proof of Theorem 14.Corollary 15.
IfH(t,s)=(t-s)m, m>1 in Theorem 14, and (32) and (33) are replaced by (69)limsupt→∞1tm∫t0tpss-t021-αt-sm-2dt0αs<∞,and (70)limsupt→∞1tm∫Ttt-smqs-m2ps4s-t021-αt-s2m-2dt0αs≥AT,then (1) is oscillatory provided (34) holds.Theorem 16.
LetH and h be as in Theorem 10. Moreover, suppose that (31) and(71)liminft→∞1Ht,t0∫t0tHt,sqsdt0αs<∞hold. If there exists a continuous function A on [t0,∞) such that(72)
l
i
m
i
n
f
t
→
∞
1
H
t
,
T
∫
T
t
H
t
,
s
q
s
-
1
4
p
s
s
-
t
0
2
1
-
α
h
2
t
,
s
d
t
0
α
s
≥
A
Tfor every T≥t0, then (1) is oscillatory provided (34) is satisfied.Proof.
Suppose on the contrary that there exists a prepared solutiony of (1) which is not oscillatory. Suppose that y(t)≠0, for all t≥t0. Let V(t)=p(t)y(α)(t)/y(t), on [t0,∞); as in the proof of Theorem 14, we get (35) for any t,T0,T with t≥T≥T0>t0. Thus (73)liminft→∞1Ht,T∫TtHt,sqs-14s-t02-2αpsh2t,sdt0αs=VT-limsupt→∞1Ht,T∫TtHt,sVsps+s-t01-α2psht,s2dt0αs.Using condition (72), we get (74)VT≥AT+limsupt→∞1Ht,T∫TtHt,sVsps+s-t01-α2psht,s2dt0αs.By (39), we have (75)limsupt→∞1Ht,T∫TtHt,sVsps+s-t01-α2psht,s2dt0αs<∞,T≥T0.Hence, we have (76)∞>limsupt→∞1Ht,T0∫T0tHt,sVsps+s-t01-α2psht,s2dt0αs=limsupt→∞1Ht,T0∫T0tHt,sV2spsdt0αs+1Ht,T0∫T0ts-t01-αht,sHt,sVsdt0αs,i.e.,(77)limsupt→∞Pt+Qt<∞,where P and Q are defined by (43) and (44). Using condition (72), we obtain (78)At0≤liminft→∞1Ht,t0∫t0tHt,sqsdt0αs-14liminft→∞1Ht,t0∫t0ts-t021-αpsh2t,sdt0αsand by condition (71), we get (79)liminft→∞1Ht,t0∫t0ts-t021-αpsh2t,sdt0αs<∞.Hence there exists a sequence {ti},i=1,2,⋯ in the interval (T0,∞) with limi→∞ti=∞ and such that(80)liminfi→∞1Hti,t0∫t0tis-t021-αpsh2ti,sdt0αs<∞.If (47) holds, following the procedure of the proof of Theorem 14, we conclude that (51) is satisfied. By (77), there exists a constant C0, such that (53) is fulfilled. Then, as in the proof of Theorem 14, we can arrive at (65), which contradicts (80). This proves that (45) holds. The remainder of the proof proceeds as in the proof of Theorem 14.Corollary 17.
IfH(t,s)=(t-s)m, m>1 in Theorem 16, and (71) and (72) are replaced by (81)liminft→∞1tm∫t0tt-smqsdt0αs<∞,and (82)liminft→∞1tm∫Ttt-smqs-m24s-t021-αt-sm-2psdt0αs≥ATfor every T≥t0, then (1) is oscillatory provided (34) is satisfied.Remark 18.
Choosing different functionsH(t,s) and h(t,s) in Theorems 10, 14, and 16, we can obtain various oscillation criteria for (1). For example, H(t,s)≡ρ(t-s), where ρ(u) is a continuously differentiable function on [0,∞), ρ(0)=0, ρ(u)>0, ρ′(u)≥0 for u>0; H(t,s)=ln⁡t/sm-1, t≥s≥t0; H(t,s)=∫stdz/ρ(z)n-1, t≥s≥t0, where n is an integer with n>2 and ρ is a positive continuous function on [t0,∞) such that ∫t0∞1/ρ(z)dz=∞.Remark 19.
Ifα≡1, then Theorem 10, Theorem 14, and Theorem 16 reduce to Theorem 1.1, Theorem 1.2, and Theorem 1.3 in [26], respectively. Our results improve the results mentioned above, since they can handle the cases not covered by known results.Remark 20.
Theorems10–16 are not suitable for differential equations with Caputo or Riemann-Liouville fractional derivatives in general cases.Example 21.
Consider the following conformable fractional differential equation:(83)γt-t0λyαtα+yt=0,t≥t0,0<α≤1,where γ, λ are constants. Using Corollary 12, we can verify that (84)1tm∫T1tt-sm-m2γ4t-sm-2s-t02-2αs-t0λdt0αs=1tm∫T1tt-sms-t0α-1ds-1tm∫T1tm2γ4t-sm-2s-t0λ-α+1ds≥1tm∫T1tt-sms-T1α-1ds-m2γ4tm∫t0tt-sm-2s-t0λ-α+1ds=1tmt-T1m+αBm+1,α-m2γ4tmt-t0m+λ-αBm-1,λ-α+2,where B(·,·) is the classical Beta function defined by (85)Bm,n=∫011-sm-1sn-1ds.By (86)limt→∞1tmt-T1m+αBm+1,α=∞and (87)limt→∞m2γ4tmt-t0m+λ-αBm-1,λ-α+2=m2γ4Bm-1,2,λ=α,0,λ<α.we get (88)
∞
≥
l
i
m
s
u
p
t
→
∞
1
t
m
∫
T
1
t
t
-
s
m
-
m
2
γ
4
t
-
s
m
-
2
s
-
t
0
2
-
2
α
s
-
t
0
λ
d
t
0
α
s
≥
l
i
m
s
u
p
t
→
∞
1
t
m
t
-
T
1
m
+
α
B
m
+
1
,
α
-
l
i
m
s
u
p
t
→
∞
m
2
γ
4
t
m
t
-
t
0
m
+
λ
-
α
B
m
-
1
,
λ
-
α
+
2
=
∞
.Hence by Corollary 12, we obtain that the conformable fractional differential equation (83) is oscillatory.Example 22.
Consider the following conformable fractional differential equation:(89)yαtα+tγyt=0,t≥0,0<α≤1,where γ is constant. Since p(t)≡1, using Corollary 13, we can verify that (90)limt→∞1tm∫1tsγt-smdt0αs=limt→∞1tm∫01t-smsγ+α-1ds+limt→∞1tm∫1tt-smsγ+α-1ds=limt→∞1tmBm+1,α+γtm+α+γ=+∞forγ>-α.Hence by Corollary 13, we conclude that the conformable fractional differential equation (89) is oscillatory for γ>-α.Example 23.
Consider the following conformable fractional differential equation:(91)tλyαtα+tγcos⁡tααyt=0,t≥0,0<α≤1,where γ,λ are constants. We conclude that (91) is oscillatory for λ≤α and 2γ-λ+α≥0. In fact, taking H(t,s)=(t-s)2, then direct calculation implies h(t,s)=2. Using Corollary 15, we can verify that (92)limsupt→∞1t2∫0tsλs21-αdt0αs=limsupt→∞1t2∫0tsλ+1-αds=limt→∞1λ+2-αt2tλ+2-α<+∞forλ≤α.1tm∫Ttt-smqs-m2ps4s-t021-αt-s2m-2dt0αs=1t2∫Ttt-s2sγcos⁡tαα-sλ+21-αdt0αs≥-Tγsin⁡Tαα-k,where k is a positive constant. Set A(s)=-sγsinsα/α-k; there is an integer N>0 such that for n>N and (2n+1)π+π/4≤s≤2(n+1)π-π/4, A(s)=-sγsinsα/α-k>εsγ, where ε is a small constant. Noticing 2γ-λ+α≥0, we have (93)limt→∞∫1tA+2spsdt0αs≥∑n=N∞∫2n+1π+π/42n+1π-π/4A+2spsdt0αs≥ε2∑n=N∞∫2n+1π+π/42n+1π-π/4s2γ-λdt0αs=ε2∑n=N∞∫2n+1π+π/42n+1π-π/4s2γ-λ+α-1ds≥ε2∑n=N∞∫2n+1π+π/42n+1π-π/4s-1ds=∞.Hence by Corollary 15, we get that the conformable fractional differential equation (91) is oscillatory for λ≤α and 2γ-λ+α≥0.



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*Source: 2310185-2019-05-02.xml*