# Oscillation for a Class of Right Fractional Differential Equations on the Right Half Line with Damping

Authors: Hui Liu; Run Xu
Journal: Discrete Dynamics in Nature and Society
 (2019)

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## Abstract

In this paper, we discuss a class of fractional differential equations of the formD-α+1y(t)·D-αy(t)-p(t)f(D-αy(t))+q(t)h∫t∞(s-t)-αy(s)ds=0.D-αy(t) is the Liouville right-sided fractional derivative of order α∈(0,1). We obtain some oscillation criteria for the equation by employing a generalized Riccati transformation technique. Some examples are given to illustrate the significance of our results.


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## Body

## 1. Introduction

The theory of fractional derivatives was originated from G.W. Leibniz’s conjecture. To this day, the theory about fractional calculus and fractional differential equation have been well developed; see [1–7]. In the beginning, the theory of fractional derivatives developed mainly as a pure theoretical filed of mathematics, which can be used only for mathematicians. However, in the past few decades, fractional differential equations were widely used in many fields, such as fluid flow, rheology, electrical networks, and many other branches of science. Great attention was paid to study the properties of solutions of fractional differential equations.Because only few differential equations can be solved, many researches focus on the analysis of qualitative theory for fractional differential equations, such as the existence, uniqueness of solutions, numerical solutions, stability, and oscillation of solutions; see [8–34] and the references therein. Among them, there have been many results for the oscillation of solutions for fractional differential equations.In 2013, Chen [16] studied oscillatory behavior of the fractional differential equation in the form of(1)D-α+1yt-ptD-αyt+qtf∫t∞v-t-αyvdv=0,for t>0, where D-αy is the Liouville right-sided fractional derivative of order α∈(0,1).In 2013, Han [17] brought up the oscillation of fractional differential equations(2)rtgD-αyt′-ptf∫t∞s-t-αys=0,for t>0, where 0<α<1 is a real number, and D-αy is the Liouville right-sided fractional derivative of y.In 2013, Xu [18] studied the oscillation of nonlinear fractional differential equations of the form(3)atrtD-αxt′η′-Ft,∫t∞v-t-αxvdv=0,t≥t0>0,where α∈(0,1) is a constant, and η is a ratio of two odd positive integers.In 2013, based on the modified Riemann-Liouville derivative, Qin and Zheng [19] discussed the oscillation of a class of fractional differential equations with damping term as follows:(4)DtαatDtαrtDtαxt+ptDtαrtDtαxt+qtxt=0,for t≥t0>0,0<α<1, where Dtα(·) denotes the modified Riemann-Liouville derivative regarding the variable t, the function a∈Cα([t0,∞),R+),r∈C2α([t0,∞),R+),p,q∈Cα([t0,∞),R+), and Cα denotes continuous derivative of order.In 2014, Jehad Alzabut and Thabet Abdeljawad [20] studied the oscillatory theory of fractional difference equations in the form(5)∇aq-1qxt+f1t,xt=rt+f2t,xt,t∈Naq,∇aq-1-1-qxtt=aq=xaq=c,c∈R,where m-1<q<m,m∈N,∇a(q)q is the Riemann-Liouville difference operator of order q and ∇a(q)-q is the Riemann-Liouville sum operator where Na(q)={a(q)+1,a(q)+2,…},a(q)=a+m-1,m=[q]+1 and [q] is the greatest integer less than or equal to q.In 2017, B. Abdalla, K. Abodayeh, T. Abdeljawad, J. Alzabut [21] studied the oscillation of solutions of nonlinear forced fractional difference equations in the form(6)∇aq-1qxt+f1t,xt=rt+f2t,xt,t∈Naq,∇aq-1-m-qxtt=aq=xaq=c,c∈R,where q>0,m=[q]+1,m∈N,[q] is the greatest integer less than or equal to q, Na(q)={a(q)+1,a(q)+2,…},a(q)=a+m-1,fi:Na(q)×R→R(i=1,2), and ∇a(q)-q and ∇a(q)q are the Riemann-Liouville sum and difference operators.In 2018, Bai and Xu [22] discussed the oscillation problem of a class of nonlinear fractional difference equations with the damping term in the from(7)ΔctΔrtΔαxtγ+ptΔrtΔαxtγ+qtf∑s=t0t-1+αt-s-1-αxs=0,t∈Nt0,where γ≥1 is a quotient of two odd positive integers, 0<α≤1 is a constant, Δα denotes the Riemann-Liouville fractional difference operator of order α, and Nt0={t0,t0+1,t0+2,…}.In 2018, Bahaaeldin Abdalla and Thabet Abdeljawad [23] studied the oscillation of Hadamard fractional differential equation of the form(8)Daαxt+f1t,x=rt+f2t,x,t>a,limt→a+Daα-jxt=bjj=1,2,…,nwhere n=⌈α⌉, Daα is the left-fractional Hadamard derivative of order α∈C, Re⁡(α)≥0 in the Riemann-Liouville setting.In 2018, J. Alzabut, T. Abdeljawad, H. Alrabaiah [24] considered the following forced and damped nabla fractional difference equation(9)1-pn∇∇0αyn+pn∇0αyn+qnfyn=gn,n∈N1,∇0-1-αy1=y1=c,where ∇0αy and ∇0-αy are the Riemann-Liouville fractional difference and sum operators of y of order α, respectively, α is a real number, c is constant, N1=1,2,… and p,q are real sequences from N1→R,p(n)<1,q is a positive real sequence from N1→R+ and f:R→R such that f(s)/s>0 for all s≠0.In 2018, B. Abdalla, J. Alzabut, T. Abdeljawad [25] investigated the oscillation of solutions for fractional difference equations with mixed nonlinearities in forms(10)∇aα-1αxt-ptxt+∑i=1nqitxtλi-1xt=vt,t∈Naα+1,∇aα-1-m-axtt=aα=xaα=c,c∈R,and(11)c∇aααxt-ptxt+∑i=1nqitxtλi-1xt=vt,t∈Naα,∇kxaα=bk,k∈R,k=0,1,2,…,m-1,where m=[α]+1,α>0,p(t),v(t) and qi(t)(1≤i≤n) are functions defined from Na(α) to R, and λi(1≤i≤n) are ratios of odd positive integers with λ1>…>λl>1>λl+1>…>λn.Inspired by the above results, in this paper, we discuss the oscillatory behavior of the fractional differential equational with damping(12)D-α+1yt·D-αyt-ptfD-αyt+qth∫t∞s-t-αysds=0,t>0,where 0<α<1 is a real number. D-αy is the Liouville right-sided fractional derivative of y. We always assume that the following conditions are valid.( A 1 )  p(t)≥0 and q(t)≥0 are continuous functions on t∈[t0,∞), t0>0.( A 2 )  h,f:R→R are continuous functions with xh(x)>0, xf(x)>0 for x≠0, and there exist positive constants k1,k2 such that h(x)/x≥k1, x/f(x)≥k2 for all x≠0.( A 3 )  f′(u)≤u, f-1(u)∈C(R,R) are continuous functions with f-1(u)>0 for u≠0, and there exists some positive constant α1 such that f-1(uv)≥α1f-1(u)f-1(v) for uv≠0.


## 2. Preliminaries

For convenience, some background materials from fractional calculus are given.From [4], we can get the definition for Liouville right-side fractional integral and Liouville right-side fractional derivative on the whole axis R of order β for a function g:R+→R as follows,(13)I-βgt≔1Γβ∫t∞v-tβ-1gvdv,t>0,(14)D-βgt≔-1βdβdtβI-β-βgt=-1β1Γβ-β∫t∞v-tβ-β-1gvdv,t>0,provided the right hand side is pointwise defined on R+, where Γ(·) is the gamma function defined by Γ(t)≔∫t∞st-1e-sds, and △≔min⁡{z∈Z:z≥β} is the ceiling function.Ifβ∈(0,1), we have(15)D-αyt≔-1Γ1-αddt∫t∞s-t-αysds,for t∈R+≔(0,∞).The following relations also existed:(16)D-1+αyt=-D-αy′t,α∈0,1,t>0.Set(17)Gt≔∫t∞v-t-αyvdv,α∈0,1,and then(18)G′t=-Γ1-αD-αyt,α∈0,1.


## 3. Main Results

First, we study the oscillation of (12) under the following condition:(19)∫t0∞f-1exp⁡-∫t0spvdvds=∞.Theorem 1.
Suppose that(A1)-(A3) and (19) hold; furthermore, assume that there exists a positive function r(t)∈C1[t0,∞] such that(20)limsupt→∞∫t0tvsk1rsqs-r′s24k2Γ1-αrsds=∞,where k1, k2 are defined as in (A2), and(21)vs≔exp⁡∫t0spvdv,s≥t0.Then every solution of (12) is oscillatory.Proof.
Suppose thaty(t) is a nonoscillation solution of (12); without loss of generality, we may assume that y(t) is an eventually positive solution of (12). Then there exists t1∈[t0,∞] such that y(t)>0 and G(t)>0 for t∈[t1,∞], where G is defined in (16). From (A3), (12), and (16) we have(22)fD-αytvt′=-f′D-αytvtD-α+1yt+fD-αytvtpt=fD-αytvtpt-f′D-αytD-α+1ytvt≥fD-αytvtpt-D-α+1ytD-αytvt=qthGtvt>0,t∈t0,∞.Thus f(D-αy(t))v(t) is strictly increasing on [t0,∞]. Since v(t)>0 for t∈[t0,∞], and from (A3), we see that D-αy(t) is eventually of one sign. Now we can claim(23)D-αyt<0,t∈t1,∞.If not, then there exists t2∈[t1,∞] such that D-αy(t2)>0. Since f(D-αy(t))v(t) is strictly increasing on [t1,∞], it is clear that f(D-αy(t))v(t)≥f(D-αy(t2))v(t2)≔c>0 for t∈[t2,∞]. Therefore, from (18), we have(24)-G′tΓ1-α=D-αyt≥f-1cvt=f-1c·exp⁡-∫t0tpvdv≥α1f-1cf-1exp-∫t0tpvdv,t∈t2,∞.Then, we get(25)f-1exp⁡-∫t0tpvdv≤-G′tα1f-1cΓ1-α,t∈t2,∞.Integrating the above inequality from t2 to t, we have(26)∫t2tf-1exp⁡-∫t0spvdvds≤-∫t2tG′sα1f-1cΓ1-αds=-Gt-Gt2α1f-1cΓ1-α<Gt2α1f-1cΓ1-α,t∈t2,∞.Letting t→∞, we see(27)∫t2∞f-1exp⁡-∫t0spvdvds≤Gt2α1f-1cΓ1-α<∞.This is in contradiction with (19). Hence, (23) holds.
Define the functionw as generalized Riccati substitution (28)wt=rt-vtfD-αytGt,t∈t1,∞.Then we have w(t)>0 for t∈[t1,∞]. From (18), (22), (28), and (A1)-(A3), it follows that(29)w′t=r′t-fD-αytvtGt+rt-fD-αytvt′Gt+fD-αytvtG′tG2t≤r′t-fD-αytvtGt+rt-qthGtvtGt+fD-αytvtG′tG2t=r′trtwt+rt-qthGtvtGt+rtfD-αytvt-Γ1-αD-αytG2t≤r′trtwt-k1rtqtvt+w2t-Γ1-αD-αytrtfD-αytvt≤r′trtwt-k1rtqtvt-k2Γ1-αrtvtw2t.That is,(30)w′t≤-k1rtqtvt+r′trtwt-k2Γ1-αrtvtw2t.Taking x=w(t), a=k2Γ(1-α)/r(t)v(t)  (a≠0), and b=r′(t)/r(t), from ax2+bx≤-b2/4a and (30) we could conclude that(31)w′t≤-k1rtqtvt+vtr′t24k2Γ1-αrt.Integrating both sides of inequality (31) from t0 to t, we obtain (32)∞>wt0>wt0-wt≥∫t0t-k1rsqsvs+vsr′s24k2Γ1-αrtds.Taking the limit supremum of both sides of the above inequality as t→∞, we get (33)limsupt→∞∫t0t-k1rsqsvs+vsr′s24k2Γ1-αrsds<wt0<∞,which is in contradiction with (20).
Ify(t) is an eventually negative solution of (12), the proof is similar; hence we omit it.
The proof is complete.Theorem 2.
Suppose that(A1)-(A3) and (19) hold. Furthermore, assume that there exists a positive function r(t)∈C1[t0,∞] such that(34)limsupt→∞∫t0tk1rsqs-r′s-rsps24k2Γ1-αrsds=∞,where v(s) are defined in Theorem 1. Then every solution of (12) is oscillatory.Proof.
Suppose thaty(t) is a nonoscillation solution of (12); without loss of generality, we may assume that y(t) is an eventually positive solution of (12). Proceeding the same as in the proof of Theorem 1, we get (23). Define the function w(t) as follows (35)wt=rt-fD-αytGt,t∈t1,∞.Then we have w(t)>0 for t∈[t1,∞]. From (16), (18), (22), (35), and (A1)-(A3), it follows that(36)w′t=r′t-fD-αytGt+rt-fD-αytGt′=r′t-fD-αytGt+rtf′D-αytD-α+1ytGt+fD-αytG′tG2t=r′trtwt+rtf′D-αytD-α+1ytGt+rtfD-αyt-Γ1-αD-αytG2t≤r′trtwt+rtD-αytD-α+1ytGt-k2Γ1-αrtw2t=r′trtwt+rtptfD-αyt-qthGtGt-k2Γ1-αrtw2t≤r′trtwt-ptwt-k1rtqt-k2Γ1-αrtw2t.That is,(37)w′t≤-k1rtqt+r′t-rtptrtwt-k2Γ1-αrtw2t.Taking x=w(t), a=k2Γ(1-α)/r(t)  (a≠0), and b=(r′(t)-r(t)p(t))/r(t), from ax2+bx≤-b2/4a and (37) we conclude(38)w′t≤-k1rsqs+r′t-rtpt24k2Γ1-αrt.Integrating both sides of inequality (38) from t0 to t, we obtain(39)∞>wt0>wt0-wt≥∫t0t-k1rsqs+r′s-rsps24k2Γ1-αrsds.Taking the limit supremum of both sides of the above inequality as t→∞, we get (40)limsupt→∞∫t0tk1rsqs-r′s-rsps24k2Γ1-αrsds<wt0<∞,which is in contradiction with (34).
Ify(t) is an eventually negative solution of (12), the proof is similar; here we omit it.
The proof is complete.We define a function classG; set D≔{(t,s)≔t≥s≥t0}, D0≔{(t,s)≔t>s≥t0}. We say H(t,s)∈G, if H(t,s) satisfy(41)Ht,t=0,t≥t0,Ht,s>0,t,s∈D0,and H has a nonpositive continuous partial derivative Hs′(t,s)≔∂H(t,s)/∂s on D0 with respect to the second variable.Theorem 3.
Suppose that(A1)-(A3) and (19) hold. Furthermore, assume that there exists a positive function r(t)∈C1[t0,∞] and a function H∈G satisfies (42)limsupt→∞1Ht,t0∫t0tHt,srsqs-r′s24k1k2Γ1-αrsds=∞.Then every solution of (12) is oscillatory.Proof.
Suppose thaty is a nonoscillation solution of (12); without loss of generality, we may assume that y is an eventually positive solution of (12). Proceeding as in the proof of Theorem 2 we get (37).
Multiplying (37) by H(t,s) and integrating from t1 to t, we get(43)∫t1tk1rsqsHt,sds≤-∫t1tHt,sw′sds+-∫t1tHt,swsr′srsds-∫t1tHt,sw2sk2Γ1-αrsds,t∈t1,∞.Using the formula integration by parts, we obtain(44)-∫t1tHt,sw′sds=-Ht,swss=t1s=t+∫t1tHs′t,swsds=Ht,t1wt1+∫t1tHs′t,swsds,t∈t1,∞.Substituting (44) with (43), we have(45)k1∫t1trsqsHt,sds≤Ht,t1wt1+∫t1tHs′t,s+Ht,sr′srsws-k2Γ1-αHt,srsw2sds.Taking x=w(t), a=-k2Γ(1-α)(H(t,s)/r(s))  (a≠0), and b=Hs′(t,s)+H(t,s)(r′(s)/r(s)), from ax2+bx≤-b2/4a we get(46)∫t1tHs′t,s+Ht,sr′srsws-k2Γ1-αHt,srsw2sds≤∫t1tHs′t,s2+H2t,sr′s2/r2s+2Hs′t,sHt,sr′s/rs4k2Γ1-αHt,s/rsds≤∫t1tHs′t,s2+H2t,sr′s2/r2srs4k2Γ1-αHt,sds=∫t1tHs′t,srs2+Ht,sr′s24k2Γ1-αHt,sds≤∫t1tHs′t,srs+Ht,sr′s24k2Γ1-αHt,sds≤∫t1tHt,sr′s24k2Γ1-αHt,sds=∫t1tHt,sr′s24k2Γ1-αrsds.Substituting (46) in (45), we have(47)k1∫t1trsqsHt,sds≤Ht,t1wt1+∫t1tHt,sr′s24k2Γ1-αrsds.Since Hs′(t,s)≤0 for t>s≥t0, we have 0<H(t,t1)≤H(t,t0) for t>t1≥t0. Therefore, from the previous inequality, we get(48)∫t1trsqsHt,sds-Ht,sr′s24k2Γ1-αrsds≤k-1Ht,t1wt1≤k-1Ht,t0wt1,t∈t1,∞.Since 0<H(t,t1)≤H(t,t0) for t>s≥t0, we have 0<H(t,s)/H(t,t0)≤1 for t>s≥t0. Hence, it follows from (48) that we have(49)
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.Letting t→∞, we obtain(50)limsupt→∞1Ht,t0∫t0tHt,srsqs-r′s24k1k2Γ1-αrsds≤∫t0t1rsqsds+k-1wt1<∞,which yields a contradiction to (42). The proof is complete.Second, we study the oscillation of (12) under the following condition:(51)∫t0∞f-1exp⁡-∫t0spvdvds<∞.Theorem 4.
Suppose that(A1)-(A3) and (51) hold, and there exists a positive function r(t)∈C1[t0,∞] such that (20) holds. Furthermore, assume that, for every constant T≥t0,(52)∫T∞f-11vt∫Ttqsvsdsdt=∞,where v(s) are defined as in Theorem 1. Then every solution of (12) is oscillatory or satisfies limt→∞∫t0∞(s-t)-αy(s)ds=0.Proof.
Suppose thaty is a nonoscillation solution of (12); without loss of generality, we may assume that y is an eventually positive solution of (12). Proceeding as in the proof of Theorem 1, we know that D-αy(t) is eventually one sign; then there are two cases for the sign of D-αy(t).
IfD-αy(t) is eventually negative, similar to Theorem 1, we have the oscillation of (12). Next, if D-αy(t) is eventually positive, then there exists t2≥t1 such that D-αy(t)>0 for t≥t2. From (18), we get G′(t)<0 for t≥t2. Thus, we get limt→∞G(t)≔L≥0 and G(t)≥L. We now claim that L=0. Assuming not, that is, L>0, then from (23) and (A3) we get(53)fD-αytvt′≥qthGtvt≥k1qtGtvt≥k1Lqtvt,t∈t2,∞.Integrating both sides of the above inequality from t2 to t, we have(54)fD-αytvt≥fD-αyt2vt2+k1L∫t2tqsvsds>k1L∫t2tqsvsds,t∈t2,∞.Hence, from (17), we get(55)-G′tΓ1-α=D-αyt≥f-1k1L∫t2tqsvsdsvt≥α1f-1k1Lf-1∫t2tqsvsdsvt,t∈t2,∞.Integrating both sides of the last inequality from t2 to t, we obtain(56)Gt≤Gt2-α1Γ1-αf-1k1L∫t2tf-1∫t2tqsvsdsvudu,t∈t2,∞.Letting t→∞, from (52), we get limt→∞G(t)=-∞; this is in contradiction with G(t)>0. Therefore, we have L=0, that is, limt→∞G(t)=0. The proof is complete.Theorem 5.
Suppose that(A1)-(A3) and (51) hold. Let r(t) and H(t,s) be defined as in Theorem 3 such that (42) holds. Furthermore, assume that, for every constant T≥t0, (52) holds. Then every solution of (12) is oscillatory or satisfies limt→∞∫t0∞(s-t)-αy(s)ds=0.From Theorem3, proceeding as in the proof of Theorem 4, we get the results of the theorem.


## 4. Examples

Example 1.
Consider the fractional differential equation(57)D-3/2yt·D-1/2yt-1t2D-1/2yt+1t∫t∞s-t-αysds=0,t>0.In (57), α=1/2, p(t)=1/t2, q(t)=1/t, and f(x)=h(x)=x.
Since(58)∫t0∞f-1exp⁡-∫t0tpvdvdt=∫t0∞exp⁡-∫t0tpvdvdt=∫t0∞exp⁡1t-1t0dt≥∫t0∞exp⁡-1t0dt=∞,then (19) holds.
Takingt0=1, k1=k2=1. It is clear that conditions (A1)-(A3) hold. Furthermore, taking r(t)=t2, we have(59)limsupt→∞∫1tk1rsqs-r′s-rsps24k2Γ1-αrsds=limsupt→∞∫1t1ss2-2s-s21/s224πs2ds=limsupt→∞∫1ts-14π2s-1s2ds=∞,which shows that (34) holds. Therefore, by Theorem 2, every solution of (12) is oscillatory.Example 2.
Consider the fractional differential equation(60)D-5/4yt·D-1/4yt-1t2D-1/4yt+t∫t∞s-t-αysds=0,t>0.In (60), α=1/4, p(t)=1/t2, q(t)=t, and f(x)=h(x)=x.
Proceeding the same process as Example1, we see that (19) holds. Taking t0=1, k1=k2=1, r(t)=1. It is clear that conditions (A1)-(A3) hold. Furthermore, taking H(t,s)=(t-s)1/4, it meets Hs′(t,s)=-(1/4)(t-s)<0 for (t,s)∈D0, D0≔{(t,s):t>s≥t0}.
Since(61)limsupt→∞1Ht,t0∫t0tHt,srsqs-r′s24k1k2Γ1-αrsds=limsupt→∞1t-t01/4∫t0tst-s1/4ds=limsupt→∞1t-t01/445t-t05/4t0+1645t-t09/4=limsupt→∞45t-t0t0+1645t-t02=∞,which shows that (42) holds, by Theorem 3, every solution of (12) is oscillatory.


## 5. Conclusion

In the paper, by using the generalized Riccati transformation and inequality technique, we study a class of2α+1 order fractional differential equations in the form (12), which contains the damping term and has not been studied before. The oscillation criteria of (12) are obtained and some examples are given to reinforce our results.



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