# The Existence and Uniqueness of Solutions and Lyapunov-Type Inequality for CFR Fractional Differential Equations

Authors: Xia Wang; Run Xu
Journal: Journal of Function Spaces
 (2018)

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## Abstract

In this paper, we research CFR fractional differential equations with the derivative of order3<α<4. We prove existence and uniqueness theorems for CFR-type initial value problem. By Green’s function and its corresponding maximum value, we obtain the Lyapunov-type inequality of corresponding equations. As for application, we study the eigenvalue problem in the sense of CFR.


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## Body

## 1. Introduction

In the past decades, fractional calculus arose ([1–3]) and received extensive attention of many researchers. In recent years, it becomes more important because the subject of fractional calculus frequently appears in various fields such as science and engineering. Recently, some new fractional differential definitions have been created. With the rising of fractional computation, the research on the quantitative and qualitative properties of fractional differential equations has become a hot topic; see papers [4–15] and the references therein.In [4–7], the authors presented new fractional derivatives with Mittsg-Leffler kernels and exponential-type kernels. The boundary value problem of fractional equations emerged as new branch in the field of differential equation due to its wide applications. In [8], Abdeljawad defined the higher order fractional derivative in the sense of Abdon and Baleanu and obtained a Lyapunov-type inequality for ABR fractional boundary value problem (1)aABRDαyt+qtyt=0,a<t<b,2<α<3,ya=yb=0,and if y is a nontrivial solution, then (2)∫abTsds>4b-a,where(3)Ts=3-αBα-2qt+α-2Bα-2aIα-2qst.In [5], Abdeljawad defined the higher order fractional derivative in the sense of Caputo and Fabrizio and obtained a Lyapunov-type inequality for CFR fractional boundary value problem (4)aCFRDαyt+qtyt=0,a<t<b,2<α<3,ya=yb=0,and if y is a nontrivial solution, then (5)∫abRsds>4b-a,where (6)Rs=3-αBα-2qt+α-2Bα-2∫abqsds.In this paper, we study the existence and uniqueness for initial problems(7)aCFRDαyt=ft,yt,a<t<b,2<α<3,ya=c1,y′a=c2,and(8)aCFRDαyt=ft,yt,a<t<b,3<α<4,ya=c1,y′a=c2,y′′a=c3.We also consider the following boundary value problem:(9)aCFRDαyt+qtyt=0,a<t<b,3<α<4,ya=y′a=yb=0,where isaCFRDα the Riemann-Liouville fractional derivative in the sense of Caputo and Fabrizio and q:[a,b]→R is a continuous function.Let us introduce the concepts of the Riemann-Liouville fractional integral and the Riemann-Liouville fractional derivative in the sense of Caputo and Fabrizio.Definition 1 (see [1]).
Letα>0 and f be a real function defined on [a,b]. The Riemann-Liouville fractional integral of order α is defined by (10)aIαft=1Γα∫att-sα-1fsds,α>0,t∈a,b,where Γ(α) is defined by (11)Γα=∫0∞tα-1e-tdt,α>0.This is fractionalizing of the n-iterated integral (12)aInft=1n-1!∫att-sn-1fsds,t∈a,b.Definition 2 (see [5]).
Letf∈H1[a,b], a<b, α∈[0,1), then the Riemann-Liouville fractional derivative in the sense of Caputo and Fabrizio is defined by(13)aCFRDαft=Bα1-αddt∫atfxe-αt-xα/1-αdx.The associated fractional integral is (14)aCFIαft=1-αBαft+αBα∫atfsds,where B(α)>0 is a normalization function satisfying B(0)=B(1)=1.Definition 3 (see [5]).
Letn<α≤n+1 and f be such that f(n)∈H1[a,b], a<b. Set β=α-n, β∈(0,1), then the Riemann-Liouville fractional derivative in the sense of Caputo and Fabrizio has the following form: (15)aCFRDαft=aCFRDβfnt.The associated fractional integral is (16)aCFIαft=IanaCFIβft.We also give a proposition which will be used in this article.Proposition 4 (see [5]).
Foru(t)∈[a,b] and n≤α≤n+1, we have (17)aCFRDα·aCFIαut=ut,aCFIα·aCFRDαut=ut-∑k=0n-1ukak!t-ak.


## 2. Existence and Uniqueness Theorems

In this section, we establish existence and uniqueness theorems for CFR-type initial value problem and give corresponding proofs. We make some conditions as the mark(A0): (18)A0ft,y1-ft,y2≤Ay1-y2,A>0,f:a,b×R→R,y:a,b→R.Theorem 5.
Consider the initial value problem (7). Suppose that (A0) holds; if (19)ABα-23-αb-a22+α-2b-a36<1,then system (7) has a unique solution of the form (20)yt=c1+c2t-a+aCFIαft,yt.Proof.
First, applyingaCFIα to system (7) and using Proposition 4 with β=α-2, then we have (20). On the other hand, if we apply aCFRDα to (20) and using Proposition 4, then we obtain (7). It is clear that y(t) satisfies the system (7) if and only if it satisfies (20).
We endow the setC[a,b] with the norm x=maxt∈[a,b]x(t). We define the linear operator T: (21)Txt=c1+c2t-a+aCFIαft,xt.Then, for arbitrary x1,x2∈[a,b], β=α-2, we have (22)Tx1t-Tx2t=aCFIαft,x1t-ft,x2t≤AaCFIαx1-x2=AaI2aCFIβx1-x2=AaI21-βBβx1-x2+βBβ∫atx1-x2ds=A1-βBβ∫att-sx1-x2ds+β2Bβ∫att-s2x1-x2ds=A3-αBα-2∫att-sx1-x2ds+α-22Bα-2∫att-s2x1-x2ds≤A3-αBα-2∫att-sds+α-22Bα-2∫att-s2dsx1-x2≤ABα-23-αb-a22+α-2b-a36x1-x2≤x1-x2.Hence T is a contraction. Form the Banach contraction principle, there exists a unique x such that Tx=x. The proof is complete.Theorem 6.
Consider the initial value problem (8). Suppose that (A0) holds; if (23)ABα-34-αb-a36+α-3b-a424<1,then system (8) has a unique solution of the form (24)yt=c1+c2t-a+c3t-a2+aCFIαft,yt.Proof.
ApplyingaCFIα to system (8) and using Proposition 4 with β=α-3, then we have (24). On the other hand, if we apply aCFRDα to (24) and using Proposition 4, then we obtain (8). It is clear that y(t) satisfies system (8) if and only if it satisfies (24).
We endow the setC[a,b] with the norm x=maxt∈a,b⁡|x(t)|. We define the linear operator T: (25)Txt=c1+c2t-a+c3t-a2+aCFIαft,yt.Then, for arbitrary x1,x2∈[a,b], β=α-3, we have (26)Tx1t-Tx2t=aCFIαft,x1t-ft,x2t≤AaCFIαx1-x2=AaI3aCFIβx1-x2=AaI31-βBβx1-x2+βBβ∫atx1-x2ds=A1-β2Bβ∫att-s2x1-x2ds+β6Bβ∫att-s3x1-x2ds=A4-α2Bα-3∫att-s2x1-x2ds+α-36Bα-3∫att-s3x1-x2ds≤A4-α2Bα-3∫att-s2ds+α-36Bα-3∫att-s3dsx1-x2≤ABα-34-αb-a36+α-3b-a424x1-x2≤x1-x2,Hence T is a contraction. Form the Banach contraction principle, there exists a unique x such that Tx=x. The proof is complete.


## 3. Lyapunov Inequality for the CFR Boundary Value Problem

In this section, we establish some results for the CFR boundary value problem and give corresponding proofs.Theorem 7.
y ∈ C [ a , b ] is a solution of the boundary value problem (9) if and only if y satisfies the integral equation(27)yt=∫abGt,sTs,ysds,where G(t,s) is Green’s function defined as(28)Gt,s=t-a22b-a2b-s2,a≤t≤s≤b,t-a22b-a2b-s2-t-s22,a≤s≤t≤b,and(29)Ts,ys=aCFIβq·y·s=1-βBβqsys+βBβaI1q·y·s,β=α-3.Proof.
From (9), we have(30)aCFRDαyt=-qtyt.Apply integral aCFIα on the (30); we have (31)aCFIαaCFRDαyt=aCFIα-qtyt.Then, by Proposition 4 and Definitions 2 and 3, we obtain (32)yt+c1+c2t-a+c3t-a2=-aI3·aCFIβqtyt=-aI31-βBβqtyt+βBβaI1q·y·s=-12∫abt-s2Ts,ysdsThen, we have(33)yt=c1+c2t-a+c3t-a2-12∫att-s2Ts,ysds,for some real constants ci∈R(i=1,2,3).
Fromy(a)=0, we get immediately that c1=0. By the boundary condition y′(a)=0, we can obtain that c2=0. Hence, (34)yt=c3t-a2-12∫att-s2Ts,ysds.Using the boundary condition y(b)=0 yields (35)c3=12b-a2∫abb-s2Ts,ysds.Hence, equality (33) becomes (36)yt=t-a22b-a2∫abb-s2Ts,ysds-12∫att-s2Ts,ysds.By splitting the integral as follows: (37)∫abb-s2Ts,ysds=∫atb-s2Ts,ysds+∫tbb-s2Ts,ysds.We have that (27) holds. The proof is completed.Corollary 8.
Fors∈[a,b], the function G in (28) satisfied the following properties: (38)maxt∈a,b⁡Gt,s=Gt∗,s,maxs∈a,b⁡Gt∗,s=b-s2s-a22b-s-a,where(39)t∗=ab-s/b-a2-sb-s/b-a2-1.Proof.
First, we define the function(40)g1t,s=t-a22b-a2b-s2,and(41)g2t,s=t-a22b-a2b-s2-t-s22.Differentiating g1(t,s) with respect to t, we get (42)∂g1t,s=b-s2b-a2t-a≥0.Hence, g1(t,s) is an increasing function on t.
Then,(43)g2t,s=t-a22b-a2b-s2-t-s22=12b-sb-a2t-a2-t-s2=12b-sb-a2-1t2+2s-2ab-sb-a2t+b-sb-a2a2-s2=b-s/b-a2-12t2+s-ab-sb-a2t+b-s/b-a2a2-s22.Choose(44)t∗=-b2a=ab-s/b-a2-sb-s/b-a2-1.Then, we have (45)t∗-a=a-sb-s/b-a2-1>0and(46)t∗-b=a-bb-s/b-a2-s-bb-s/b-a2-1<s-bb-s/b-a2-s-bb-s/b-a2-1=s-b<0.It is clear that t∗∈[a,b]. Then, (47)t∗-s=ab-s/b-a2-sb-s/b-a2-1-s=a-sb-s/b-a2b-s/b-a2-1>0.Clearly, t∗>s. Hence, we get the maximum at t∗,(48)maxt∈a,b⁡Gt,s=Gt∗,s=12b-s2b-a2ab-s/b-a2-sb-s/b-a2-1-a2-12ab-s/b-a2-sb-s/b-a2-1-s2=12b-s2b-a2a-sb-s/b-a2-12-12a-sb-s/b-a2b-s/b-a2-12=12b-s2b-a2a-s2b-a4b-s2-b-a22-12a-s2b-s4b-s2-b-a22=12a-s2b-s2b-s2-b-a22b-a2-b-s2=b-s2s-a22b-s-a.The proof is complete.From (29), we have the following Corollary 9.Corollary 9.
Fory∈C[a,b], 3<α<4, and β=α-3, for any t∈[a,b], we have(49)Tt,yt≤Rty,where(50)Rt=4-αBα-3qt+α-3Bα-3∫atqsds.Proof.
Form (29), we have (51)Ts,ys=1-βBβqsys+βBβaI1q·y·s≤1-βBβqsys+βBβ∫atqsdsys≤1-βBβqsys+βBβ∫atqsdsys=1-βBβqs+βBβ∫atqsdsys.By β=α-3, we obtain (49). The proof is complete.Theorem 10.
If the boundary value problem (9) has a nontrivial continuous solution, then(52)∫abb-ss-aRsds≥4.Proof.
Lety∈Y=C[a,b] be a nontrivial solution of the boundary value problem (9) and (53)y=supt∈a,b⁡yt.From Theorem 7, y satisfies the integral equation (54)yt=∫abGt,sTs,ysds.t∈a,b,where G(t,s) is defined in (28) and T(s,y(s)) is defined in (29).
By (48), we have (55)yt≤∫absupt∈a,bGt,sTs,ysds≤∫abb-s2s-a22b-s-aTs,ysds≤∫abb-s2s-a4b-sTs,ysds=14∫abb-ss-aTs,ysds.Form Corollary 9, we obtain (56)y≤14∫abb-ss-aRsdsy⁡,then we get the inequality in (52). This completes the proof.Theorem 11.
If the boundary value problem (9) has a nontrivial continuous solution, then (57)∫abRsds≥16a-b2,where R(t) is defined in (50).Proof.
Define the function(58)Hs=b-ss-a=-s2+a+bs-ab,s∈a,b.Then, we have (59)H′s=-2s+a+b.Observe that H′(s)=0 if and only if(60)s=s∗=a+b2.It is easily to see that s∗∈[a,b].
Hence, we get(61)maxs∈a,b⁡Hs=Hs∗=-a+b22+a+b22-ab=a-b24.Applying the result in (52), we have (62)∫abRsds≥4Hs∗=16a-b2.The proof is complete.Example 12.
Consider the following fractional differential equation:(63)aCFRDαyt+λyt=0,0<t<1,3<α<4,y0=y′0=y1=0.If λ is an eigenvalue to the boundary value problem (9), then (64)λ≥164-αBα-3+α-32Bα-3-1.Proof.
By using Corollary9, we have (65)Rt=4-αBα-3λ+α-3Bα-3∫01λdt=λ4-αBα-3+α-3Bα-3t.From Theorem 11, we have (66)∫01Rsds=λ4-αBα-3+α-32Bα-3>16.Hence, we have (67)λ≥164-αBα-3+α-32Bα-3-1.That concludes the proof.


## 4. Conclusions

In this paper, compared with existing results of fractional differential equations, we extend the order fromα∈(2,3) to α∈(3,4). We prove existence and uniqueness theorems for initial value problem in the frame of CFR-type derivative of order 2<α<3 and 3<α<4 by using Banach Contraction Theorem. Then, we use our extension to obtain new Lyapunov-type inequality for CFR fractional boundary value problem with order 3<α<4 by Green’s function and its corresponding maximum value. As for application, we give an example on the eigenvalue problem.



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