# Arbitrary Order Fractional Difference Operators with Discrete Exponential Kernels and Applications

Authors: Thabet Abdeljawad; Qasem M. Al-Mdallal; Mohamed A. Hajji
Journal: Discrete Dynamics in Nature and Society
 (2017)

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## Abstract

Recently, Abdeljawad and Baleanu have formulated and studied the discrete versions of the fractional operators of order0<α≤1 with exponential kernels initiated by Caputo-Fabrizio. In this paper, we extend the order of such fractional difference operators to arbitrary positive order. The extension is given to both left and right fractional differences and sums. Then, existence and uniqueness theorems for the Caputo (CFC) and Riemann (CFR) type initial difference value problems by using Banach contraction theorem are proved. Finally, a Lyapunov type inequality for the Riemann type fractional difference boundary value problems of order 2<α≤3 is proved and the ordinary difference Lyapunov inequality then follows as α tends to 2 from right. Illustrative examples are discussed and an application about Sturm-Liouville eigenvalue problem in the sense of this new fractional difference calculus is given.


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## Body

## 1. Introduction

In the last few decades, the continuous and discrete fractional differential equations have received considerable interest due to their importance in many scientific fields; see, by way of example not exhaustive enumeration, [1–7].In [8], the authors introduced a fractional derivative with an exponential kernel which tends to the ordinary derivative as α tends to 1. More properties of this fractional derivative have been studied in [9], where the correspondent fractional integral operator was formulated. Then, the authors in [7] defined the left and right fractional derivatives with exponential kernel in the Riemann sense and formulated the right fractional derivatives in the sense of Caputo-Fabrizio with complete investigation to the correspondent fractional integrals and all the discrete versions with integration and summation by parts applied in the fractional and discrete fractional variational calculus. Then, very recently, the same authors proved an interesting monotonicity result in the sense of this new fractional difference calculus in [10].In the same direction, for the purpose of providing more fractional derivatives with different nonsingular kernels, the authors in [11] defined a fractional operator with Mittag-Leffler kernel and in [12, 13] the complete details and discrete versions have been studied. The exponential kernel fractional derivatives and hence their discrete counterparts are quite different from the Mittag-Leffler kernel fractional operators. For example, the integral operator corresponding to exponential kernel fractional derivatives consists of a multiple of the function f added to a multiple of the integration of f, whereas the Mittag-Leffler kernel correspondent integral operator consists of a multiple of f and a Riemann-Liouville fractional integral of the same order. Also, the monotonicity coefficient of the CFR fractional difference operator of order 0<α≤1 is α as shown in [10], whereas for the discrete Mittag-Leffler CFR operator is α2 as proven in [14].Motivated, by what we mentioned above, we extend the order of fractional difference type operators with discrete exponential kernels to arbitrary positive order, prove existence and uniqueness theorems for the fractional initial value difference problems, and finally prove a Lyapunov type inequality for theCFR fractional difference operators of order 2<α≤3. The ordinary discrete Lyapunov inequality is then confirmed as α tends to 2 from the right not as in the case of the classical fractional difference as α tends to 2 from the left [15]. For various fractional Lyapunov extensions we refer, for example, to [16–29]. All the authors there were motivated by the following theorem on ordinary Lyapunov inequality.Theorem 1 (see [30]).
If the boundary value problem(1)y′′t+qtyt=0,t∈a,b,ya=yb=0,has a nontrivial solution, where q is a real continuous function; then(2)∫abqsds>4b-a.Notice that inequality (2) is known as the classical Lyapunov inequality. It is worth mentioning that Cheng [31] had pointed out that Lyapunov neither stated nor proved Theorem 1 but he only stated the following result.Theorem 2 (see [26]).
Letqt be a nontrivial, continuous, and nonnegative function with period ω and let(3)∫0ωqsds≤4ω.Then the roots of the characteristic equation corresponding to Hills equation (4)x′′t+qtxt=0,-∞<t<∞,are purely imaginary with modulus one.For the classical fractional calculus which is behind many extensions, we refer the reader to [32–35] and for the sake of comparison with the classical discrete fractional case we refer to [36] and the references therein. In addition, for the discrete fractional operators and their duality we refer to [37–39].The article will be organised as follows: In the remaining part of this section we shall give some basics about the discreteCFC and CFR fractional differences and their correspondent sums as used in [7, 10]. In Section 2, we extend the order of CFC and CFR fractional differences and their correspondent sums to arbitrary positive order and give some illustrative examples. In Section 3, we prove some existence and uniqueness theorems by means of Banach fixed point theorem and give some illustrative examples. In Section 4, we prove a Lyapunov type inequality for a fractional CFR difference boundary value problem of order 2<α≤3 and give an application to the fractional difference Sturm-Liouville Eigenvalue problem (SLEP) to enrich the applicability of our proven Lyapunov inequality in the frame of fractional difference operators with discrete exponential kernels.


## 2. Preliminaries

Definition 3 (see [36]).
Forα>0, a∈R, ρs=s-1, and f a real-valued function defined on Na=a,a+1,…, the left Riemann-Liouville fractional sum of order α>0 is defined by (5)∇a-αft=1Γα∑s=a+1tt-ρsα-1¯fs.This is fractionalising of the n-iterated nabla sum (6)∇a-nft=1n-1!∑a+1tt-ρsn-1¯fs.The right fractional integral ending at b, where usually we assume that a≡bmod1, is defined by (7)∇b-αft=1Γα∑s=tb-1s-ρtα-1¯fs,where tα¯=Γt+α/Γt and Γt is the well-known gamma special function of t.Definition 4 (see [7, 10]).
Forα∈0,1 and f defined on Na, or Nb=b,b-1,… in right case, we have the following definitions:
(i) The left (nabla)CFC fractional difference is given by(8)∇aCFCαft=Bα1-α∑s=a+1t∇sfs1-αt-ρs=Bα∑s=a+1t∇sfs1-αt-s.
(ii) The right (nabla)CFC fractional difference has the following form:(9)∇CFCbαft=Bα1-α∑s=tb-1-Δsfs1-αs-ρt=Bα∑s=tb-1-Δsfs1-αs-t.
(iii) The left (nabla)CFR fractional difference is written as(10)∇aCFRαft=Bα1-α∇t∑s=a+1tfs1-αt-ρs=Bα∇t∑s=a+1tfs1-αt-s.
(iv) The right (nabla)CFR fractional difference is given by(11)∇CFRbαft=Bα1-α-Δt∑s=tb-1fs1-αs-ρt=Bα-Δt∑s=tb-1fs1-αs-t,where Bα is a normalization positive constant depending on α satisfying B0=B1=1, ∇gt=gt-gt-1, and Δgt=gt+1-gt.In [7, 8], it was verified that (∇aCF-α∇aCFRαf)t=ft and (∇aCFRα∇aCF-αf)t=ft. Also, in the right case ICFbα∇CFRbαft=ft and ∇CFRbα∇CFb-αft=ft. From [7, 8] we recall the relation between the CFC and CFR fractional differences as(12)∇aCFCαft=∇aCFRαft-Bα1-αfa1-αt-a,and for the right case by(13)∇CFCbαft=∇CFRbαft-Bα1-αfb1-αb-t.Notice that we extend Definition 4 to arbitrary α>0 in the next section.Lemma 5 (see [7]).
For0<α<1, we have(14)∇aCF-α∇aCFCαft=ft-fa,∇CFb-α∇CFCbαft=ft-fb.Notation. For a positive integer n, we have(i)
∇nft=∇∇⋯∇︸ntimesft.(ii)
Δnft=ΔΔ⋯Δ︸ntimesft.(iii)
Δ⊖nft=-1nΔnft.


## 3. Higher Order Fractional Differences and Sums

Definition 6.
Letn<α≤n+1 and f be defined on Na∩Nb. Set β=α-n and define(15)∇aCFCαft=∇aCFCβ∇nft,∇aCFRαft=∇aCFRβ∇nft.The associated fractional sum is given by(16)∇aCF-αft=∇a-n∇aCF-βft.Note that if we use the convention that∇a-0ft=ft, then for the case 0<α≤1 we have β=α and hence we obtain Definition 4. Also, the convention ∇0ft=ft leads to (∇aCFRαf)t and (∇aCFCαf)(t) as in Definition 4 for 0<α≤1.Remark 7.
In Definition6, if we let α=n+1 then β=1 and hence (∇aCFRαf)t=(∇aCFR1∇nf)t=∇n+1ft. Also, by noting that (∇aCF-1f)t=∇a-1ft, we see that for α=n+1 we have (∇aCF-αf)t=∇a-n+1ft. Also, for 0<α≤1 we reobtain the concepts defined in Definition 4. Therefore, our generalization to higher order case is confirmed.Analogously, in the right case we have the following extension.Definition 8.
Letn<α≤n+1 and f be defined on Na∩Nb. Set β=α-n. Then β∈0,1 and we define(17)∇CFCbαft=∇CFCbβΔ⊖nft,∇CFRbαft=∇CFRbβΔ⊖nft.The associated fractional integral is given by(18)∇CFb-αft=∇b-n∇CFb-βft.An immediate extension of (12) and (13) by using Definition 6 is the following.Proposition 9.
Forf defined on Na∩Nb and n<α≤n+1, we have(19)∇aCFCαft=∇aCFRαft-Bα1-α∇nfa1-αt-a,and for the right case(20)∇CFCbαft=∇CFRbαft-Bα1-αΔ⊖nfb1-αb-t.Next proposition explains the action of the arbitrary order sum operator∇aCF-α on the arbitrary order CFR and CFC differences (and vice versa) and the action of the CFR difference on the CF correspondent sum operator.Proposition 10.
Forut defined on Na∩Nb and for some n∈N0 with n<α≤n+1, we have (i)
∇aCFRα∇aCF-αut=ut.(ii)
∇aCF-α∇aCFRαut=ut-∑k=0n-1(∇kua/k!)t-ak¯.(iii)
∇aCF-α∇aCFCαut=ut-∑k=0n(∇kua/k!)t-ak¯.Proof.
(i) By Definition6 and the statement after Definition 4, we have(21)∇aCFRα∇aCF-αut=∇aCFRβ∇n∇a-n∇aCF-βut=∇aCFRβ∇aCF-βut=ut.
(ii) By Definition6 and the statement after Definition 4, we have(22)∇aCF-α∇aCFRαut=∇a-n∇aCF-β∇aCFRβ∇nut=∇a-n∇nut=ut-∑k=0n-1∇kuak!t-ak¯,where β=α-n.
(iii) By Lemma5 applied to ft=∇nut, we have(23)∇aCF-α∇aCFCαut=∇a-n∇a-β∇aCFCβ∇nut=∇a-n∇nut-∇nua=ut-∑k=0n-1∇kuak!t-ak¯-∇nuat-an¯n!=ut-∑k=0n∇kuak!t-ak¯.Using the facts thatΔ⊖n∇b-ngt=gt, (24)∇bnΔ⊖ngt=gt-∑k=0n-1Δ⊖kgbk!b-tk¯,and making use of Lemma 5 and the statement after Definition 4, we can state, for the right case, the following.Proposition 11.
Forut defined on Na,b and α∈n,n+1, for some n∈N0, we have (i)
∇CFRbα∇CFb-αut=ut.(ii)
∇CFb-α∇CFRbαut=ut-∑k=0n-1(Δ⊖kub/k!)b-tk¯.(iii)
∇CFb-α∇CFCbαut=ut-∑k=0n(Δ⊖kub/k!)b-tk¯.Example 12.
Consider the initial value problem:(25)∇aCFCαyt=Ft,where Ft is defined on Na,b=Na∩Nb. Let us consider two cases depending on the order α>0: (i)
Assume0<α≤1, ya=c, and Fa=0. By applying ∇aCF-α and making use of Proposition 10, we get the solution (26)yt=c+1-αBαFt+αBα∑s=a+1tFs.Notice that the conditionFa=0 verifies the initial condition ya=c. In addition, when α→1 we obtain the solution of the ordinary difference initial value problem ∇yt=Ft, ya=c.(ii)
Assume1<α≤2, Fa=0, ya=c1, and ∇ya=c2. By applying ∇aCF-α and making use of Proposition 10 and Definition 6 with β=α-1, we obtain the solution (27)yt=c1+c2t-a+2-αBα-1∑s=a+1tKs+α-1Bα-1∑s=a+1tt-ρsFs.Notice that the solutionyt verifies ya=c1 without the use of Fa=0. However, it verifies ∇ya=c2 under the assumption Fa=0. Also, note that when α→2 we recover the solution of the second-order ordinary difference initial value problem ∇2yt=Ft, ya=c1, and ∇ya=c2.In the next section, we prove existence and uniqueness theorems for some types ofCFC and CFR initial value difference problems.Example 13.
Consider theCFC difference boundary value problem(28)∇aCFCαyt+qtyt=0,1<α≤2,t∈Na,b,ya=yb=0.Then β=α-1 and by Proposition 10 if we apply the operator ∇aCF-α, we obtain the solution (29)yt=c1+c2t-a-∇aCF-αq·y·t.But (30)∇aCF-αq·y·t=1-βBβ∑s=a+1tqsys+βBβ∇a-2qtyt.Hence, the solution has the form (31)yt=c1+c2t-a-2-αBα-1∑s=a+1tqsys-α-1Bα-1∑s=a+1tt-ρsqsys.The boundary conditions imply that c1=0 and (32)c2=2-αb-aBα-1∑s=a+1bqsys+α-1b-aBα-1∑s=a+1bb-ρsqsys.Hence,(33)yt=2-αt-ab-aBα-1∑s=a+1bqsys-α-1t-ab-aBα-1∑s=a+1bb-ρsqsys-2-αBα-1∑s=a+1tqsys-α-1Bα-1∑s=a+1tt-ρsqsys.


## 4. Existence and Uniqueness Theorems for the Initial Value Problem Types

In this section we prove existence and uniqueness theorems forCFC and CFR type initial value problems.Theorem 14.
Consider the system(34)∇aCFCαyt=ft,yt,t∈Na,b,0<α≤1,ya=c,such that fa,ya=0, A1-α/Bα+αb-a/Bα<1, and ft,y1-ft,y2≤Ay1-y2, A>0, where f:Na,b×R→R and y:Na,b→R. Then, system (34) has a unique solution of the form(35)yt=c+∇aCF-αft,yt.Proof.
First, by the help of Proposition10, (12), and taking into account the fact that fa,ya=0, it is straight forward to prove that yt satisfies system (34) if and only if it satisfies (35).
LetX=x:maxt∈Na,b⁡xt<∞ be the Banach space endowed with the norm x=maxt∈Na,b⁡xt. On X define the linear operator (36)Txt=c+∇aCF-αft,xt.Then, for arbitrary x1,x2∈X and t∈Na,b, we have by assumption that(37)Tx1t-Tx2t=∇aCF-αft,x1t-ft,x2t≤A1-αBα+αb-aBαx1-x2,and hence T is a contraction. By Banach fixed point theorem, there exists a unique x∈X such that Tx=x and hence the proof is complete.Remark 15.
Similar existence and uniqueness theorems can be proved for system (34) with higher order by making use of Proposition 10. The condition fa,ya=0 always can not be avoided as we have seen in Example 12 with ft,yt=Ft. As a result of Theorem 14, we conclude that the fractional difference linear initial value problem (38)∇aCFCαyt=ryt,r∈R,t∈Na,b,0<α≤1,ya=c,can have only the trivial solution unless α=1. Indeed, the solution satisfies yt=c+r(1-α/Bα)yt+(αr/Bα)∑s=a+1tys. This solution is only verified at a if 1-αya=0.Theorem 16.
Consider the system(39)∇aCFRαyt=ft,yt,t∈Na,b,1<α≤2,ya=c,such that (A/Bα-1)2-αb-a+α-1b-a2¯/2<1, and ft,y1-ft,y2≤Ay1-y2, A>0, where f:Na,b×R→R and y:Na,b→R. Then, system (34) has a unique solution of the form(40)yt=c+∇aCF-αft,yt=c+2-αBα-1∑s=a+1tfs,ys+α-1Bα-1∇a-2f·,y·t.Proof.
If we apply∇aCF-α to system (39) and make use of Proposition 10 with β=α-1 then we reach at the representation (40). Conversely, if we apply ∇aCFRα, make use of Proposition 10 and by noting that (41)∇aCFRα=∇aCFRβ∇tc=0,we obtain system (39). Hence, yt satisfies system (39) if and only if it satisfies (40).
LetX=x:maxt∈Na,b⁡xt<∞ be the Banach space endowed with the norm x=maxt∈Na,b⁡xt. On X define the linear operator (42)Txt=c+∇aCF-αft,xt.Then, for arbitrary x1,x2∈X and t∈Na,b, we have by assumption that(43)Tx1t-Tx2t=∇aCF-αft,x1t-ft,x2t≤ABα-12-αb-a+α-1b-a2¯2x1-x2,and hence T is a contraction. By Banach Contraction Principle, there exists a unique x∈X such that Tx=x and hence the proof is complete.


## 5. The Lyapunov Inequality for the CFR Difference Boundary Value Problem

In this section, we prove a Lyapunov inequality for aCFR boundary value difference problem of order 2<α≤3.Consider the boundary value problem(44)∇aCFRαyt+qtyρt=0,2<α≤3,t∈Na+1,b-1,ya=yb=0,where, yρt=yρt=yt-1.Lemma 17.
y
t is a solution of the boundary value problem (44) if and only if it satisfies the equation(45)yt=∑s=a+1bGt,sTs,ys,where(46)Gt,s=t-ab-ρsb-a,t+1≤s,t,s∈Na,b,t-ab-ρsb-a-t-ρs,s-1≤t,t,s∈Na,b,Tt,yt=∇aCFβq·yρ·t=1-βBβqtyρt+βBβ∇a-1q·yρ·t,β=α-2.Proof.
Apply the integral∇aCF-α to (44) and make use of Definition 6 and Proposition 10 with n=2 and β=α-2 to reach(47)yt=c1+c2t-a-∇a-2T·,y·t=c1+c2t-a-∑s=a+1tt-ρsTs,ys.
The conditionya=0 implies that c1=0 and the condition yb=0 implies that c2=(1/(b-a))∑s=a+1bb-ρsTs,ys, and hence (48)yt=t-ab-a∑s=a+1bb-ρsTs,ys-∑s=a+1tt-ρsqsTs,ysds.Then, the result follows by splitting the summation (49)∑s=a+1bb-ρsTs,ys=∑s=a+1tb-ρsTs,ys+∑s=t+1bb-ρsTs,ys.Lemma 18.
Given thatb≡amod1, the Green function Gt,s defined in Lemma 17 has the following properties:(1)
Gt,s≥0 for all t,s∈Na,b.(2)
maxt∈Na,b⁡Gt,s=Gρs,s for s∈Na+1,b-1.(3)
fs=Gρs,s=ρs-ab-ρs/b-a has a unique maximum, given by (50)maxs∈Na,b⁡fs=fa+b+22=b-a4ifb-aisevenfa+b+32=b-a2-14b-aifb-aisodd.Hence, in either casesmaxs∈Na,b⁡fs≤(b-a)/4.Proof.
(1) It is clear that g1t,s=t-ab-ρs/(b-a)≥0. Regarding the part g2t,s=t-ab-ρs/(b-a)-t-ρs we see that t-ρs=((t-a)/b-a)b-a+ρs-ab-a/t-a and that a+ρs-ab-a/t-a≥ρs if and only if ρsb-t+at-b≥0 if and only if s≥a+1. Hence, we conclude that g2t,s≥0 as well.
(2) Clearly, g1t,s is an increasing function in t. Applying ∇ to g2 with respect to t for every fixed s≥a+1 we see that ∇tgt,s=b-a-1a-ρs and hence g2 is a decreasing function in t.
(3) Let fs=Gρs,s=ρs-ab-ρs/b-a. Then, (51)∇fs=a+b-2s+3b-a=0,if s=(a+b+3)/2 and hence for a,b∈N, f attains its maximum at s1=(a+b+3)/2 if a+b (or b-a) is odd and at (a+b+2)/2 if a+b (or b-a) is even. More generally, if a and b are such that b≡amod1, we see that s1≡amod1 if b-a is odd and s2≡amod1 if b-a is even. Finally, fs1=(b-a2-1)/4b-a and fs2=(b-a)/4.In next lemma, we estimateTt,yt for a function y∈BNa,b, the Banach space of all Banach-valued finite sequences on Na,b with y=maxt∈Na,b⁡yt, where · is the norm in the Banach space.Lemma 19.
Fory∈BNa,b and 2<α≤3, β=α-2, we have, for any t∈Na,b, (52)Tt,yt≤Rty,where (53)Rt=3-αBα-2qt+α-2Bα-2∑s=a+1tqs.Theorem 20 (theCFR fractional difference Lyapunov inequality).
If the boundary value problem (44) has a nontrivial solution, where qt is a real-valued bounded function on Na,b, then(54)∑s=a+1bRs>4b-a.Proof.
Assumey∈Y=BNa,b is a nontrivial solution of the boundary value problem (44). By Lemma 17, y must satisfy(55)yt=∑s=a+1bGt,sTs,ys.Then, by using the properties of the Green function Gt,s proved in Lemmas 18 and 19, we come to the conclusion that (56)y<b-a4∑s=a+1bRsyfrom which (54) follows.Remark 21.
Note that ifα→2+, then Rt tends to qt and hence we obtain the classical nabla discrete version of the Lyapunov inequality (2). For the sake of more comparisons of Lyapunov inequalities on time scales we refer to [40].Example 22.
Consider the followingCFR Sturm-Liouville difference eigenvalue problem (SLDEP) of order 2<α≤3(57)∇0CFRαyt+λyρt=0,t∈N1,b-1,b∈N,y0=yb=0.If λ is an eigenvalue of (57), then by Theorem 20 with qt=λ, we have(58)Tt=3-αBα-2λ+α-2Bα-2∇0-1λt=λ3-αBα-2+α-2Bα-2t.Hence, we must have(59)∑s=1bTs=λb3-αBα-2+b2¯α-22Bα-2>4b.Notice that the limiting case α→2+ implies that λ>4/b2 which is the lower bound for the eigenvalues of the ordinary difference eigenvalue problem: (60)∇2yt+λyρt=0,t∈N1,b-1,y0=yb=0.


## 6. Conclusions

Fractional differences and their correspondent fractional sum operators are of importance in discrete modeling of various problems in science. We extended the fractional difference calculus whose difference operators depend on a discrete exponential function kernel to arbitrary positive order. The correspondent arbitrary order fractional sum operators have been defined as well and applied to solve fractional initial and boundary value difference problems. The extension for right fractional differences and sums is also achieved. To set up the basic concepts, we proved existence and uniqueness theorems by means of Banach fixed point theorem for initial value problems in the frame ofCFC and CFR fractional differences. We have come to the conclusion that the condition fa,ya=0 is necessary to guarantee the existence of solution and hence fractional linear difference initial value problem with constant coefficients results in the trivial solution unless the order is positive integer. We used our extension to arbitrary order to prove a Lyapunov type inequality for a CFR boundary value problem of order 2<α≤3 and then obtain the classical ordinary case when α tends to 2 from right. This proves different behavior from the classical fractional difference case, where the Lyapunov inequality was proved for a fractional difference boundary problem of order 1<α≤2 and the classical ordinary case was then recovered when α tends to 2 from left.



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