Arbitrary Order Fractional Difference Operators with Discrete Exponential Kernels and Applications

Copyright © 2017 Thabet Abdeljawad et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Recently, Abdeljawad and Baleanu have formulated and studied the discrete versions of the fractional operators of order 0 < 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 asα 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.


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][2][3][4][5][6][7].
In [8], the authors introduced a fractional derivative with an exponential kernel which tends to the ordinary derivative as  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  added to a multiple of the integration of , whereas the Mittag-Leffler kernel correspondent integral operator consists of a multiple of  and a Riemann-Liouville fractional integral of the same order.Also, the monotonicity coefficient of the CFR fractional difference operator of order 0 <  ≤ 1 is  as shown in [10], whereas for the discrete Mittag-Leffler CFR operator is  2 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 the CFR fractional difference operators of order 2 <  ≤ 3. The ordinary discrete Lyapunov inequality is then confirmed as  tends to 2 from the right not as in the case of the classical fractional difference as  tends to 2 from the left [15].For various fractional Lyapunov extensions we refer, for example, to [16][17][18][19][20][21][22][23][24][25][26][27][28][29].All the authors there were motivated by the following theorem on ordinary Lyapunov inequality.
Theorem 2 (see [26]).Let () be a nontrivial, continuous, and nonnegative function with period  and let Then the roots of the characteristic equation corresponding to Hills equation are purely imaginary with modulus one.
For the classical fractional calculus which is behind many extensions, we refer the reader to [32][33][34][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][38][39].
The article will be organised as follows: In the remaining part of this section we shall give some basics about the discrete CFC 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 <  ≤ 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.
In [7,8], it was verified that and Also, in the right case . From [7,8] we recall the relation between the CFC and CFR fractional differences as and for the right case by Notice that we extend Definition 4 to arbitrary  > 0 in the next section.

Higher Order Fractional Differences and Sums
Definition 6.Let  <  ≤  + 1 and  be defined on N  ∩  N.
Analogously, in the right case we have the following extension.
Definition 8. Let  <  ≤  + 1 and  be defined on N  ∩  N. Set  =  − .Then  ∈ (0, 1] and we define The associated fractional integral is given by An immediate extension of ( 12) and ( 13) by using Definition 6 is the following.Proposition 9.For  defined on N  ∩  N and  <  ≤  + 1, we have and for the right case Next proposition explains the action of the arbitrary order sum operator CF  ∇ − 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.For () defined on N  ∩  N and for some  ∈ N 0 with  <  ≤  + 1, we have Proof.(i) By Definition 6 and the statement after Definition 4, we have (ii) By Definition 6 and the statement after Definition 4, we have where  =  − .
(iii) By Lemma 5 applied to () = (∇  )(), we have Using the facts that ⊖ Δ  ∇ −  () = (), and making use of Lemma 5 and the statement after Definition 4, we can state, for the right case, the following.
In the next section, we prove existence and uniqueness theorems for some types of CFC and CFR initial value difference problems.

Example 13. Consider the CFC difference boundary value problem
Then  =  − 1 and by Proposition 10 if we apply the operator CF  ∇ − , we obtain the solution But

Existence and Uniqueness Theorems for the Initial Value Problem Types
In this section we prove existence and uniqueness theorems for CFC and CFR type initial value problems.
Theorem 14.Consider the system Proof.First, by the help of Proposition 10, (12), and taking into account the fact that (, ()) = 0, it is straight forward to prove that () satisfies system (34) if and only if it satisfies (35).
Let  = { : max ∈N , |()| < ∞} be the Banach space endowed with the norm ‖‖ = max ∈N , |()|.On  define the linear operator Then, for arbitrary  1 ,  2 ∈  and  ∈ N , , we have by assumption that and hence  is a contraction.By Banach fixed point theorem, there exists a unique  ∈  such that  =  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 (, ()) = 0 always can not be avoided as we have seen in Example 12 with (, ()) = ().As a result of Theorem 14, we conclude that the fractional difference linear initial value problem can have only the trivial solution unless  = 1.Indeed, the solution satisfies () =  + ((1 − )/())() + (/()) ∑  =+1 ().This solution is only verified at  if (1 − )() = 0. Theorem 16.Consider the system Proof.If we apply CF  ∇ − to system (39) and make use of Proposition 10 with  =  − 1 then we reach at the representation (40).Conversely, if we apply CFR  ∇  , make use of Proposition 10 and by noting that we obtain system (39).Hence, () satisfies system (39) if and only if it satisfies (40).
Let  = { : max ∈N , |()| < ∞} be the Banach space endowed with the norm ‖‖ = max ∈N , |()|.On  define the linear operator Then, for arbitrary  1 , and hence  is a contraction.By Banach Contraction Principle, there exists a unique  ∈  such that  =  and hence the proof is complete.

The Lyapunov Inequality for the CFR Difference Boundary Value Problem
In this section, we prove a Lyapunov inequality for a CFR boundary value difference problem of order 2 <  ≤ 3.

Lemma 17. 𝑦(𝑡) is a solution of the boundary value problem (44) if and only if it satisfies the equation
where Proof.Apply the integral CF  ∇ − to (44) and make use of Lemma 18.Given that  ≡  (mod 1), the Green function (, ) defined in Lemma 17 has the following properties: (1) (, ) ≥ 0 for all ,  ∈ N , .