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Fractional difference operators with discrete generalized Mittag–Leffler kernels
Chaos, Solitons and Fractals 126 (2019) 315–324
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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Fractional difference operators with discrete generalized Mittag–Leffler kernels Thabet Abdeljawad Department of Mathematics and General Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi Arabia
a r t i c l e
i n f o
Article history: Received 26 April 2019 Revised 9 June 2019 Accepted 12 June 2019 Available online 27 June 2019 Keywords: Discrete generalized Mittag–Leffler function Discrete nabla Laplace transform Convolution AB Fractional sums ABR Fractional difference ABC Fractional difference Iterated AB sum-differences Higher order
a b s t r a c t In this article we define fractional difference operators with discrete generalized Mittag–Leffler kernels of γ the form (E (λ, t − ρ (s )) for both Riemann type (ABR) and Caputo type (ABC) cases, where AB stands θ ,μ
for Atangana–Baleanu. Then, we employ the discrete Laplace transforms to formulate their corresponding AB−fractional sums, and prove useful and applicable versions of their semi-group properties. The action of fractional sums on the ABC type fractional differences is proved and used to solve the ABC−fractional difference initial value problems. The nonhomogeneous linear ABC fractional difference equation with constant coefficient is solved by both the discrete Laplace transforms and the successive approximation, and the Laplace transform method is remarked for the continuous counterpart. In fact, for the case μ = 1, we obtain a nontrivial solution for the homogeneous linear ABC− type initial value problem with constant coefficient. The relation between the ABC and ABR fractional differences are formulated by using the discrete Laplace transform. We iterate the fractional sums of order −(θ , μ, 1 ) to generate fractional sumdifferences for which a semigroup property is proved. The nabla discrete transforms for the AB−fractional sums and the AB−iterated fractional sum-differences are calculated. Examples and remarks are given to clarify and confirm the obtained results and some of their particular cases are highlighted. Finally, the discrete extension to the higher order case is discussed. © 2019 Elsevier Ltd. All rights reserved.
1. Introduction The question about the calculus of differentiating or integrating up to arbitrary order is old [1–3] while many recent investigations regarding its applications and theoretical and numerical methods are still under development [4–8]. The answer to this question brought up many kinds of nonlocal fractional integrals and derivatives gradually. With the appearance of the time scale calculus [9], the theory of discrete fractional derivatives (fractional differences), discrete fractional integrals (fractional sums) and the q-fractional calculus has started to be developed in the last two decades or so [10–19]. During the last few years, and for the sake of having fractional operators (or derivatives) with nonsingular kernels the authors in [20] defined the so called Caputo–Fabrizio fractional derivatives by imposing a nonsingular exponential kernel. Later, the authors in [21] replaced the exponential kernel by the one parameter Mittag–Leffler function. The fractional integral operators that cancel the fractional derivatives with such nonsingular kernels consist of two parts, of which the first part is a multiple of the function itself. Such a structure for the fractional integrals
E-mail address: [email protected] https://doi.org/10.1016/j.chaos.2019.06.012 0960-0779/© 2019 Elsevier Ltd. All rights reserved.
leads to the loss of the semigroup property so that it becomes slightly difficult to solve the fractional dynamical systems by using the successive approximation method. This structure also results in having the trivial solution for the linear system with constant coefficients within Caputo fractional derivatives with Mittag–Leffler kernel (the ABC-fractional linear system with constant coefficients). In fact, we request the right hand side of the ABC-fractional system to vanish at the starting point a [22]. These properties for the AB−fractional integrals have been investigated for the discrete counter parts in [23]. For more about the higher order fractional operators with nonsingular kernels and their discrete counterparts we refer to [24–27] and for the monotonicity analysis of the fractional differences with discrete exponential kernels and discrete Mittag–Leffler kernels we refer to the recent manuscripts [28–30]. More developments on the light of applications of fractional operators with nonsingular kernels we refer to [31–44]. In [45] the authors introduced fractional operators with generalized Mittag– Leffler functions and derived their fractional integrals and then the author in [46] continue to show the advantages of such extension in gaining a certain semigroup property for the fractional integrals and in proving existence and uniqueness for the solution of the initial value problems without the need of assuming that the right hand side is zero at the starting point a.
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T. Abdeljawad / Chaos, Solitons and Fractals 126 (2019) 315–324
In this article we define and investigate the fractional differences with discrete generalized Mittag–Leffler kernels, find their corresponding fractional sums, prove their semigroup properties and their action on the ABC-fractional differences. Indeed, this work will be organized as follows: Next section, Section 2, will contain preliminaries about discrete generalized Mittag–Leffler functions and their discrete Laplace transforms. Section 3 is about the ABR and ABC discrete generalized fractional derivatives and their fractional sums. Section 4 is devoted to the application in the discrete ABC-initial value problems by investigating the action of the AB-fractional differences on the ABC-fractional differences and proving semigroup type properties. An existence and uniqueness theorem is proved and an example for the linear case with constant coefficients is solved by both discrete Laplace transforms and the successive approximation. Section 5 is about the iterated fractional sum-differences. Section 6 deals with the extension to higher order and finally the last section will contain our conclusions. 2. Preliminaries
•
(μ + 1 ) ( b − t )θ +μ (5) (μ + 1 + θ ) Lemma 2 [14]. Let θ1 , θ2 ∈ R and η be defined on Na . Then, the fol-
∇b−θ (b − t )μ =
lowing hold: a
∇ −θ1 a ∇ −θ2 η (t ) = a ∇ −(θ1 +θ2 ) η (t ),
(6)
and
∇b−θ1 ∇b−θ2 η (t ) = ∇b−(θ1 +θ2 ) η (t ),
(7)
The Q −operator action, (Q η )(t ) = η (a + b − t ), which was used in [16,17] to relate left and right fractional sums and differences, is a useful tool in confirming the right definition cases and hence setting a proper integration by parts. It was shown that
(∇a−θ Q η )(t ) = Q b ∇ −θ η (t ). Since in general it is not true that (t θ )β = t θ β and (ab)θ = aθ bθ in general, the author in [16,17], motivated by the time scale calculus notations, defined the following versions of nabla discrete Mittag–Leffler functions.
This section contains some basic concepts about discrete fractional operators, discrete generalized Mittag–Leffler functions and the discrete Laplace transforms into nabla.
Definition 3 (Nabla Discrete Mittag–Leffler see [16,17]). For λ ∈ R, |λ| < 1 and θ , β , ρ , z ∈ C with Re(θ ) > 0, the nabla discrete Mittag–Leffler functions are defined by
2.1. The rising factorial, the nabla fractional sums and the nabla discrete Mittag–Leffler functions in three indices
E
For the details about the basic given in this subsection refer to [14–17]. Definition 1. (i) For a natural number m, the m rising (ascending) factorial of t is defined by
t
m
=
m −1
(t + k ),
0
t = 1.
(1)
k=0
(t + θ ) , (t )
t ∈ R \ {. . . , −2, −1, 0}, 0θ = 0
(2)
The following property of the rising factorial function can be observed:
∇ (t θ ) = θ t θ −1 ,
(3)
hence t θ is increasing on N0 , where ρ (t ) = t − 1. Definition 2 (See [16,17]). Let ρ (t ) = t − 1 be backward jump operator. Then,for a function η : Na = {a, a + 1, a + 2, .} → R, the nabla left fractional sum of order θ > 0 (starting from a) by a
∇ −θ η (t ) =
t
1
(θ ) s=a+1
(t − ρ (s ))θ −1 η (s ), t ∈ Na+1 ;
=
1
b−1
(θ )
s=t
1
b−1
(θ )
s=t
(8)
For β = ρ = 1, it is written that
Eθ (λ, z ) Eθ1,1 (λ, z ) =
∞ k=0
λk
z kθ , (θ k + 1 )
(9)
where (ρ )k = ρ (ρ + 1 ) . . . (ρ + k − 1 ). Notice that (1 )k = k! so that E 1 (λ, z ) = Eθ ,β (λ, z ). θ ,β
Direct properties of the discrete Mittag–Leffler functions are:
(10)
and
∇ Eθ (λ, z ) = λEθ ,θ (λ, z ).
(11)
Also, by the help of Lemma 1 we have a
∇ −ν Eθρ,β (λ, z ) = Eθρ,β +ν (λ, z )
(12)
2.2. The nabla discrete Laplace of fractional sums and Mittag–Leffler functions and the discrete convolution Definition 4 [14,47]. The nabla discrete Laplace transform for a discrete function f defined on Na is given by ∞
(1 − z )t−a−1 f (t )
(13)
t=a+1
Lemma 3 [47]. Let θ ∈ R \ {. . . , −2, −1, 0}. Then, we have Za ((t − a )θ −1 )(z ) = (θθ ) , |1 − z| < 1, Definition 5 (See [48] also). Assume f, g : Na → R be discrete functions. Then, the nabla discrete convolution of f with g is defined by
(σ (s ) − t )θ −1 η (s ), t ∈ b−1 N.
( f ∗ g)(t ) =
(μ + 1 ) (t − a )θ +μ (μ + 1 + θ )
t
g(t − ρ (s ) + a ) f (s ).
(14)
s=a+1
Proposition 1 [14]. Assume f, g are discrete functions defined on Na . Then, we have
•
∇ −θ (t − a )μ =
zkθ +β −1 (ρ )k . (θ k + β )k!
z
(s − ρ (t ))θ −1 η (s )
Lemma 1 [15–17]. Let θ > 0, μ > −1. Then,
a
λk
k=0
(Za f (t ))(z ) =
the nabla right fractional sum of order θ > 0 (ending at b)for η : b N = {b, b − 1, b − 2, .} → R by
∇b−θ η (t ) =
∞
∇ n Eθρ,β +n (λ, z ) = Eθρ,β (λ, z ), n = 1, 2, . . .
(ii) For any real number the θ rising function is defined by
tθ =
ρ (λ, z ) = θ ,β
(4)
(Za ( f ∗ g))(z ) = (Za f )(z )(Za g)(z ).
(15)
T. Abdeljawad / Chaos, Solitons and Fractals 126 (2019) 315–324
Lemma 4. Let f be defined on Na . Then
(Za ∇ ( f (t ))(z ) = z(Za f )(z ) − f (a ).
(16)
derived fractional sums, and construct an example of a nonconstant function whose ABR and ABC fractional differences are zeros. The relation between the ABR and the ABC fractional differences are demonstrated.
(17)
Definition 6. The discrete generalized ABR and ABC fractional γ derivatives with kernel E (λ, t ), where 0 < θ < 1/2, Re(μ ) >
More generally,
(Za ∇ n f )(z ) = zn (Za f )(z ) −
n−1
zn−1−i ∇ i f (a + 1 ).
i=0
θ ,μ
0, γ ∈ R, are defined respectively by
Lemma 5 [47]. For any real number ν , we have
(Za a ∇ −ν ) f (s ) = s−ν (Za f )(s ).
θ ,μ,γ f )(t ) = (ABC a ∇
zθ −1 . z θ −λ 1 = θ z −λ
(ABC ∇bθ ,μ,γ f )(t ) =
The proof of the following lemma is straight forward and its continuous counterpart can be found in [49] (see (2.19) there and Section 2 in [45] for the modified Mittag–Leffler version). Lemma 7. For γ , θ , β , λ ∈ C (Re(β ) > 0), and s ∈ C with Re(s ) > 0, |λs−θ | < 1, we have
Za E
k=0
1
f or 1 − λs−θ
(Za Eθγ,β (λ, t − a ))(s ) =
∞ k=0
= s −β
(18)
=s
λk (γ )k
k!sθ k+β ∞ (λs−θ )k (γ )k k!
−β
[1 − λs−θ ]−γ .
(19)
In fact by differentiating (18) γ −times we see that
( γ )k k!
=
1
(1 − λs−θ )γ
f or
|λs−θ | < 1.
s=a+1
γ −B(θ ) t E (λ, s − ρ (t )) f (s ), t ∈ θ ,μ 1−θ s=t b−1
−θ 1−θ
b N,
.
θ ,μ,γ f (t ) = u (t ) with γ = 1 Now, we solve the equation ABR a ∇ to find the discrete fractional integral operator of two parameters. Apply the discrete Laplace transform Za to both sides, use the discrete convolution theorem and make use of Lemma 7 to get ABR a
B (θ ) (Za ∇ [ f (t ) ∗ Eθ ,β (λ, t )] )(s ) 1−θ B ( θ ) −μ = s.s [1 − λs−θ ]−1 F (s ) 1−θ = U ( s ),
∇ θ ,μ,1 f (t )(s ) =
(20)
Lemma 8. Let ρ , μ, γ , ν, σ , λ ∈ C (Re(ρ ), Re(μ), Re(ν ) > 0), then γ γ +σ Eρ ,μ (λ, t − ρ (s ))Eρσ,ν (λ, s − a ) = Eρ ,μ+ν (λ, t − a ).
where λ =
( Za
Since the Pochhammer symbol (γ )k is valid for any γ ∈ C then γ can be arbitrary complex number.
t
s=a+1
Remark 1. It is clear that definitions above obey the action of the discrete Q−operator. Also, notice that to guarantee the convergence of the kernels of the above defined fractional difference operators, we need 0 < θ < 1/2 to have |λ| < 1 for λ = 1−−θθ .
k=0
(λs−θ )k
t B (θ ) γ E (λ, t − ρ (s )) f (s ), t ∈ Na ∇t θ ,μ 1−θ
(25)
|λs−θ | < 1,
we have
k=0
θ ,μ,γ f )(t ) = (ABR a ∇
(ABR ∇bθ ,μ,γ f )(t ) =
Proof. On the light of Lemma 3 and that
∞
and
the right one by
(λs−θ )k =
b N,
(24)
Za Eθ ,β (λ, t − a ) (s ) = s−β [1 − λs−θ ]−1 .
∞
b−1 −B(θ ) γ E (λ, s − ρ (t )) s f (s ), t ∈ 1 − θ s=t θ ,μ
(23)
γ (λ, t − a ) (s ) = s−β [1 − λs−θ ]−γ . θ ,β
In particular
(22) the right one by
Lemma 6. Let 0 < θ ≤ 1 and f be defined on Na . Then, (ii) (Za Eθ ,θ (λ, t − a ))(z )
t B (θ ) γ E (λ, t − ρ (s ))∇s f (s ), t ∈ Na , θ ,μ 1−θ s=a+1
The following lemma is a modification to what was proved in [48]. (i) (Za Eθ (λ, t − a ))(z ) =
317
(21)
Then, the proof follows by applying the discrete Laplace Za and its inverse by making use of Lemma 7 and Proposition 1. γ
where U (s ) = (Za u(t )(s ), F (s ) = (Za f (t )(s ) and λ = which it follows that
F (s ) =
−θ 1−θ
(26) . From
1 − θ μ−1 1−θ θ s −θ s [1 − λs−θ ]U (s ) = sμ−1 + U ( s ). B (θ ) B (θ ) B (θ )
Apply the discrete inverse Laplace and make use of Lemma 5 to see that
1−θ θ (a ∇ −(1−μ) u )(t ) + (a ∇ −(1−μ+ θ ) u )(t ). B (θ ) B (θ )
Proof. The left hand side of (21) is Eρ ,μ (λ, t − a ) ∗ Eρσ,ν (λ, t − a ).
f (t ) =
3. The ABR and ABC discrete generalized fractional derivatives and their fractional sums
Definition 7. The discrete left and right AB− fractional integrals of two parameters θ and μ are defined by:
In this section, we define fractional differences with three parameter discrete Mittag–Leffler kernels in the sense of Riemann and Caputo, derive their corresponding fractional sums, study the Laplace transforms for the defined fractional differences and their
−(θ ,μ ) (AB η )(t ) = a ∇
Hence, we have the following definition.
1−θ (a ∇ −(1−μ) η )(t ) B (θ ) +
θ
B (θ )
(a ∇ −(1−μ+θ ) η )(t ).
(27)
318
T. Abdeljawad / Chaos, Solitons and Fractals 126 (2019) 315–324
and
(AB ∇b−(θ ,μ) η )(t ) =
1−θ θ (∇b−(1−μ) η )(t ) + (∇ −(1−μ+ θ ) η )(t ). B (θ ) B (θ ) b (28)
Remark 2. Note that if in the above definition μ tends to −(θ ,1 ) η )(t ) = (AB ∇ −θ η )(t ) and (AB ∇ −(θ ,1 ) η )(t ) = 1, we have (AB a ∇ a b
(AB ∇b−θ η )(t ).
B (θ ) B ( θ ) γ +σ E (λ, t − a ), ∇t [Eθγ,+μσ+ν (λ, t − a )] = 1−θ 1 − θ θ ,ν +μ−1
(29)
∇ θ ,μ,γ [Eθσ,ν (λ, t − a )]
(s ) = (Za η )(s ) and λ = where η we have
Za
AB a
∇ −θ η (t ) (s ) =
=
B ( θ ) γ +σ E (λ, t − a ). 1 − θ θ ,ν +μ−1
(30)
If we apply the Q−operator on (29) and (30), we get the following right versions, respectively. θ ,μ,γ
B ( θ ) γ +σ [Eθσ,ν (λ, b − t )] = E (λ, b − t ), 1 − θ θ ,ν +μ−1
θ ,μ,γ
B ( θ ) γ +σ [Eθσ,ν (λ, b − t )] = E (λ, b − t ). 1 − θ θ ,ν +μ−1
∇b
(31)
and
∇b
follows
by
(AB ∇b−θ η )(t ). The case of finding explicit formulas for the left and right AB fractional integrals of order θ , μ, γ when γ = 1 has not been considered above. However, it is possible to formulate the particular cases γ = 2, 3, 4 . . . by the help of Laplace transforms. In fact, the AB fractional integrals of order 0 < θ ≤ 1, μ > 0, γ = 1, 2, . . . are given by
( ∇
−(θ ,μ,γ )
η )(t ) =
γ
γ i
i=0
θi
B(θ )(1 − θ )i−1
(a ∇
−(θ i+1−μ )
η )(t ). (32)
and
(AB ∇b−(θ ,μ,γ ) η )(t ) =
γ
γ
θi
B(θ )(1 − θ )i−1
(∇b−(θ i+1−μ) η )(t ). (33)
On the light of Remark 3 and the generalized binomial theorem we have the following definition for the AB fractional sums. Definition 8. Assume η(t) is defined on Na and a ≡ b (mod 1). Then, the AB fractional sums of order 0 < θ ≤ 1, μ > 0, γ > 0 are given by
( ∇
−(θ ,μ,γ )
η )(t ) =
∞
γ i
i=0
θi
B(θ )(1 − θ )
( ∇ −(θ i+1−μ) η )(t ). i−1 a (34)
and
(AB ∇b−(θ ,μ,γ ) η )(t ) =
∞ i=0
. In particular, when μ = γ = 1
Lemma
5
and
(37) the
binomial
B (θ ) f (a )Eθ (λ, t − a ) 1−θ
(38)
γ i
θi
B(θ )(1 − θ )
i−1
(∇b−(θ i+1−μ) η )(t ).
(35)
B (θ ) f (b)Eθ (λ, b − t ) 1−θ
(39)
The following theorem is a generalization to (38) and (39) above. Theorem 2. For any 0 < θ < 1/2, μ > 0, γ ∈ R, and χ defined on Na with b ≡ a (mod 1), we have θ ,μ,γ χ )(t ) = (ABR ∇ θ ,μ,γ χ )(t ) − B(θ ) χ (a )E γ (λ, t − a ). • (ABC a ∇ a 1−θ θ ,μ ,γ
• (ABC ∇b
χ )(t ) = (ABR ∇bθ χ )(t ) − −θ 1−θ
θ ,μ γ B (θ ) χ ( b ) E ( λ, b − t ). 1−θ θ ,μ
.
Proof. From the relations, which are deducted from Lemma 7 and by the help of the discrete convolution theorem stated in Proposition 1, θ ,μ,γ χ )(t )} (s ) = Za {(ABR a ∇
B ( θ ) 1 −μ
(s )[1 − λs−θ ]−γ , s χ 1−θ
(40)
and θ ,μ,γ χ )(t )} (s ) = Za {(ABC a ∇
B ( θ ) 1 −μ
(s )[1 − λs−θ ]−γ s χ 1−θ B (θ ) − χ (a )s−μ [1 − λs−θ ]−γ , 1−θ
(41)
we deduce that θ ,μ,γ χ )(t )} (s ) = Z { (ABR Za {(ABC a a ∇
−
i
i=0
AB a
(36)
According to [12] we recall the following
Above λ =
Remark 3. Note that if μ tends to 1 in Definition 7, we −(θ ,1 ) η )(t ) = (AB ∇ −θ η )(t ) have (AB and (AB ∇b−(θ ,1) η )(t ) = a ∇ a
AB a
−θ 1−θ
1−θ θ −θ
(s ). + s η B (θ ) B (θ )
(ABC ∇bθ f )(t ) = (ABR ∇bθ f )(t ) −
s=1
ABC
γ 1−θ
(s ), ∇ −(θ ,μ,γ ) η (t ) (s ) = 1 − λs−θ s−(1−μ) η B (θ )
and the b− right fractional version is
t B (θ ) γ = E (λ, t − ρ (s ))∇s [Eθσ,ν (λ, s − a )] θ ,μ 1−θ
ABR
AB a
θ ABR θ (ABC a ∇ f )(t ) = (a ∇ f )(t ) −
and ABC a
Za
Proof. The proof theorem.
∇ θ ,μ,γ [Eθσ,ν (λ, t − a )]
=
Example 1. By the help of Lemma 8 and the property (10) we conclude that ABR a
Theorem 1 (Nabla discrete Laplace transforms for the AB fractional sums). Let θ ∈ (0, 1/2 ), μ, γ ∈ C and η a discrete function defined on Na . Then,
a
∇ θ ,μ,γ χ )(t )}(s )
B (θ ) χ (a )s−μ [1 − λs−θ ]−γ . 1−θ
(42)
Applying the inverse Laplace to (42) finishes the first part. The second part can be proved by the help of the first part as well as the Q−operator action. θ ,μ,γ 1 )(t ) = Remark 4. From Theorem 2 we conclude that (ABR a ∇ B (θ ) γ E (λ, t 1−θ θ ,μ
θ ,μ ,γ
− a ) and (ABR ∇b
θ ∈ (0, 1/2), and μ, γ ∈ C.
γ 1 )(t ) = B1(−θθ) E (λ, b − t ) for any θ ,μ
4. Application: the ABC type fractional initial value problems 4.1. The action of the left and right fractional sums on the left and right ABC fractional differences Using Theorem 2, (27), (28) and the identity (12), we can conclude that following important tool for the particular case γ = 1. Proposition 2. For 0 < θ < 1, μ > 0, γ = 1, one has −(θ ,μ,1 ) (AB a ∇
ABC a
∇ θ ,μ,1 η )(t ) = η (t ) − η (a ).
T. Abdeljawad / Chaos, Solitons and Fractals 126 (2019) 315–324
Similarly,
( ∇ AB
−(θ ,μ,1 ) ABC b
θ ,μ,1
∇b
η )(t ) = η (t ) − η (b)
= η (t ) − η (a )
(43)
i=0
Proof. Using Theorem 2, (27) and the identity (12), we have −(θ ,μ,1 ) (AB a ∇
= η (t ) − η (a )
∇ θ ,μ,1 η )(t ) B (θ ) −(θ ,μ,1 ) = η (t ) − Eθ ,μ (λ, t − a ) η (a ) AB a ∇ 1−θ = η (t ) − η (a )Eθ ,1 (λ, t − a ) θ − η (a )Eθ ,θ +1 (λ, t − a ) 1−θ = η (t ) − η (a ). ABC a
λ=
= η (t ) − η (a ) = η (t ) − η (a )
⎢ ⎢ ⎣
×⎢ 1 +
−θ . 1−θ
−(θ ,μ,2 ) (AB a ∇
γ = 2, and λ =
−θ 1−θ
, we
m (−1 )i γi (γ )m−i λm (t − a )θ m (θ m + 1 ) i=0 ( m − i )! m=0 ∞
⎤ i γ
m (−1 ) i (γ )m−i ⎥ λm (t − a )θ m ⎥ ⎥ ( θ m + 1) ( m − i )! ⎦ m=1 i=0 ∞
= η (t ) − η (a ).
(50)
Finally, the proof of (49) follows by applying the Q−operator to (48).
(AB ∇b−(θ ,μ,γ )
ABC
∇bθ ,μ,γ η )(t ) = η (t ) − η (b).
(51)
ABC a
∇ θ ,μ,2 η )(t ) = η (t ) − η (a ).
(45)
ABC
∇bθ ,μ,2 η )(t ) = η (t ) − η (b).
(46)
and
(AB ∇b−(θ ,μ,2)
k!(θ (k + i ) + 1 )
=0 by 47
(44)
In what follows we prove Proposition 2 for other values of γ .
μ > 0,
γ ∞ ∞ (−1 )i λi+k (t − a )θ (k+i) (γ )k i
⎡
Proposition 3. For 0 < θ < 1, have
(−1 )i λi Eθγ,θ i+1 (λ, t − a )
i=0 k=0
Above we have made use of the identity
Eθ ,1 (λ, t − a ) − λEθ ,θ +1 (λ, t − a ) = 1,
∞
319
Remark 5. Regarding the orders of the fractional differences and the positivity of their real parts when be complex, we invite the reader to notice Remark 3 in [46].
Proof. Using Theorem 2, (34) with γ = 2, and the identity (12), we
4.2. Semigroup properties for the AB fractional sums
−(θ ,μ,2 ) ABC θ ,μ,2 (AB η )(t ) a ∇ a ∇ = η (t ) − η (a )[Eθ2,1 (λ, t − a ) − 2λEθ2,θ +1 (λ, t − a )
The following identity which was used in [46] to prove a semigroup property is also essential to proceed for the discrete case in this article. m
+λ2 Eθ2,2θ +1 (λ, t − a )
= η (t ) − η (a )
⎡
i=0
⎤
AB a
=0
= η (t ) − η (a ).
AB
The following identity, which was proved and used in [46], will help in generalizing Propositions 2 and 3 for any arbitrary γ .
i γ
(γ )m−i i = 0, ∀ m = 1, 2, . . . and γ ∈ C. ( m − i )!
Theorem 3. For 0 < θ < 1, μ > 0, γ ∈ C, and λ = −(θ ,μ,γ ) (AB a ∇
−θ 1−θ
(47) , we have
ABC a
∇ θ ,μ,γ η )(t ) = η (t ) − η (a ).
(48)
ABC
∇bθ ,μ,γ η )(t ) = η (t ) − η (b).
(49)
and
(AB ∇b−(θ ,μ,γ )
Proof. Using Theorem 2, (34), and the identity (12), we have −(θ ,μ,γ ) ABC θ ,μ,γ (AB η )(t ) a ∇ a ∇ −(θ ,μ,γ ) ABR θ ,μ,γ = AB ∇ ∇ η (t ) a a B (θ ) −(θ ,μ,γ ) γ − E (λ, t − a ) η (a ) AB a ∇ θ ,μ 1−θ γ i ∞ θ B (θ ) i = η (t ) − η (a ) 1−θ ( 1 − θ )i−1 B(θ ) i=0
×∇ −(θ i+1−μ) E
γ +σ
.
m
(52)
∇ −(θ ,1,γ )
AB a
∇ −(θ ,1,σ ) η (t ) =
1−θ B (θ )
AB a
∇ −(θ ,1,γ +σ ) η (t ).
(53)
AB
∇b−(θ ,1,σ ) η (t ) =
1−θ B (θ )
AB
∇b−(θ ,1,γ +σ ) η (t ).
(54)
and
To prove (46) we use (45) and the Q−operator.
i=0
m−i
i
=
Theorem 4. Assume 0 < θ < 1/2, μ, β ∈ C, σ , γ ∈ C and η(t) be defined on Na , b ≡ a (mod 1 ). If μ = β = 1, then we have
∞ ⎢ (t − a )θ k (2 )k 2(2 )k−1 (2 )k−2 ⎥ ×⎢ 1 + λk ( − + )⎥ ⎣ (θ k + 1 ) k! ( k − 1 )! ( k − 2 )! ⎦ k=2
m (−1 )
γ σ
γ (λ, t − a ) θ ,μ
∇b−(θ ,1,γ )
Proof. On the light of Definition 8 and by applying Lemma 2 one can have AB a
∇ −(θ ,μ,γ )
∇ −(θ ,β ,σ ) η (t ) i γ ∞ θ i −(θ i+1−μ ) AB −(θ ,β ,σ ) = η (t ) a∇ a ∇ B(θ )(1 − θ )i−1 i=0 γ i σ k ∞ ∞ θ θ k i = B(θ )(1 − θ )i−1 B(θ )(1 − θ )k−1 i=0 k=0 AB a
∇ −(θ (k+i)+1−μ+1−β ) η (t ) γ σ i+k ∞ ∞ θ 1 − θ k i = B (θ ) B(θ )(1 − θ )i+k−1 i=0 k=0 ×
a
×a ∇ −(θ (k+i )+1−β +1−μ) η (t ) ∞ 1−θ θm = B (θ ) B(θ )(1 − θ )m m=0
×a ∇ −(θ m+1−μ+1−β ) η (t )
m i=0
γ σ i
m−i (55)
In order to prove our claim (53) we just make use of the identity (52) by substituting μ = β = 1 above in (55). As always, the
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Q−operator is the tool to prove the right case by being applied to the left case. Depending on the proof of Theorem 4 we can state the following version of a semigroup property. Theorem 5 (A semigroup property). Assume 0 < θ < 1/2, μ, β ∈ C, σ , γ ∈ C and η(t) be defined on Na , b ≡ a (mod 1 ). Then we have AB a
∇ −(θ ,μ,γ )
AB a
∇ −(θ ,β ,σ ) η (t ) =
1−θ B (θ )
AB a
Above we have used that E
for 0 < θ < 1/2 by the ratio test.
θ ,μ,γ 1 )(t ) = (ABR a ∇
=
∇b−(θ ,μ,γ )
AB
∇b−(θ ,β ,σ ) η (t ) =
1−θ B (θ )
AB
∇b−(θ ,μ+β ,γ +σ ) ∇b−1 η (t ).
∞ B (θ ) 1−θ
λk (γ )k (t − a )ν +θ k+μ−1 k! (ν + θ k + μ ) k=0
B (θ ) γ E (λ, t − a ). 1 − θ θ ,μ
More generally, for μ > 0 we have
and AB
Remark 6. A particular case of (59) is when η (t ) = 1 = (t − a )0 . Indeed, by the help of the identity (4), we have
∇ −(θ ,μ+β ,γ +σ ) a ∇ −1 η (t ). (56)
λk ( γ ) k γ (λ, 1 ) = ∞ , which is finite k=0 k! θ ,μ
ABR a
∇ θ ,μ,γ (t − a )ν =
∞ B(θ ) λk (γ )k (ν + 1 ) (t − a )ν +θ k+μ−1 . 1−θ (k! )(μ + θ k + ν ) k=0
(60)
(57) 4.3. The ABC fractional difference systems
From Remark 4 we that that θ ,μ,γ 1 )(t ) = (ABR a ∇
B (θ ) γ E (λ, t − a ). 1 − θ θ ,μ
(58)
The following representation for ABR fractional differences can be used to provide an alternative proof for the identity (58) : Theorem 6. For any 0 < θ < we have θ ,μ,γ η )(t ) = (ABR a ∇
∞ B (θ ) 1−θ
1 2,
μ, γ ∈ C and η(t) defined on Na ,
λk (γ )k
k=0
k!
(a ∇ −(θ k+μ−1) η )(t ).
(59)
Proof. From definition and the identity
∇t
t
G(t, s ) = G(t , t ) +
s=a+1
t−1
∇t G(t, s ),
s=a+1
we have θ ,μ,γ η )(t ) (ABR a ∇ t−1 B (θ ) γ γ = E (λ, 1 )η (t ) + ∇t Eθ ,μ (λ, t − ρ (s ))η (s ) θ ,μ 1−θ s=a+1 t−1 ∞ B (θ ) λk (t − ρ (s ))θ k+μ−1 (γ )k γ = E (λ, 1 )η (t ) + ∇t η (s ) θ ,μ 1−θ (θ k + μ )k! s=a+1 k=0 ∞ B (θ ) λk ( γ ) k γ = E (λ, 1 )η (t ) + θ ,μ 1−θ (θ k + μ − 1 )k! k=0
×
t−1
(t − ρ (s ))θ k+μ−2 η (s )
∞ B ( θ ) λk ( γ ) k B (θ ) γ E (λ, 1 )η (t ) + 1 − θ θ ,μ 1−θ k!
1 × (θ k + μ − 1 )
η (s )
s=a+1
∞ B (θ ) γ B ( θ ) λk ( γ ) k E (λ, 1 )η (t ) + 1 − θ θ ,μ 1−θ k! k=0
∞ B ( θ ) λk ( γ ) k − η (t ) 1−θ k! k=0
=
∞ B ( θ ) λk ( γ ) k 1−θ k! k=0
a
∇ −(θ k+μ−1) η (t ).
0<θ <
1 . 2
(61)
θ (ABC a ∇ η )(t ) = r η (t ),
y ( a ) = a0 ,
0<θ <
1 , 2
(62)
has the trivial solution unless θ = 1. Therefore, linearity with constant coefficient results in the trivial solution. The next theorem outlines the existence and uniqueness of solution for the ABC fractional difference systems of order (θ , μ, γ ), where the case μ = 1, without the need of the restrictive condition f (a, η (a )) = 0. Theorem 7. Assume f : Na,b × R → R, η : Na,b → R and consider the system θ ,μ,γ η (t )(t ) = f (t, η (t )), t ∈ N (ABC η (a ) = c a,b = {a, a + 1, . . . b} a ∇
(63) such that b ≡ a (mod 1 ), f (a, η (a )) = 0 if μ = 1,
A
∞ i=0
γ
θ i (b − a )θ i+μ−1 < 1, B(θ )(1 − θ )i−1 (2 + θ i − μ ) i
AB a
(64) A > 0. Then, the system
∇ −(θ ,μ,γ ) f (t , η (t )).
(65)
Proof. By Theorem 2, Theorem 3, the identity
(t − ρ (s ))θ k+μ−2 =
−(θ k + μ − 1 )η (t ) =
η ( a ) = a0 ,
In order to verify the initial condition they requested the assumption that f (a, η (a )) = 0. In particular, they showed that the system:
η (t ) = c +
k=0
t
θ (ABC a ∇ η )(t ) = f (t , η (t )),
and | f (t, η1 ) − f (t, η2 )| ≤ A|η1 − η2 )|, (63) has a unique solution of the form
s=a+1
=
In [23], the authors proved an existence and uniqueness theorem for the solution of the system
a
∇ θ k+μ−1 η (t )
ABR a
∇ (θ ,μ,γ )
AB a
∇ −(θ ,μ,γ ) η (t ) = η (t ),
and taking into account that f (a, y(a )) = 0 in the case μ = 1, it can be shown that y(t) satisfies the system (63) if and only if it −(θ ,μ,γ ) h )(a ) = satisfies (65). In context, one has to note that (AB a ∇ −(θ ,1,γ ) h )(a ) = 0 with the need of h (a ) = 0. 0 for μ = 1 and (AB a ∇ Let ϒ = {η : maxt∈Na,b |η (t )| < ∞} be the Banach space endowed with the norm η = maxt∈Na,b |η (t )|. On ϒ , define the linear operator
( η )(t ) = c +
AB a
∇ −(θ ,μ,γ ) f (t, η (t )).
Then, for arbitrary η1 , η2 ∈ X and t ∈ Na,b , we have, by assumption, that
T. Abdeljawad / Chaos, Solitons and Fractals 126 (2019) 315–324
∇ −θ [ f (t, η1 (t )) − f (t, η2 (t ))]| γ i ∞ θ (b − a )θ i+μ−1 i ≤A B(θ )(1 − θ )i−1 (2 + θ i − μ ) i=0
|( η1 )(t ) − ( η2 )(t )| = |
AB a
η1 − η2 .
×
Now, we solve the below system by following the method of successive approximation outlined in Theorem 7. θ ,μ,γ η )(t ) = r η (t ) + g(t ), (ABC a ∇
(66)
and, hence, is a contraction. Now by Banach Contraction Principle, there exists a unique η ∈ X such that η = η and hence the proof is complete. The solution η(t) can be then obtained by means of the successive process
ηm (t ) = η (a ) +
AB a
∇ −(θ ,μ,γ ) f (t , ηm−1 (t )),
for m = 1, 2, . . . , with η0 (t ) = c.
0<θ
Remark 7. It is to be noted that Theorem 2 in [23] is a particular case of Theorem 7 above with μ = γ = 1 but with the need of the assumption that f (a, η (a )) = 0.
ηm (t ) = a0 + r
μ, γ ∈ C, t ∈ Na+1 .
(67)
η (a ) + (1 − λs−θ )γ
g( s )
B ( θ ) 1 −μ s [1 1 −θ
− λs−θ ]−γ − r
(s )}(t ) η (t ) = Za−1 {η
(68)
, (69)
(70)
is our nontrivial solution for μ = 1. The case μ = γ = 1, g(t ) = 0 ends with the trivial solution following the discussion before Theorem 7.
(s ) in To find the above discrete Laplace inverse, we expand η the form
(s ) = η
+
g( s )
s ∞
rj
j=0
rj
j=0
1−θ B (θ )
1−θ B (θ )
ηm (t ) = a0 + a0
η (t ) = η (a )
∞
rj
j=0
+ g(t ) ∗
∞
r
j=0
=
η (a ) + η (a )
+ g(t ) ∗
∞ j=0
rj
r
i−1
∞
r
1−θ B (θ )
r
∞
r
i−1
1−θ B (θ )
1−θ B (θ )
j+1 E
(76)
i−1
i−1
−(θ ,iμ,iγ ) −(i−1 ) (AB 1 )(t ) a∇ a ∇
−(θ ,iμ,iγ ) −(i−1 ) (AB g)(t ). a∇ a ∇
(77)
η ( a ) = a0 , (78)
will have the explicit solution
η (t ) = η (a )
∞
+ g(t ) ∗
r
j
∞
1−θ B (θ )
r
j
=
η (a ) + η (a )
∞
j
+ g(t ) ∗
∞
r
j
−γ j
Eθ , j (1−μ)+1 (λ, t − a )
1−θ B (θ )
rj
j=1
−γ j (λ, t − a ) θ , j (1−μ )+1
1−θ B (θ )
−(θ ,iμ,iγ ) −(i−1 ) (AB g)(t ) a∇ a ∇
θ ,μ,γ η )(t ) = r η (t ) + g(t ), (ABC a D 0 < θ ≤ 1, μ, γ ∈ C, t ≥ a,
j=0
E
j
j=0
j+1
i−1
−(θ ,iμ,iγ ) −(i−1 ) (AB 1 )(t ) a∇ a ∇
Remark 8. If by referring to Section D in [46], in the case of Laplace method, we follow similar steps as above, then the continuous system
s−(1−μ)( j+1) [1 − λs−θ ]γ ( j+1) , |x| < 1, (71)
E
i−1
1−θ B (θ )
1−θ B (θ )
i
j=0
j
(75)
−(θ ,iμ,iγ ) ∇ −(i−1 ) 1 )(t ) can be calculated by makThe terms (AB a a ∇ ing use of Lemma 1. Also, it is to be noted that η (a ) = a0 . Moreover, it is of interest to compare with the obtained solution by means of the discrete Laplace transforms mentioned above.
.s−(1−μ) j [1 − λs−θ ]γ j
j=1
ri
i=1
j
1−θ B (θ )
∞
m
j+1
1−θ B (θ )
j
m
η (t ) = a0 + a0
where x = r B1(−θθ) sμ−1 [1 − λs−θ ]γ and 0 < μ < 1. If we make use of Lemma 7, we reach at the solution
∇ −(θ ,μ,γ ) g(t ), m = 1, 2, . . . ,
If we proceed inductively, and use the semigroup property (56) in Theorem 5, we reach the formula
i=1
and hence
AB a
where η0 (t ) = a0 . For m = 1 we have
+
(s ) = Za {η (t )}(s ),
η g(s ) = Za {g(t )}(s ). From which it follows that
∞ η (a )
∇ −(θ ,μ,γ ) ηm−1 (t ) +
If letting m → ∞ and on the light of the condition (64) satisfied by b, we obtain the solution representation
η ( a ) = a0 ,
B (θ ) B ( θ ) 1 −μ
(s )[1 − λs−θ ]−γ − s η η (a )s−μ [1 − λs−θ ]−γ 1−θ 1−θ
(s ) +
= rη g( s ) ,
s−
AB a
i=1
r B1(−θθ) sμ
(73)
(74)
+
If we apply the discrete Laplace transform Za to the system (67) above we get
(s ) = η
μ = 1, t ∈ Na+1 .
i=1
Consider the system
1 , 2
η ( a ) = a0 ,
Indeed, we have
4.4. The ABC fractional linear difference system with constant coefficient
0<θ <
1 < , γ ∈ C, 2
−(θ ,μ,γ ) −(θ ,μ,γ ) η1 (t ) = a0 + ra0 (AB 1 )(t ) + (AB g)(t ). a ∇ a ∇
θ ,μ,γ η )(t ) = r η (t ) + g(t ), (ABC a ∇
321
j+1
1−θ B (θ )
1−θ B (θ )
−γ ( j+1 )
Eθ ,( j+1)(1−μ) (λ, t − a )
j
j+1
−γ j
Eθ , j (1−μ)+1 (λ, t − a ) −γ ( j+1 )
Eθ ,( j+1)(1−μ) (λ, t − a ),
(79)
where the continuous counterpart of Lemma 7 will be used instead [49]. It is of interest to compare between the solution (79) and the solution given by Eq. (66) in [46], when the successive approximation is applied.
−γ ( j+1 ) (λ, t − a ) θ ,( j+1 )(1−μ )
5. The iterated fractional sums when γ = 1
j E
−γ j
θ , j (1−μ )+1
(λ, t − a )
−γ ( j+1 ) (λ, t − a ). θ ,( j+1 )(1−μ )
On the light of Definition 8 or Remark 3 for γ = 1 consider the fractional sum operator
(72)
AB a
∇ −(θ ,μ,1) η (t ) =
1−θ B (θ )
a
∇ −(1−μ) η (t )
322
T. Abdeljawad / Chaos, Solitons and Fractals 126 (2019) 315–324
+
θ −(θ +1−μ ) η (t ), θ ∈ (0, 1/2 ), μ ∈ C. a∇ B (θ )
If we iterate
[
AB ∇ −(θ ,μ,1 ) η (t ) a
∇ −(θ ,μ,1) ]n η (t ) =
AB a
n
n
k
=
(1 − θ )
n−k
θ
k a
n
n
k
k=0
∇ −(kθ +1−μ) η (t )
(1 − θ )n−k θ k −(kθ +1−μ ) η (t ). (81) a∇ B ( θ )n
−(θ ,μ,1,n ) η (t ). We write [ = AB a ∇ By means of (81) and the binomial theorem we formulate the following definition.
Definition 9. For θ ∈ (0, 1/2) and σ , μ ∈ R define the following iterated sum-differ operators: AB a
∇ −(θ ,μ,1),σ η (t ) =
∞
k
k=0
∇
−(θ ,μ,1 ),−σ
η (t ) =
(1 − θ )σ −k θ k −(kθ +1−μ ) η (t ), (82) a∇ B ( θ )σ
∞
− σ k
a
∇
−(kθ +1−μ )
η (t ).
(83)
AB a
−(θ ,μ,1 ),1 ∇ −(θ ,μ,1),0 η (t ) = a ∇ −(1−μ) η (t ), AB η (t ) a ∇ −(θ ,μ,1 ) = AB ∇ η ( t ) , a
AB a
∇ −(θ ,μ+β ,1),σ +ν a ∇ −1 η (t ).
(87)
This finishes the proof of (85). The proof for (86) is similar.
Remark 10. If we let μ = β = 1 in the proof of Theorem 8, then we obtain the following semigroup property AB a
∇ −(θ ,1,1),σ
AB a
∇ −(θ ,1,1),ν f (t ) =
AB a
∇ −(θ ,1,1),σ +ν η (t ),
(88)
∇b−(θ ,1,1),σ
AB
∇b−(θ ,1,1),ν f (t ) =
AB
∇b−(θ ,1,1),σ +ν η (t ).
(89)
Remark 11. From (85) with σ = ν = 1 we have
∇ −(θ ,μ,1),1
AB a
∇ −(θ ,β ,1),1 η (t ) =
AB a
∇ −(θ ,μ+β ,1),2 a ∇ −1 η (t ), (90)
∇ −(θ ,μ,1),n η (t ) = [
AB a
θ ,μ,1 η )(t ) = (ABR a ∇
where
θ ,1,1 η )(t ) = (ABR a ∇
AB a
AB a
∇
−(θ ,μ,1 ),−1
However, for
∇ −(θ ,1,1),−1 η (t ),
which is according to [13] the same as tually, we have
η (t ) = (
ABR a
AB ∇ (−θ ,−1 ) η (t ). a
∇ θ ,2−μ,1 η )(t )
Ac-
(84)
Theorem 8. Let θ ∈ [0, 1] and μ, β , σ , ν ∈ R and η be a function defined on Na . Then for any t ∈ Na we have
∇ −(θ ,μ,1),σ
AB a
∇ −(θ ,β ,1),ν η (t ) =
AB a
∇ −(θ ,μ+β ,1),σ +ν a ∇ −1 η (t ), (85)
and for any t ∈ b N we have −(θ ,μ,1 ),σ AB b
∇
−(θ ,β ,1,ν ) b
f (t ) =
∇
−(θ ,μ+β ,1 ),σ +ν b
∇ η (t ). −1 b
Proof. By the help of the identity (52) and that the left fractional sums have the semigroup property, we have −(θ ,μ,1 ),σ AB a
∇ σ ∞
k=0
k
AB a
∇ −(θ ,μ+β ,2) a ∇ −1 η (t ). conclude that possible to ver-
∇
(−θ ,γ )
η (t ) =
γ ∞ (1 − θ )γ −k θ k k B ( θ )γ
k=0
a
∇ −kθ η (t )
(92)
The following result can be observed from definitions. Theorem 9 (The relation between the iterated AB sumdifferences discussed in [13] and the iterated ones here). For θ ∈ (0, 1/2 ), μ, γ ∈ C and η define on Na we have AB a
∇ −(θ ,μ,1),γ η (t ) =
∇ (−θ ,γ ) a ∇ −(1−μ) η (t ).
(93)
∇ (−θ ,γ ) η (t ).
(94)
AB a
In particular, we have AB a
∇ −(θ ,1,1),γ η (t ) =
AB a
Example 2. Consider the fractional difference-sum equation of the form ABR 0
∇ θ ,2−μ,1 η (t ) = −A 0 ∇ −(1−μ) η (t ) + b(t ), A > 0,
(95)
where, AB
(86)
AB a
1−θ B (θ )
ify this directly.
AB a
k=0
which is different from μ = 1 we have
∇ −(θ ,β ,1) η (t ) =
From [13] recall that the AB iterated sum-differences obtained by iterating fractional sums correspond to fractional differences with discrete Mitag–Leffler kernels are given by
∞ B (θ ) k λ (a ∇ −(θ k+μ−1) η )(t ), 1−θ
AB ∇ −(θ ,μ,1 ),−1 η (t ). a
AB a
(91)
∇ −(θ ,μ,1) ]n η (t ).
∞ AB ∇ −(θ ,μ,1 ),−1 η (t ) = B (θ ) k −(θ k+1−μ ) η (t ), a k=0 λ a ∇ 1−θ −1 −θ λ = 1−θ and we have used that k = (−1 )k .
∇ −(θ ,μ,1)
AB ∇ −(θ ,μ,1 ),1 f (t ) = AB ∇ −(θ ,μ,1 ) f (t ), we Since, a a AB ∇ −(θ ,μ,1 ),2 f (t ) = 1−θ AB ∇ −(θ ,μ,2 ) f (t ). It is also a B (θ ) a
3. If we let γ = 1 in Theorem 6, we notice that
=
AB a
AB a
and
∇
=
and from (56) with σ = γ = 1 we have
1.
AB
a
AB a
Remark 9 (Particular cases).
AB a
∇ −(mθ +1−μ+1−β ) η (t ) σ +μ m ∞ θ (1 − θ )σ +ν −m m −(mθ +1−(μ+β )) −1 = η (t ) a∇ a∇ B ( θ )σ +ν m=0
×
AB
B ( θ )σ θ k
( 1 − θ )σ +k
k=0
2.
a
and for any t ∈ b N we have
and AB a
B ( θ )σ +ν
∇ −((k+i)θ +1−μ+1−β ) η (t ) ∞ m θ m (1 − θ )σ +ν −m ν σ = B ( θ )σ +ν i m−i m=0 i=0
×
AB ∇ −(θ ,μ,1 ) ]n η (t ) a
σ
=
σ ν ∞ ∞ ( 1 − θ )σ +ν − ( k+i ) θ i+k k i k=0 i=0
n−times we have
B(θ )n−k B(θ )k
k=0
(80)
−(θ ,β ,1 ),ν
∇ η (t ) σ −k k (1 − θ ) θ −(kθ +1−μ ) −(θ ,β ,1 ),ν η (t ) a∇ a∇ B ( θ )σ
b(t ) =
∞
bs t θ s .
s=0
From (84) and Theorem 9, Eq. (95) is equivalent to AB 0
∇ (−θ ,−1) 0 ∇ −(1−μ) η (t ) = −A 0 ∇ −(1−μ) η (t ) + b(t ). ∇ −(1−μ) η (t )
Let u(t ) = 0 Eq. (96) becomes AB 0
and hence η (t ) =
∇ (−θ ,−1) u(t ) = −Au(t ) + b(t ).
0
∇ 1−μ u(t )
(96) and
(97)
T. Abdeljawad / Chaos, Solitons and Fractals 126 (2019) 315–324
β = θ − n. Then β ∈ (0, 1] and by means of Definitions 6 and 8 we
From Remark—in [13], Eq. (97) has the solution b(t )
u(t ) =
A+
B (θ ) − 1−θ
where c0 = mined from cm =
∞
A+
m=1
b0 A+( B1(−θθ) )
bm
B (θ ) − 1−θ
t mθ
∞ k=1
m
cm−k (−1 )k θ k B(θ )( (m − k )θ + 1 )
(1 − θ )1+k (mθ + 1 ) A +
k=1
, B (θ ) 1−θ
and the coefficients ci , i ∈ N, can be deter-
−1
θ k B(θ )((m − k )θ + 1 ) . (1 − θ )1+k (mθ + 1 )(mθ + 1 ) A + B1(−θθ) cm−k
k
(98)
η (t ) =
d (t ) A+ ×
B (θ ) − 1 −θ
mθ
(mθ + 1 )t (mθ + μ ) m=1
m cm−k (−1 )k θ k B(θ )((m − k )θ + 1 ) k=1
where, d (t ) =
(1 − θ )1+k (mθ + 1 ) A +
B (θ ) 1 −θ
,
∞
bs (θ s+1 ) θ s+μ−1 . s=0 (θ s+μ ) t
Theorem 10 (Nabla discrete Laplace transforms for the iterated AB sum-differences). Let θ ∈ (0, 1/2 ), μ, γ ∈ C and η be defined on Na . Then, we have
Za
AB a
∇ (−θ ,μ,1),σ η (t ) (s ) =
1−θ θ −θ + s B (θ ) B (θ )
σ
define::
(ABC ∇bθ ,μ,γ η )(t ) = (ABC ∇bβ ,μ,γ ( n η )(t ).
(s ), s − ( 1 −μ ) η (99)
and in the right Riemann–Liouville sense has the following form:
(ABR ∇bθ ,μ,γ η )(t ) = (ABRR ∇bβ ,μ,γ ( n η )(t ). (AB ∇b−(θ ,μ,γ ) η )(t ) = (∇b−n Above,
n
means
AB
∇b−(β ,μ,γ ) η )(t ).
Za
AB a
∇ (−θ ,μ,1),σ η (s ) =
Theorem 11 (The action of the AB-sum on the ABC-difference in higher order case). Assume η is defined on a proper domain and θ , μ, γ , β and n are as above, then −(θ ,μ,γ ) (AB a ∇
ABC a
∇ θ ,μ,γ η )(t ) = η (t ) −
=
1−θ θ −θ + s B (θ ) B (θ )
σ ∞ (1 − θ )σ −k θ k k
σ
n j=0
∇ j η (a ) j!
(t − a ) j
(108)
and ABC
∇bθ ,μ,γ η )(t ) = η (t ) −
n (−1 ) j j f (b) (b − t ) j j! j=0
B ( θ )σ
k=0
(107)
(−1 )n n .
Proof. By Lemma 5 we have
(106)
The associated fractional integral by
(AB ∇b−(θ ,μ,γ )
(s ) = (Za η )(s ). where η
(105)
Remark 12. In the ABC-case we requested the function η to be defined on Na−n in the left case and on b+n N in the right case and the ABC-fractional difference will start from a or end at b. Alternatively, we can fixed η to be defined on Na and the left ABCdifference will start at a + n, and in the right case we fixed η on b N and the ABC-fractional difference will end at b − n. Regarding the index θ , for the convergence purposes in the difference-operator case, we request θ so that β = n − θ ∈ (0, 1/2].
Therefore, ∞
323
(109)
(s ) s−(kθ +1−μ) η
(100)
(s ). s − ( 1 −μ ) η
(101)
Proof. The proof is similar to that in Propositions 2.1 and 2.2 in [25]. However, we make use of Theorem 3 in this article. Theorem 12 (The action of the AB-sum and the ABR-difference in higher order case on each other). Assume η is defined on a proper domain and θ , μ, γ , β and n are as above, then −(θ ,μ,γ ) (AB a ∇
ABR a
∇ θ ,μ,γ η )(t ) = η (t ) −
n−1 j=0
6. The higher order case
∇ j f (a ) j!
(t − a ) j
(110)
and In this section we give the definitions to the AB fractional differences and sums with discrete generalized Mittag–Leffler kernels. In above, the value of the index θ has been taken between 0 and 1. Below, we extend the definitions when θ is arbitrary such that
(θ ) > 0. In our extension we shall obey a similar approach to that in [24–27]. Definition 10. Let n < θ ≤ n + 1, μ, γ ∈ R and η be defined on Na−n in the Caputo case, otherwise it is enough on Na . Set β = θ − n. Then β ∈ (0, 1] and by means of Definitions 6 and 8 we define: θ ,μ,γ η )(t ) = (ABC ∇ β ,μ,γ ∇ n η )(t ), t ∈ N . (ABC a a ∇ a
AB a
∇ −(β ,μ,γ ) η )(t ), t ∈ Na .
∇bθ ,μ,γ η )(t ) = η (t ) −
n−1 (−1 ) j j f (b) (b − t ) j j! j=0
(111) On the other way, θ ,μ,γ (ABR a ∇
(
ABR
AB a
∇ −(θ ,μ,γ ) η )(t ) = η (t ),
∇bθ ,μ,γ AB ∇b−(θ ,μ,γ ) η )(t ) =
η (t ).
(112)
(102)
(103)
7. Conclusions
The associated fractional sum by −(θ ,μ,γ ) (AB η )(t ) = (a ∇ −n a ∇
ABR
Proof. The proof is similar to that in Propositions 2.1 and 2.2 in [25]. However, we make use of the definitions for the higher order case of generalized Mittag–Leffler kernel discrete operators in this article.
and the left Riemann–Liouville sense has the following form: θ ,μ,γ η )(t ) = (ABR ∇ β ,μ,γ ∇ n η )(t ), t ∈ N . (ABR a a ∇ a
(AB ∇b−(θ ,μ,γ )
(104)
For the right case we have the following: Definition 11. Let n < θ ≤ n + 1, μ, γ ∈ R and η be defined on b+n N in the Caputo case, otherwise it is enough on b N. Set
1. We defined the left and right ABR and ABC fractional differences and their corresponding fractional sums, where the kernel of the discrete fractional operators is a discrete generalized Mittag–Leffler function. The action of the ABfractional sum on the ABC-fractional differences has been considered.
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T. Abdeljawad / Chaos, Solitons and Fractals 126 (2019) 315–324
2. Semigroup type properties have been proven for the derived AB−fractional sums. Making use of such semigroup property we solved the ABC-linear case with constant coefficient. Indeed, by noting that the restriction f (a, ξ (a )) = 0 can be released for the case μ = 1 we have found a nontrivial solution for a nontrivial initial condition. 3. We have found the discrete Laplace transform for the defined and analyzed discrete operators and we used it to solve an example for the linear case. 4. The Laplace transforms have been used to obtain a solution of convolution type for both the continuous and discrete ABC fractional linear nonhomogeneous initial value problem with constant coefficient. The successive approximation method has been also discussed for the discrete system. 5. We obtained the iterated sum-differences by only iter−(θ ,μ,1 ) which consist of the addiating the sums AB a ∇ tion of two terms. However, iterating the sums of order −(θ , μ, γ ), γ = 2, 3, . . . is still without investigation. It will be complicated since we will use the binomial theorem applied to more than two terms. In this case and on the light of Remark 11 it will be of interest to find relations between fractional sums with three parameter-order and the iterated fractional sum-differences with the fourth parameter. 6. Alternative representations for the discrete ABR-operators have been formulated by using the expansion of the generalized discrete Mittag–Leffler kernels and some comparisons have been made to the defined iterative sum-differences. 7. We considered the extension to the higher order case. 8. We have considered the case of the Z time sale. The generalization to the hZ time scale can be done by following an approach as in [10]. Acknowledgment The author would like to thank Prince Sultan University for funding this work through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM) group number RGDES-2017-01-17. References [1] Podlubny I. Fractional differential equations. San Diego CA: Academic Press; 1999. [2] Samko G Kilbas AA, Marichev. Fractional integrals and derivatives: theory and applications. Yverdon: Gordon and Breach; 1993. [3] Kilbas A, Srivastava MH, Trujillo JJ. Theory and application of fractional differential equations, 204. North Holland Mathematics Studies; 2006. [4] Al-Mdallal QM, Omer ASA. Fractional-order Legendre-Collocation method for solving fractional initial value problems. Appl Math Comput 2018;321:74–84. [5] M Al-Mdallal Q. An efficient method for solving fractional Sturmliouville problems. Chaos Solitons Fractals 2009;40.1:183–9. [6] M Al-Mdall Q, I Syam M. An efficient method for solving non-linear singularly perturbed two points boundary-value problems of fractional order. Commun Nonlinear Sci Numer Simul 2012;17.6:2299–308. [7] Al-Mdallal QM, Hajji MA. A convergent algorithm for solving higher-order nonlinear fractional boundary value problems. Fract Calculus Appl Anal 2015;18.6:1423–40. [8] Al-Mdallal Q, Kashif AA, Khan I. Analytical solutions of fractional walters b fluid with applications. Complexity 2018;2018. [9] Bohner M, Peterson A. Advances in dynamic equations on time scales. Boston-Basel-Berlin: Birkhäuser; 2003. [10] Abdeljawad T. Different type kernel h–fractional differences and their fractional hsums. Chaos Solitons Fractals 2018;116:146–56. [11] Suwan I, Owies S, Abdeljawad T. Monotonicity results for h-discrete fractional operators and application. Adv Differ Equ 2018;2018:207. [12] Abdeljawad T, Baleanu D. Discrete fractional differences with nonsingular discrete Mittag-Leffler kernels. Adv Difference Equations 2016;2016:232. doi:10. 1186/s13662- 016- 0949- 5. [13] Abdeljawad T., Fernandez A.. On a new class of fractional differencesum operators based on discrete Atangana-Baleanu sums. 2019. arXiv:1901. 08268v1 [math.CA]. [14] Goodrich C, Peterson A. Discrete fractional calculus. Springer; 2015. [15] Abdeljawad T, Atici F. On the definitions of nabla fractional differences. Abstr Appl Anal 2012;2012:13. doi:10.1155/2012/406757. Article ID 406757
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Contents lists available at ScienceDirect
Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Fractional difference operators with discrete generalized Mittag–Leffler kernels Thabet Abdeljawad Department of Mathematics and General Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi Arabia
a r t i c l e
i n f o
Article history: Received 26 April 2019 Revised 9 June 2019 Accepted 12 June 2019 Available online 27 June 2019 Keywords: Discrete generalized Mittag–Leffler function Discrete nabla Laplace transform Convolution AB Fractional sums ABR Fractional difference ABC Fractional difference Iterated AB sum-differences Higher order
a b s t r a c t In this article we define fractional difference operators with discrete generalized Mittag–Leffler kernels of γ the form (E (λ, t − ρ (s )) for both Riemann type (ABR) and Caputo type (ABC) cases, where AB stands θ ,μ
for Atangana–Baleanu. Then, we employ the discrete Laplace transforms to formulate their corresponding AB−fractional sums, and prove useful and applicable versions of their semi-group properties. The action of fractional sums on the ABC type fractional differences is proved and used to solve the ABC−fractional difference initial value problems. The nonhomogeneous linear ABC fractional difference equation with constant coefficient is solved by both the discrete Laplace transforms and the successive approximation, and the Laplace transform method is remarked for the continuous counterpart. In fact, for the case μ = 1, we obtain a nontrivial solution for the homogeneous linear ABC− type initial value problem with constant coefficient. The relation between the ABC and ABR fractional differences are formulated by using the discrete Laplace transform. We iterate the fractional sums of order −(θ , μ, 1 ) to generate fractional sumdifferences for which a semigroup property is proved. The nabla discrete transforms for the AB−fractional sums and the AB−iterated fractional sum-differences are calculated. Examples and remarks are given to clarify and confirm the obtained results and some of their particular cases are highlighted. Finally, the discrete extension to the higher order case is discussed. © 2019 Elsevier Ltd. All rights reserved.
1. Introduction The question about the calculus of differentiating or integrating up to arbitrary order is old [1–3] while many recent investigations regarding its applications and theoretical and numerical methods are still under development [4–8]. The answer to this question brought up many kinds of nonlocal fractional integrals and derivatives gradually. With the appearance of the time scale calculus [9], the theory of discrete fractional derivatives (fractional differences), discrete fractional integrals (fractional sums) and the q-fractional calculus has started to be developed in the last two decades or so [10–19]. During the last few years, and for the sake of having fractional operators (or derivatives) with nonsingular kernels the authors in [20] defined the so called Caputo–Fabrizio fractional derivatives by imposing a nonsingular exponential kernel. Later, the authors in [21] replaced the exponential kernel by the one parameter Mittag–Leffler function. The fractional integral operators that cancel the fractional derivatives with such nonsingular kernels consist of two parts, of which the first part is a multiple of the function itself. Such a structure for the fractional integrals
E-mail address: [email protected] https://doi.org/10.1016/j.chaos.2019.06.012 0960-0779/© 2019 Elsevier Ltd. All rights reserved.
leads to the loss of the semigroup property so that it becomes slightly difficult to solve the fractional dynamical systems by using the successive approximation method. This structure also results in having the trivial solution for the linear system with constant coefficients within Caputo fractional derivatives with Mittag–Leffler kernel (the ABC-fractional linear system with constant coefficients). In fact, we request the right hand side of the ABC-fractional system to vanish at the starting point a [22]. These properties for the AB−fractional integrals have been investigated for the discrete counter parts in [23]. For more about the higher order fractional operators with nonsingular kernels and their discrete counterparts we refer to [24–27] and for the monotonicity analysis of the fractional differences with discrete exponential kernels and discrete Mittag–Leffler kernels we refer to the recent manuscripts [28–30]. More developments on the light of applications of fractional operators with nonsingular kernels we refer to [31–44]. In [45] the authors introduced fractional operators with generalized Mittag– Leffler functions and derived their fractional integrals and then the author in [46] continue to show the advantages of such extension in gaining a certain semigroup property for the fractional integrals and in proving existence and uniqueness for the solution of the initial value problems without the need of assuming that the right hand side is zero at the starting point a.
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T. Abdeljawad / Chaos, Solitons and Fractals 126 (2019) 315–324
In this article we define and investigate the fractional differences with discrete generalized Mittag–Leffler kernels, find their corresponding fractional sums, prove their semigroup properties and their action on the ABC-fractional differences. Indeed, this work will be organized as follows: Next section, Section 2, will contain preliminaries about discrete generalized Mittag–Leffler functions and their discrete Laplace transforms. Section 3 is about the ABR and ABC discrete generalized fractional derivatives and their fractional sums. Section 4 is devoted to the application in the discrete ABC-initial value problems by investigating the action of the AB-fractional differences on the ABC-fractional differences and proving semigroup type properties. An existence and uniqueness theorem is proved and an example for the linear case with constant coefficients is solved by both discrete Laplace transforms and the successive approximation. Section 5 is about the iterated fractional sum-differences. Section 6 deals with the extension to higher order and finally the last section will contain our conclusions. 2. Preliminaries
•
(μ + 1 ) ( b − t )θ +μ (5) (μ + 1 + θ ) Lemma 2 [14]. Let θ1 , θ2 ∈ R and η be defined on Na . Then, the fol-
∇b−θ (b − t )μ =
lowing hold: a
∇ −θ1 a ∇ −θ2 η (t ) = a ∇ −(θ1 +θ2 ) η (t ),
(6)
and
∇b−θ1 ∇b−θ2 η (t ) = ∇b−(θ1 +θ2 ) η (t ),
(7)
The Q −operator action, (Q η )(t ) = η (a + b − t ), which was used in [16,17] to relate left and right fractional sums and differences, is a useful tool in confirming the right definition cases and hence setting a proper integration by parts. It was shown that
(∇a−θ Q η )(t ) = Q b ∇ −θ η (t ). Since in general it is not true that (t θ )β = t θ β and (ab)θ = aθ bθ in general, the author in [16,17], motivated by the time scale calculus notations, defined the following versions of nabla discrete Mittag–Leffler functions.
This section contains some basic concepts about discrete fractional operators, discrete generalized Mittag–Leffler functions and the discrete Laplace transforms into nabla.
Definition 3 (Nabla Discrete Mittag–Leffler see [16,17]). For λ ∈ R, |λ| < 1 and θ , β , ρ , z ∈ C with Re(θ ) > 0, the nabla discrete Mittag–Leffler functions are defined by
2.1. The rising factorial, the nabla fractional sums and the nabla discrete Mittag–Leffler functions in three indices
E
For the details about the basic given in this subsection refer to [14–17]. Definition 1. (i) For a natural number m, the m rising (ascending) factorial of t is defined by
t
m
=
m −1
(t + k ),
0
t = 1.
(1)
k=0
(t + θ ) , (t )
t ∈ R \ {. . . , −2, −1, 0}, 0θ = 0
(2)
The following property of the rising factorial function can be observed:
∇ (t θ ) = θ t θ −1 ,
(3)
hence t θ is increasing on N0 , where ρ (t ) = t − 1. Definition 2 (See [16,17]). Let ρ (t ) = t − 1 be backward jump operator. Then,for a function η : Na = {a, a + 1, a + 2, .} → R, the nabla left fractional sum of order θ > 0 (starting from a) by a
∇ −θ η (t ) =
t
1
(θ ) s=a+1
(t − ρ (s ))θ −1 η (s ), t ∈ Na+1 ;
=
1
b−1
(θ )
s=t
1
b−1
(θ )
s=t
(8)
For β = ρ = 1, it is written that
Eθ (λ, z ) Eθ1,1 (λ, z ) =
∞ k=0
λk
z kθ , (θ k + 1 )
(9)
where (ρ )k = ρ (ρ + 1 ) . . . (ρ + k − 1 ). Notice that (1 )k = k! so that E 1 (λ, z ) = Eθ ,β (λ, z ). θ ,β
Direct properties of the discrete Mittag–Leffler functions are:
(10)
and
∇ Eθ (λ, z ) = λEθ ,θ (λ, z ).
(11)
Also, by the help of Lemma 1 we have a
∇ −ν Eθρ,β (λ, z ) = Eθρ,β +ν (λ, z )
(12)
2.2. The nabla discrete Laplace of fractional sums and Mittag–Leffler functions and the discrete convolution Definition 4 [14,47]. The nabla discrete Laplace transform for a discrete function f defined on Na is given by ∞
(1 − z )t−a−1 f (t )
(13)
t=a+1
Lemma 3 [47]. Let θ ∈ R \ {. . . , −2, −1, 0}. Then, we have Za ((t − a )θ −1 )(z ) = (θθ ) , |1 − z| < 1, Definition 5 (See [48] also). Assume f, g : Na → R be discrete functions. Then, the nabla discrete convolution of f with g is defined by
(σ (s ) − t )θ −1 η (s ), t ∈ b−1 N.
( f ∗ g)(t ) =
(μ + 1 ) (t − a )θ +μ (μ + 1 + θ )
t
g(t − ρ (s ) + a ) f (s ).
(14)
s=a+1
Proposition 1 [14]. Assume f, g are discrete functions defined on Na . Then, we have
•
∇ −θ (t − a )μ =
zkθ +β −1 (ρ )k . (θ k + β )k!
z
(s − ρ (t ))θ −1 η (s )
Lemma 1 [15–17]. Let θ > 0, μ > −1. Then,
a
λk
k=0
(Za f (t ))(z ) =
the nabla right fractional sum of order θ > 0 (ending at b)for η : b N = {b, b − 1, b − 2, .} → R by
∇b−θ η (t ) =
∞
∇ n Eθρ,β +n (λ, z ) = Eθρ,β (λ, z ), n = 1, 2, . . .
(ii) For any real number the θ rising function is defined by
tθ =
ρ (λ, z ) = θ ,β
(4)
(Za ( f ∗ g))(z ) = (Za f )(z )(Za g)(z ).
(15)
T. Abdeljawad / Chaos, Solitons and Fractals 126 (2019) 315–324
Lemma 4. Let f be defined on Na . Then
(Za ∇ ( f (t ))(z ) = z(Za f )(z ) − f (a ).
(16)
derived fractional sums, and construct an example of a nonconstant function whose ABR and ABC fractional differences are zeros. The relation between the ABR and the ABC fractional differences are demonstrated.
(17)
Definition 6. The discrete generalized ABR and ABC fractional γ derivatives with kernel E (λ, t ), where 0 < θ < 1/2, Re(μ ) >
More generally,
(Za ∇ n f )(z ) = zn (Za f )(z ) −
n−1
zn−1−i ∇ i f (a + 1 ).
i=0
θ ,μ
0, γ ∈ R, are defined respectively by
Lemma 5 [47]. For any real number ν , we have
(Za a ∇ −ν ) f (s ) = s−ν (Za f )(s ).
θ ,μ,γ f )(t ) = (ABC a ∇
zθ −1 . z θ −λ 1 = θ z −λ
(ABC ∇bθ ,μ,γ f )(t ) =
The proof of the following lemma is straight forward and its continuous counterpart can be found in [49] (see (2.19) there and Section 2 in [45] for the modified Mittag–Leffler version). Lemma 7. For γ , θ , β , λ ∈ C (Re(β ) > 0), and s ∈ C with Re(s ) > 0, |λs−θ | < 1, we have
Za E
k=0
1
f or 1 − λs−θ
(Za Eθγ,β (λ, t − a ))(s ) =
∞ k=0
= s −β
(18)
=s
λk (γ )k
k!sθ k+β ∞ (λs−θ )k (γ )k k!
−β
[1 − λs−θ ]−γ .
(19)
In fact by differentiating (18) γ −times we see that
( γ )k k!
=
1
(1 − λs−θ )γ
f or
|λs−θ | < 1.
s=a+1
γ −B(θ ) t E (λ, s − ρ (t )) f (s ), t ∈ θ ,μ 1−θ s=t b−1
−θ 1−θ
b N,
.
θ ,μ,γ f (t ) = u (t ) with γ = 1 Now, we solve the equation ABR a ∇ to find the discrete fractional integral operator of two parameters. Apply the discrete Laplace transform Za to both sides, use the discrete convolution theorem and make use of Lemma 7 to get ABR a
B (θ ) (Za ∇ [ f (t ) ∗ Eθ ,β (λ, t )] )(s ) 1−θ B ( θ ) −μ = s.s [1 − λs−θ ]−1 F (s ) 1−θ = U ( s ),
∇ θ ,μ,1 f (t )(s ) =
(20)
Lemma 8. Let ρ , μ, γ , ν, σ , λ ∈ C (Re(ρ ), Re(μ), Re(ν ) > 0), then γ γ +σ Eρ ,μ (λ, t − ρ (s ))Eρσ,ν (λ, s − a ) = Eρ ,μ+ν (λ, t − a ).
where λ =
( Za
Since the Pochhammer symbol (γ )k is valid for any γ ∈ C then γ can be arbitrary complex number.
t
s=a+1
Remark 1. It is clear that definitions above obey the action of the discrete Q−operator. Also, notice that to guarantee the convergence of the kernels of the above defined fractional difference operators, we need 0 < θ < 1/2 to have |λ| < 1 for λ = 1−−θθ .
k=0
(λs−θ )k
t B (θ ) γ E (λ, t − ρ (s )) f (s ), t ∈ Na ∇t θ ,μ 1−θ
(25)
|λs−θ | < 1,
we have
k=0
θ ,μ,γ f )(t ) = (ABR a ∇
(ABR ∇bθ ,μ,γ f )(t ) =
Proof. On the light of Lemma 3 and that
∞
and
the right one by
(λs−θ )k =
b N,
(24)
Za Eθ ,β (λ, t − a ) (s ) = s−β [1 − λs−θ ]−1 .
∞
b−1 −B(θ ) γ E (λ, s − ρ (t )) s f (s ), t ∈ 1 − θ s=t θ ,μ
(23)
γ (λ, t − a ) (s ) = s−β [1 − λs−θ ]−γ . θ ,β
In particular
(22) the right one by
Lemma 6. Let 0 < θ ≤ 1 and f be defined on Na . Then, (ii) (Za Eθ ,θ (λ, t − a ))(z )
t B (θ ) γ E (λ, t − ρ (s ))∇s f (s ), t ∈ Na , θ ,μ 1−θ s=a+1
The following lemma is a modification to what was proved in [48]. (i) (Za Eθ (λ, t − a ))(z ) =
317
(21)
Then, the proof follows by applying the discrete Laplace Za and its inverse by making use of Lemma 7 and Proposition 1. γ
where U (s ) = (Za u(t )(s ), F (s ) = (Za f (t )(s ) and λ = which it follows that
F (s ) =
−θ 1−θ
(26) . From
1 − θ μ−1 1−θ θ s −θ s [1 − λs−θ ]U (s ) = sμ−1 + U ( s ). B (θ ) B (θ ) B (θ )
Apply the discrete inverse Laplace and make use of Lemma 5 to see that
1−θ θ (a ∇ −(1−μ) u )(t ) + (a ∇ −(1−μ+ θ ) u )(t ). B (θ ) B (θ )
Proof. The left hand side of (21) is Eρ ,μ (λ, t − a ) ∗ Eρσ,ν (λ, t − a ).
f (t ) =
3. The ABR and ABC discrete generalized fractional derivatives and their fractional sums
Definition 7. The discrete left and right AB− fractional integrals of two parameters θ and μ are defined by:
In this section, we define fractional differences with three parameter discrete Mittag–Leffler kernels in the sense of Riemann and Caputo, derive their corresponding fractional sums, study the Laplace transforms for the defined fractional differences and their
−(θ ,μ ) (AB η )(t ) = a ∇
Hence, we have the following definition.
1−θ (a ∇ −(1−μ) η )(t ) B (θ ) +
θ
B (θ )
(a ∇ −(1−μ+θ ) η )(t ).
(27)
318
T. Abdeljawad / Chaos, Solitons and Fractals 126 (2019) 315–324
and
(AB ∇b−(θ ,μ) η )(t ) =
1−θ θ (∇b−(1−μ) η )(t ) + (∇ −(1−μ+ θ ) η )(t ). B (θ ) B (θ ) b (28)
Remark 2. Note that if in the above definition μ tends to −(θ ,1 ) η )(t ) = (AB ∇ −θ η )(t ) and (AB ∇ −(θ ,1 ) η )(t ) = 1, we have (AB a ∇ a b
(AB ∇b−θ η )(t ).
B (θ ) B ( θ ) γ +σ E (λ, t − a ), ∇t [Eθγ,+μσ+ν (λ, t − a )] = 1−θ 1 − θ θ ,ν +μ−1
(29)
∇ θ ,μ,γ [Eθσ,ν (λ, t − a )]
(s ) = (Za η )(s ) and λ = where η we have
Za
AB a
∇ −θ η (t ) (s ) =
=
B ( θ ) γ +σ E (λ, t − a ). 1 − θ θ ,ν +μ−1
(30)
If we apply the Q−operator on (29) and (30), we get the following right versions, respectively. θ ,μ,γ
B ( θ ) γ +σ [Eθσ,ν (λ, b − t )] = E (λ, b − t ), 1 − θ θ ,ν +μ−1
θ ,μ,γ
B ( θ ) γ +σ [Eθσ,ν (λ, b − t )] = E (λ, b − t ). 1 − θ θ ,ν +μ−1
∇b
(31)
and
∇b
follows
by
(AB ∇b−θ η )(t ). The case of finding explicit formulas for the left and right AB fractional integrals of order θ , μ, γ when γ = 1 has not been considered above. However, it is possible to formulate the particular cases γ = 2, 3, 4 . . . by the help of Laplace transforms. In fact, the AB fractional integrals of order 0 < θ ≤ 1, μ > 0, γ = 1, 2, . . . are given by
( ∇
−(θ ,μ,γ )
η )(t ) =
γ
γ i
i=0
θi
B(θ )(1 − θ )i−1
(a ∇
−(θ i+1−μ )
η )(t ). (32)
and
(AB ∇b−(θ ,μ,γ ) η )(t ) =
γ
γ
θi
B(θ )(1 − θ )i−1
(∇b−(θ i+1−μ) η )(t ). (33)
On the light of Remark 3 and the generalized binomial theorem we have the following definition for the AB fractional sums. Definition 8. Assume η(t) is defined on Na and a ≡ b (mod 1). Then, the AB fractional sums of order 0 < θ ≤ 1, μ > 0, γ > 0 are given by
( ∇
−(θ ,μ,γ )
η )(t ) =
∞
γ i
i=0
θi
B(θ )(1 − θ )
( ∇ −(θ i+1−μ) η )(t ). i−1 a (34)
and
(AB ∇b−(θ ,μ,γ ) η )(t ) =
∞ i=0
. In particular, when μ = γ = 1
Lemma
5
and
(37) the
binomial
B (θ ) f (a )Eθ (λ, t − a ) 1−θ
(38)
γ i
θi
B(θ )(1 − θ )
i−1
(∇b−(θ i+1−μ) η )(t ).
(35)
B (θ ) f (b)Eθ (λ, b − t ) 1−θ
(39)
The following theorem is a generalization to (38) and (39) above. Theorem 2. For any 0 < θ < 1/2, μ > 0, γ ∈ R, and χ defined on Na with b ≡ a (mod 1), we have θ ,μ,γ χ )(t ) = (ABR ∇ θ ,μ,γ χ )(t ) − B(θ ) χ (a )E γ (λ, t − a ). • (ABC a ∇ a 1−θ θ ,μ ,γ
• (ABC ∇b
χ )(t ) = (ABR ∇bθ χ )(t ) − −θ 1−θ
θ ,μ γ B (θ ) χ ( b ) E ( λ, b − t ). 1−θ θ ,μ
.
Proof. From the relations, which are deducted from Lemma 7 and by the help of the discrete convolution theorem stated in Proposition 1, θ ,μ,γ χ )(t )} (s ) = Za {(ABR a ∇
B ( θ ) 1 −μ
(s )[1 − λs−θ ]−γ , s χ 1−θ
(40)
and θ ,μ,γ χ )(t )} (s ) = Za {(ABC a ∇
B ( θ ) 1 −μ
(s )[1 − λs−θ ]−γ s χ 1−θ B (θ ) − χ (a )s−μ [1 − λs−θ ]−γ , 1−θ
(41)
we deduce that θ ,μ,γ χ )(t )} (s ) = Z { (ABR Za {(ABC a a ∇
−
i
i=0
AB a
(36)
According to [12] we recall the following
Above λ =
Remark 3. Note that if μ tends to 1 in Definition 7, we −(θ ,1 ) η )(t ) = (AB ∇ −θ η )(t ) have (AB and (AB ∇b−(θ ,1) η )(t ) = a ∇ a
AB a
−θ 1−θ
1−θ θ −θ
(s ). + s η B (θ ) B (θ )
(ABC ∇bθ f )(t ) = (ABR ∇bθ f )(t ) −
s=1
ABC
γ 1−θ
(s ), ∇ −(θ ,μ,γ ) η (t ) (s ) = 1 − λs−θ s−(1−μ) η B (θ )
and the b− right fractional version is
t B (θ ) γ = E (λ, t − ρ (s ))∇s [Eθσ,ν (λ, s − a )] θ ,μ 1−θ
ABR
AB a
θ ABR θ (ABC a ∇ f )(t ) = (a ∇ f )(t ) −
and ABC a
Za
Proof. The proof theorem.
∇ θ ,μ,γ [Eθσ,ν (λ, t − a )]
=
Example 1. By the help of Lemma 8 and the property (10) we conclude that ABR a
Theorem 1 (Nabla discrete Laplace transforms for the AB fractional sums). Let θ ∈ (0, 1/2 ), μ, γ ∈ C and η a discrete function defined on Na . Then,
a
∇ θ ,μ,γ χ )(t )}(s )
B (θ ) χ (a )s−μ [1 − λs−θ ]−γ . 1−θ
(42)
Applying the inverse Laplace to (42) finishes the first part. The second part can be proved by the help of the first part as well as the Q−operator action. θ ,μ,γ 1 )(t ) = Remark 4. From Theorem 2 we conclude that (ABR a ∇ B (θ ) γ E (λ, t 1−θ θ ,μ
θ ,μ ,γ
− a ) and (ABR ∇b
θ ∈ (0, 1/2), and μ, γ ∈ C.
γ 1 )(t ) = B1(−θθ) E (λ, b − t ) for any θ ,μ
4. Application: the ABC type fractional initial value problems 4.1. The action of the left and right fractional sums on the left and right ABC fractional differences Using Theorem 2, (27), (28) and the identity (12), we can conclude that following important tool for the particular case γ = 1. Proposition 2. For 0 < θ < 1, μ > 0, γ = 1, one has −(θ ,μ,1 ) (AB a ∇
ABC a
∇ θ ,μ,1 η )(t ) = η (t ) − η (a ).
T. Abdeljawad / Chaos, Solitons and Fractals 126 (2019) 315–324
Similarly,
( ∇ AB
−(θ ,μ,1 ) ABC b
θ ,μ,1
∇b
η )(t ) = η (t ) − η (b)
= η (t ) − η (a )
(43)
i=0
Proof. Using Theorem 2, (27) and the identity (12), we have −(θ ,μ,1 ) (AB a ∇
= η (t ) − η (a )
∇ θ ,μ,1 η )(t ) B (θ ) −(θ ,μ,1 ) = η (t ) − Eθ ,μ (λ, t − a ) η (a ) AB a ∇ 1−θ = η (t ) − η (a )Eθ ,1 (λ, t − a ) θ − η (a )Eθ ,θ +1 (λ, t − a ) 1−θ = η (t ) − η (a ). ABC a
λ=
= η (t ) − η (a ) = η (t ) − η (a )
⎢ ⎢ ⎣
×⎢ 1 +
−θ . 1−θ
−(θ ,μ,2 ) (AB a ∇
γ = 2, and λ =
−θ 1−θ
, we
m (−1 )i γi (γ )m−i λm (t − a )θ m (θ m + 1 ) i=0 ( m − i )! m=0 ∞
⎤ i γ
m (−1 ) i (γ )m−i ⎥ λm (t − a )θ m ⎥ ⎥ ( θ m + 1) ( m − i )! ⎦ m=1 i=0 ∞
= η (t ) − η (a ).
(50)
Finally, the proof of (49) follows by applying the Q−operator to (48).
(AB ∇b−(θ ,μ,γ )
ABC
∇bθ ,μ,γ η )(t ) = η (t ) − η (b).
(51)
ABC a
∇ θ ,μ,2 η )(t ) = η (t ) − η (a ).
(45)
ABC
∇bθ ,μ,2 η )(t ) = η (t ) − η (b).
(46)
and
(AB ∇b−(θ ,μ,2)
k!(θ (k + i ) + 1 )
=0 by 47
(44)
In what follows we prove Proposition 2 for other values of γ .
μ > 0,
γ ∞ ∞ (−1 )i λi+k (t − a )θ (k+i) (γ )k i
⎡
Proposition 3. For 0 < θ < 1, have
(−1 )i λi Eθγ,θ i+1 (λ, t − a )
i=0 k=0
Above we have made use of the identity
Eθ ,1 (λ, t − a ) − λEθ ,θ +1 (λ, t − a ) = 1,
∞
319
Remark 5. Regarding the orders of the fractional differences and the positivity of their real parts when be complex, we invite the reader to notice Remark 3 in [46].
Proof. Using Theorem 2, (34) with γ = 2, and the identity (12), we
4.2. Semigroup properties for the AB fractional sums
−(θ ,μ,2 ) ABC θ ,μ,2 (AB η )(t ) a ∇ a ∇ = η (t ) − η (a )[Eθ2,1 (λ, t − a ) − 2λEθ2,θ +1 (λ, t − a )
The following identity which was used in [46] to prove a semigroup property is also essential to proceed for the discrete case in this article. m
+λ2 Eθ2,2θ +1 (λ, t − a )
= η (t ) − η (a )
⎡
i=0
⎤
AB a
=0
= η (t ) − η (a ).
AB
The following identity, which was proved and used in [46], will help in generalizing Propositions 2 and 3 for any arbitrary γ .
i γ
(γ )m−i i = 0, ∀ m = 1, 2, . . . and γ ∈ C. ( m − i )!
Theorem 3. For 0 < θ < 1, μ > 0, γ ∈ C, and λ = −(θ ,μ,γ ) (AB a ∇
−θ 1−θ
(47) , we have
ABC a
∇ θ ,μ,γ η )(t ) = η (t ) − η (a ).
(48)
ABC
∇bθ ,μ,γ η )(t ) = η (t ) − η (b).
(49)
and
(AB ∇b−(θ ,μ,γ )
Proof. Using Theorem 2, (34), and the identity (12), we have −(θ ,μ,γ ) ABC θ ,μ,γ (AB η )(t ) a ∇ a ∇ −(θ ,μ,γ ) ABR θ ,μ,γ = AB ∇ ∇ η (t ) a a B (θ ) −(θ ,μ,γ ) γ − E (λ, t − a ) η (a ) AB a ∇ θ ,μ 1−θ γ i ∞ θ B (θ ) i = η (t ) − η (a ) 1−θ ( 1 − θ )i−1 B(θ ) i=0
×∇ −(θ i+1−μ) E
γ +σ
.
m
(52)
∇ −(θ ,1,γ )
AB a
∇ −(θ ,1,σ ) η (t ) =
1−θ B (θ )
AB a
∇ −(θ ,1,γ +σ ) η (t ).
(53)
AB
∇b−(θ ,1,σ ) η (t ) =
1−θ B (θ )
AB
∇b−(θ ,1,γ +σ ) η (t ).
(54)
and
To prove (46) we use (45) and the Q−operator.
i=0
m−i
i
=
Theorem 4. Assume 0 < θ < 1/2, μ, β ∈ C, σ , γ ∈ C and η(t) be defined on Na , b ≡ a (mod 1 ). If μ = β = 1, then we have
∞ ⎢ (t − a )θ k (2 )k 2(2 )k−1 (2 )k−2 ⎥ ×⎢ 1 + λk ( − + )⎥ ⎣ (θ k + 1 ) k! ( k − 1 )! ( k − 2 )! ⎦ k=2
m (−1 )
γ σ
γ (λ, t − a ) θ ,μ
∇b−(θ ,1,γ )
Proof. On the light of Definition 8 and by applying Lemma 2 one can have AB a
∇ −(θ ,μ,γ )
∇ −(θ ,β ,σ ) η (t ) i γ ∞ θ i −(θ i+1−μ ) AB −(θ ,β ,σ ) = η (t ) a∇ a ∇ B(θ )(1 − θ )i−1 i=0 γ i σ k ∞ ∞ θ θ k i = B(θ )(1 − θ )i−1 B(θ )(1 − θ )k−1 i=0 k=0 AB a
∇ −(θ (k+i)+1−μ+1−β ) η (t ) γ σ i+k ∞ ∞ θ 1 − θ k i = B (θ ) B(θ )(1 − θ )i+k−1 i=0 k=0 ×
a
×a ∇ −(θ (k+i )+1−β +1−μ) η (t ) ∞ 1−θ θm = B (θ ) B(θ )(1 − θ )m m=0
×a ∇ −(θ m+1−μ+1−β ) η (t )
m i=0
γ σ i
m−i (55)
In order to prove our claim (53) we just make use of the identity (52) by substituting μ = β = 1 above in (55). As always, the
320
T. Abdeljawad / Chaos, Solitons and Fractals 126 (2019) 315–324
Q−operator is the tool to prove the right case by being applied to the left case. Depending on the proof of Theorem 4 we can state the following version of a semigroup property. Theorem 5 (A semigroup property). Assume 0 < θ < 1/2, μ, β ∈ C, σ , γ ∈ C and η(t) be defined on Na , b ≡ a (mod 1 ). Then we have AB a
∇ −(θ ,μ,γ )
AB a
∇ −(θ ,β ,σ ) η (t ) =
1−θ B (θ )
AB a
Above we have used that E
for 0 < θ < 1/2 by the ratio test.
θ ,μ,γ 1 )(t ) = (ABR a ∇
=
∇b−(θ ,μ,γ )
AB
∇b−(θ ,β ,σ ) η (t ) =
1−θ B (θ )
AB
∇b−(θ ,μ+β ,γ +σ ) ∇b−1 η (t ).
∞ B (θ ) 1−θ
λk (γ )k (t − a )ν +θ k+μ−1 k! (ν + θ k + μ ) k=0
B (θ ) γ E (λ, t − a ). 1 − θ θ ,μ
More generally, for μ > 0 we have
and AB
Remark 6. A particular case of (59) is when η (t ) = 1 = (t − a )0 . Indeed, by the help of the identity (4), we have
∇ −(θ ,μ+β ,γ +σ ) a ∇ −1 η (t ). (56)
λk ( γ ) k γ (λ, 1 ) = ∞ , which is finite k=0 k! θ ,μ
ABR a
∇ θ ,μ,γ (t − a )ν =
∞ B(θ ) λk (γ )k (ν + 1 ) (t − a )ν +θ k+μ−1 . 1−θ (k! )(μ + θ k + ν ) k=0
(60)
(57) 4.3. The ABC fractional difference systems
From Remark 4 we that that θ ,μ,γ 1 )(t ) = (ABR a ∇
B (θ ) γ E (λ, t − a ). 1 − θ θ ,μ
(58)
The following representation for ABR fractional differences can be used to provide an alternative proof for the identity (58) : Theorem 6. For any 0 < θ < we have θ ,μ,γ η )(t ) = (ABR a ∇
∞ B (θ ) 1−θ
1 2,
μ, γ ∈ C and η(t) defined on Na ,
λk (γ )k
k=0
k!
(a ∇ −(θ k+μ−1) η )(t ).
(59)
Proof. From definition and the identity
∇t
t
G(t, s ) = G(t , t ) +
s=a+1
t−1
∇t G(t, s ),
s=a+1
we have θ ,μ,γ η )(t ) (ABR a ∇ t−1 B (θ ) γ γ = E (λ, 1 )η (t ) + ∇t Eθ ,μ (λ, t − ρ (s ))η (s ) θ ,μ 1−θ s=a+1 t−1 ∞ B (θ ) λk (t − ρ (s ))θ k+μ−1 (γ )k γ = E (λ, 1 )η (t ) + ∇t η (s ) θ ,μ 1−θ (θ k + μ )k! s=a+1 k=0 ∞ B (θ ) λk ( γ ) k γ = E (λ, 1 )η (t ) + θ ,μ 1−θ (θ k + μ − 1 )k! k=0
×
t−1
(t − ρ (s ))θ k+μ−2 η (s )
∞ B ( θ ) λk ( γ ) k B (θ ) γ E (λ, 1 )η (t ) + 1 − θ θ ,μ 1−θ k!
1 × (θ k + μ − 1 )
η (s )
s=a+1
∞ B (θ ) γ B ( θ ) λk ( γ ) k E (λ, 1 )η (t ) + 1 − θ θ ,μ 1−θ k! k=0
∞ B ( θ ) λk ( γ ) k − η (t ) 1−θ k! k=0
=
∞ B ( θ ) λk ( γ ) k 1−θ k! k=0
a
∇ −(θ k+μ−1) η (t ).
0<θ <
1 . 2
(61)
θ (ABC a ∇ η )(t ) = r η (t ),
y ( a ) = a0 ,
0<θ <
1 , 2
(62)
has the trivial solution unless θ = 1. Therefore, linearity with constant coefficient results in the trivial solution. The next theorem outlines the existence and uniqueness of solution for the ABC fractional difference systems of order (θ , μ, γ ), where the case μ = 1, without the need of the restrictive condition f (a, η (a )) = 0. Theorem 7. Assume f : Na,b × R → R, η : Na,b → R and consider the system θ ,μ,γ η (t )(t ) = f (t, η (t )), t ∈ N (ABC η (a ) = c a,b = {a, a + 1, . . . b} a ∇
(63) such that b ≡ a (mod 1 ), f (a, η (a )) = 0 if μ = 1,
A
∞ i=0
γ
θ i (b − a )θ i+μ−1 < 1, B(θ )(1 − θ )i−1 (2 + θ i − μ ) i
AB a
(64) A > 0. Then, the system
∇ −(θ ,μ,γ ) f (t , η (t )).
(65)
Proof. By Theorem 2, Theorem 3, the identity
(t − ρ (s ))θ k+μ−2 =
−(θ k + μ − 1 )η (t ) =
η ( a ) = a0 ,
In order to verify the initial condition they requested the assumption that f (a, η (a )) = 0. In particular, they showed that the system:
η (t ) = c +
k=0
t
θ (ABC a ∇ η )(t ) = f (t , η (t )),
and | f (t, η1 ) − f (t, η2 )| ≤ A|η1 − η2 )|, (63) has a unique solution of the form
s=a+1
=
In [23], the authors proved an existence and uniqueness theorem for the solution of the system
a
∇ θ k+μ−1 η (t )
ABR a
∇ (θ ,μ,γ )
AB a
∇ −(θ ,μ,γ ) η (t ) = η (t ),
and taking into account that f (a, y(a )) = 0 in the case μ = 1, it can be shown that y(t) satisfies the system (63) if and only if it −(θ ,μ,γ ) h )(a ) = satisfies (65). In context, one has to note that (AB a ∇ −(θ ,1,γ ) h )(a ) = 0 with the need of h (a ) = 0. 0 for μ = 1 and (AB a ∇ Let ϒ = {η : maxt∈Na,b |η (t )| < ∞} be the Banach space endowed with the norm η = maxt∈Na,b |η (t )|. On ϒ , define the linear operator
( η )(t ) = c +
AB a
∇ −(θ ,μ,γ ) f (t, η (t )).
Then, for arbitrary η1 , η2 ∈ X and t ∈ Na,b , we have, by assumption, that
T. Abdeljawad / Chaos, Solitons and Fractals 126 (2019) 315–324
∇ −θ [ f (t, η1 (t )) − f (t, η2 (t ))]| γ i ∞ θ (b − a )θ i+μ−1 i ≤A B(θ )(1 − θ )i−1 (2 + θ i − μ ) i=0
|( η1 )(t ) − ( η2 )(t )| = |
AB a
η1 − η2 .
×
Now, we solve the below system by following the method of successive approximation outlined in Theorem 7. θ ,μ,γ η )(t ) = r η (t ) + g(t ), (ABC a ∇
(66)
and, hence, is a contraction. Now by Banach Contraction Principle, there exists a unique η ∈ X such that η = η and hence the proof is complete. The solution η(t) can be then obtained by means of the successive process
ηm (t ) = η (a ) +
AB a
∇ −(θ ,μ,γ ) f (t , ηm−1 (t )),
for m = 1, 2, . . . , with η0 (t ) = c.
0<θ
Remark 7. It is to be noted that Theorem 2 in [23] is a particular case of Theorem 7 above with μ = γ = 1 but with the need of the assumption that f (a, η (a )) = 0.
ηm (t ) = a0 + r
μ, γ ∈ C, t ∈ Na+1 .
(67)
η (a ) + (1 − λs−θ )γ
g( s )
B ( θ ) 1 −μ s [1 1 −θ
− λs−θ ]−γ − r
(s )}(t ) η (t ) = Za−1 {η
(68)
, (69)
(70)
is our nontrivial solution for μ = 1. The case μ = γ = 1, g(t ) = 0 ends with the trivial solution following the discussion before Theorem 7.
(s ) in To find the above discrete Laplace inverse, we expand η the form
(s ) = η
+
g( s )
s ∞
rj
j=0
rj
j=0
1−θ B (θ )
1−θ B (θ )
ηm (t ) = a0 + a0
η (t ) = η (a )
∞
rj
j=0
+ g(t ) ∗
∞
r
j=0
=
η (a ) + η (a )
+ g(t ) ∗
∞ j=0
rj
r
i−1
∞
r
1−θ B (θ )
r
∞
r
i−1
1−θ B (θ )
1−θ B (θ )
j+1 E
(76)
i−1
i−1
−(θ ,iμ,iγ ) −(i−1 ) (AB 1 )(t ) a∇ a ∇
−(θ ,iμ,iγ ) −(i−1 ) (AB g)(t ). a∇ a ∇
(77)
η ( a ) = a0 , (78)
will have the explicit solution
η (t ) = η (a )
∞
+ g(t ) ∗
r
j
∞
1−θ B (θ )
r
j
=
η (a ) + η (a )
∞
j
+ g(t ) ∗
∞
r
j
−γ j
Eθ , j (1−μ)+1 (λ, t − a )
1−θ B (θ )
rj
j=1
−γ j (λ, t − a ) θ , j (1−μ )+1
1−θ B (θ )
−(θ ,iμ,iγ ) −(i−1 ) (AB g)(t ) a∇ a ∇
θ ,μ,γ η )(t ) = r η (t ) + g(t ), (ABC a D 0 < θ ≤ 1, μ, γ ∈ C, t ≥ a,
j=0
E
j
j=0
j+1
i−1
−(θ ,iμ,iγ ) −(i−1 ) (AB 1 )(t ) a∇ a ∇
Remark 8. If by referring to Section D in [46], in the case of Laplace method, we follow similar steps as above, then the continuous system
s−(1−μ)( j+1) [1 − λs−θ ]γ ( j+1) , |x| < 1, (71)
E
i−1
1−θ B (θ )
1−θ B (θ )
i
j=0
j
(75)
−(θ ,iμ,iγ ) ∇ −(i−1 ) 1 )(t ) can be calculated by makThe terms (AB a a ∇ ing use of Lemma 1. Also, it is to be noted that η (a ) = a0 . Moreover, it is of interest to compare with the obtained solution by means of the discrete Laplace transforms mentioned above.
.s−(1−μ) j [1 − λs−θ ]γ j
j=1
ri
i=1
j
1−θ B (θ )
∞
m
j+1
1−θ B (θ )
j
m
η (t ) = a0 + a0
where x = r B1(−θθ) sμ−1 [1 − λs−θ ]γ and 0 < μ < 1. If we make use of Lemma 7, we reach at the solution
∇ −(θ ,μ,γ ) g(t ), m = 1, 2, . . . ,
If we proceed inductively, and use the semigroup property (56) in Theorem 5, we reach the formula
i=1
and hence
AB a
where η0 (t ) = a0 . For m = 1 we have
+
(s ) = Za {η (t )}(s ),
η g(s ) = Za {g(t )}(s ). From which it follows that
∞ η (a )
∇ −(θ ,μ,γ ) ηm−1 (t ) +
If letting m → ∞ and on the light of the condition (64) satisfied by b, we obtain the solution representation
η ( a ) = a0 ,
B (θ ) B ( θ ) 1 −μ
(s )[1 − λs−θ ]−γ − s η η (a )s−μ [1 − λs−θ ]−γ 1−θ 1−θ
(s ) +
= rη g( s ) ,
s−
AB a
i=1
r B1(−θθ) sμ
(73)
(74)
+
If we apply the discrete Laplace transform Za to the system (67) above we get
(s ) = η
μ = 1, t ∈ Na+1 .
i=1
Consider the system
1 , 2
η ( a ) = a0 ,
Indeed, we have
4.4. The ABC fractional linear difference system with constant coefficient
0<θ <
1 < , γ ∈ C, 2
−(θ ,μ,γ ) −(θ ,μ,γ ) η1 (t ) = a0 + ra0 (AB 1 )(t ) + (AB g)(t ). a ∇ a ∇
θ ,μ,γ η )(t ) = r η (t ) + g(t ), (ABC a ∇
321
j+1
1−θ B (θ )
1−θ B (θ )
−γ ( j+1 )
Eθ ,( j+1)(1−μ) (λ, t − a )
j
j+1
−γ j
Eθ , j (1−μ)+1 (λ, t − a ) −γ ( j+1 )
Eθ ,( j+1)(1−μ) (λ, t − a ),
(79)
where the continuous counterpart of Lemma 7 will be used instead [49]. It is of interest to compare between the solution (79) and the solution given by Eq. (66) in [46], when the successive approximation is applied.
−γ ( j+1 ) (λ, t − a ) θ ,( j+1 )(1−μ )
5. The iterated fractional sums when γ = 1
j E
−γ j
θ , j (1−μ )+1
(λ, t − a )
−γ ( j+1 ) (λ, t − a ). θ ,( j+1 )(1−μ )
On the light of Definition 8 or Remark 3 for γ = 1 consider the fractional sum operator
(72)
AB a
∇ −(θ ,μ,1) η (t ) =
1−θ B (θ )
a
∇ −(1−μ) η (t )
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T. Abdeljawad / Chaos, Solitons and Fractals 126 (2019) 315–324
+
θ −(θ +1−μ ) η (t ), θ ∈ (0, 1/2 ), μ ∈ C. a∇ B (θ )
If we iterate
[
AB ∇ −(θ ,μ,1 ) η (t ) a
∇ −(θ ,μ,1) ]n η (t ) =
AB a
n
n
k
=
(1 − θ )
n−k
θ
k a
n
n
k
k=0
∇ −(kθ +1−μ) η (t )
(1 − θ )n−k θ k −(kθ +1−μ ) η (t ). (81) a∇ B ( θ )n
−(θ ,μ,1,n ) η (t ). We write [ = AB a ∇ By means of (81) and the binomial theorem we formulate the following definition.
Definition 9. For θ ∈ (0, 1/2) and σ , μ ∈ R define the following iterated sum-differ operators: AB a
∇ −(θ ,μ,1),σ η (t ) =
∞
k
k=0
∇
−(θ ,μ,1 ),−σ
η (t ) =
(1 − θ )σ −k θ k −(kθ +1−μ ) η (t ), (82) a∇ B ( θ )σ
∞
− σ k
a
∇
−(kθ +1−μ )
η (t ).
(83)
AB a
−(θ ,μ,1 ),1 ∇ −(θ ,μ,1),0 η (t ) = a ∇ −(1−μ) η (t ), AB η (t ) a ∇ −(θ ,μ,1 ) = AB ∇ η ( t ) , a
AB a
∇ −(θ ,μ+β ,1),σ +ν a ∇ −1 η (t ).
(87)
This finishes the proof of (85). The proof for (86) is similar.
Remark 10. If we let μ = β = 1 in the proof of Theorem 8, then we obtain the following semigroup property AB a
∇ −(θ ,1,1),σ
AB a
∇ −(θ ,1,1),ν f (t ) =
AB a
∇ −(θ ,1,1),σ +ν η (t ),
(88)
∇b−(θ ,1,1),σ
AB
∇b−(θ ,1,1),ν f (t ) =
AB
∇b−(θ ,1,1),σ +ν η (t ).
(89)
Remark 11. From (85) with σ = ν = 1 we have
∇ −(θ ,μ,1),1
AB a
∇ −(θ ,β ,1),1 η (t ) =
AB a
∇ −(θ ,μ+β ,1),2 a ∇ −1 η (t ), (90)
∇ −(θ ,μ,1),n η (t ) = [
AB a
θ ,μ,1 η )(t ) = (ABR a ∇
where
θ ,1,1 η )(t ) = (ABR a ∇
AB a
AB a
∇
−(θ ,μ,1 ),−1
However, for
∇ −(θ ,1,1),−1 η (t ),
which is according to [13] the same as tually, we have
η (t ) = (
ABR a
AB ∇ (−θ ,−1 ) η (t ). a
∇ θ ,2−μ,1 η )(t )
Ac-
(84)
Theorem 8. Let θ ∈ [0, 1] and μ, β , σ , ν ∈ R and η be a function defined on Na . Then for any t ∈ Na we have
∇ −(θ ,μ,1),σ
AB a
∇ −(θ ,β ,1),ν η (t ) =
AB a
∇ −(θ ,μ+β ,1),σ +ν a ∇ −1 η (t ), (85)
and for any t ∈ b N we have −(θ ,μ,1 ),σ AB b
∇
−(θ ,β ,1,ν ) b
f (t ) =
∇
−(θ ,μ+β ,1 ),σ +ν b
∇ η (t ). −1 b
Proof. By the help of the identity (52) and that the left fractional sums have the semigroup property, we have −(θ ,μ,1 ),σ AB a
∇ σ ∞
k=0
k
AB a
∇ −(θ ,μ+β ,2) a ∇ −1 η (t ). conclude that possible to ver-
∇
(−θ ,γ )
η (t ) =
γ ∞ (1 − θ )γ −k θ k k B ( θ )γ
k=0
a
∇ −kθ η (t )
(92)
The following result can be observed from definitions. Theorem 9 (The relation between the iterated AB sumdifferences discussed in [13] and the iterated ones here). For θ ∈ (0, 1/2 ), μ, γ ∈ C and η define on Na we have AB a
∇ −(θ ,μ,1),γ η (t ) =
∇ (−θ ,γ ) a ∇ −(1−μ) η (t ).
(93)
∇ (−θ ,γ ) η (t ).
(94)
AB a
In particular, we have AB a
∇ −(θ ,1,1),γ η (t ) =
AB a
Example 2. Consider the fractional difference-sum equation of the form ABR 0
∇ θ ,2−μ,1 η (t ) = −A 0 ∇ −(1−μ) η (t ) + b(t ), A > 0,
(95)
where, AB
(86)
AB a
1−θ B (θ )
ify this directly.
AB a
k=0
which is different from μ = 1 we have
∇ −(θ ,β ,1) η (t ) =
From [13] recall that the AB iterated sum-differences obtained by iterating fractional sums correspond to fractional differences with discrete Mitag–Leffler kernels are given by
∞ B (θ ) k λ (a ∇ −(θ k+μ−1) η )(t ), 1−θ
AB ∇ −(θ ,μ,1 ),−1 η (t ). a
AB a
(91)
∇ −(θ ,μ,1) ]n η (t ).
∞ AB ∇ −(θ ,μ,1 ),−1 η (t ) = B (θ ) k −(θ k+1−μ ) η (t ), a k=0 λ a ∇ 1−θ −1 −θ λ = 1−θ and we have used that k = (−1 )k .
∇ −(θ ,μ,1)
AB ∇ −(θ ,μ,1 ),1 f (t ) = AB ∇ −(θ ,μ,1 ) f (t ), we Since, a a AB ∇ −(θ ,μ,1 ),2 f (t ) = 1−θ AB ∇ −(θ ,μ,2 ) f (t ). It is also a B (θ ) a
3. If we let γ = 1 in Theorem 6, we notice that
=
AB a
AB a
and
∇
=
and from (56) with σ = γ = 1 we have
1.
AB
a
AB a
Remark 9 (Particular cases).
AB a
∇ −(mθ +1−μ+1−β ) η (t ) σ +μ m ∞ θ (1 − θ )σ +ν −m m −(mθ +1−(μ+β )) −1 = η (t ) a∇ a∇ B ( θ )σ +ν m=0
×
AB
B ( θ )σ θ k
( 1 − θ )σ +k
k=0
2.
a
and for any t ∈ b N we have
and AB a
B ( θ )σ +ν
∇ −((k+i)θ +1−μ+1−β ) η (t ) ∞ m θ m (1 − θ )σ +ν −m ν σ = B ( θ )σ +ν i m−i m=0 i=0
×
AB ∇ −(θ ,μ,1 ) ]n η (t ) a
σ
=
σ ν ∞ ∞ ( 1 − θ )σ +ν − ( k+i ) θ i+k k i k=0 i=0
n−times we have
B(θ )n−k B(θ )k
k=0
(80)
−(θ ,β ,1 ),ν
∇ η (t ) σ −k k (1 − θ ) θ −(kθ +1−μ ) −(θ ,β ,1 ),ν η (t ) a∇ a∇ B ( θ )σ
b(t ) =
∞
bs t θ s .
s=0
From (84) and Theorem 9, Eq. (95) is equivalent to AB 0
∇ (−θ ,−1) 0 ∇ −(1−μ) η (t ) = −A 0 ∇ −(1−μ) η (t ) + b(t ). ∇ −(1−μ) η (t )
Let u(t ) = 0 Eq. (96) becomes AB 0
and hence η (t ) =
∇ (−θ ,−1) u(t ) = −Au(t ) + b(t ).
0
∇ 1−μ u(t )
(96) and
(97)
T. Abdeljawad / Chaos, Solitons and Fractals 126 (2019) 315–324
β = θ − n. Then β ∈ (0, 1] and by means of Definitions 6 and 8 we
From Remark—in [13], Eq. (97) has the solution b(t )
u(t ) =
A+
B (θ ) − 1−θ
where c0 = mined from cm =
∞
A+
m=1
b0 A+( B1(−θθ) )
bm
B (θ ) − 1−θ
t mθ
∞ k=1
m
cm−k (−1 )k θ k B(θ )( (m − k )θ + 1 )
(1 − θ )1+k (mθ + 1 ) A +
k=1
, B (θ ) 1−θ
and the coefficients ci , i ∈ N, can be deter-
−1
θ k B(θ )((m − k )θ + 1 ) . (1 − θ )1+k (mθ + 1 )(mθ + 1 ) A + B1(−θθ) cm−k
k
(98)
η (t ) =
d (t ) A+ ×
B (θ ) − 1 −θ
mθ
(mθ + 1 )t (mθ + μ ) m=1
m cm−k (−1 )k θ k B(θ )((m − k )θ + 1 ) k=1
where, d (t ) =
(1 − θ )1+k (mθ + 1 ) A +
B (θ ) 1 −θ
,
∞
bs (θ s+1 ) θ s+μ−1 . s=0 (θ s+μ ) t
Theorem 10 (Nabla discrete Laplace transforms for the iterated AB sum-differences). Let θ ∈ (0, 1/2 ), μ, γ ∈ C and η be defined on Na . Then, we have
Za
AB a
∇ (−θ ,μ,1),σ η (t ) (s ) =
1−θ θ −θ + s B (θ ) B (θ )
σ
define::
(ABC ∇bθ ,μ,γ η )(t ) = (ABC ∇bβ ,μ,γ ( n η )(t ).
(s ), s − ( 1 −μ ) η (99)
and in the right Riemann–Liouville sense has the following form:
(ABR ∇bθ ,μ,γ η )(t ) = (ABRR ∇bβ ,μ,γ ( n η )(t ). (AB ∇b−(θ ,μ,γ ) η )(t ) = (∇b−n Above,
n
means
AB
∇b−(β ,μ,γ ) η )(t ).
Za
AB a
∇ (−θ ,μ,1),σ η (s ) =
Theorem 11 (The action of the AB-sum on the ABC-difference in higher order case). Assume η is defined on a proper domain and θ , μ, γ , β and n are as above, then −(θ ,μ,γ ) (AB a ∇
ABC a
∇ θ ,μ,γ η )(t ) = η (t ) −
=
1−θ θ −θ + s B (θ ) B (θ )
σ ∞ (1 − θ )σ −k θ k k
σ
n j=0
∇ j η (a ) j!
(t − a ) j
(108)
and ABC
∇bθ ,μ,γ η )(t ) = η (t ) −
n (−1 ) j j f (b) (b − t ) j j! j=0
B ( θ )σ
k=0
(107)
(−1 )n n .
Proof. By Lemma 5 we have
(106)
The associated fractional integral by
(AB ∇b−(θ ,μ,γ )
(s ) = (Za η )(s ). where η
(105)
Remark 12. In the ABC-case we requested the function η to be defined on Na−n in the left case and on b+n N in the right case and the ABC-fractional difference will start from a or end at b. Alternatively, we can fixed η to be defined on Na and the left ABCdifference will start at a + n, and in the right case we fixed η on b N and the ABC-fractional difference will end at b − n. Regarding the index θ , for the convergence purposes in the difference-operator case, we request θ so that β = n − θ ∈ (0, 1/2].
Therefore, ∞
323
(109)
(s ) s−(kθ +1−μ) η
(100)
(s ). s − ( 1 −μ ) η
(101)
Proof. The proof is similar to that in Propositions 2.1 and 2.2 in [25]. However, we make use of Theorem 3 in this article. Theorem 12 (The action of the AB-sum and the ABR-difference in higher order case on each other). Assume η is defined on a proper domain and θ , μ, γ , β and n are as above, then −(θ ,μ,γ ) (AB a ∇
ABR a
∇ θ ,μ,γ η )(t ) = η (t ) −
n−1 j=0
6. The higher order case
∇ j f (a ) j!
(t − a ) j
(110)
and In this section we give the definitions to the AB fractional differences and sums with discrete generalized Mittag–Leffler kernels. In above, the value of the index θ has been taken between 0 and 1. Below, we extend the definitions when θ is arbitrary such that
(θ ) > 0. In our extension we shall obey a similar approach to that in [24–27]. Definition 10. Let n < θ ≤ n + 1, μ, γ ∈ R and η be defined on Na−n in the Caputo case, otherwise it is enough on Na . Set β = θ − n. Then β ∈ (0, 1] and by means of Definitions 6 and 8 we define: θ ,μ,γ η )(t ) = (ABC ∇ β ,μ,γ ∇ n η )(t ), t ∈ N . (ABC a a ∇ a
AB a
∇ −(β ,μ,γ ) η )(t ), t ∈ Na .
∇bθ ,μ,γ η )(t ) = η (t ) −
n−1 (−1 ) j j f (b) (b − t ) j j! j=0
(111) On the other way, θ ,μ,γ (ABR a ∇
(
ABR
AB a
∇ −(θ ,μ,γ ) η )(t ) = η (t ),
∇bθ ,μ,γ AB ∇b−(θ ,μ,γ ) η )(t ) =
η (t ).
(112)
(102)
(103)
7. Conclusions
The associated fractional sum by −(θ ,μ,γ ) (AB η )(t ) = (a ∇ −n a ∇
ABR
Proof. The proof is similar to that in Propositions 2.1 and 2.2 in [25]. However, we make use of the definitions for the higher order case of generalized Mittag–Leffler kernel discrete operators in this article.
and the left Riemann–Liouville sense has the following form: θ ,μ,γ η )(t ) = (ABR ∇ β ,μ,γ ∇ n η )(t ), t ∈ N . (ABR a a ∇ a
(AB ∇b−(θ ,μ,γ )
(104)
For the right case we have the following: Definition 11. Let n < θ ≤ n + 1, μ, γ ∈ R and η be defined on b+n N in the Caputo case, otherwise it is enough on b N. Set
1. We defined the left and right ABR and ABC fractional differences and their corresponding fractional sums, where the kernel of the discrete fractional operators is a discrete generalized Mittag–Leffler function. The action of the ABfractional sum on the ABC-fractional differences has been considered.
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T. Abdeljawad / Chaos, Solitons and Fractals 126 (2019) 315–324
2. Semigroup type properties have been proven for the derived AB−fractional sums. Making use of such semigroup property we solved the ABC-linear case with constant coefficient. Indeed, by noting that the restriction f (a, ξ (a )) = 0 can be released for the case μ = 1 we have found a nontrivial solution for a nontrivial initial condition. 3. We have found the discrete Laplace transform for the defined and analyzed discrete operators and we used it to solve an example for the linear case. 4. The Laplace transforms have been used to obtain a solution of convolution type for both the continuous and discrete ABC fractional linear nonhomogeneous initial value problem with constant coefficient. The successive approximation method has been also discussed for the discrete system. 5. We obtained the iterated sum-differences by only iter−(θ ,μ,1 ) which consist of the addiating the sums AB a ∇ tion of two terms. However, iterating the sums of order −(θ , μ, γ ), γ = 2, 3, . . . is still without investigation. It will be complicated since we will use the binomial theorem applied to more than two terms. In this case and on the light of Remark 11 it will be of interest to find relations between fractional sums with three parameter-order and the iterated fractional sum-differences with the fourth parameter. 6. Alternative representations for the discrete ABR-operators have been formulated by using the expansion of the generalized discrete Mittag–Leffler kernels and some comparisons have been made to the defined iterative sum-differences. 7. We considered the extension to the higher order case. 8. We have considered the case of the Z time sale. The generalization to the hZ time scale can be done by following an approach as in [10]. Acknowledgment The author would like to thank Prince Sultan University for funding this work through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM) group number RGDES-2017-01-17. References [1] Podlubny I. Fractional differential equations. San Diego CA: Academic Press; 1999. [2] Samko G Kilbas AA, Marichev. Fractional integrals and derivatives: theory and applications. Yverdon: Gordon and Breach; 1993. [3] Kilbas A, Srivastava MH, Trujillo JJ. Theory and application of fractional differential equations, 204. North Holland Mathematics Studies; 2006. [4] Al-Mdallal QM, Omer ASA. Fractional-order Legendre-Collocation method for solving fractional initial value problems. Appl Math Comput 2018;321:74–84. [5] M Al-Mdallal Q. An efficient method for solving fractional Sturmliouville problems. Chaos Solitons Fractals 2009;40.1:183–9. [6] M Al-Mdall Q, I Syam M. An efficient method for solving non-linear singularly perturbed two points boundary-value problems of fractional order. Commun Nonlinear Sci Numer Simul 2012;17.6:2299–308. [7] Al-Mdallal QM, Hajji MA. A convergent algorithm for solving higher-order nonlinear fractional boundary value problems. Fract Calculus Appl Anal 2015;18.6:1423–40. [8] Al-Mdallal Q, Kashif AA, Khan I. Analytical solutions of fractional walters b fluid with applications. Complexity 2018;2018. [9] Bohner M, Peterson A. Advances in dynamic equations on time scales. Boston-Basel-Berlin: Birkhäuser; 2003. [10] Abdeljawad T. Different type kernel h–fractional differences and their fractional hsums. Chaos Solitons Fractals 2018;116:146–56. [11] Suwan I, Owies S, Abdeljawad T. Monotonicity results for h-discrete fractional operators and application. Adv Differ Equ 2018;2018:207. [12] Abdeljawad T, Baleanu D. Discrete fractional differences with nonsingular discrete Mittag-Leffler kernels. Adv Difference Equations 2016;2016:232. doi:10. 1186/s13662- 016- 0949- 5. [13] Abdeljawad T., Fernandez A.. On a new class of fractional differencesum operators based on discrete Atangana-Baleanu sums. 2019. arXiv:1901. 08268v1 [math.CA]. [14] Goodrich C, Peterson A. Discrete fractional calculus. Springer; 2015. [15] Abdeljawad T, Atici F. On the definitions of nabla fractional differences. Abstr Appl Anal 2012;2012:13. doi:10.1155/2012/406757. Article ID 406757
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