A Correlation Between Solutions of Uncertain Fractional Forward Difference Equations and Their Paths

Abstract

We consider the comparison theorems for the fractional forward h-difference equations in the context of discrete fractional calculus. Moreover, we consider the existence and uniqueness theorem for the uncertain fractional forward h-difference equations. After that the relations between the solutions for the uncertain fractional forward h-difference equations with symmetrical uncertain variables and their α-paths are established and verified using the comparison theorems and existence and uniqueness theorem. Finally, two examples are provided to illustrate the relationship between the solutions.

1. Introduction

The study of fractional calculus and fractional differential equations has received recent attention from both applied and theoretical disciplines. Indeed, it was observed that the use of them are very useful for modeling many problems in mathematical analysis, medical labs, engineering sciences, and integral inequalities (see for e.g., [1–14]). There is much interesting research on what is usually called integer-order difference equations (see for e.g., [15, 16]). Discrete fractional calculus and fractional difference equations represent a new branch of fractional calculus and fractional differential equations, respectively. Also, for scientists, they represent new areas that have, in their early stages, developed slowly. Some works are dedicated to boundary value problems, initial value problems, chaos, and stability for the fractional difference equations (see for e.g., [17–23]).

Besides the discrete fractional calculus, the uncertain fractional differential and difference equations have been introduced and investigated in order to model the continuous or discrete systems with memory effects and human uncertainty (see for e.g., [24–28]). In Lu and Zhu [27], the relations between uncertain fractional differential equations and the associated fractional differential equations have been created via comparison theorems for fractional differential equations of Caputo type in Lu and Zhu [26]. Lu et al. [28] presented analytic solutions to a type of special linear uncertain fractional difference equation (UFDE) by the Picard iteration method. Moreover, they provided an existence and uniqueness theorem for the solutions by applying the Banach contraction mapping theorem. After that, Mohammed [29] generalized the above work.

Nowadays, discrete fractional calculus shows incredible performance in the fields of physical and mathematical modeling. The motivation behind solving the fractional difference equations relies on fast investigation of the properties within models of fractional sum and difference operators (see for e.g., [20, 30–36]).

Motivated by the aforementioned results, we will try to create a link between uncertain fractional forward h-difference equations (UFFhDEs) and associated fractional forward h-difference equations (FFhDEs) in the sense of Riemann–Liouville fractional operators via the comparison theorems and existence and uniqueness theorem.

The rest of our article is designed as follows. In section 2, we presented the preliminary definitions and important features that are useful in the accomplishment of this study. In section 3, the comparison theorems of the fractional differences are pointed out. Inverse uncertainty distribution, the existence and uniqueness theorem, the relation between UFFhDEs and associated FFhDEs, and some related examples are pointed out in section 4. Finally, the future scope and concluding remarks are summarized in section 5.

2. Preliminaries

In what follows, we recall some results in discrete fractional calculus that has been developed in the last few years; for more details, we refer to references [24–28, 28, 29, 37, 38] and the related references therein.

Definition 2.1 ([39]). The forward difference operator on hℤ is defined by

and the backward difference operator on hℤ is defined by

For h = 1, we get the classical forward and backward difference operators Δψ(η) = ψ(η + 1) − ψ(η) and ∇ψ(η) = ψ(η) − ψ(η − h), respectively. The forward jumping operator on hℤ is σ(r) = r + h and the backward jumping operator is ρ(r) = r − h.

For a, b ∈ ℝ with and 0 < h ≤ 1, we use the notations ℕ_a,h = {a, a + h, a + 2h, ...}, _b,hℕ = {b, b − h, b − 2h, ...}.

Definition 2.2 ([39]). Let η, θ ∈ ℝ and 0 < h ≤ 1, the delta h-factorial of η is defined by

where we use the convention that division at a pole yields zero and θ is the falling delta h-factorial order of η. It is worth mentioning that is a function of η for given θ and h.

Definition 2.3 ([37, 38, 40]). Let f be defined on ℕ_a,h for the left case and _b,hℕ for the right case. Then, the left delta h-fractional sum of order θ > 0 is defined by

and the right delta h-fractional sum is defined by

Lemma 2.1 ([40]). Let θ, μ > 0, h > 0, and p be defined on ∈ ℕ_a,h. We then have

for all η ∈ ℕ_{a + (θ + μ)h,h}.

Lemma 2.2 ([40]). Let θ > 0 and ψ be defined on ℕ_a,hand_b,hℕ, respectively. Then the left and right delta h-fractional differences of order θ are defined by

where m = [θ] + 1.

Lemma 2.3 ([40]). Let ψ be defined on ℕ_a,h, then, for any θ > 0, we have

Lemma 2.4 ([40]). Let θ > 0, μ > 0, and h > 0, and we then have

Lemma 2.5 ([40]). Let θ ∈ ℝ and q be any positive integer, then

for η ∈ ℕ_{a + θ h,h}.

Lemma 2.6 ([38]). Suppose that, then we have

for each η ∈ ℕ_a+θh,h.

Lemma 2.7. Let ψ be defined on ℕ_a,hand m be a positive integer with 0 < m − 1 < μ ≤ m. The definition of the fractional h-difference (2.3) is then equivalent to

for η ∈ ℕ_a,h.

Motivated by the definition of nth order forward sum for uncertain sequence ξ_η, we define the θth order forward sum for uncertain sequence ξ_η as follows:

Definition 2.4. Let θ be a positive real number, a ∈ ℝ, and ξ_η be an uncertain sequence indexed by η ∈ ℕ_a,h. Then,

is called the θth order forward fractional sum of uncertain sequence ξ_η, where σ(r) = r + h.

Definition 2.5. The fractional Riemann–Liouville-like forward difference for uncertain sequence ξ_η is defined by

where θ > 0 and 0 ≤ n − 1 < μ ≤ n, n represents a positive integer.

3. The Comparison Theorems

Consider the following FFhDEs:

subject to the initial conditions

where denotes a fractional Riemann–Liouville forward h-difference with 0 ≤ n − 1 < θ ≤ n, g is a real-valued function defined on [0, ∞) × ℝ, η ∈ ℕ_0,h, and ψ_i ∈ ℝ for i = 0, 1, ..., n − 1.

Now, by applying the operator to Equation (3.1), then the initial value problem (3.1) and (3.2) is equivalent to the following fractional sum equation:

where we have used Lemma 2.1, Lemma 2.5, and the fact that .

First, a comparison theorem for Riemann–Liouville fractional h-difference equations with θ ∈ (0, 1] will be presented.

Theorem 3.1. Suppose g(η, ψ) and k(η, ψ) are two real-value functions defined on [0, ∞] × ℝ. Function k is Lipschitz continuous in y with Lipschitz constant L_kthat has. If ψ₁(η) and ψ₂(η) are, respectively, unique solutions of the following IVPs

and

• if g(η, ψ) ≤ k(η, ψ), then ψ₁(η) ≤ ψ₂(η) for each η ∈ ℕ_{(θ − 1)h,h},

• if g(η, ψ) > k(η, ψ), then ψ₁(η) > ψ₂(η) for each η ∈ ℕ_{θ h,h}.

Proof:(1) Assume that the condition ψ₁(η) ≤ ψ₂(η) is not valid; there thus exists η₀ ∈ ℕ_(θ−1)h,h such that ψ₁(η₀) > ψ₂(η₀). Let η₁ = min{η ∈ ℕ_(θ−1)h,h; ψ₁(η) > ψ₂(η)} and X(η) = ψ₁(η) − ψ₂(η). Then, we have

Considering the fractional sum equations equivalent to IVPs (3.4) and (3.5), we have

Subtracting these and then making use of h^θ > 0 for h > 0, θ ∈ (0, 1], and g(η, ψ) ≤ k(η, ψ), we get

This verifies that η₁ > θ h. From this and since η₁ ∈ ℕ_{(θ − 1)h,h}, we can write η₁ = (θ + ℓ)h, l = 1, 2, .... By Lemma 2.6, we then get

That is,

Now, by using the Lipschitz continuity of k in y, g(η, x) ≤ k(η, x), and (3.7), we get

Denoting , it follows that

This gives

Thus, Equation (3.8) becomes

We write r = v − 1 + i, i = 0, 1, ..., ℓ − 1 to obtain

Since θ ∈ (0, 1] and h^−θ−1 > 0, so

Considering , h^−θ > 0 and Equations (3.9)–(3.11), it follows that

This implies that X(η₁) ≤ 0, which contradicts with (3.6).

(2) By the same technique of (1), we assume that the condition ψ₁(η) > ψ₂(η) is not valid. There thus exists η₂ ∈ ℕ_θh,h, such that ψ₁(η₂) ≤ ψ₂(η₂). Let η₃ = min {η ∈ ℕ_θh,h; ψ₁(η) ≤ ψ₂(η)} and z(η) = ψ₂(η) − ψ₁(η). We then have

Considering the fractional sum equations equivalent to IVPs (3.4) and (3.5), h^θ > 0 and g(η, ψ) > k(η, ψ), we find ψ₁(θ h) > ψ₂(θ h). That is; η₃ > θ h. If we write η₃ = (θ + ℓ)h, l = 1, 2, ..., then, by Lemma 2.6, we get

or equivalently,

Now, by using the Lipschitz continuity of k in y, g(η, z) > k(η, z), and (3.13), we get

Denoting , it follows that

This gives

Equation (3.14) thus becomes

Similarly for θ ∈ (0, 1] and h^−θ−1 > 0, we can show that

Considering , h^−θ > 0 and Equations (3.15)–(3.17), it follows that

This implies that z(η₃) ≤ 0, which contradicts with (3.12). The proof of Theorem 3.1 is thus completed.     □

In the sequel, we will extend a comparison theorem for Riemann-Liouville fractional h-difference equations of the order θ with 0 ≤ n − 1 < θ ≤ n.

Theorem 3.2. Suppose g(η, ψ), and k(η, ψ) are two real-value functions defined on [0, ∞] × ℝ. Function k is Lipschitz continuous in y with a Lipschitz constant L_kthat has. If ψ₁(η) and ψ₂(η) are, respectively, unique solutions of the following IVPs

and

• if g(η, ψ) ≤ k(η, ψ), then ψ₁(η) ≤ ψ₂(η) for each η ∈ ℕ_(θ−n)h,h,

• if g(η, ψ) > k(η, ψ), then ψ₁(η) > ψ₂(η) for each.

Proof:(1) For μ = θ − n + 1 ∈ (0, 1] and η ∈ ℕ_0,h, we have By using Lemma 2.5, the IVPs (3.18) and (3.19) can be easily converted to the following IVPs, respectively,

and

Denote

and

for η ∈ ℕ_(θ−1)h,h. These give

Since g(η, ψ) ≤ k(η, ψ) and

it follows from (3.22) that for η ∈ ℕ_(θ−1)h,h. Then, by applying Theorem 3.1 for the above findings, we get ψ₁(η) ≤ ψ₂(η) for η ∈ ℕ_(θ−n)h,h. Hence, the proof of the first item is completed.

(2) Analogously, we can obtain the proof of this item, and thus our proof is completely done.     □

4. Inverse Uncertainty Distribution

In this section, we make a link between the solution for an UFFhDE and the solution for the associated FFhDE; we firstly define a symmetrical uncertain variable and α-path for an UFFhDE in view of Lu and Zhu [27]. After that, we state and verify a theorem that demonstrates a link between solution for the UFFhDE with symmetrical uncertain variables and its α-path via the comparison theorems in section 3. To understand the theory of inverse uncertainty distribution, we advise the readers to read [41] carefully.

First, we recall the inverse uncertainty distribution theory:

Definition 4.1 ([41]). An uncertainty distribution Ψ is called regular if it is a continues and strictly increasing function and satisfies

Definition 4.2 ([41]). Let ξ be an uncertain variable with a regular uncertainty distribution Ψ. Then, the inverse function Ψ⁻¹ is called the inverse uncertainty distribution of ξ.

Example 4.1. From Definition 4.2, we deduce that

• the inverse uncertainty distribution of a linear uncertain variable is given by

• the inverse uncertainty distribution of a normal uncertain variable is given by

• and the inverse uncertainty distribution of a normal uncertain variable is given by

Definition 4.3 ([41]). We say that an uncertain variable ξ is symmetrical if

where Ψ(x) is a regular uncertainty distribution of ξ.

Remark 4.1. From definition 4.3, we can deduce that the symmetrical uncertain variable has the inverse uncertainty distribution Ψ⁻¹(θ), which satiates

Example 4.2. From definition 4.3, we deduce the following:

• the linear uncertain variable is symmetrical for any positive real number a.

• The normal uncertain variable is symmetrical.

Consider the following UFFhDE with Riemann-Liouville-like forward difference:

subject to the crisp initial conditions

where denotes a fractional Riemann–Liouville forward h-difference with 0 ≤ n − 1 < θ ≤ n, M, N are two real-valued functions defined on [0, ∞) × ℝ, η ∈ ℕ_0,h ∩ [0, Th], X_k ∈ ℝ for k = 0, 1, ..., n − 1, and ξ_{(θ − n)h}, ξ_(θ−n+1)h, ⋯, ξ_η+(θ−n)h are i.i.d. uncertain variables with symmetrical uncertainty distribution .

Definition 4.4 ([41]). An UFFhDE (4.7) with crisp initial conditions (4.8) is said to have an α-path if it is the solution of the corresponding FFhDE

with the same initial conditions (4.8), where Ψ⁻¹(θ) is the inverse uncertainty distribution of uncertain variables ξ_η for η ∈ ℕ_(θ−n)h,h ∩ [0, Th].

Theorem 4.1. Let η ∈ ℕ_0,h ∩ [0, Th], n ∈ ℕ, λ ∈ (0, 1) and θ ∈ (0, 1]. The linear UFFhDE:

with the initial conditions

has a solution

where ξ_ηis an uncertain sequence with the uncertainty distribution, and

and

Proof: By making the use of Lemma 2.5, we can easily prove this theorem by the similar technique of [29, Theorem 3.1], so it is omitted.     □

Example 4.3. Consider the following UFFhDE:

where ξ_(θ−1)h, ξ_{θ h}, …, ξ_η+(θ−1)h are i.i.d linear uncertain variable , which has the inverse uncertainty distribution Ψ⁻¹(θ) = 4θ − 2 by (4.2).

By Theorem 4.1, the associated FFhDE of (4.10) with its initial condition

has a solution

The UFFhDE (4.10) has an α-path

with the initial condition .

Example 4.4. Consider the following UFFhDE:

where ξ_(θ−2)h, ξ_(θ−1)h, …, andξ_η+(θ−2)h are the i.i.d normal uncertain variable , which has the inverse uncertainty distribution by (4.2).

By Theorem 4.1, the associated FFhDE of (4.11) with its initial condition

has a solution

The UFFhDE (4.11) has an α-path

with the initial condition .

In the following theorem, we make a relationship between uncertain fractional forward h-difference equations (UFFhDEs) and fractional h-difference equations (FFhDEs) based on the comparison theorems in section 3.

Theorem 4.2. IfX_ηandare the unique solution and α-path of UFFhDE (4.7) with the initial conditions (4.8), respectively. Assume thatF + |G|Ψ⁻¹(θ) is a Lipschitz continues function in x with a Lipschitz constant L_kthat has. Assume that ξ_ηis the i.i.d. symmetrical uncertain variable for, then

• ifforandfor, where

  and

• ifforandfor.

Proof: First, we let for . Then η ∈ ℕ_{(θ−(n−1))h,h} ∩ [0, Th] and G(η, x) ≥ 0. Therefore,

Moreover, if for , we have η ∈ ℕ_{(θ−(n−1))h,h} ∩ [0, Th] and G(η, x) < 0. Since ξ_η is symmetrical, we have Ψ⁻¹(θ) + Ψ⁻¹(1 − θ) = 0. Thus,

Since X_η(γ) and are the unique solution and α-path of UFFhDE (4.7) with the initial conditions (4.8), respectively, we have

Hence, by use of Theorem 3.2 with (4.12)–(4.15), we get the proof of item (i). The proof of the second item (ii) is similar to (i). Thus, the proof of Theorem 4.2 is completed.     □

Theorem 4.3 (Existence and Uniqueness). Assume thatF(η, x) andG(η, x) satisfy the Lipschitz condition

and there is a positive number L that satisfies the following inequality:

where Q = |a| ∨ |b|. Then UFFhDE (4.7) with the initial conditions (4.8) has a unique solutionX(η) for η ∈ ℕ_θh,h ∩ [0, Th].

Proof: Proof of this theorem is similar to the existence and uniqueness theorem [29, Theorem 3.2], and it is therefore omitted.     □

Example 4.5. Consider the following UFFhDE:

where ξ₋₁, ξ₁, ξ₃, ξ₅, ξ₇ are 5 i.i.d. linear uncertain variables with linear uncertainty distribution .

In this example h = 2, θ = 0.5, T = 4,

and

Thus, the existence and uniqueness Theorem 4.3 confirms that UFFhDE (4.18) has a unique solution.

Now, since

we deduce that F(η, x) + |G(η, x)|Ψ⁻¹(θ) is Lipschitz continues in x with Lipschitz constant L = 0.02 < 0.35 = θ h^−θ.

We see that G(η, x) = 1 > 0, and, from example 4.2, we see is symmetrical. Hence, by Theorem 4.2, we deduce the following link between unique solution and α-path of UFFhDE (4.18):

• if ξ_η ≤ 4θ − 2,

• if ξ_η > 4θ − 2.

Example 4.6. Consider the following UFFhDE:

where are 4 i.i.d. linear uncertain variables with linear uncertainty distribution .

In this example h = 0.5, θ = 0.25, T = 3,

and

Thus, the existence and uniqueness Theorem 4.3 confirms that UFFhDE (4.19) has a unique solution.

Now, since

we deduce that F(η, x) + |G(η, x)|Ψ⁻¹(θ) is Lipschitz, continued in x with Lipschitz constant L = 0.1 < 0.3 = θ h^−θ.

We see that G(η, x) = 1 > 0, and, from example 4.2, we see is symmetrical. Hence, by use of Theorem 4.2, we deduce that if ξ_η ≤ 6θ − 3 and if ξ_η > 6θ − 3. This is a link between unique solution and α-path of UFFhDE (4.19).

5. Conclusions

We have considered the fractional forward h-difference equations and uncertain fractional forward h-difference equations in the context of discrete fractional calculus. The comparison theorems and existence and uniqueness theorem for the FFhDEs and UFFhDEs have been found. From a theoretical point of view, we have created a strong relationship between the solutions for UFFhDEs with the symmetrical uncertain variables and the solutions for associated UFFhDEs (namely the α-path of UFFhDEs).

Our presented results are in the sense of Riemann-Liouville fractional operator. It is important to point out the future scope of our results. There is an important task here that the researchers will be able to consider in the future. What is the task? The interested readers can extend the ideas that were presented in this article to the two well-known models of fractional calculus that were defined by operators similar to the Riemann-Liouville fractional operator but with Mittag-Leffler functions in the kernel, namely the Atangana–Baleanu (or briefly AB) [42, 43] and Prabhakar [44] models.

References

• 1.

  MillerKSRossB. An Introduction to the Fractional Calculus and Fractional Differential Equations. New York, NY: John Wiley & Sons (1993).

  • Google Scholar

• 2.

  PodlubnyI. Fractional Differential Equations. San Diego, CA: Academic Press (1999).

  • Google Scholar

• 3.

  KilbasAASrivastavaHMTrujilloJJ. Theory and Applications of Fractional Differential Equations. Amsterdam: Elsevier B.V. (2006).

  • Google Scholar

• 4.

  DiethelmK. The Analysis of Fractional Differential Equations. Berlin: Springer (2010).

  • Google Scholar

• 5.

  HamasalhFKMohammedPO. Generalized quartic fractional spline approximation function with applications. Math Sci Lett. (2016) 5:131–36.

  • Google Scholar

• 6.

  MartinezFMohammedPOValdesJEN. Non-conformable fractional Laplace transform. Kragujevac J Math. (2022) 46:341–54.

  • Google Scholar

• 7.

  MohammedPO. Some integral inequalities of fractional quantum type. Malaya J Mat. (2016) 4:93–9.

  • Google Scholar

• 8.

  MohammedPO. Hermite-Hadamard inequalities for Riemann-Liouville fractional integrals of a convex function with respect to a monotone function. Math Meth Appl Sci. (2019) 1–11. 10.1002/mma.5784

  • CrossRef
  • Google Scholar

• 9.

  MohammedPOAbdeljawadT. Modification of certain fractional integral inequalities for convex functions. Adv Differ Equ. (2020) 2020:69. 10.1186/s13662-020-2541-2

  • CrossRef
  • Google Scholar

• 10.

  MohammedPOBrevikI. A new version of the Hermite-Hadamard inequality for Riemann-Liouville fractional integrals. Symmetry. (2020) 12:610. 10.3390/sym12040610

  • CrossRef
  • Google Scholar

• 11.

  MohammedPOSarikayaMZ. Hermite-Hadamard type inequalities for F-convex function involving fractional integrals. J Inequal Appl. (2018) 2018:359. 10.1186/s13660-018-1950-1

  • Pubmed Abstract
  • CrossRef
  • Google Scholar

• 12.

  MohammedPOSarikayaMZ. On generalized fractional integral inequalities for twice differentiable convex functions. J Comput Appl Math. (2020) 372:112740. 10.1016/j.cam.2020.112740

  • CrossRef
  • Google Scholar

• 13.

  MohammedPOSarikayaMZBaleanuD. On the generalized Hermite-Hadamard inequalities via the tempered fractional integrals. Symmetry. (2020) 12:595. 10.3390/sym12040595

  • CrossRef
  • Google Scholar

• 14.

  QiFMohammedPOYaoJCYaoYH. Generalized fractional integral inequalities of Hermite–Hadamard type for (α, m)-convex functions. J Inequal Appl. (2019) 2019:135. 10.1186/s13660-019-2079-6

  • CrossRef
  • Google Scholar

• 15.

  AgarwalRP. Difference Equations and Inequalities: Theory, Methods, and Application. New York, NY: Marcel Dekker (2000).

  • Google Scholar

• 16.

  BohnerMPetersonAC. Advances in Dynamic Equations on Time Scales. Boston, MA: Birkhauser (2003).

  • Google Scholar

• 17.

  AticiFEloeP. A transform method in discrete fractional calculus. IJDE. (2007) 2:165–76.

  • Google Scholar

• 18.

  AticiFEloeP. Initial value problems in discrete fractional calculus. P Am Math Soc. (2009) 137:981–89. 10.1090/S0002-9939-08-09626-3

  • CrossRef
  • Google Scholar

• 19.

  GoodrichC. Existence of a positive solution to a system of discrete fractional boundary value problems. Appl Math Comput. (2011) 217:4740–53. 10.1016/j.amc.2010.11.029

  • CrossRef
  • Google Scholar

• 20.

  GoodrichCPetersonA. Discrete Fractional Calculus. Berlin: Springer (2015).

  • Google Scholar

• 21.

  WuGBaleanuD. Discrete chaos in fractional delayed logistic maps. Nonlin Dyn. (2015) 80:1697–703. 10.1007/s11071-014-1250-3

  • CrossRef
  • Google Scholar

• 22.

  WuGBaleanuDLuoW. Lyapunov functions for Riemann–Liouville-like fractional difference equations. Appl Math Comput. (2017) 314:228–36. 10.1016/j.amc.2017.06.019

  • CrossRef
  • Google Scholar

• 23.

  WuGBaleanuDZengS. Finite-time stability of discrete fractional delay systems: Gronwall inequality and stability criterion. Commun Nonlin Sci. (2018) 57:299–308. 10.1016/j.cnsns.2017.09.001

  • CrossRef
  • Google Scholar

• 24.

  ZhuY. Uncertain fractional differential equations and an interest rate model. Math Meth Appl Sci. (2015) 38:3359–68. 10.1002/mma.3335

  • CrossRef
  • Google Scholar

• 25.

  ZhuY. Existence and uniqueness of the solution to uncertain fractional differential equation. J Uncertain Anal Appl. (2015) 3:1–11. 10.1186/s40467-015-0028-6

  • CrossRef
  • Google Scholar

• 26.

  LuZZhuY. Comparison principles for fractional differential equations with the Caputo derivatives. Adv Differ Equat. (2018) 237:1–11. 10.1186/s13662-018-1691-y

  • CrossRef
  • Google Scholar

• 27.

  LuZZhuY. Numerical approach for solution to an uncertain fractional differential equation. Appl Math Comput. (2019) 343:137–48. 10.1016/j.amc.2018.09.044

  • CrossRef
  • Google Scholar

• 28.

  LuQZhuYLuZ. Uncertain fractional forward difference equations for Riemann–Liouville type. Adv Differ Equ. (2019) 2019:147. 10.1186/s13662-019-2093-5

  • CrossRef
  • Google Scholar

• 29.

  MohammedPO. A generalized uncertain fractional forward difference equations of Riemann-Liouville Type. J Math Res. (2019) 11:43–50. 10.5539/jmr.v11n4p43

  • CrossRef
  • Google Scholar

• 30.

  AticiFSengulS. Modeling with fractional difference equations. J Math Anal Appl. (2010) 369:1–9. 10.1016/j.jmaa.2010.02.009

  • CrossRef
  • Google Scholar

• 31.

  AbdeljawadTBaleanuD. Discrete fractional differences with nonsingular discrete Mittag-Leffler kernels. Adv Differ. Equat. (2016) 2016:232. 10.1186/s13662-016-0949-5

  • CrossRef
  • Google Scholar

• 32.

  AbdeljawadT. Fractional difference operators with discrete generalized Mittag-Leffler kernels. Chaos Soliton Fract. (2019) 126:315–24. 10.1016/j.chaos.2019.06.012

  • CrossRef
  • Google Scholar

• 33.

  ShahKJaradFAbdeljawadT. On a nonlinear fractional order model of dengue fever disease under Caputo-Fabrizio derivative. Alex Eng J. (2020) 59:2305–13. 10.1016/j.aej.2020.02.022

  • CrossRef
  • Google Scholar

• 34.

  AbdeljawadTBaleanuD. Monotonicity results for fractional difference operators with discrete exponential kernels. Adv Differ Equat. (2017) 2017:78. 10.1186/s13662-017-1126-1

  • CrossRef
  • Google Scholar

• 35.

  AbdeljawadT. On delta and Nabla Caputo fractional differences and dual identities. Discrete Dyn Nat Soc. (2013) 2013:12. 10.1155/2013/406910

  • CrossRef
  • Google Scholar

• 36.

  AbdeljawadT. Dual identities in fractional difference calculus within Riemann. Adv Differ Equat. (2017) 2017:36. 10.1186/1687-1847-2013-36

  • Pubmed Abstract
  • CrossRef
  • Google Scholar

• 37.

  NROBastosFerreiraRACTorresDFM. Discrete-time fractional variational problems. Signal Process (2011) 91:513–24. 10.1016/j.sigpro.2010.05.001

  • CrossRef
  • Google Scholar

• 38.

  FerreiraRACTorresDFM. Fractional h-difference equations arising from the calculus of variations. Appl Anal Discrete Math. (2011) 5:110–2. 10.2298/AADM110131002F

  • CrossRef
  • Google Scholar

• 39.

  SuwanIAbdeljawadTJaradF. Monotonicity analysis for nabla h-discrete fractional Atangana-Baleanu differences. Chaos Solit Fract. (2018) 117:50–9. 10.1016/j.chaos.2018.10.010

  • CrossRef
  • Google Scholar

• 40.

  AbdeljawadT. Different type kernel h–fractional differences and their fractional h–sums. Chaos Solit Fract. (2018) 116:146–56. 10.1016/j.chaos.2018.09.022

  • CrossRef
  • Google Scholar

• 41.

  LiuB. Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty. Berlin: Springer (2010).

  • Google Scholar

• 42.

  AtanganaABaleanuD. New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model. Thermal Sci. (2016) 20:763–9. 10.2298/TSCI160111018A

  • CrossRef
  • Google Scholar

• 43.

  BaleanuDFernandezA. On some new properties of fractional derivatives with Mittag-Leffler kernel. Commun Nonlin Sci. (2018) 59:444–62. 10.1016/j.cnsns.2017.12.003

  • CrossRef
  • Google Scholar

• 44.

  PrabhakarTR. A singular integral equation with a generalized Mittag Leffler function in the kernel. Yokohama Math J. (1971) 19:7–15.

  • Google Scholar