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.

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 a<b,b-ah∈ℕ 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 ηh(θ) 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,h and _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 μh,μh+θ∈ℝ\{...,-2,-1}, then we have

for each η ∈ ℕ_a+θh,h.

Lemma 2.7. Let ψ be defined on ℕ_a,h and 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.

Consider the following FFhDEs:

subject to the initial conditions

where ​(θ-n)hΔhθ 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 ​0Δh-θ 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 ΔhnΔh-nψ(η)=ψ(η).

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_k that has 0<Lk≤h-θθ. If ψ₁(η) and ψ₂(η) are, respectively, unique solutions of the following IVPs

and

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 ω(η1-h):=(θ-1)hΔhθX(η1-θ h)+LkX(η1-h), 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 Lk<θh-θ, 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 w(η3-h):=(θ-1)hΔhθz(η3-θ h)+Lkz(η3-h), 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 Lk<θ h-θ, 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_k that has 0<Lk≤h-θθ. If ψ₁(η) and ψ₂(η) are, respectively, unique solutions of the following IVPs

and

Proof: (1) For μ = θ − n + 1 ∈ (0, 1] and η ∈ ℕ_0,h, we have Δ​(θ−n)h​hθψ(η)=Δhn−1Δ​(μ−1)h​hμψ(η). 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 g¯(η,ψ)≤k̄(η,ψ) 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.     □

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

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:

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

subject to the crisp initial conditions

where ​(θ-n)hΔhθ 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 L(a,b).

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_η is an uncertain sequence with the uncertainty distribution L(a·eθ,λ;h(η),b·eθ,λ;h(η)), 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 L(-2,2), 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 ​(θ-1)hΔhθ-1X(η)|t=0=X0.