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

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

$\begin{array}{l}{\Delta }_{h}f(\eta )=\frac{f(\eta +h)-f(\eta )}{h},\end{array}$

and the backward difference operator on hℤ is defined by

$\begin{array}{l}{\nabla }_{h}f(\eta )=\frac{f(\eta )-f(\eta -h)}{h}.\end{array}$

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 $a<b,\frac{b-a}{h}\in \mathbb{N}$ 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

$\begin{array}{ll}{\eta }_h^{(\theta )}=\frac{\Gamma \left(\frac{\eta }{h}+1\right)}{\Gamma \left(\frac{\eta }{h}+1-\theta \right)},& (2.1)\end{array}$

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 ${\eta }_h^{(\theta )}$ 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

$\begin{array}{l}\left({}_a\Delta {\text{}}_h^{-\theta }\psi \right)(\eta )={\displaystyle {\int }_a^{\sigma (\eta -\theta h)}(\eta -\sigma (}\tau ){)}_h^{(\theta -1)}\psi (\tau ){\Delta }_h\tau \\ =\frac{1}{\Gamma (\theta )}{\displaystyle \sum _{r=\frac{a}{h}}^{\frac{\eta }{h}-\theta }{(\eta -\sigma (rh))}_h^{(\theta -1)}}\psi (rh)h,\eta \in {\mathbb{N}}_{a+\theta h,h},\end{array}$

and the right delta h-fractional sum is defined by

$\begin{array}{l}\left({}_h\Delta {\text{}}_b^{-\theta }\psi \right)(\eta )={\displaystyle {\int }_{\rho (\eta +\theta h)}^b(\rho (}\tau )-\eta {)}_h^{(\theta -1)}\psi (\tau ){\nabla }_h\tau \\ =\frac{1}{\Gamma (\theta )}{\displaystyle \sum _{r=\frac{\eta }{h}+\theta }^{\frac{b}{h}}(rh-\sigma (}\eta ){)}_h^{(\theta -1)}\psi (rh)h,\\ \eta \in {}_{b-\theta h,h}\mathbb{N}.\end{array}$

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

$\begin{array}{l}\left({}_{a+\mu h}\Delta {\text{}}_h^{-\theta }{}_a\Delta {\text{}}_h^{-\mu }p\right)(\eta )=\left({}_{{\text{}}_a}\Delta {\text{}}_h^{-(\mu +\theta )}p\right)(\eta )\\ =\left({}_{{\text{}}_{a+\theta h}}\Delta {\text{}}_h^{-\mu }{}_{{\text{}}_a}\Delta {\text{}}_h^{-\theta }p\right)(\eta ),& (2.2)\end{array}$

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

$\begin{array}{ll}\left({}_a\Delta {\text{}}_h^{\mu }\psi \right)(\eta )=\left({\Delta }_h^m{}_{{\text{}}_a}\Delta {\text{}}_h^{-(m-\mu )}\psi \right)(\eta ),& (2.3)\end{array}$

$\begin{array}{ll}\left({}_h\Delta {\text{}}_b^{\mu }\psi \right)(\eta )={(-1)}^m\left({\nabla }_h^m{}_h\Delta {\text{}}_b^{-(m-\mu )}\psi \right)(\eta ),& (2.4)\end{array}$

where m = [θ] + 1.

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

$\begin{array}{ll}\left({}_a\Delta {\text{}}_h^{-\theta }{\Delta }_h\psi \right)(\eta )={\Delta }_h{}_a\Delta {\text{}}_h^{-\theta }\psi (\eta )-\frac{{(\eta -a)}_h^{(\theta -1)}}{\Gamma (\theta )}\psi (a).& (2.5)\end{array}$

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

$\begin{array}{l}{}_{{\text{}}_{a+\mu h}}\Delta {\text{}}_h^{\theta }{(\eta -a)}_h^{(\mu )}=\frac{\Gamma (\mu +1)}{\Gamma (\mu +\theta +1)}{(\eta -a)}_h^{(\theta +\mu )},\\ {}_{{\text{}}_h}\Delta {\text{}}_{b-\mu h}^{\theta }{(b-\eta )}_h^{(\theta )}=\frac{\Gamma (\mu +1)}{\Gamma (\mu +\theta +1)}{(b-\eta )}_h^{(\theta +\mu )}.\end{array}$

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

$\begin{array}{l}\left({}_a\Delta {\text{}}_h^{-\theta }{\Delta }_h^q\psi \right)(\eta )=\left({\Delta }_h^q{}_{{\text{}}_a}\Delta {\text{}}_h^{-\theta }\psi \right)(\eta )\\ -{\displaystyle \sum _{k=0}^{q-1}\frac{{(\eta -a)}_h^{(v-q+k)}}{\Gamma (v-q+k+1)}}{\Delta }_h^k\psi (a),& (2.6)\end{array}$

for η ∈ ℕ_{a + θ h,h}.

Lemma 2.6 ([38]). Suppose that $\frac{\mu }{h},\frac{\mu }{h}+\theta \in ℝ\backslash \left\{...,-2,-1\right\}$, then we have

${}_a\Delta {\text{}}_h^{-\theta }{(\eta -a+\mu )}_h^{\left(\frac{\mu }{h}\right)}=\frac{\Gamma \left(\frac{\mu }{h}+1\right)}{\Gamma \left(\frac{\mu }{h}+\theta +1\right)}{(\eta -a+\mu )}_h^{\left(\frac{\mu }{h}+\theta \right)},$

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

$\begin{array}{l}\left({}_a\Delta {\text{}}_h^{-\mu }\psi \right)(\eta )=\\ \{\begin{array}{cc}\frac{1}{\Gamma (-\mu )}{\displaystyle \sum _{r=\frac{a}{h}}^{\frac{\eta }{h}+\mu }{(\eta -\sigma (rh))}_h^{(-\mu -1)}}\psi (rh)h,\hfill & m-1<\mu <m,\hfill \\ {}_{{}_a\text{}}\Delta {\text{}}_h^{m}p(\eta ),\hfill & \mu =m,\hfill \end{array}\hfill \end{array}$

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,

${}_{{\text{}}_a}\Delta {\text{}}_h^{-\theta }{\xi }_{\eta }=\frac{1}{\Gamma (\theta )}{\displaystyle \sum _{r=\frac{a}{h}}^{\frac{\eta }{h}-\theta }{(\eta -\sigma (rh))}_h^{(\theta -1)}}{\xi }_{rh}h$

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

${}_{{\text{}}_a}\Delta {\text{}}_h^{\mu }{\xi }_{\eta }={\Delta }_h^n\left({}_a\Delta {\text{}}_h^{-(n-\mu )}{\xi }_{\eta }\right),$

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

3. The Comparison Theorems

Consider the following FFhDEs:

$\begin{array}{ll}{}_{{\text{}}_{(\theta -n)h}}\Delta {\text{}}_h^{\theta }\psi (\eta )=g(\eta +(\theta -n)h,\psi (\eta +(\theta -n)h)),& (3.1)\end{array}$

subject to the initial conditions

$\begin{array}{ll}{}_{{\text{}}_{(\theta -n)h}}\Delta {\text{}}_h^{\theta -n+i}\psi (\eta ){|}_{t=0}={\psi }_i,i=0,1,\dots ,n-1,& (3.2)\end{array}$

where ${\text{}}_{(\theta -n)h}{\Delta }_h^{\theta }$ 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 ${\text{}}_0{\Delta }_h^{-\theta }$ to Equation (3.1), then the initial value problem (3.1) and (3.2) is equivalent to the following fractional sum equation:

$\begin{array}{l}\psi (\eta )={\displaystyle \sum _{i=0}^{n-1}}\frac{{(\eta )}_h^{(\theta -n+i)}}{\Gamma (\theta -n+i+1)}{\psi }_i\\ \begin{array}{l}+\frac{1}{\Gamma (\theta )}{\displaystyle \sum _{r=0}^{\frac{\eta }{h}-\theta }}{(\eta -\sigma (rh))}_h^{(\theta -1)}g(r+(\theta -n)h,\psi (r+(\theta -n)h))h,\end{array}& (3.3)\end{array}$

where we have used Lemma 2.1, Lemma 2.5, and the fact that ${\Delta }_h^n{\Delta }_h^{-n}\psi (\eta )=\psi (\eta )$.

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 $0<L_k\le h^{-\theta }\theta$. If ψ₁(η) and ψ₂(η) are, respectively, unique solutions of the following IVPs

$\begin{array}{ll}\{\begin{array}{c}{}_{(\theta -1)h}\Delta {\text{}}_h^{\theta }\psi (\eta )=g(\eta +(\theta -1)h,\psi (\eta +(\theta -1)h)),\eta \in {\mathbb{N}}_0,\hfill \\ {}_{{\text{}}_{(\theta -1)h}}\Delta {\text{}}_h^{\theta -1}\psi (\eta ){|}_{t=0}={\text{X}}_0,\hfill \end{array}& (3.4)\end{array}$

and

$\begin{array}{ll}\{\begin{array}{c}{}_{{\text{}}_{(\theta -1)h}}\Delta {\text{}}_h^{\theta }\psi (\eta )=k(\eta +(\theta -1)h,\psi (\eta +(\theta -1)h)),\eta \in {\mathbb{N}}_0,\hfill \\ {}_{{\text{}}_{(\theta -1)h}}\Delta {\text{}}_h^{\theta -1}\psi (\eta ){|}_{t=0}={\psi }_0.\hfill \end{array}& (3.5)\end{array}$

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

2. 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

$\begin{array}{ll}\text{X}({\eta }_1)>0,& (3.6)\end{array}$

$\begin{array}{ll}\text{X}(\eta )\le 0,\eta \in {\mathbb{N}}_{(\theta -1)h,h}\cap [0,{\eta }_1-h].& (3.7)\end{array}$

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

$\begin{array}{l}{\psi }_1(\theta h)=\theta h^{v-1}{\psi }_0+h^{\theta }g((\theta -1)h,{\text{X}}_0),\\ {\psi }_2(\theta h)=\theta h^{v-1}{\psi }_0+h^{\theta }k((\theta -1)h,{\text{X}}_0).\end{array}$

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

$\begin{array}{l}{\psi }_1(\theta h)-{\psi }_2(\theta h)=h^v\left(g((\theta -1)h,{\text{X}}_0)-k((\theta -1)h,{\text{X}}_0)\right)\le 0.\end{array}$

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

$\begin{array}{l}{\text{}}_{(\theta -1)h}{\Delta }_h^{\theta }\text{X}({\eta }_1-\theta h)\\ =\frac{1}{\Gamma (-\theta )}{\displaystyle \sum _{r=\theta -1}^{\frac{{\eta }_1}{h}}{\left({\eta }_1-\theta h-\sigma (rh)\right)}_h^{(-\theta -1)}}\text{X}(rh)h\\ =\frac{1}{\Gamma (-\theta )}{\displaystyle \sum _{r=\theta -1}^{\theta +\ell }{\left(\ell h-\sigma (rh)\right)}_h^{(-\theta -1)}}\text{X}(rh)h\\ =h^{-\theta }\text{X}((\theta +\ell )h)-\theta h^{-\theta }\text{X}((\theta +\ell -1)h)\\ +\frac{1}{\Gamma (-\theta )}{\displaystyle \sum _{r=\theta -1}^{\theta +\ell -2}}{\left(\ell h-\sigma (rh)\right)}_h^{(-\theta -1)}\text{X}(rh)h.\end{array}$

That is,

$\begin{array}{l}{h}^{-\theta }\text{X}((\theta +\ell )h){=}_{(\theta -1)h}{\Delta }_h^{\theta }\text{X}({\eta }_1-\theta h)+\theta h^{-\theta }\text{X}((\theta +\ell -1)h)\\ \begin{array}{l}-\frac{1}{\Gamma (-\theta )}{\displaystyle \sum _{r=\theta -1}^{\theta +\ell -2}}{\left(\ell h-\sigma (rh)\right)}_h^{(-\theta -1)}\text{X}(rh)h.\end{array}& (3.8)\end{array}$

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

$\begin{array}{l}{}_{{\text{}}_{(\theta -1)h}}\Delta {\text{}}_h^{\theta }\text{X}({\eta }_1-\theta h)={}_{{\text{}}_{(\theta -1)h}}\Delta {\text{}}_h^{\theta }{\psi }_1({\eta }_1-\theta h)\\ -{}_{{\text{}}_{(\theta -1)h}}\Delta {\text{}}_h^{\theta }{\psi }_2({\eta }_1-\theta h)\\ =g({\eta }_1-h,{\psi }_1({\eta }_1-h))\\ -k({\eta }_1-h,{\psi }_2({\eta }_1-h))\\ \le k({\eta }_1-h,{\psi }_1({\eta }_1-h))\\ -k({\eta }_1-h,{\psi }_2({\eta }_1-h))\\ \le -L_k\left({\psi }_1({\eta }_1-h)-{\psi }_2({\eta }_1-h)\right)\\ \le -L_k\text{X}({\eta }_1-h).\end{array}$

Denoting $\omega ({\eta }_1-h):{=}_{(\theta -1)h}{\Delta }_h^{\theta }\text{X}({\eta }_1-\theta h)+L_k\text{X}({\eta }_1-h)$, it follows that

$\begin{array}{ll}\omega ((\theta +\ell -1)h)\le 0.& (3.9)\end{array}$

This gives

${}_{(\theta -1)h}\Delta {\text{}}_h^{\theta }\text{X}({\eta }_1-\theta h)=-L_k\text{X}((\theta +\ell -1)h)+\omega ((\theta +\ell -1)h).$

Thus, Equation (3.8) becomes

$\begin{array}{l}{h}^{-\theta }\text{X}((\theta +\ell )h)=\left(\theta h^{-\theta }-L_k\right)\text{X}((\theta +\ell -1)h)+\omega ((\theta +\ell -1)h)\\ \begin{array}{l}-\frac{1}{\Gamma (-\theta )}{\displaystyle \sum _{r=\theta -1}^{\theta +\ell -2}}{\left(\ell h-\sigma (rh)\right)}_h^{(-\theta -1)}\text{X}(rh)h.\end{array}& (3.10)\end{array}$

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

$\begin{array}{l}\frac{{\left(\ell h-\sigma (rh)\right)}_h^{(-\theta -1)}}{\Gamma (-\theta )}=\frac{{(\ell h-(\theta +i)h)}_h^{(-\theta -1)}}{\Gamma (-\theta )}\\ =h^{-\theta -1}\frac{\Gamma (\ell -i+1-\theta )}{\Gamma (-\theta )\Gamma (\ell -i+2)}\\ =h^{-\theta -1}\frac{(\ell -i-\theta )(\ell -i-1-\theta )\cdots (-\theta )\Gamma (-\theta )}{\Gamma (-\theta )\Gamma (\ell -i+2)}\\ =h^{-\theta -1}\frac{(-\theta )(-\theta +1)\cdots (\ell -i-1-\theta )(\ell -i-\theta )}{\Gamma (\ell -i+2)}\\ =h^{-\theta -1}\frac{(-\theta )(-\theta +1)\cdots (-\theta -1+c)(-\theta +c)}{\Gamma (c+1)},\\ \text{ where }c=\ell -i.\end{array}$

Since θ ∈ (0, 1] and h^−θ−1 > 0, so

$\begin{array}{ll}\frac{{\left(\ell h-\sigma (rh)\right)}_h^{(-\theta -1)}}{\Gamma (-\theta )}\le 0.& (3.11)\end{array}$

Considering $L_k<\theta h^{-\theta }$, h^−θ > 0 and Equations (3.9)–(3.11), it follows that

$\begin{array}{l}{h}^{-\theta }\text{X}((\theta +\ell )h)\le 0.\end{array}$

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

$\begin{array}{ll}z({\eta }_3)\ge 0,& (3.12)\end{array}$

$\begin{array}{ll}z(\eta )<0,\eta \in {\mathbb{N}}_{\theta h,h}\cap [0,{\eta }_3-h].& (3.13)\end{array}$

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

$\begin{array}{l}{}_{{\text{}}_{(\theta -1)h}}\Delta {\text{}}_h^{\theta }z({\eta }_3-\theta h)\\ =\frac{1}{\Gamma (-\theta )}{\displaystyle \sum _{r=\theta -1}^{\frac{{\eta }_3}{h}}{\left({\eta }_3-\theta h-\sigma (rh)\right)}_h^{(-\theta -1)}}z(rh)h\\ =h^{-\theta }z((\theta +\ell )h)-\theta h^{-\theta }z((\theta +\ell -1)h)\\ +\frac{1}{\Gamma (-\theta )}{\displaystyle \sum _{r=\theta -1}^{\theta +\ell -2}{\left(\ell h-\sigma (rh)\right)}_h^{(-\theta -1)}}z(rh)h,\end{array}$

or equivalently,

$\begin{array}{l}{h}^{-\theta }z((\theta +\ell )h){=}_{(\theta -1)h}{\Delta }_h^{\theta }z({\eta }_3-\theta h)+\theta h^{-\theta }z((\theta +\ell -1)h)\\ \begin{array}{l}-\frac{1}{\Gamma (-\theta )}{\displaystyle \sum _{r=\theta -1}^{\theta +\ell -2}}{\left(\ell h-\sigma (rh)\right)}_h^{(-\theta -1)}z(rh)h.\end{array}& (3.14)\end{array}$

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

$\begin{array}{l}{}_{{\text{}}_{(\theta -1)h}}\Delta {\text{}}_h^{\theta }z({\eta }_3-\theta h)={}_{(\theta -1)h}\Delta {\text{}}_h^{\theta }{\psi }_1({\eta }_3-\theta h)\\ -{}_{{\text{}}_{(\theta -1)h}}\Delta {\text{}}_h^{\theta }{\psi }_2({\eta }_3-\theta h)\\ =w({\eta }_3-h,{\psi }_2({\eta }_3-h))\\ -k({\eta }_3-h,{\psi }_1({\eta }_3-h))\\ \le k({\eta }_3-h,{\psi }_2({\eta }_3-h))\\ -k({\eta }_3-h,{\psi }_1({\eta }_3-h))\\ \le -L_k\left({\psi }_2({\eta }_3-h)-{\psi }_1({\eta }_3-h)\right)\\ \le -L_{k}z({\eta }_3-h).\end{array}$

Denoting $w({\eta }_3-h):{=}_{(\theta -1)h}{\Delta }_h^{\theta }z({\eta }_3-\theta h)+L_{k}z({\eta }_3-h)$, it follows that

$\begin{array}{ll}w((\theta +\ell -1)h)\le 0.& (3.15)\end{array}$

This gives

${}_{{\text{}}_{(\theta -1)h}}\Delta {\text{}}_h^{\theta }z({\eta }_3-\theta h)=-L_{k}z((\theta +\ell -1)h)+w((\theta +\ell -1)h).$

Equation (3.14) thus becomes

$\begin{array}{l}{h}^{-\theta }z((\theta +\ell )h)=\left(\theta h^{-\theta }-L_k\right)z((\theta +\ell -1)h)+w((\theta +\ell -1)h)\\ \begin{array}{l}-\frac{1}{\Gamma (-\theta )}{\displaystyle \sum _{r=\theta -1}^{\theta +\ell -2}}{\left(\ell h-\sigma (rh)\right)}_h^{(-\theta -1)}z(rh)h.\end{array}& (3.16)\end{array}$

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

$\begin{array}{ll}\frac{{\left(\ell h-\sigma (rh)\right)}_h^{(-\theta -1)}}{\Gamma (-\theta )}\le 0.& (3.17)\end{array}$

Considering $L_k<\theta h^{-\theta }$, h^−θ > 0 and Equations (3.15)–(3.17), it follows that

$\begin{array}{l}{h}^{-\theta }z((\theta +\ell )h)\le 0.\end{array}$

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 $0<L_k\le h^{-\theta }\theta$. If ψ₁(η) and ψ₂(η) are, respectively, unique solutions of the following IVPs

$\begin{array}{ll}\{\begin{array}{c}{}_{(\theta -n)h}{\Delta }_h^{\theta }\psi (\eta )=g(\eta +(\theta -n)h,\psi (\eta +(\theta -n)h)),\eta \in {\mathbb{N}}_0,\hfill \\ {}_{(\theta -n)h}{\Delta }_h^{\theta -n+i}\psi (\eta ){|}_{t=0}={\psi }_i,i=0,1,\dots ,n-1\hfill \end{array}& (3.18)\end{array}$

and

$\begin{array}{ll}\{\begin{array}{c}{}_{(\theta -n)h}{\Delta }_h^{\theta }\psi (\eta )=k(\eta +(\theta -n)h,\psi (\eta +(\theta -n)h)),\eta \in {\mathbb{N}}_0,\hfill \\ {}_{(\theta -n)h}{\Delta }_h^{\theta -n+i}\psi (\eta ){|}_{t=0}={\psi }_i,i=0,1,\dots ,n-1.\hfill \end{array}& (3.19)\end{array}$

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

2. if g(η, ψ) > k(η, ψ), then ψ₁(η) > ψ₂(η) for each $\eta \in {\mathbb{N}}_{(\theta -n+1)h}^h$.

Proof: (1) For μ = θ − n + 1 ∈ (0, 1] and η ∈ ℕ_0,h, we have ${}_{{\text{}}_{(\theta -n)h}}\Delta {\text{}}_h^{\theta }\psi (\eta )={\Delta }_h^{n-1}{}_{{\text{}}_{(\mu -1)h}}\Delta {\text{}}_h^{\mu }\psi (\eta ).$ By using Lemma 2.5, the IVPs (3.18) and (3.19) can be easily converted to the following IVPs, respectively,

$\begin{array}{ll}\{\begin{array}{c}{}_{(\mu -1)h}{\Delta }_h^{\mu }\psi (\eta )=\frac{1}{\Gamma (n-1)}{\displaystyle \sum _{r=0}^{\frac{\eta }{h}-(n-1)}{\left(\eta -\sigma (rh)\right)}_h^{(\mu -2)}}g(r+(\mu -1)h,\hfill \\ \psi (r+(\mu -1)h))h+{\displaystyle \sum _{i=0}^{n-2}\frac{{(\eta )}_h^{(i)}}{\Gamma (i+1)}}{\psi }_{i+1},\hfill \\ {}_{(\mu -1)h}{\Delta }_h^{\mu -1}\psi (\eta ){|}_{t=0}={\psi }_0,\hfill \end{array}& (3.20)\end{array}$

and

$\begin{array}{ll}\{\begin{array}{c}{}_{(\mu -1)h}{\Delta }_h^{\mu }\psi (\eta )=\frac{1}{\Gamma (n-1)}{\displaystyle \sum _{r=0}^{\frac{\eta }{h}-(n-1)}{\left(\eta -\sigma (rh)\right)}_h^{(\mu -2)}}k(r+(\mu -1)h,\hfill \\ \psi (r+(\mu -1)h))h+{\displaystyle \sum _{i=0}^{n-2}\frac{{(\eta )}_h^{(i)}}{\Gamma (i+1)}}{\psi }_{i+1},\hfill \\ {}_{(\mu -1)h}{\Delta }_h^{\mu -1}\psi (\eta ){|}_{t=0}={\psi }_0.\hfill \end{array}& (3.21)\end{array}$

Denote

$\begin{array}{c}\overline{g}(\eta ,x)=\frac{1}{\Gamma (n-1)}{\displaystyle \sum _{r=0}^{\frac{\eta }{h}-(n-1)}{\left(\eta -\sigma (rh)\right)}_h^{(n-2)}}g(r+(\mu -1)h,\\ \psi (r+(\mu -1)h))h+{\displaystyle \sum _{i=0}^{n-2}\frac{{(\eta )}_h^{(i)}}{\Gamma (i+1)}}{\psi }_{i+1},\end{array}$

and

$\begin{array}{c}\overline{k}(\eta ,x)=\frac{1}{\Gamma (n-1)}{\displaystyle \sum _{r=0}^{\frac{\eta }{h}-(n-1)}{\left(\eta -\sigma (rh)\right)}_h^{(n-2)}}k(r+(\mu -1)h,\\ \psi (r+(\mu -1)h))h+{\displaystyle \sum _{i=0}^{n-2}\frac{{(\eta )}_h^{(i)}}{\Gamma (i+1)}}{\psi }_{i+1},\end{array}$

for η ∈ ℕ_(θ−1)h,h. These give

$\begin{array}{l}\overline{g}(\eta ,x)-\overline{k}(\eta ,x)=\frac{1}{\Gamma (n-1)}{\displaystyle \sum _{r=0}^{\frac{\eta }{h}-(n-1)}}{\left(\eta -\sigma (rh)\right)}_h^{(n-2)}\\ \times [g(r+(\mu -1)h,\psi (r+(\mu -1)h))-k(r+(\mu -1)h,\\ \psi (r+(\mu -1)h))]h.& (3.22)\end{array}$

Since g(η, ψ) ≤ k(η, ψ) and

$\begin{array}{l}\frac{{\left(\eta -\sigma (rh)\right)}_h^{(n-2)}}{\Gamma (n-1)}=\frac{{\left(\eta -(r+1)h\right)}_h^{(n-2)}}{\Gamma (n-1)}\\ =h^{-n-2}\frac{\Gamma \left(\frac{\eta }{h}-r\right)}{\Gamma (n-1)\Gamma \left(\frac{\eta }{h}-r-n+2\right)}\\ =h^{-n-2}\frac{\Gamma (c)}{\Gamma (n-1)\Gamma (c-n+2)},\\ \text{ where }c=\frac{\eta }{h}-r,r=0,1,...,\frac{\eta }{h}-n+1>0,\end{array}$

it follows from (3.22) that $\overline{g}(\eta ,\psi )\le \bar{k}(\eta ,\psi )$ 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

$\begin{array}{ll}{\displaystyle \underset{x\to -\infty }{\text{lim}}}\Psi (x)=0,{\displaystyle \underset{x\to +\infty }{\text{lim}}}\Psi (x)=1.& (4.1)\end{array}$

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

(i) the inverse uncertainty distribution of a linear uncertain variable ${L}(a,b)$ is given by

$\begin{array}{ll}{\Psi }^{-1}(\theta )=(1-\theta )a+\theta b;& (4.2)\end{array}$

(ii) the inverse uncertainty distribution of a normal uncertain variable ${N}(e,\sigma )$ is given by

$\begin{array}{ll}{\Psi }^{-1}(\theta )=e+\frac{\sqrt{3}\sigma }{\pi }\text{ln}\left(\frac{\theta }{1-\theta }\right);& (4.3)\end{array}$

(iii) and the inverse uncertainty distribution of a normal uncertain variable ${L}{O}{G}{N}(e,\sigma )$ is given by

$\begin{array}{ll}{\Psi }^{-1}(\theta )=\text{exp}(e)+{\left(\frac{\theta }{1-\theta }\right)}^{\frac{\sqrt{3}\sigma }{\pi }}.& (4.4)\end{array}$

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

$\begin{array}{ll}\Psi (x)+\Psi (-x)=1,& (4.5)\end{array}$

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

$\begin{array}{ll}{\Psi }^{-1}(\theta )+{\Psi }^{-1}(1-\theta )=0.& (4.6)\end{array}$

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

1. the linear uncertain variable ${L}(-a,a)$ is symmetrical for any positive real number a.

2. The normal uncertain variable ${N}(0,1)$ is symmetrical.

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

$\begin{array}{ll}\begin{array}{l}{\text{}}_{\left(\theta -n\right)h}{\Delta }_h^{\theta }\text{X}(\eta )=\text{F}(\eta +(\theta -n)h,\text{X}(\eta +(\theta -n)h))\hfill \\ +\text{G}(\eta +(\theta -n)h,\text{X}(\eta +(\theta -n)h)){\xi }_{\eta +(\theta -n)h},\hfill \end{array}& (4.7)\end{array}$

subject to the crisp initial conditions

$\begin{array}{ll}\begin{array}{l}{\text{}}_{\left(\theta -n\right)h}{\Delta }_h^{\theta -n-k}\text{X}(\eta ){|}_{t=0}={\text{X}}_k,k=0,1,\dots ,n-1,\hfill \end{array}& (4.8)\end{array}$

where ${\text{}}_{(\theta -n)h}{\Delta }_h^{\theta }$ 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

$\begin{array}{l}{}_{{\text{}}_{(\theta -n)h}}\Delta {\text{}}_h^{\theta }\text{X}(\eta )=\text{F}(\eta +(\theta -n)h,\text{X}(\eta +(\theta -n)h))\\ +|\text{G}(\eta +(\theta -n)h,\text{X}(\eta +(\theta -n)h))|{\Psi }^{-1}(\theta )& (4.9)\end{array}$

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:

${}_{(\theta -n)h}\Delta {\text{}}_h^{\theta }\text{X}(\eta )=\lambda \text{X}(\eta +(\theta -n)h)+\lambda {\xi }_{\eta +(\theta -n)h},$

with the initial conditions

${\text{}}_{(\theta -n)h}{\Delta }_h^{\theta -n-i}\text{X}(\eta ){|}_{t=0}={\text{X}}_i,i=0,1,\dots ,n-1,$

has a solution

$\begin{array}{l}\text{X}(\eta )={\text{X}}_i{F}_{\mu ,\lambda ;h}(\eta )+{\xi }_{\eta },,i=0,1,...,n-1,\end{array}$

where ξ_η is an uncertain sequence with the uncertainty distribution ${L}(a·e_{\theta ,\lambda ;h}(\eta ),b·e_{\theta ,\lambda ;h}(\eta ))$, and

$\begin{array}{l}{F}_{\theta ,\lambda ;h}(\eta )={\displaystyle \sum _{k=0}^{\infty }}{\lambda }^k{\displaystyle \sum _{i=0}^{n-1}}\frac{{(\eta +k(\theta -n)h)}_h^{((k+1)\theta h-nh+i)}}{\Gamma ((k+1)\theta -n+i+1)},\end{array}$

and

$\begin{array}{l}{e}_{\theta ,\lambda ;h}(\eta )={\displaystyle \sum _{k=1}^{\infty }}{\lambda }^k\frac{{(\eta +(k-1)(\theta -n)h)}_h^{(k\theta )}}{\Gamma (k\theta +1)}.\end{array}$

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:

$\begin{array}{l}{\text{}}_{(\theta -1)h}{\Delta }_h^{\theta }\text{X}(\eta )=\lambda \text{X}(\eta +(\theta -1)h)+\lambda {\xi }_{\eta +(\theta -1)h},\\ \eta \in {\mathbb{N}}_{0,h}\cap [0,Th],\lambda \in (0,1),\theta \in (0,1],& (4.10)\end{array}$

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

$\begin{array}{l}{}_{\left(\theta -1\right)h}{\Delta }_h^{\theta }\text{X}(\eta )=\lambda \text{X}(\eta +(\theta -1)h)+\lambda {\Psi }^{-1}(\theta ),\\ {\displaystyle {\text{}}_{(\theta -1)h}{\Delta }_h^{\theta -1}\text{X}(\eta ){|}_{t=0}={\text{X}}_0}\end{array}$

has a solution

$\begin{array}{l}\text{X}(\eta )={\text{X}}_0{\displaystyle \sum _{k=0}^{\infty }}{\lambda }^k\frac{{(\eta +k(\theta -1)h)}_h^{((k+1)\theta -1)}}{\Gamma ((k+1)\theta )}\\ +{\displaystyle \sum _{k=1}^{\infty }}{\lambda }^k\frac{{(\eta +(k-1)(\theta -1)h)}_h^{(k\theta )}}{\Gamma (k\theta +1)}(4\theta -2).\end{array}$

The UFFhDE (4.10) has an α-path

$\begin{array}{l}{\text{X}}_{\eta }^{\theta }={\text{X}}_0{\displaystyle \sum _{k=0}^{\infty }}{\lambda }^k\frac{{(\eta +k(\theta -1)h)}_h^{((k+1)\theta -1)}}{\Gamma ((k+1)\theta )}\\ +{\displaystyle \sum _{k=1}^{\infty }}{\lambda }^k\frac{{(\eta +(k-1)(\theta -1)h)}_h^{(k\theta )}}{\Gamma (k\theta +1)}(4\theta -2).\end{array}$

with the initial condition ${\text{}}_{(\theta -1)h}{\Delta }_h^{\theta -1}\text{X}(\eta ){|}_{t=0}={\text{X}}_0$.

Example 4.4. Consider the following UFFhDE:

$\begin{array}{l}{\text{}}_{\left(\theta -2\right)h}{\Delta }_h^{\theta }\text{X}(\eta )=q\text{X}(\eta +(\theta -2)h)+q{\xi }_{\eta +(\theta -1)h},\\ \eta \in {\mathbb{N}}_{0,h}\cap [0,Th],q\in (0,1),\theta \in (0,1],& (4.11)\end{array}$

where ξ_(θ−2)h, ξ_(θ−1)h, …, andξ_η+(θ−2)h are the i.i.d normal uncertain variable ${N}(0,1)$, which has the inverse uncertainty distribution ${\Psi }^{-1}(\theta )=\frac{\sqrt{3}}{\pi }\text{ln}\left(\frac{\theta }{1-\theta }\right)$ by (4.2).

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

$\begin{array}{l}{}_{{}_{(\theta -2)h}}\Delta {}_h^{\theta }\text{X}(\eta )=q\text{X}(\eta +(\theta -2)h)+q{\Psi }^{-1}(\theta ),\\ {\displaystyle {}_{{}_{(\theta -2)h}}\Delta {}_h^{\theta -2+i}{\text{X}}_i(\eta ){|}_{t=0}={\text{X}}_i,i=0,1}\end{array}$

has a solution

$\begin{array}{l}\text{X}(\eta )={\displaystyle \sum _{k=0}^{\infty }}{q}^k{\displaystyle \sum _{i=0}^1}{\text{X}}_i\frac{{(\eta +k(\theta -2)h)}_h^{((k+1)\theta h-2h+i)}}{\Gamma ((k+1)\theta -1+i)}\\ +\frac{\sqrt{3}}{\pi }\text{ln}\left(\frac{\theta }{1-\theta }\right){\displaystyle \sum _{k=1}^{\infty }}{q}^k\frac{{(\eta +(k-1)(\theta -2)h)}_h^{(k\theta )}}{\Gamma (k\theta +1)}.\end{array}$

The UFFhDE (4.11) has an α-path

$\begin{array}{l}{\text{X}}_{\eta }^{\theta }={\displaystyle \sum _{k=0}^{\infty }}{q}^k{\displaystyle \sum _{i=0}^1}{\text{X}}_i\frac{{(\eta +k(\theta -2)h)}_h^{((k+1)\theta h-2h+i)}}{\Gamma ((k+1)\theta -1+i)}\\ +\frac{\sqrt{3}}{\pi }\text{ln}\left(\frac{\theta }{1-\theta }\right){\displaystyle \sum _{k=1}^{\infty }}{q}^k\frac{{(\eta +(k-1)(\theta -2)h)}_h^{(k\theta )}}{\Gamma (k\theta +1)}.\end{array}$

with the initial condition ${\text{}}_{(\theta -2)h}{\Delta }_h^{\theta -2+i}{\text{X}}_i(\eta ){|}_{t=0}={\text{X}}_i,i=0,1$.

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. If X_η and ${\text{X}}_{\eta }^{\theta }$ are the unique solution and α-path of UFFhDE (4.7) with the initial conditions (4.8), respectively. Assume that F + |G|Ψ⁻¹(θ) is a Lipschitz continues function in x with a Lipschitz constant L_k that has $0<L_k<\theta h^{-\theta }$. Assume that ξ_η is the i.i.d. symmetrical uncertain variable for $\eta \in {\mathbb{N}}_{(\theta -(n-1))h,h}^h\cap [0,Th]$, then

(i) ${\text{X}}_{\eta }\le {\text{X}}_{\eta }^{\theta }$ if ${\xi }_{\eta }(\gamma )\le {\Psi }^{-1}(\theta )$ for $\eta \in {{D}}^{+}$ and ${\xi }_{\eta }(\gamma )\ge {\Psi }^{-1}(1-\theta )$ for $\eta \in {{D}}^{-}$, where

$\begin{array}{l}{{D}}^{+}=\left\{\eta \in {\mathbb{N}}_{(\theta -(n-1))h,h}\cap [0,Th];\text{G}(\eta ,x)\ge 0\right\},\end{array}$

and

$\begin{array}{l}{{D}}^{-}=\left\{\eta \in {\mathbb{N}}_{(\theta -(n-1))h,h}\cap [0,Th];\text{G}(\eta ,x)<0\right\},\end{array}$

(ii) ${\text{X}}_{\eta }>{\text{X}}_{\eta }^{\theta }$ if ${\xi }_{\eta }(\gamma )>{\Psi }^{-1}(\theta )$ for $\eta \in {{D}}^{+}$ and ${\xi }_{\eta }(\gamma )<{\Psi }^{-1}(1-\theta )$ for $\eta \in {{D}}^{-}$.

Proof: First, we let ${\xi }_{\eta }(\gamma )\le {\Psi }^{-1}(\theta )$ for $\eta \in {{D}}^{+}$. Then η ∈ ℕ_{(θ−(n−1))h,h} ∩ [0, Th] and G(η, x) ≥ 0. Therefore,

$\begin{array}{ll}\text{G}(\eta ,x){\xi }_{\eta }(\gamma )\le |\text{G}(\eta ,x)|{\Psi }^{-1}(\theta ).& (4.12)\end{array}$

Moreover, if ${\xi }_{\eta }(\gamma )\ge {\Psi }^{-1}(1-\theta )$ for $\eta \in {{D}}^{-}$, 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,

$\begin{array}{l}\text{G}(\eta ,x){\xi }_{\eta }(\gamma )\le \text{G}(\eta ,x){\Psi }^{-1}(1-\theta )=-\text{G}(\eta ,x){\Psi }^{-1}(\theta )\\ \begin{array}{l}=|\text{G}(\eta ,x)|{\Psi }^{-1}(\theta ).\end{array}& (4.13)\end{array}$

Since X_η(γ) and ${\text{X}}_{\eta }^{\theta }$ are the unique solution and α-path of UFFhDE (4.7) with the initial conditions (4.8), respectively, we have

$\begin{array}{l}{}_{{\text{}}_{(\theta -n)h}}\Delta {\text{}}_h^{\theta }\text{X}(\eta )=\text{F}(\eta +(\theta -n)h,\text{X}(\eta +(\theta -n)h))\\ +\text{G}(\eta +(\theta -n)h,\text{X}(\eta +(\theta -n)h)){\xi }_{\eta +(\theta -n)h}(\gamma ),& (4.14)\end{array}$

$\begin{array}{l}{}_{{\text{}}_{(\theta -n)h}}\Delta {\text{}}_h^{\theta }\text{X}(\eta )=\text{F}(\eta +(\theta -n)h,\text{X}(\eta +(\theta -n)h))\\ +|\text{G}(\eta +(\theta -n)h,\text{X}(\eta +(\theta -n)h))|{\Psi }^{-1}(\theta ).& (4.15)\end{array}$

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 that F(η, x) and G(η, x) satisfy the Lipschitz condition

$\begin{array}{ll}|\text{F}(\eta ,x)-\text{F}(\eta ,\psi )|+|\text{G}(\eta ,x)-\text{G}(\eta ,\psi )|\le L|x-y|,& (4.16)\end{array}$

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

$\begin{array}{ll}L<h^{-\theta -1}\frac{\Gamma (\theta +1)\Gamma (T+1-\theta )}{\Gamma (T+1)(Q+1)},& (4.17)\end{array}$

where Q = |a| ∨ |b|. Then UFFhDE (4.7) with the initial conditions (4.8) has a unique solution X(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:

$\begin{array}{ll}{\text{}}_{-1}\begin{array}{l}{\Delta }_2^{0.5}\text{X}(\eta )=\frac{\text{sin }\text{X}(\eta -1)}{50+{(\eta -1)}^2}+{\xi }_{\eta -1},\eta \in {\mathbb{N}}_0^2\cap [0,8],\hfill \end{array}& (4.18)\end{array}$

where ξ₋₁, ξ₁, ξ₃, ξ₅, ξ₇ are 5 i.i.d. linear uncertain variables with linear uncertainty distribution ${L}(-2,2)$.

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

$\begin{array}{l}|\text{F}(\eta ,x)-\text{F}(\eta ,\psi )|+|\text{G}(\eta ,x)-\text{G}(\eta ,\psi )|\le \frac{1}{50}|x-y|=0.02|x-y|,\end{array}$

and

$\begin{array}{l}{h}^{-\theta -1}\frac{\Gamma (\theta +1)\Gamma (T+1-\theta )}{\Gamma (T+1)(Q+1)}=2^{-1.5}\frac{\Gamma (0.5+1)\Gamma (4+1-0.5)}{3\Gamma (4+1)}\\ \approx 0.05>0.02.\end{array}$

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

Now, since

$\begin{array}{l}\text{F}(\eta ,x)+|\text{G}(\eta ,x)|{\Psi }^{-1}(\theta )=\frac{\text{sin }x}{50+{(\eta -1)}^2}+4\theta -2,\end{array}$

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 ${L}(-2,2)$ is symmetrical. Hence, by Theorem 4.2, we deduce the following link between unique solution and α-path of UFFhDE (4.18):

(i) ${\text{X}}_{\eta }\le {\text{X}}_{\eta }^{\theta }$ if ξ_η ≤ 4θ − 2,

(ii) ${\text{X}}_{\eta }>{\text{X}}_{\eta }^{\theta }$ if ξ_η > 4θ − 2.

Example 4.6. Consider the following UFFhDE:

$\begin{array}{ll}{}_{{\text{}}_{-\frac{3}{8}}}\Delta {\text{}}_{\frac{1}{2}}^{\frac{1}{4}}\text{X}(\eta )=0.025{\text{X}}^2\left(\eta -\frac{3}{8}\right)+{\xi }_{\eta -\frac{3}{8}},\eta \in {\mathbb{N}}_0^{\frac{1}{2}}\cap \left[0,\frac{3}{2}\right],& (4.19)\end{array}$

where ${\xi }_{-\frac{3}{8}},{\xi }_{\frac{1}{8}},{\xi }_{\frac{5}{8}},{\xi }_{\frac{9}{8}}$ are 4 i.i.d. linear uncertain variables with linear uncertainty distribution ${L}(-3,3)$.

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

$\begin{array}{l}|\text{F}(\eta ,x)-\text{F}(\eta ,\psi )|+|\text{G}(\eta ,x)-\text{G}(\eta ,\psi )|\le 0.025|x+y||x-y|\\ =0.1|x-y|,\text{ for }x\in [-2,2],\end{array}$

and

$\begin{array}{l}{h}^{-\theta -1}\frac{\Gamma (\theta +1)\Gamma (T+1-\theta )}{\Gamma (T+1)(Q+1)}={\left(\frac{1}{2}\right)}^{-\frac{5}{4}}\frac{\Gamma (0.25+1)\Gamma (3+1-0.25)}{4\Gamma (3+1)}\\ \approx 0.4>0.1.\end{array}$

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

Now, since

$\begin{array}{l}\text{F}(\eta ,x)+|\text{G}(\eta ,x)|{\Psi }^{-1}(\theta )=0.025x^2+6\theta -3,\end{array}$

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 ${L}(-3,3)$ is symmetrical. Hence, by use of Theorem 4.2, we deduce that ${\text{X}}_{\eta }\le {\text{X}}_{\eta }^{\theta }$ if ξ_η ≤ 6θ − 3 and ${\text{X}}_{\eta }>{\text{X}}_{\eta }^{\theta }$ 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.

Data Availability Statement

The original contributions presented in the study are included in the article/supplementary materials, further inquiries can be directed to the corresponding author/s.

Author Contributions

All authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication.

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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