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.


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][2][3][4][5][6][7][8][9][10][11][12][13][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][18][19][20][21][22][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][25][26][27][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][31][32][33][34][35][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.

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 2.2 ([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 η (θ ) h 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,h N for the right case. Then, the left delta h-fractional sum of order θ > 0 is defined by h ψ(rh)h, η ∈ N a+θ h,h , 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,h N, respectively. Then the left and right delta h-fractional differences of order θ are defined by

Lemma 2.5 ([40]
). Let θ ∈ R and q be any positive integer, then Motivated by the definition of nth order forward sum for uncertain sequence ξ η , we define the θ th 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 θ th order forward fractional sum of uncertain sequence ξ η , where σ (r) = r + h. where θ > 0 and 0 ≤ n − 1 < µ ≤ n, n represents a positive integer.
(2) Analogously, we can obtain the proof of this item, and thus our proof is completely done.

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:  (ii) the inverse uncertainty distribution of a normal uncertain variable N (e, σ ) is given by (iii) and the inverse uncertainty distribution of a normal uncertain variable LOGN (e, σ ) is given by (4.4)

Theorem 4.2.
If X η and X θ η are the unique solution and α-path of UFFhDE (4.7) with the initial conditions (4.8), respectively. Assume that F + |G| −1 (θ ) is a Lipschitz continues function in x with a Lipschitz constant L k that has 0 < L k < θ h −θ . Assume that ξ η is the i.i.d. symmetrical uncertain variable for Proof: First, we let ξ η (γ ) ≤ −1 (θ ) for η ∈ D + . Then η ∈ N (θ −(n−1))h,h ∩ [0, Th] and G(η, x) ≥ 0. Therefore, Since X η (γ ) and X θ η 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. 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 solution X(η) for η ∈ N θ h,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.

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.