Existence and Uniqueness of Uncertain Fractional Backward Difference Equations of Riemann–Liouville Type

In this article, we consider the analytic solutions of the uncertain fractional backward diﬀerence equations in the sense of Riemann–Liouville fractional operators which are solved by using the Picard successive iteration method. Also, we consider the existence and uniqueness theorem of the solution to an uncertain fractional backward diﬀerence equation via the Banach contraction ﬁxed-point theorem under the conditions of Lipschitz constant and linear combination growth. Finally, we point out some examples to conﬁrm the validity of the existence and uniqueness of the solution.


Introduction
Fractional calculus is based on an old idea that has become important and popular in applications only recently. e idea is to generalize integration and differentiation to noninteger orders in order to develop and extend the theory of calculus and to describe a more extensive range of possible doings in reality. During the past decades, fractional differential equations have been widely employed in many fields: mathematical analysis, optics and thermal systems, control engineering, and robotics, see, for example, [1][2][3][4][5][6][7][8][9].
In recent years, uncertain fractional differential and difference equations and discrete difference equations have become popular in both theory and applications. ese represent a new area for researchers which was developed slowly in their early stages. By using modeling techniques with discrete fractional calculus, some researchers established the existence, uniqueness, monotonicity, multiplicity, and qualitative properties of solutions to uncertain fractional difference equations (UFDEs) in the sense of Riemann-Liouville, Caputo, and AB operators; for further details, see [10][11][12][13][14][15][16][17][18][19][20][21][22] and the references cited therein. e aim of this attempt is to investigate the existence and uniqueness of fractional difference equations in the sense of Riemann-Liouville-like difference operator with assuming Lipschitz condition on its nonlinear term. Our findings are partial continuation of some results obtained in [23][24][25]. It is worth mentioning that the uncertainty theory of fractional difference equations is used to make the problems have a unique solution almost surely.
Definition 1 (see [26]). For any function f: N a ⟶ R, the backward difference operator is defined by while the backward sum is given by Definition 2 (see [26][27][28][29]). For any natural number j, the ∇-rising factorial function of t is defined by Moreover, for any ] ∈ R, the ∇-rising factorial function is defined by for t ∈ R\ . . . , − 2, − 1, 0 { }. Also, note that the division by negative integer poles of the gamma function gives zero.
A major property of the rising factorial function is as follows: is implies that t ] is increasing on N 0 such that ] > 0.

UFBDE and Existence and Uniqueness Theorem
First, we recall the inverse uncertainty distribution theory.
Definition 7. (see [11]). An uncertainty distribution Ψ is called regular if it is a continuous and strictly increasing function and satisfies Definition 8. (see [11]). Let ξ be an uncertain variable with a regular uncertainty distribution Ψ. en, the inverse function Ψ − 1 is called the inverse uncertainty distribution of ξ. Example 1. From Definition 8, we deduce that the following: (i) e inverse uncertainty distribution of a linear uncertain variable L(a, b) is given by (ii) e inverse uncertainty distribution of a normal uncertain variable L(e, σ) is given by (iii) e inverse uncertainty distribution of a normal uncertain variable LOGN(e, σ) is given by Definition 9. (see [11]). We say that an uncertain variable ξ is symmetrical if where Ψ(x) is a regular uncertainty distribution of ξ.
Remark 1. From Definition 9, we can deduce that the symmetrical uncertain variable has the inverse uncertainty distribution Ψ − 1 (α) that satiates Example 2. From Definition 9, we deduce that the following: (1) e linear uncertain variable L(− a, a) is symmetrical for any positive real number a (2) e normal uncertain variable L(0, 1) is symmetrical Definition 10. (see [11]; i.i.d. definition). In statistics and probability theory, a collection of random variables ξ i s is independent and identically distributed (or briefly, i.i.d.) if each random variable ξ i has the same probability distribution as the others and all are mutually independent. en, we state the definition of the UFBDE.
Definition 11. An uncertain fractional difference equation is a fractional difference equation which is driven by an uncertain sequence. Moreover, an uncertain fractional backward difference equation for the Riemann-Liouville type is the uncertain fractional difference equation with Riemann-Liouville-like backward difference. Consider the following generalized Riemann-Liouville fractional difference equation: subject to the initial condition (i.c.) where ∇ α α− 1 denotes fractional Riemann-Liouville-like backward difference with 0 < α < 1, G, H are two real-valued a crisp number, and ξ 1 , ξ 2 , . . . , ξ T+1 are (T + 1)-i.i.d. uncertain variables with symmetrical uncertainty distribution L(a, b).

Remark 2.
Observe that the i.i.d. uncertain variables are those uncertain variables that are independent and have the same uncertainty distribution. See [11] for more detail. Lemma 6. Initial value problem (22) with i.c. (23) is equivalent to the following uncertain fractional sum equation: (t − ρ(r)) α− 1 G(r, y(r)) + H(r, y(r))ξ r ,

Remark 3.
e following identity is useful in proving the upcoming theorem. From Definition 3 and Lemma 5, we can deduce for any real number a where α > 0 and β > − 1.
Proof. Applying the operator ∇ − α α to equation (25), we get Making use of Lemma 2 and Lemma 3 to the left-hand side of (29), we get It follows from this and equation (29) that which is the solution of UFBDE (28).
To derive the solution, we use the Picard approximation recurrence formula with a starting point y 0 (t) � . e other components can be determined by using the following recurrence relation: for t ∈ N α+1 ∩ [1, T + 1] and j ∈ N 1 . Since ξ 1 , ξ 2 , . . . , ξ T+1 are i.i.d. uncertain variables, we write ξ t � ξ in distribution. By using Lemma 5, Remark 3, and the fact that the linear combination of finite independent uncertain variables is an uncertain variable with a positive linear combination coefficient (see eorems 1.21-1.24 of [11]), we can deduce and so on, continuing the process up to the jth term to get are absolutely convergent for |λ| < 1 by the d'Alembert ratio comparison test, and the limitation Y(t) ≔ lim j⟶∞ y j exists. us, we have On the contrary, taking the limit on both sides of (32) yields at is, Y(t) satisfies equation (31), and hence, Y(t) is a solution of equation (25) subject to the initial condition (26).
us, our proof is completed. e following theorem provides and confirms the existence and uniqueness of the solution of UFBDEs.

Theorem 2. Assume that G(t, x) and H(t, x) satisfy the Lipschitz condition
and there is a positive number L that satisfies the following inequality: where Q � |a| ∨ |b|. en, UFBDE (25) subject to the initial condition (26) has a unique solution y(t) for t ∈ N α+1 ∩ [1, T + 1] almost surely.
Proof. Define where x(t) { } k α are finite real sequences which have k terms. It is clear that (l k α , ‖·‖) is a Banach space (see [31], Chapter 4). Now, for any y t ∈ l k α , we define the operator P as follows: (t − ρ(r)) α− 1 G r, y r + H r, y r ξ r .
According to Lemma 6 with α � (1/2), the inverse uncertainty distribution of the solution for UFBDE (50) is the solution of the following sum equation: Consequently, UFBDE (50) has a unique solution almost surely by eorem 2.

Conclusion
We have presented analytical solutions to a special type of linear UFBDEs. Moreover, a Lipschitz condition with its constant is given to provide a unique solution almost surely to an UFBDE. It can be seen that our obtained results pave the way for the future works, that is, to investigate the stability analysis and applications of UFBDEs.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.