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

Authors: Pshtiwan Othman Mohammed; Thabet Abdeljawad; Fahd Jarad; Yu-Ming Chu
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

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.


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## Body

## 1. Introduction

Fractional calculus is based on an old idea that has become important and popular in applications only recently. The 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, [1–9].In recent years, uncertain fractional differential and difference equations and discrete difference equations have become popular in both theory and applications. These 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 [10–22] and the references cited therein.The 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 [23–25]. It is worth mentioning that the uncertainty theory of fractional difference equations is used to make the problems have a unique solution almost surely.


## 2. Preliminaries

This section presents some preliminaries, definitions, and facts in the field of uncertainty theory and discrete fractional calculus, see, e.g., [12, 26–28]. Throughout the article, we consider ℕa=a,a+1,a+2,…, for a∈ℝ and the backward jump operator ρr≔r−1 for r∈ℕa.Definition 1 (see [26]).
For any functionf:ℕa⟶ℝ, the backward difference operator is defined by(1)∇ft=ft−fρt,t∈ℕa,while the backward sum is given by(2)∇a−1ft=∑r=a+1tfr,t∈ℕa+1.Definition 2 (see [26–29]).
For any natural numberj, the ∇-rising factorial function of t is defined by(3)tj¯=∏ℓ=0j−1t+ℓ,t0¯=1.
Moreover, for anyν∈ℝ, the ∇-rising factorial function is defined by(4)tν¯=Γt+νΓt,0ν¯=0,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:(5)∇tν¯=νtν−1¯.
This implies thattν¯ is increasing on ℕ0 such that ν>0.Definition 3 (see [13, 14, 26–29]).
For any functionf:ℕa⟶ℝ, the nabla fractional sum of order ν>0 in the sense of Riemann–Liouville is defined by(6)∇a−νft=1Γν∑r=a+1tt−ρrν−1¯fr,∇a0ft=ft.Lemma 1 (see [13, 14, 26–28]).
For any functionf defined on ℕa and any ν,α>0, we have(7)∇a−ν∇a−αft=∇a−α+νft=∇a−α∇a−νft.Lemma 2 (see [13, 14, 26–28]).
For any functionf defined on ℕa and any ν>0, we have(8)∇aνft=∇∇a−1−νft.Lemma 3 (see [13, 14, 26–28]).
For any functionf defined on ℕa and any ν>0, we have(9)∇a−ν∇ft=∇∇a−νft−t−aν−1¯Γνfa.Lemma 4 (see [13, 14, 26–28]).
For any functionf defined on ℕa and any ν>0, we have(10)∇a−ν∇νft=∇aν∇−νft=ft,ν∉ℕ,∇a−ν∇νft=ft−∑k=0ν−1t−ak¯k!∇kfa,ν∈ℕ.Lemma 5 (see [13, 14, 26–28]).
For anya∈ℝ and ν,α>0, we have(11)∇a−νt−aα¯=Γα+1Γα+ν+1t−aα+ν¯.Motivated by the definition of thenth-order backward sum for uncertain sequence ξt, we define the νth-order backward sum for uncertain sequence ξt as follows:Definition 4 (see [13, 14, 28]).
Letν>0,a∈ℝ, and ξt be an uncertain sequence indexed by t∈ℕa. Then, we have(12)∇a−νξt=1Γν∑r=a+1tt−ρrν−1¯ξr,which is called the νth-order backward fractional sum of uncertain sequence ξt.Definition 5 (see [13, 14, 28]).
For anyν>0, the fractional Riemann–Liouville-like backward difference for uncertain sequence ξt is defined by(13)∇aνξt=∇a∇a−1−νξt.
Next, we recall the definition of nabla discrete Mittag-Leffler (ML).Definition 6 (see [28]).
For anyλ∈ℝ and ν,μ,η∈ℂ with Reν>0, the two-parameter discrete ML function is defined by(14)Eν,μ¯λ,η≔∑ℓ=0∞λℓηℓν+μ−1¯Γℓν+μ,λ<1.
Particularly, ifμ=1, we get the one-parameter discrete ML function:(15)Eν¯λ,η≔∑ℓ=0∞λℓηℓν¯Γℓν+1,λ<1.


## 3. 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(16)limx⟶−∞Ψx=0,limx⟶+∞Ψx=1.Definition 8.
(see [11]). 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 1.
From Definition8, we deduce that the following:(i)
The inverse uncertainty distribution of a linear uncertain variableℒa,b is given by(17)Ψ−1α=1−αa+αb.(ii)
The inverse uncertainty distribution of a normal uncertain variableLe,σ is given by(18)Ψ−1α=e+3σπlnα1−α.(iii)
The inverse uncertainty distribution of a normal uncertain variableLOGNe,σ is given by(19)Ψ−1α=expe+α1−α3σ/π.Definition 9.
(see [11]). We say that an uncertain variable ξ is symmetrical if(20)Ψx+Ψ−x=1,where Ψx is a regular uncertainty distribution of ξ.Remark 1.
From Definition9, we can deduce that the symmetrical uncertain variable has the inverse uncertainty distribution Ψ−1α that satiates(21)Ψ−1α+Ψ−11−α=0.Example 2.
From Definition9, we deduce that the following:(1)
The linear uncertain variableℒ−a,a is symmetrical for any positive real number a(2)
The normal uncertain variableL0,1 is symmetricalDefinition 10.
(see [11]; i.i.d. definition). In statistics and probability theory, a collection of random variables ξis is independent and identically distributed (or briefly, i.i.d.) if each random variable ξi has the same probability distribution as the others and all are mutually independent.
Then, 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:(22)∇α−1αyt=Gt,yt+Ht,ytξt,subject to the initial condition (i.c.)(23)∇α−1−1−αyt|t=α=yα,where ∇α−1α denotes fractional Riemann–Liouville-like backward difference with 0<α<1, G,H are two real-valued functions defined on 1,∞×ℝ,t∈ℕ1∩1,T+1, yα∈ℝ is a crisp number, and ξ1,ξ2,…,ξT+1 are T+1-i.i.d. uncertain variables with symmetrical uncertainty distribution ℒa,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:(24)yt=t−α+1α−1¯Γαyα+1Γα∑r=α+1tt−ρrα−1¯Gr,yr+Hr,yrξr,for t∈ℕα+1∩1,T+1.Proof.
The proof is very similar to [29], Lemma 5.1, and [30], Theorem 2, hereby applying the operator ∇α−α to IVP (22) with Definition 3, Lemma 2, and Lemma 3, so we omit this.
In this paper, the following special linear UFBDE will be considered:(25)∇α−1αyt=λyt+λξt,(26)∇α−1−1−αyt|t=α=yα,for t∈ℕ1∩1,T+1,n∈ℕ1, and λ∈0,1.Remark 3.
The following identity is useful in proving the upcoming theorem. From Definition3 and Lemma 5, we can deduce for any real number a(27)∇a−αt−a+1β¯=Γβ+1Γα+β+1t−a+1α+β¯−Γβ+1Γαt−a+1α−1¯,where α>0 and β>−1.Theorem 1.
For anyt∈ℕα+1∩1,T+1 and λ<1, linear UFBDE (25) with the initial condition (26) has a solution(28)yt=1−λEα,α¯λ,t−α+1yα+ξt,where ξt is an uncertain sequence with the uncertainty distribution ℒa⋅λEα,α+1¯λ,t−α,b⋅λEα,α+1¯λ,t−α.Proof.
Applying the operator∇α−α to equation (25), we get(29)∇α−α∇α−1αyt=λ∇α−αyt+λ∇α−αξt,t∈ℕα+1∩1,T+1.
Making use of Lemma2 and Lemma 3 to the left-hand side of (29), we get(30)∇α−α∇α−1αyt=∇α−α∇α∇α−1−1−αyt,=∇α∇α−α∇α−1−1−αyt−t−α+1α−1¯Γαyα,=yt−t−α+1α−1¯Γαyα,t∈ℕα+1∩1,T+1.
It follows from this and equation (29) that(31)yt=t−α+1α−1¯Γαyα+λ∇α−αyt+λ∇α−αξt,t∈ℕα+1∩1,T+1,which is the solution of UFBDE (28).
To derive the solution, we use the Picard approximation recurrence formula with a starting pointy0t=t−α+1α−1¯/Γαyα for each t∈ℕα−1∩1,T+1. The other components can be determined by using the following recurrence relation:(32)yjt=t−α+1α−1¯Γαyα+λ∇α−αyj−1t+λ∇α−αξt,for t∈ℕα+1∩1,T+1 and j∈ℕ1. Since ξ1,ξ2,…,ξT+1 are i.i.d. uncertain variables, we write ξt=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 Theorems 1.21–1.24 of [11]), we can deduce(33)y1t=t−α+1α−1¯Γαyα+λ∇α−αy0t+λ∇α−αξ=t−α+1α−1¯Γαyα+λt−α+12α−1¯Γ2α−t−α+1α−1¯Γαyα+λt−αα¯Γα+1ξ,y2t=t−α+1α−1¯Γαyα+λ∇α−αy1t+λ∇α−αξ=t−α+1α−1¯Γαyα+λt−α+12α−1¯Γ2α−t−α+1α−1¯Γαyα+λ2t−α+13α−1¯Γ3α−t−α+12α−1¯Γ2αyα+λt−αα¯Γα+1+λ2t−α2α¯Γ2α+1ξ,⋮and so on, continuing the process up to the jth term to get(34)yjt=∑k=0jλkt−α+1kα+α−1¯Γk+1α−t−α+1kα−1¯Γkαyα+∑k=1jλkt−αkα¯Γkα+1ξ,for each t∈ℕα+1∩1,T+1. Observe that the two series(35)∑k=0∞λkt−α+1kα+α−1¯Γk+1α−t−α+1kα−1¯Γkα=Eα,α¯λ,t−α+1−λEα,α¯λ,t−α+1=1−λEα,α¯λ,t−α+1,∑k=1∞λkt−αkα¯Γkα+1=λEα,α+1¯λ,t−α,are absolutely convergent for λ<1 by the d’Alembert ratio comparison test, and the limitation Y¯t≔limj⟶∞yj exists. Thus, we have(36)Y¯t=λEα,α+1¯λ,t−αξ+1−λEα,α¯λ,t−α+1yα,t∈ℕα+1∩1,T+1.
On the contrary, taking the limit on both sides of (32) yields(37)Y¯t=t−α+1α−1¯Γαyα+λ∇α−αY¯t+λ∇α−αξt,t∈ℕα+1∩1,T+1.
That is,Y¯t satisfies equation (31), and hence, Y¯t is a solution of equation (25) subject to the initial condition (26). Thus, our proof is completed.
The following theorem provides and confirms the existence and uniqueness of the solution of UFBDEs.Theorem 2.
Assume thatGt,x and Ht,x satisfy the Lipschitz condition(38)Gt,x−Gt,y+Ht,x−Ht,y≤Lx−y,and there is a positive number L that satisfies the following inequality:(39)L<Γα+1ΓT+1−αΓT+1Q+1,where Q=a∨b. Then, UFBDE (25) subject to the initial condition (26) has a unique solution yt for t∈ℕα+1∩1,T+1 almost surely.Proof.
Define(40)lαk≔x;x=xtα+1k,k∈ℕ1,x,≔maxt∈ℕα+1∩1,T+1xt,where xtαk are finite real sequences which have k terms. It is clear that lαk,⋅ is a Banach space (see [31], Chapter 4). Now, for any yt∈lαk, we define the operator P as follows:(41)Pyt=t−αα−1¯Γαyα+1Γα∑r=α+1tt−ρrα−1¯Gr,yr+Hr,yrξr.
Sinceξtt∈ℕα+1∩1,T+1 is an uncertain variable at each time t with the linear uncertainty distribution ℒa,b, we have ℳξt<a∪ξt>b=0. The inequality ξtγ≤Q (where Q=a∨b) holds almost surely for any γ∈χξt<a∪ξt>b;t∈ℕα+1∩1,T+1, where χ represents the universal set on the uncertainty space. Then, by making use of the assumptions and Lemma 5, we have, for any xt,yt∈lαk,(42)Pxtγ−Pytγ=maxt∈ℕα+1∩1,T+1Pxtγ−Pytγ,≤1Γαmaxt∈ℕα+1∩1,T+1∑r=α+1tt−ρrα−1¯,×Gr,xrγ−Gr,yrγ+Hr,xrγ−Hr,yrγξr,≤1Γαmaxt∈ℕα+1∩1,T+1∑r=α+1tt−ρrα−1¯,×Gr,xrγ−Gr,yrγ+QHr,xrγ−Hr,yrγ,≤L1+Q1Γαmaxt∈ℕα+1∩1,T+1∑r=α+1tt−ρrα−1¯xrγ−yrγ,≤L1+Qxtγ−ytγmaxt∈ℕα+1∩1,T+1∇α−αt−α0¯,=L1+Qxtγ−ytγmaxt∈ℕα+1∩1,T+11Γα+1t−αα¯,≤L1+QT+1−αα¯Γα+1xtγ−ytγ,=L1+QΓT+1Γα+1ΓT−α+1xtγ−ytγ.
Now, we can observe that the mappingP is a contraction in lαk almost surely with 0<L<Γα+1ΓT−α+1/1+QΓT+1 (see [31], Chapter 4). Then, by using the Banach contraction mapping theorem (see [31], Chapter 4), we get a unique fixed point ytγ of P in lαk almost surely. Moreover, ytγ=limj⟶∞ytjγ, where ytjγ=Pytj−1γ, with yt0γ=t−αα−1¯/Γαyα.
For any givent∈ℕα+1∩1,T+1, as G and H are Lipschitz continuous functions, the operator P is measurable. Since yt1γ,yt2γ,…,ytjγ,… are uncertain variables and yt0γ is a real-valued measurable function of uncertain variables, yt0γ is an uncertain variable by [11], Theorem 1.10. Hence, yt=limj⟶∞ytj is an uncertain variable (see [21], Theorem 3).
Consequently, UFBDE (25) with i.c. (26) has a unique solution yt for t∈ℕα+1∩1,T+1 almost surely.


## 4. Example Illustrations

This section deals with some examples to confirm the validity of Theorem2.Example 3.
Consider the following UFBDE:(43)∇−1/21/2yt=lnyt+12t3+0.5ξt,t∈ℕ1∩1,4,y12=1,where ξ1,…,ξ4 are four i.i.d. uncertain variables with uncertainty distribution ℒ−1,2.
According to Lemma6 with α=1/2, the inverse uncertainty distribution of the solution for UFBDE (43) is the solution of the following sum equation:(44)yt=t+1/2−0.5¯Γ0.5yα+1Γ0.5∑r=3/2tt−ρr−0.5¯lnyr+14r3+0.25ξr,t∈ℕ0.5∩1,4.
Then, fort∈ℕ0.5∩1,4, we have(45)Gt,x−Gt,y+Ht,x−Ht,y=lnx+12t3−lny+12t3,=12t3lnx+1−lny+1,≤123/23x−y≤4x−y27,Γ0.5+1Γ4+1−0.53Γ4+1≈0.1636>427=0.1481.
Thus, UFBDE (43) has a unique solution almost surely by Theorem 2.Example 4.
We consider the following UFBDE:(46)∇−0.250.25yt=y2t40+ξt,t∈ℕ1∩1,4,where ξ1,ξ2,ξ3,ξ4 are four i.i.d. linear uncertain variables with linear uncertainty distribution ℒ−3,3.
According to Lemma6 with α=1/4, the inverse uncertainty distribution of the solution for UFBDE (46) is the solution of the following sum equation:(47)yt=t+1/2−3/4¯Γ0.25yα+1Γ0.25∑r=1.25tt−ρr−3/4¯y2r40+ξr.
Observe thatGt,x−Gt,y+Ht,x−Ht,y is Lipschitz-continuous in −20,20 with Lipschitz constant 0.1 as follows:(48)Gt,x−Gt,y+Ht,x−Ht,y≤140x+yx−y=0.1x−y.
Also, we have(49)Γ0.25+1Γ3+1−0.254Γ3+1≈0.167>0.1.
Consequently, UFBDE (50) has a unique solution almost surely by Theorem 2.Example 5.
Consider the following UFBDE:(50)∇−0.50.5yt=sinty10+t2+0.1ξt,t∈ℕ1∩1,4,where ξ1,ξ2,ξ3,ξ4 are 4 i.i.d. linear uncertain variables with linear uncertainty distribution ℒ−1,1.
According to Lemma6 with α=1/2, the inverse uncertainty distribution of the solution for UFBDE (50) is the solution of the following sum equation:(51)yt=t+1/2−0.5¯πyα+1π∑r=1.5tt−ρr−0.5¯sinry10+r2+0.1ξr.
Then, we can directly verify that(52)Gt,x−Gt,y+Ht,x−Ht,y≤110x−y,Γ0.5+1Γ3+1−0.52Γ3+1≈0.2454>110=0.1.
Consequently, UFBDE (50) has a unique solution almost surely by Theorem 2.


## 5. 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.



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