Mathematical Problems in Engineering
Volume 2020 (2020), Article ID 6598682, 8 pages
https://doi.org/10.1155/2020/6598682

Research Article

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

Pshtiwan Othman Mohammed,¹ Thabet Abdeljawad,2,3,4 Fahd Jarad,⁵ and Yu-Ming Chu6,7

¹Department of Mathematics, College of Education, University of Sulaimani, Kurdistan Region, Sulaimani, Iraq
²Department of Mathematics and General Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi Arabia
³Department of Medical Research, China Medical University, Taichung 40402, Taiwan
⁴Department of Computer Science and Information Engineering, Asia University, Taichung, Taiwan
⁵Department of Mathematics, Cankaya University, Etimesgut, Ankara 06790, Turkey
⁶Department of Mathematics, Huzhou University, Huzhou 313000, China
⁷Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, Changsha University of Science & Technology, Changsha 410114, China

Correspondence should be addressed to Thabet Abdeljawad; tabdeljawad@psu.edu.sa and Yu-Ming Chu; chuyuming@zjhu.edu.cn

Received 6 August 2020; Revised 6 September 2020; Accepted 29 September 2020; Published 17 October 2020

Academic Editor: A. M. Abd-Alla

Copyright © 2020 Pshtiwan Othman Mohammed et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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 for and the backward jump operator for .

Definition 1 (see [26]). For any function , the backward difference operator is defined by

while the backward sum is given by

Definition 2 (see [26–29]). For any natural number , the -rising factorial function of is defined by

Moreover, for any , the -rising factorial function is defined by

for . 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:

This implies that is increasing on such that .

Definition 3 (see [13, 14, 26–29]). For any function , the nabla fractional sum of order in the sense of Riemann–Liouville is defined by

Lemma 1 (see [13, 14, 26–28]). For any function defined on and any , we have

Lemma 2 (see [13, 14, 26–28]). For any function defined on and any , we have

Lemma 3 (see [13, 14, 26–28]). For any function defined on and any , we have

Lemma 4 (see [13, 14, 26–28]). For any function defined on and any , we have

Lemma 5 (see [13, 14, 26–28]). For any and , we have

Motivated by the definition of the th-order backward sum for uncertain sequence , we define the th-order backward sum for uncertain sequence as follows:

Definition 4 (see [13, 14, 28]). Let , and be an uncertain sequence indexed by . Then, we have

which is called the th-order backward fractional sum of uncertain sequence .

Definition 5 (see [13, 14, 28]). For any , the fractional Riemann–Liouville-like backward difference for uncertain sequence is defined by

Next, we recall the definition of nabla discrete Mittag-Leffler (ML).

Definition 6 (see [28]). For any and with , the two-parameter discrete ML function is defined by

Particularly, if , we get the one-parameter discrete ML function:

3. UFBDE and Existence and Uniqueness Theorem

First, we recall the inverse uncertainty distribution theory.

Definition 7. (see [11]). An uncertainty distribution is called regular if it is a continuous and strictly increasing function and satisfies

Definition 8. (see [11]). Let be an uncertain variable with a regular uncertainty distribution . Then, the inverse function is called the inverse uncertainty distribution of .

Example 1. From Definition 8, we deduce that the following:

(i)The inverse uncertainty distribution of a linear uncertain variable is given by

(ii)The inverse uncertainty distribution of a normal uncertain variable is given by

(iii)The inverse uncertainty distribution of a normal uncertain variable is given by

Definition 9. (see [11]). We say that an uncertain variable is symmetrical if

where is a regular uncertainty distribution of .

Remark 1. From Definition 9, we can deduce that the symmetrical uncertain variable has the inverse uncertainty distribution that satiates

Example 2. From Definition 9, we deduce that the following:

(1)The linear uncertain variable is symmetrical for any positive real number (2)The normal uncertain variable is symmetrical

Definition 10. (see [11]; i.i.d. definition). In statistics and probability theory, a collection of random variables is independent and identically distributed (or briefly, i.i.d.) if each random variable 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:

subject to the initial condition (i.c.)

where denotes fractional Riemann–Liouville-like backward difference with , are two real-valued functions defined on , is a crisp number, and are -i.i.d. uncertain variables with symmetrical uncertainty distribution .

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:

for .

Proof. The proof is very similar to [29], Lemma 5.1, and [30], Theorem 2, hereby applying the operator 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:

for , and .

Remark 3. The following identity is useful in proving the upcoming theorem. From Definition 3 and Lemma 5, we can deduce for any real number

where and .

Theorem 1. For any and , linear UFBDE (25) with the initial condition (26) has a solution

where is an uncertain sequence with the uncertainty distribution .

Proof. Applying the operator 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 for each . The other components can be determined by using the following recurrence relation:

for and . Since are i.i.d. uncertain variables, we write 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

and so on, continuing the process up to the th term to get

for each . Observe that the two series

are absolutely convergent for by the d’Alembert ratio comparison test, and the limitation exists. Thus, we have

On the contrary, taking the limit on both sides of (32) yields

That is, satisfies equation (31), and hence, 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 that and satisfy the Lipschitz condition

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

where . Then, UFBDE (25) subject to the initial condition (26) has a unique solution for almost surely.

Proof. Define

where are finite real sequences which have terms. It is clear that is a Banach space (see [31], Chapter 4). Now, for any , we define the operator as follows:

Since is an uncertain variable at each time with the linear uncertainty distribution , we have . The inequality (where ) holds almost surely for any , where represents the universal set on the uncertainty space. Then, by making use of the assumptions and Lemma 5, we have, for any ,

Now, we can observe that the mapping is a contraction in almost surely with (see [31], Chapter 4). Then, by using the Banach contraction mapping theorem (see [31], Chapter 4), we get a unique fixed point of in almost surely. Moreover, , where , with .
For any given , as and are Lipschitz continuous functions, the operator is measurable. Since are uncertain variables and is a real-valued measurable function of uncertain variables, is an uncertain variable by [11], Theorem 1.10. Hence, is an uncertain variable (see [21], Theorem 3).
Consequently, UFBDE (25) with i.c. (26) has a unique solution for almost surely.

4. Example Illustrations

This section deals with some examples to confirm the validity of Theorem 2.

Example 3. Consider the following UFBDE:

where are four i.i.d. uncertain variables with uncertainty distribution .
According to Lemma 6 with , the inverse uncertainty distribution of the solution for UFBDE (43) is the solution of the following sum equation:

Then, for , we have

Thus, UFBDE (43) has a unique solution almost surely by Theorem 2.

Example 4. We consider the following UFBDE:

where are four i.i.d. linear uncertain variables with linear uncertainty distribution .
According to Lemma 6 with , the inverse uncertainty distribution of the solution for UFBDE (46) is the solution of the following sum equation:

Observe that is Lipschitz-continuous in with Lipschitz constant 0.1 as follows:

Also, we have

Consequently, UFBDE (50) has a unique solution almost surely by Theorem 2.

Example 5. Consider the following UFBDE:

where are 4 i.i.d. linear uncertain variables with linear uncertainty distribution .
According to Lemma 6 with , the inverse uncertainty distribution of the solution for UFBDE (50) is the solution of the following sum equation:

Then, we can directly verify that

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.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally and significantly to writing this article. All authors read and approved the final manuscript.

Acknowledgments

The first author would like to thank Prince Sultan University for funding this work through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM) (group number: RG-DES-2017-01-17).

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