#### Abstract

We investigate the Hyers-Ulam stability, the generalized Hyers-Ulam stability, and the -Hyers-Ulam stability of a linear fractional nabla difference equation using discrete Laplace transform. We provide a few examples to illustrate the applicability of established results.

#### 1. Introduction

Ulam [1] posed the following problem on the stability of functional equations in 1940.

Ulam’s Problem (see [1]). Given a group , a metric group , and a positive number , does there exist such that if a mapping satisfies the inequality for all , then there exists a homomorphism such that for all ?

Hyers [2] solved the problem for additive functions defined on Banach spaces in 1941 as follows.

Hyer’s Theorem (see [2]). Let be a normed vector space and a Banach space and suppose that the mapping satisfies the inequality for all , where is a constant. Then the limit exists for each and is the unique additive mapping satisfying for all .

Rassias [3] provided a generalization of the Hyers theorem for linear mappings. Later, many mathematicians have extended Ulam’s problem in different directions. Recently, a generalization of Ulam’s problem on the stability of differential equations was proposed.

Let be a normed space and be an open interval. The differential equation is Hyers-Ulam stable, if, for given and a function such that , there exists a solution of the differential equation such that for any , where is an expression of only. If the above statement is also true when we replace and by and , where are functions not depending on and explicitly, then we say that the corresponding differential equation has the generalized Hyers-Ulam stability. For a detailed discussion on the Hyers-Ulam stability, refer to [4, 5].

Recently, Rezaei et al. [6] obtained the Hyers-Ulam stability of a linear differential equation using Laplace transforms. Motivated by this article, Wang and Xu [7, 8] and Wang and Li [9] investigated the same for a class of linear fractional differential equations involving both Riemann-Liouville and Caputo type fractional derivatives. In this article, we extend this study to linear fractional nabla difference equations.

#### 2. Preliminaries

Throughout this article, we use the following notation, definitions, and known results of fractional nabla calculus [10]: Denote the set of all real numbers and complex numbers by and , respectively. Define for any . Assume that empty sums and products are taken to be 0 and 1, respectively.

Definition 1 (rising factorial function). For any , such that , the rising factorial function is defined by

Definition 2. Let and .(1)(Fractional nabla sum) [11]: the th-order nabla sum of is given by(2)(R-L fractional nabla difference) [11]: the th-order nabla difference of is given by(3)(Caputo fractional nabla difference) [11]: the th-order nabla difference of is given by

Nagai [12] and Atici and Eloe [13] defined the one- and two-parameter Mittag-Leffler functions of fractional nabla calculus as follows.

Definition 3 (see [12, 13]). The one- and two-parameter nabla Mittag-Leffler functions are defined by where , , and .

Estimates of nabla Mittag-Leffler functions are provided in Lemma 4.

Lemma 4. Let . The functions and are nonnegative and for any and ,

Definition 5 (see [13]). Let . The -transform of is defined by for each for which the series converges.

Definition 6 (see [13]). Let . The convolution of and is defined by

Atici and Eloe [13] developed the following properties of -transforms.

Theorem 7 (see [13]). Assume that the following functions are well defined: (1).(2).(3).(4).(5).

#### 3. Main Results

The main purpose of this section is to discuss the Hyers-Ulam stability of the following difference equation:where and is a constant.

Let and for . Using Theorem 7, we havewhich impliesSetClearly . Applying the transform on both sides of (17), we get which implies Since is one-to-one, it follows that , so is a solution of (14). From (16) and (18), we get

Since is one-to-one, it follows that

First, we establish the generalized Hyers-Ulam stability of (14) as follows.

Theorem 8. Let and . If then, there exists a solution of (14) and such that

Proof. Using (21) and Lemma 4, we have

Now, consider a particular case of Theorem 8 which we define as the -Hyers-Ulam stability of (14).

Corollary 9. Let , , and . If then, there exists a solution of (14) such that where

Next, we investigate the Hyers-Ulam stability of (14).

Theorem 10. Let and . If there exists a solution of (14) such that where

Proof. Using (21) and Lemma 4, we have

Finally, we discuss the Hyers-Ulam stability of the following Caputo type linear fractional nabla difference equation:Using Definition 2 in (32), we getwhich is similar to (14). Here

Example 11. Consider the following fractional nabla difference equation:

For and for all , we haveLet . Then, the exact solution of (35) is given by Consequently, for , we getwhere Thus, (35) is Hyers-Ulam stable on . Furthermore, we illustrate these concepts numerically in Table 1.

Example 12. Consider the following fractional nabla difference equation:

For , we have Let . Then, the exact solution of (40) is given by Consequently, we get whereThus, (40) is generalized Hyers-Ulam stable. Furthermore, we illustrate these concepts numerically in Table 2.

#### Competing Interests

The author declares that they have no competing interests.