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.