Research Article | Open Access

Volume 2016 |Article ID 7265307 | https://doi.org/10.1155/2016/7265307

Jagan Mohan Jonnalagadda, "Hyers-Ulam Stability of Fractional Nabla Difference Equations", International Journal of Analysis, vol. 2016, Article ID 7265307, 5 pages, 2016. https://doi.org/10.1155/2016/7265307

# Hyers-Ulam Stability of Fractional Nabla Difference Equations

Accepted28 Aug 2016
Published21 Sep 2016

#### 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  posed the following problem on the stability of functional equations in 1940.

Ulam’s Problem (see ). 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  solved the problem for additive functions defined on Banach spaces in 1941 as follows.

Hyer’s Theorem (see ). 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  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.  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  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 : 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) : the th-order nabla sum of is given by(2)(R-L fractional nabla difference) : the th-order nabla difference of is given by(3)(Caputo fractional nabla difference) : the th-order nabla difference of is given by

Nagai  and Atici and Eloe  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 ). Let . The -transform of is defined by for each for which the series converges.

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

Atici and Eloe  developed the following properties of -transforms.

Theorem 7 (see ). 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.

 0 1 0.09 0.1 1.0000 0 1.1111 1 1 0.09 0.1 1.1364 0.1364 1.1111 2 1 0.09 0.1 1.1389 0.1389 1.1111 3 1 0.09 0.1 1.1346 0.1346 1.1111 4 1 0.09 0.1 1.1292 0.1292 1.1111 5 1 0.09 0.1 1.1238 0.1238 1.1111 6 1 0.09 0.1 1.1188 0.1188 1.1111 7 1 0.09 0.1 1.1140 0.1140 1.1111 8 1 0.09 0.1 1.1096 0.1096 1.1111 9 1 0.09 0.1 1.1054 0.1054 1.1111 10 1 0.09 0.1 1.1015 0.1015 1.1111 11 1 0.09 0.1 1.0979 0.0979 1.1111 12 1 0.09 0.1 1.0944 0.0944 1.1111

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.

 0 0.9064 1.5142 2.2257 0.9064 0 0.0362 1 1.1330 1.0452 1.4527 0.5711 0.5619 1.4980 2 1.2746 0.9846 1.3020 0.4384 0.8362 2.4424 3 1.3808 0.9751 1.2443 0.3618 1.0190 3.2293 4 1.4672 0.9803 1.2182 0.3110 1.1562 3.9384 5 1.5405 0.9908 1.2064 0.2746 1.2659 4.6007 6 1.6047 1.0036 1.2045 0.2471 1.3576 5.2317 7 1.6620 1.0172 1.2023 0.2257 1.4363 5.8401 8 1.7139 1.0311 1.2050 0.2084 1.5055 6.4313 9 1.7616 1.0449 1.2095 0.1942 1.5674 7.0092 10 1.8056 1.0584 1.2151 0.1824 1.6232 7.5762 11 1.8466 1.0716 1.2214 0.1723 1.6743 8.1343 12 1.8851 1.0845 1.2282 0.1636 1.7215 8.6849

#### Competing Interests

The author declares that they have no competing interests.

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