International Journal of Analysis
Volume 2016 (2016), Article ID 7265307, 5 pages
http://dx.doi.org/10.1155/2016/7265307

Research Article

Hyers-Ulam Stability of Fractional Nabla Difference Equations

Jagan Mohan Jonnalagadda

Department of Mathematics, Birla Institute of Technology and Science Pilani, Hyderabad, Telangana 500078, India

Received 21 July 2016; Accepted 28 August 2016

Academic Editor: Dong Ye

Copyright © 2016 Jagan Mohan Jonnalagadda. 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.

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 have

which implies

Set

Clearly . 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 get

which is similar to (14). Here

Example 11. Consider the following fractional nabla difference equation:

For and for all , we have

Let . Then, the exact solution of (35) is given by

Consequently, for , we get

where

Thus, (35) is Hyers-Ulam stable on . Furthermore, we illustrate these concepts numerically in Table 1.

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

where

Thus, (40) is generalized Hyers-Ulam stable. Furthermore, we illustrate these concepts numerically in Table 2.

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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