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Hyers-Ulam Stability of Fractional Nabla Difference Equations.

1. Introduction

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

Ulam's Problem (see [1]). Given a group [G.sub.1], a metric group ([G.sub.2], d), and a positive number [epsilon], does there exist [delta] > 0 such that if a mapping f: [G.sub.1] [right arrow] [G.sub.2] satisfies the inequality

d(f(xy), f(x), f(y)) < [delta], (1)

for all x, y [member of] G1, then there exists a homomorphism T : [G.sub.1] [right arrow] [G.sub.2] such that

d(f(x), T(x)) < [delta], (2)

for all x, y [member of] [G.sub.1]?

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

Hyer's Theorem (see [2]). Let [E.sub.1] be anormed vector space and [E.sub.2] a Banach space and suppose that the mapping f: [E.sub.1] [right arrow] [E.sub.2] satisfies the inequality

[parallel]f(x + y) - f(x) - f(y)[parallel] [less than or equal to] [epsilon], (3)

for all x, y [member of] [E.sub.1], where [epsilon] > 0 is a constant. Then the limit

[mathematical expression not reproducible] (4)

exists for each x [member of] [E.sub.1] and g is the unique additive mapping satisfying

[parallel]f(x) - g(x)[parallel] [less than or equal to] [epsilon], (5)

for all x [member of] [E.sub.1].

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 X be a normed space and I be an open interval. The differential equation F(t, u, u', u", ..., [u.sup.(n)] = 0 is Hyers-Ulam stable, if, for given [epsilon] > 0 and a function v : I [right arrow] X such that [parallel]F(t, v, v', v", ..., [v.sup.(n)])[parallel] [less than or equal to] [epsilon], there exists a solution u : I [right arrow] X of the differential equation such that [parallel]u(t) - v(f)[parallel] [less than or equal to] K([epsilon]) for any t [member of] I, where K([epsilon]) is an expression of [epsilon] only. If the above statement is also true when we replace [epsilon] and K([epsilon]) by [phi](t) and [psi](t), where [phi],[psi] :I [right arrow] [0, [infinity]) are functions not depending on v and u 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 R and C, respectively. Define [N.sub.a] = {a, a+ 1, a + 2,...} for any a [member of] R. Assume that empty sums and products are taken to be 0 and 1, respectively.

Definition 1 (rising factorial function). For any [alpha] [member of] R, t [member of] R \ {..., -2, -1, 0} such that (t + [alpha]) [member of] R \ {..., -2, -1,0}, the rising factorial function is defined by

[t.sup.[bar.[alpha]]] = [GAMMA](t + [alpha])/[GAMMA](t), [0.sup.[bar.[alpha]]] = 0. (6)

Definition 2. Let u : [N.sub.a] [right arrow] R and 0 < [alpha] < 1.

(1) (Fractional nabla sum) [11]: the ath-order nabla sum of u is given by

[mathematical expression not reproducible]. (7)

(2) (R-L fractional nabla difference) [11]: the [alpha]th-order nabla difference of u is given by

[mathematical expression not reproducible]. (8)

(3) (Caputo fractional nabla difference) [11]: the [alpha]th-order nabla difference of u is given by

[mathematical expression not reproducible]. (9)

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

[mathematical expression not reproducible] (10)

where [alpha], [beta] > 0, [absolute value of ([lambda])] < 1, and t [member of] [N.sub.a].

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

Lemma 4. Let 0 < [alpha], [beta] < 1. The functions [F.sub.[alpha]] and [F.sub.[alpha],[alpha]] are nonnegative and for any 0 < [lambda] < 1 and t [member of] [N.sub.0],

[F.sub.[alpha]] (-[lambda], [t.sup.[bar.[alpha]]]) [less than or equal to] 1, [F.sub.[alpha],[alpha]] (-[lambda], [t.sup.[bar.[alpha]]]) [less than or equal to] 1/[GAMMA]([alpha]). (11)

Definition 5 (see [13]). Let u : [N.sub.a] [right arrow] R. The N-transform of u is defined by

[N.sub.a] [u(t)] = [[infinity].summation over (j=a)] u(j)[(1 - z).sup.j-1], (12)

for each z [member of] C for which the series converges.

Definition 6 (see [13]). Let u, v : [N.sub.a] [right arrow] R. The convolution of u and v is defined by

(u[*.sub.a]v)(t) = [[infinity].summation over (s=a)] u(t + a - [rho](s)) v(s). (13)

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

Theorem 7 (see [13]). Assume that the following functions are well defined:

(1) [N.sub.a][(u[*.sub.a]v)(t)] = [N.sub.1][u(t + o)][N.sub.a][v(t)].

(2) [N.sub.a][[(t - a + 1).sup.[bar.[alpha]]]] = [(1 - z).sup.a-1] ([GAMMA]([alpha] + 1)/[z.sup.[alpha]+1]), [alpha] [member of] R \{..., -3, -2,-1}.

(3) [N.sub.a] [([[DELTA].sup.[alpha].sub.[alpha]]u)(t)] = [z.sup.[alpha]] [N.sub.a][u(t)], 0 < [alpha] < 1.

(4) [N.sub.a+1] [([[DELTA].sup.[alpha].sub.[alpha]])(t)] = [z.sup.[alpha]][N.sub.a][u(t)] -[(1 - z).sup.a-1] u(a), 0 < [alpha] < 1.

(5) [N.sub.a][[(t - a + 1).sup.[bar.[alpha]-1] [F.sub.[alpha],[alpha]] ([lambda], [(t - a + [alpha]).sup.[alpha])]] = [(1 - z).sup.a-1]/([z.sup.[alpha]] - [lambda]).

3. Main Results

The main purpose of this section is to discuss the Hyers-Ulam stability of the following difference equation:

([[NABLA].sup.[alpha].sub.0] u) (t) + [lambda]u(t) = f(t), 0 < [alpha] < 1, [lambda] [greater than or equal to] 0, (14)

where u, f: [N.sub.0] [right arrow] R and [lambda] is a constant.

Let v : [N.sub.0] [right arrow] R and w(t) = ([[NABLA].sup.[alpha].sub.0] v)(t) + [lambda]v(t) - f(t) for t [member of] [N.sub.0]. Using Theorem 7, we have

[mathematical expression not reproducible], (15)

which implies

[mathematical expression not reproducible]. (16)

Set

[mathematical expression not reproducible] (17)

Clearly u(0) = v(0). Applying the [N.sub.0] transform on both sides of (17), we get

[mathematical expression not reproducible], (18)

which implies

[mathematical expression not reproducible]. (19)

Since [N.sub.1] is one-to-one, it follows that ([[NABLA].sup.[alpha].sub.0] u)(t) + [lambda]u(t) = f(t), so u(t) is a solution of (14). From (16) and (18), we get

[mathematical expression not reproducible]. (20)

Since [N.sub.0] is one-to-one, it follows that

[mathematical expression not reproducible]. (21)

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

Theorem 8. Letv : [N.sub.0] [right arrow] R and [phi]: [N.sub.0] [right arrow] [0, [infinity]). If

[absolute value of ([[NABLA].sup.[alpha].sub.0] v)(t) + [lambda]v(t) - f(t))] [less than or equal to] [phi](t), t [member of] [N.sub.0], (22)

then, there exists a solution u : [N.sub.0] [right arrow] R of (14) and [psi] : [N.sub.0] [right arrow] [0, [infinity]) such that

[absolute value of (u(t) - v(t))] [less than or equal to] [psi](t), t [member of] [N.sub.0]. (23)

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

[mathematical expression not reproducible]. (24)

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

Corollary 9. Let [epsilon] > 0, 0 < [eta] < l, and v : [N.sub.0] [right arrow] R.If

[absolute value of ([[NABLA].sup.[alpha].sub.0] v)(t) + [lambda]v(t) - f(t))] [less than or equal to] [epsilon] [F.sub.[alpha]] ([eta],[t.sup.[bar.[alpha]]]), t [member of] [N.sub.0], (25)

then, there exists a solution u : [N.sub.0] [right arrow] R of (14) such that

[absolute value of (u(t) - v(t))] [less than or equal to] K([epsilon])[F.sub.[alpha]] ([eta],[t.sup.[bar.[alpha]]]), te [N.sub.0], (26)

where

K([epsilon]) = [epsilon]/[eta]. (27)

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

Theorem 10. Let [epsilon] > 0 and v : [N.sub.0] [right arrow] R. If

[absolute value of (([[NABLA].sup.[alpha].sub.0]v)(t) + [lambda]v (t) - f(t))] [less than or equal to] [epsilon], t [member of] [N.sub.0], (28)

there exists a solution u : [N.sub.0] [right arrow] R of (14) such that

[absolute value of (u(t)- v(t)] [less than or equal to] K([epsilon]), t [member of] [N.sub.0], (29)

where

K([epsilon]) = [epsilon]/[lambda]. (30)

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

[mathematical expression not reproducible]. (31)

Finally, we discuss the Hyers-Ulam stability of the following Caputo type linear fractional nabla difference equation:

([[NABLA].sup.[alpha].sub.0*] (t) + [lambda] u(t) = f (t), 0 < [alpha] < 1, [lambda] [greater than or equal to] 0. (32)

Using Definition 2 in (32), we get

([[NABLA].sup.[alpha].sub.0] u) (t) + [lambda]u (t) = g(t), t [member of] [N.sub.1], (33)

which is similar to (14). Here

g(t) = f(t) + [(t + 1).sup.[bar.-[alpha]]]/[GAMMA](1 - [alpha]) u(0), t [member of] [N.sub.0]. (34)

Example 11. Consider the following fractional nabla difference equation:

([[NABLA].sup.0.25.sub.0] u) (t) + (0.09) u(t) = [(t + 1).sup.[-bar.0.25]]/[GAMMA](0.75) t [member of] [N.sub.1]. (35)

For [epsilon] = 0.1 and v(t) = 1 for all t [member of] [N.sub.1], we have

[mathematical expression not reproducible]. (36)

Let u(0) = 1. Then, the exact solution of (35) is given by

[mathematical expression not reproducible]. (37)

Consequently, for t [member of] [N.sub.0], we get

[absolute value of (u(t) - v)] [less than or equal to] K([epsilon]), (38)

where

K ([epsilon]) = [epsilon]/[lambda] = 1.1111. (39)

Thus, (35) is Hyers-Ulam stable on [N.sub.0]. Furthermore, we illustrate these concepts numerically in Table 1.

Example 12. Consider the following fractional nabla difference equation:

([[NABLA].sup.0.75.sub.0](t) + (0.5)u(t) = (0.25)[(1/t + 1).sup.0.5],

t [member of] [N.sub.1]. (40)

For v(t) = [(t + 1).sup.[bar.0.25]], we have

[mathematical expression not reproducible]. (41)

Let u(0) = [GAMMA](l.25). Then, the exact solution of (40) is given by

[mathematical expression not reproducible]. (42)

Consequently, we get

[absolute value of (u(t) - v(t))] [less than or equal to] [psi](t), (43)

where

[mathematical expression not reproducible] (44)

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

http://dx.doi.org/10.1155/2016/7265307

Competing Interests

The author declares that they have no competing interests.

References

[1] S. M. Ulam, A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics no. 8, Interscience, New York, NY, USA, 1960.

[2] D. H. Hyers, "On the stability of the linear functional equation," Proceedings of the National Academy of Sciences of the United States of America, vol. 27, pp. 222-224, 1941.

[3] T. M. Rassias, "On the stability of the linear mapping in Banach spaces," Proceedings of the American Mathematical Society, vol. 72, no. 2, pp. 297-300, 1978.

[4] S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, Springer, New York, NY, USA, 2011.

[5] P. L. Kannappan, Functional Equations and Inequalities with Applications, Springer Monographs in Mathematics, Springer, Berline, Germany, 2009.

[6] H. Rezaei, S.-M. Jung, and T. M. Rassias, "Laplace transform and Hyers-Ulam stability of linear differential equations," Journal of Mathematical Analysis and Applications, vol. 403, no. 1, pp. 244251, 2013.

[7] C. Wang and T.-Z. Xu, "Hyers-Ulam stability of a class of fractional linear differential equations," Kodai Mathematical Journal, vol. 38, no. 3, pp. 510-520, 2015.

[8] C. Wang and T.-Z. Xu, "Hyers-Ulam stability of fractional linear differential equations involving Caputo fractional derivatives," Applications of Mathematics, vol. 60, no. 4, pp. 383-393, 2015.

[9] J. R. Wang and X. Li, "A uniform method to Ulam-Hyers stability for some linear fractional equations," Mediterranean Journal of Mathematics, vol. 13, no. 2, pp. 625-635, 2016.

[10] C. Goodrich and A. C. Peterson, Discrete Fractional Calculus, Springer International, New York, NY, USA, 2015.

[11] T. Abdeljawad, "On Riemann and Caputo fractional differences," Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1602-1611, 2011.

[12] A. Nagai, "Discrete Mittag--Leffler function and its applications," Publications of the Research Institute for Mathematical Sciences, Kyoto University, vol. 1302, pp. 1-20, 2003.

[13] F. M. Atici and P. W. Eloe, "Linear systems of fractional nabla difference equations," Rocky Mountain Journal of Mathematics, vol. 41, no. 2, pp. 353-370, 2011.

Jagan Mohan Jonnalagadda

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

Correspondence should be addressed to Jagan Mohan Jonnalagadda; j.jaganmohan@hotmail.com

Received 21 July 2016; Accepted 28 August 2016

Academic Editor: Dong Ye
Table 1

t    v(t)   [absolute value   [epsilon]    u(t)     [absolute value
              of (w(t))]                           of (u(t) - v(t))]

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

t    K([epsilon])

0       1.1111
1       1.1111
2       1.1111
3       1.1111
4       1.1111
5       1.1111
6       1.1111
7       1.1111
8       1.1111
9       1.1111
10      1.1111
11      1.1111
12      1.1111

Table 2

t     v(t)    [absolute value   [phi](t)    u(t)     [absolute value
                of (w(t))]                          of (u(t) - v(t))]

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

t    [psi](t)

0     0.0362
1     1.4980
2     2.4424
3     3.2293
4     3.9384
5     4.6007
6     5.2317
7     5.8401
8     6.4313
9     7.0092
10    7.5762
11    8.1343
12    8.6849
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Title Annotation:Research Article
Author:Jonnalagadda, Jagan Mohan
Publication:International Journal of Analysis
Article Type:Report
Date:Jan 1, 2016
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