# Hyers-Ulam Stability of Fractional Nabla Difference Equations

Authors: Jagan Mohan Jonnalagadda
Journal: International Journal of Analysis
 (2016)

---

## Abstract

We investigate the Hyers-Ulam stability, the generalized Hyers-Ulam stability, and theF
α-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.


---

## Body

## 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
1, a metric group (
G
2
,
d
), and a positive number ε, does there exist δ
>
0 such that if a mapping f
:
G
1
→
G
2 satisfies the inequality (1)
d
f
x
y
,
f
x
,
f
y
<
δ
,for all x
,
y
∈
G
1, then there exists a homomorphism T
:
G
1
→
G
2 such that (2)
d
f
x
,
T
x
<
δ
,for all x
,
y
∈
G
1?Hyers [2] solved the problem for additive functions defined on Banach spaces in 1941 as follows.Hyer’s Theorem (see [2]). Let E
1 be a normed vector space and E
2 a Banach space and suppose that the mapping f
:
E
1
→
E
2 satisfies the inequality (3)
f
x
+
y
-
f
x
-
f
y
≤
ε
,for all x
,
y
∈
E
1, where ε
>
0 is a constant. Then the limit (4)
g
x
=
lim
n
⁡
2
-
n
f
2
n
xexists for each x
∈
E
1 and g is the unique additive mapping satisfying (5)
f
x
-
g
x
≤
ε
,for all x
∈
E
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.LetX be a normed space and I be an open interval. The differential equation F
(
t
,
u
,
u
′
,
u
′
′
,
…
,
u
(
n
)
)
=
0 is Hyers-Ulam stable, if, for given ε
>
0 and a function v
:
I
→
X such that F
t
,
v
,
v
′
,
v
′
′
,
…
,
v
n
≤
ε, there exists a solution u
:
I
→
X of the differential equation such that u
t
-
v
t
≤
K
ε for any t
∈
I, where K
ε is an expression of ε only. If the above statement is also true when we replace ε and K
ε by ϕ
t and ψ
t, where ϕ
,
ψ
:
I
→
0
,
∞ 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
a
=
{
a
,
a
+
1
,
a
+
2
,
…
} for any a
∈
R. Assume that empty sums and products are taken to be 0 and 1, respectively.Definition 1 (rising factorial function).
For anyα
∈
R, t
∈
R
∖
{
…
,
-
2
,
-
1,0
} such that t
+
α
∈
R
∖
{
…
,
-
2
,
-
1,0
}, the rising factorial function is defined by(6)
t
α
¯
=
Γ
t
+
α
Γ
t
,
0
α
¯
=
0
.Definition 2.
Letu
:
N
a
→
R and 0
<
α
<
1.(1)
(Fractional nabla sum) [11]: the αth-order nabla sum of u is given by(7)
∇
a
-
α
u
t
=
1
Γ
α
∑
s
=
a
t
t
-
s
+
1
α
-
1
¯
u
s
,
t
∈
N
a
.
(2)
(R-L fractional nabla difference) [11]: the αth-order nabla difference of u is given by(8)
∇
a
α
u
t
=
1
Γ
-
α
∑
s
=
a
t
t
-
s
+
1
-
α
-
1
¯
u
s
,
t
∈
N
a
+
1
.
(3)
(Caputo fractional nabla difference) [11]: the αth-order nabla difference of u is given by(9)
∇
a
∗
α
u
t
=
∇
a
α
u
t
-
t
-
a
+
1
-
α
¯
Γ
1
-
α
u
a
,
t
∈
N
a
+
1
.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(10)
F
α
λ
,
t
-
a
α
¯
=
∑
k
=
0
∞
λ
k
t
-
a
k
α
¯
Γ
k
α
+
1
,
F
α
,
β
λ
,
t
-
a
α
¯
=
∑
k
=
0
∞
λ
k
t
-
a
k
α
¯
Γ
k
α
+
β
, where α
,
β
>
0, |
λ
|
<
1, and t
∈
N
a.Estimates of nabla Mittag-Leffler functions are provided in Lemma4.Lemma 4.
Let0
<
α
,
β
<
1. The functions F
α and F
α
,
α are nonnegative and for any 0
<
λ
<
1 and t
∈
N
0, (11)
F
α
-
λ
,
t
α
¯
≤
1
,
F
α
,
α
-
λ
,
t
α
¯
≤
1
Γ
α
.Definition 5 (see [13]).
Letu
:
N
a
→
R. The N-transform of u is defined by(12)
N
a
u
t
=
∑
j
=
a
∞
u
j
1
-
z
j
-
1
, for each z
∈
C for which the series converges.Definition 6 (see [13]).
Letu
,
v
:
N
a
→
R. The convolution of u and v is defined by(13)
u
∗
a
v
t
=
∑
s
=
a
t
u
t
+
a
-
ρ
s
v
s
.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
a
u
∗
a
v
t
=
N
1
u
t
+
a
N
a
v
t.
(2)
N
a
t
-
a
+
1
α
¯
=
1
-
z
a
-
1
(
Γ
α
+
1
/
z
α
+
1
)
,
α
∈
R
∖
…
,
-
3
,
-
2
,
-
1.
(3)
N
a
∇
a
α
u
t
=
z
α
N
a
u
t
,
0
<
α
<
1.
(4)
N
a
+
1
∇
a
α
u
t
=
z
α
N
a
u
t
-
1
-
z
a
-
1
u
a
,
0
<
α
<
1.
(5)
N
a
t
-
a
+
1
α
-
1
¯
F
α
,
α
λ
,
t
-
a
+
α
α
¯
=
1
-
z
a
-
1
/
z
α
-
λ.


## 3. Main Results

The main purpose of this section is to discuss the Hyers-Ulam stability of the following difference equation:(14)
∇
0
α
u
t
+
λ
u
t
=
f
t
,
0
<
α
<
1
,
λ
≥
0
,where u
,
f
:
N
0
→
R and λ is a constant.Letv
:
N
0
→
R and w
(
t
)
=
∇
0
α
v
t
+
λ
v
t
-
f
t for t
∈
N
0. Using Theorem 7, we have(15)
N
1
w
t
=
z
α
+
λ
N
0
v
t
-
1
+
λ
1
1
-
z
v
0
-
N
0
f
t
-
1
1
-
z
f
0
=
z
α
+
λ
N
0
v
t
-
N
0
f
t
-
1
1
-
z
1
+
λ
v
0
-
f
0
,which implies(16)
N
0
v
t
=
z
α
+
λ
-
1
N
1
w
t
+
N
0
f
t
+
1
1
-
z
1
+
λ
v
0
-
f
0
.Set(17)
u
t
=
v
0
1
+
λ
t
+
1
α
-
1
¯
F
α
,
α
-
λ
,
t
+
α
α
¯
+
∑
s
=
1
t
t
-
s
+
1
α
-
1
¯
F
α
,
α
-
λ
,
t
-
s
+
α
α
¯
f
s
.Clearly u
0
=
v
0. Applying the N
0 transform on both sides of (17), we get(18)
N
0
u
t
=
1
+
λ
v
0
-
f
0
N
0
t
+
1
α
-
1
¯
F
α
,
α
-
λ
,
t
+
α
α
¯
+
N
0
∑
s
=
0
t
t
-
s
+
1
α
-
1
¯
F
α
,
α
-
λ
,
t
-
s
+
α
α
¯
f
s
=
1
+
λ
v
0
-
f
0
N
0
t
+
1
α
-
1
¯
F
α
,
α
-
λ
,
t
+
α
α
¯
+
N
0
t
α
-
1
¯
F
α
,
α
-
λ
,
t
-
1
+
α
α
¯
∗
0
f
t
=
1
+
λ
v
0
-
f
0
1
-
z
z
α
+
λ
-
1
+
N
1
t
α
-
1
¯
F
α
,
α
-
λ
,
t
-
1
+
α
α
¯
N
0
f
t
=
1
+
λ
v
0
-
f
0
1
-
z
z
α
+
λ
-
1
+
z
α
+
λ
-
1
N
0
f
t
, which implies(19)
z
α
+
λ
N
0
u
t
=
1
+
λ
v
0
-
f
0
1
-
z
+
N
0
f
t
o
r
z
α
N
0
u
t
-
1
1
-
z
u
0
+
λ
N
0
u
t
-
1
1
-
z
u
0
=
N
0
f
t
-
1
1
-
z
f
0
o
r
N
1
∇
0
α
u
t
+
λ
u
t
=
N
1
f
t
. Since N
1 is one-to-one, it follows that ∇
0
α
u
t
+
λ
u
t
=
f
t, so u
(
t
) is a solution of (14). From (16) and (18), we get(20)
N
0
u
t
-
N
0
v
t
=
z
α
+
λ
-
1
N
1
w
t
=
z
α
+
λ
-
1
N
0
w
t
-
1
1
-
z
w
0
=
N
1
t
α
-
1
¯
F
α
,
α
-
λ
,
t
-
1
+
α
α
¯
N
0
w
t
-
w
0
N
0
t
+
1
α
-
1
¯
F
α
,
α
-
λ
,
t
+
α
α
¯
=
N
0
t
α
-
1
¯
F
α
,
α
-
λ
,
t
-
1
+
α
α
¯
∗
0
w
t
-
w
0
N
0
t
+
1
α
-
1
¯
F
α
,
α
-
λ
,
t
+
α
α
¯
=
N
0
∑
s
=
1
t
t
-
s
+
1
α
-
1
¯
F
α
,
α
-
λ
,
t
-
s
+
α
α
¯
w
s
.SinceN
0 is one-to-one, it follows that(21)
u
t
-
v
t
=
∑
s
=
1
t
t
-
s
+
1
α
-
1
¯
F
α
,
α
-
λ
,
t
-
s
+
α
α
¯
w
s
.First, we establish the generalized Hyers-Ulam stability of (14) as follows.Theorem 8.
Letv
:
N
0
→
R and ϕ
:
N
0
→
[
0
,
∞
). If (22)
∇
0
α
v
t
+
λ
v
t
-
f
t
≤
ϕ
t
,
t
∈
N
0
,then, there exists a solution u
:
N
0
→
R of (14) and ψ
:
N
0
→
[
0
,
∞
) such that (23)
u
t
-
v
t
≤
ψ
t
,
t
∈
N
0
.Proof.
Using (21) and Lemma 4, we have(24)
u
t
-
v
t
≤
∑
s
=
1
t
t
-
s
+
1
α
-
1
¯
F
α
,
α
-
λ
,
t
-
s
+
α
α
¯
w
s
≤
1
Γ
α
∑
s
=
1
t
t
-
s
+
1
α
-
1
¯
ϕ
s
=
∇
1
-
α
ϕ
t
=
ψ
t
.Now, consider a particular case of Theorem8 which we define as the F
α-Hyers-Ulam stability of (14).Corollary 9.
Letε
>
0, 0
<
η
<
1, and v
:
N
0
→
R. If (25)
∇
0
α
v
t
+
λ
v
t
-
f
t
≤
ε
F
α
η
,
t
α
¯
,
t
∈
N
0
,then, there exists a solution u
:
N
0
→
R of (14) such that (26)
u
t
-
v
t
≤
K
ε
F
α
η
,
t
α
¯
,
t
∈
N
0
,where (27)
K
ε
=
ε
η
.Next, we investigate the Hyers-Ulam stability of (14).Theorem 10.
Letε
>
0 and v
:
N
0
→
R. If (28)
∇
0
α
v
t
+
λ
v
t
-
f
t
≤
ε
,
t
∈
N
0
,there exists a solution u
:
N
0
→
R of (14) such that (29)
u
t
-
v
t
≤
K
ε
,
t
∈
N
0
,where(30)
K
ε
=
ε
λ
.Proof.
Using (21) and Lemma 4, we have(31)
u
t
-
v
t
≤
∑
s
=
0
t
t
-
s
+
1
α
-
1
¯
F
α
,
α
-
λ
,
t
-
s
+
α
α
¯
w
s
≤
ε
∑
s
=
0
t
t
-
s
+
1
α
-
1
¯
F
α
,
α
-
λ
,
t
-
s
+
α
α
¯
=
ε
∑
s
=
0
t
t
-
s
+
1
α
-
1
¯
∑
k
=
0
∞
-
λ
k
t
-
s
+
α
α
k
¯
Γ
α
k
+
α
=
ε
∑
k
=
0
∞
-
λ
k
Γ
α
k
+
α
∑
s
=
0
t
t
-
s
+
1
α
-
1
¯
t
-
s
+
α
α
k
¯
=
ε
∑
k
=
0
∞
-
λ
k
Γ
α
k
+
α
∑
s
=
0
t
t
-
s
+
1
α
k
+
α
-
1
¯
=
ε
∑
k
=
0
∞
-
λ
k
t
+
1
α
k
+
α
¯
Γ
α
k
+
α
+
1
=
ε
λ
1
-
F
α
-
λ
,
t
+
1
α
¯
≤
ε
λ
.Finally, we discuss the Hyers-Ulam stability of the following Caputo type linear fractional nabla difference equation:(32)
∇
0
∗
α
u
t
+
λ
u
t
=
f
t
,
0
<
α
<
1
,
λ
≥
0
.Using Definition 2 in (32), we get(33)
∇
0
α
u
t
+
λ
u
t
=
g
t
,
t
∈
N
1
,which is similar to (14). Here(34)
g
t
=
f
t
+
t
+
1
-
α
¯
Γ
1
-
α
u
0
,
t
∈
N
0
.Example 11.
Consider the following fractional nabla difference equation:(35)
∇
0
0.25
u
t
+
0.09
u
t
=
t
+
1
-
0.25
¯
Γ
0.75
,
t
∈
N
1
.Forε
=
0.1 and v
(
t
)
=
1 for all t
∈
N
1, we have(36)
∇
0
0.25
v
t
+
0.09
v
t
-
t
+
1
-
0.25
¯
Γ
0.75
=
t
+
1
-
0.25
¯
Γ
0.75
+
0.09
-
t
+
1
-
0.25
¯
Γ
0.75
≤
ε
.Let u
(
0
)
=
1. Then, the exact solution of (35) is given by (37)
u
t
=
1.1
t
+
1
-
0.75
¯
F
0.25,0.25
-
0.1
,
t
+
0.25
0.25
¯
+
∑
s
=
1
t
t
-
s
+
1
-
0.75
¯
F
0.25,0.25
-
0.1
,
t
-
s
+
0.25
0.25
¯
s
+
1
-
0.25
¯
,
t
∈
N
0
.Consequently, for t
∈
N
0, we get(38)
u
t
-
v
t
≤
K
ε
,where (39)
K
ε
=
ε
λ
=
1.1111
.Thus, (35) is Hyers-Ulam stable on N
0. Furthermore, we illustrate these concepts numerically in Table 1.Table 1
t
v
(
t
)
w
(
t
)
ε
u
(
t
)
u
(
t
)
-
v
(
t
)
K
(
ε
)
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
1.1111Example 12.
Consider the following fractional nabla difference equation:(40)
∇
0
0.75
u
t
+
0.5
u
t
=
0.25
1
t
+
1
0.5
,
t
∈
N
1
.Forv
t
=
t
+
1
0.25
¯, we have(41)
∇
0
0.75
u
t
+
0.5
u
t
-
0.25
1
t
+
1
0.5
=
Γ
1.25
Γ
0.75
t
+
1
-
0.5
¯
+
0.5
t
+
1
0.25
¯
-
0.25
1
t
+
1
0.5
≤
Γ
1.25
Γ
0.75
t
+
1
-
0.5
¯
+
0.5
t
+
1
0.25
¯
+
0.25
1
t
+
1
0.5
≤
0.7397
t
+
1
-
0.5
¯
+
0.5
t
+
1
0.25
¯
+
0.25
t
+
1
-
0.5
¯
=
t
+
1
-
0.5
¯
+
0.5
t
+
1
0.25
¯
=
ϕ
t
,
t
∈
N
0
. Let u
0
=
Γ
1.25. Then, the exact solution of (40) is given by (42)
u
t
=
1.5
Γ
1.25
t
+
1
-
0.25
¯
F
0.75,0.75
-
0.5
,
t
+
0.75
0.75
¯
+
0.25
∑
s
=
1
t
t
-
s
+
1
-
0.25
¯
F
0.75,0.75
-
0.5
,
t
-
s
+
0.75
0.75
¯
1
s
+
1
0.5
,
t
∈
N
0
.Consequently, we get (43)
u
t
-
v
t
≤
ψ
t
,where(44)
ψ
t
=
∇
1
-
0.75
ϕ
t
=
∇
0
-
0.75
ϕ
t
-
t
+
1
-
0.25
¯
Γ
0.75
ϕ
0
=
Γ
0.5
Γ
1.25
t
+
1
0.25
¯
+
0.5
Γ
1.25
Γ
2
t
+
1
1
¯
-
t
+
1
-
0.25
¯
Γ
0.75
ϕ
0
=
1.9954
t
+
1
0.25
¯
+
0.4532
t
+
1
-
1.8162
t
+
1
-
0.25
¯
,
t
∈
N
0
.Thus, (40) is generalized Hyers-Ulam stable. Furthermore, we illustrate these concepts numerically in Table 2.Table 2
t
v
(
t
)
w
(
t
)
ϕ
(
t
)
u
(
t
)
u
(
t
)
-
v
(
t
)
ψ
(
t
)
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



---

*Source: 7265307-2016-09-21.xml*