# On the Hyers-Ulam Stability of the First-Order Difference Equation

Authors: Soon-Mo Jung; Young Woo Nam
Journal: Journal of Function Spaces
 (2016)

---

## Abstract

We prove Hyers-Ulam stability of the first-order difference equation of the formx
i
+
1
=
F
(
i
,
x
i
), where F is a given function with some moderate features. Moreover, we introduce some conditions for the function F under which the difference equation is not stable in the sense of Hyers and Ulam.


---

## Body

## 1. Introduction

The difference equation usually describes the development of a certain phenomenon by recursively defining a sequence, each of whose terms is defined as a function of the preceding terms, once one or more initial terms are known. The difference equation often refers to a specific type of recurrence relation (see [1]).In 1940, Ulam [2] raised an important problem concerning the stability of group homomorphisms (ref. [3, 4]):given a metric group
(
G
,
·
,
d
),a positive number
ε,and a function
f
:
G
→
G
which satisfies inequality
d
(
f
(
x
y
)
,
f
(
x
)
f
(
y
)
)
≤
ε
for all
x
,
y
∈
G,do there exist a homomorphism
a
:
G
→
G
and a constant
δ
depending only on
G
and
ε
such that
d
(
a
(
x
)
,
f
(
x
)
)
≤
δ
for all
x
∈
G?If the answer to this question is affirmative, the functional equationa
(
x
y
)
=
a
(
x
)
a
(
y
) is said to be stable. A first answer to this question was given by Hyers [5] in 1941 who proved that the Cauchy additive equation is stable in Banach spaces. In general, a functional equation is said to be stable in the sense of Hyers and Ulam (or the equation has the Hyers-Ulam stability) if for each solution to the perturbed equation, there exists a solution to the equation that differs from the solution to the perturbed equation with a small error. We refer the reader to [2, 5–11] for the exact definition of Hyers-Ulam stability.Throughout this paper, we denote byN, N
0, R, and C the set of all positive integers, of all nonnegative integers, and of all real numbers and the set of all complex numbers, respectively.In this paper, we prove Hyers-Ulam stability of the first-order difference equation of the form(1)xi+1=Fi,xifor all integers i
∈
N
0, where F
:
N
0
×
C
→
C is a given function with some moderate features. More precisely, we prove that if a complex-valued sequence {
a
i
}
i
∈
N
0 satisfies inequality (2)ai+1-Fi,ai≤εfor all i
∈
N
0, then there exist a solution {
b
i
}
i
∈
N
0 to the difference equation (1) and a positive constant δ depending only on F and ε such that (3)bi-ai≤δfor all i
∈
N
0. As we know, the stability of the difference equation (1) depends on properties of the map F. We show in the last part of this paper which condition of F excludes the Hyers-Ulam stability.


## 2. Conditions for Hyers-Ulam Stability

The principle of recursive definition states that, for any functiong
:
C
→
C and any a
0
∈
C, there exists a unique function f
:
N
0
→
C such that (4)f0=a0,fi+1=gfifor all i
∈
N [12, Theorem A.5.6]. This principle assures us of the existence and the uniqueness of the sequence {
b
i
}
i
∈
N
0 mentioned in Theorem 1.First, we prove a general theorem that provides us with a powerful tool for proving the Hyers-Ulam stability of a large class of the first-order difference equations.Theorem 1.
Givenε
>
0, let F
:
N
0
×
C
→
C be a function satisfying the condition(5)Fi,u-Fi,v≤φu-vfor all i
∈
N
0 and all u
,
v
∈
C, where φ
:
[
0
,
∞
)
→
[
0
,
∞
) is a monotone increasing function. If a complex-valued sequence {
a
i
}
i
∈
N
0 satisfies inequality(6)ai+1-Fi,ai≤εfor all i
∈
N
0, then there exists a complex-valued sequence {
b
i
}
i
∈
N
0 satisfying(7)bi+1=Fi,bi,(8)bi-ai≤ψib0-a0for all i
∈
N
0, where the function ψ
:
[
0
,
∞
)
→
[
0
,
∞
) is defined by ψ
(
x
)
≔
φ
(
x
)
+
ε for all x
≥
0 and ψ
i
(
x
) denotes the value of the ith iterate of ψ at x; that is, (9)ψix=ψ∘⋯∘ψ︸ix.Proof.
In view of principle of recursive definition, the complex-valued sequence{
b
i
}
i
∈
N
0 is uniquely determined via formula (7) provided b
0 is given.
We will apply an induction oni to prove inequality (8). For i
=
1, it follows from (5), (6), and (7) that (10)b1-a1≤F0,b0-F0,a0+F0,a0-a1≤φb0-a0+ε=ψb0-a0, which we also obtain by putting i
=
1 in (8). We now assume that inequality (8) is true for some i
∈
N. Then, it follows from (5), (6), (7), and (8) that (11)bi+1-ai+1≤Fi,bi-Fi,ai+Fi,ai-ai+1≤φbi-ai+ε=ψbi-ai≤ψψib0-a0=ψi+1b0-a0 which proves the validity of (8) for all i
∈
N
0.Using Theorem1, we can prove the Hyers-Ulam stability of a class of the first-order difference equations under the condition, a
0
=
b
0 (see the statement of Corollary 2 for a
0 and b
0).Corollary 2.
Letφ
:
[
0
,
∞
)
→
[
0
,
∞
) be a monotone increasing mapping such that φ
(
0
)
=
0 and(12)φx-φy≤Kx-yfor all x
,
y
≥
0, where K is a positive real constant less than 1. Given ε
>
0, let F
:
N
0
×
C
→
C be a function satisfying condition (5) for all i
∈
N
0 and u
,
v
∈
C and let ψ
:
[
0
,
∞
)
→
[
0
,
∞
) be defined by ψ
(
x
)
≔
φ
(
x
)
+
ε for all x
≥
0. If a complex-valued sequence {
a
i
}
i
∈
N
0 satisfies inequality (6) for all i
∈
N
0, then there exists a complex-valued sequence {
b
i
}
i
∈
N
0 satisfying (7) and (13)bi-ai≤ε1-K+Ki1-Kψb0-a0-b0-a0 for all i
∈
N
0.Proof.
According to Theorem1, there exists a complex-valued sequence {
b
i
}
i
∈
N
0 satisfying (7) and(14)bi-ai≤ψib0-a0for all i
∈
N
0.
It follows from (12) that ψ is also a contraction mapping with the Lipschitz constant K. By the contraction mapping theorem, we have (15)ψib0-a0-x∗≤Ki1-Kψb0-a0-b0-a0 for all i
∈
N
0, where x
∗ is the unique fixed point of ψ, from which it follows that (16)ψib0-a0≤x∗+Ki1-Kψb0-a0-b0-a0 for each i
∈
N
0. In view of (14) and the last inequality, we obtain(17)bi-ai≤x∗+Ki1-Kψb0-a0-b0-a0for all i
∈
N
0.
We now assert thatψ
(
x
)
≤
K
x
+
ε for all x
≥
0. To prove this assertion, we assume for contradiction that there was x
0
≥
0 such that ψ
(
x
0
)
>
K
x
0
+
ε. Then we would have (18)φx0-φ0=ψx0-ψ0>Kx0-0,which is the contradiction to (12). Hence, we obtain the following inequality for the fixed point x
∗ of ψ:(19)x∗=ψx∗≤Kx∗+εorx∗≤ε1-K.
Finally, it follows from (17) and (19) that (20)bi-ai≤ε1-K+Ki1-Kψb0-a0-b0-a0 for all i
∈
N
0.Using Corollary2, we prove the Hyers-Ulam stability of the first-order difference equation (1) under a more explicit condition for F and an additional condition, a
0
=
b
0 (see the statement of Corollary 3 for a
0 and b
0).Corollary 3.
Given positive real constantsε and K with 0
<
K
<
1, let F
:
N
0
×
C
→
C be a function satisfying the condition(21)Fi,u-Fi,v≤Ku-vfor all i
∈
N
0 and u
,
v
∈
C. If a complex-valued sequence {
a
i
}
i
∈
N
0 satisfies inequality (6) for all i
∈
N
0, then there exists a complex-valued sequence {
b
i
}
i
∈
N
0 satisfying (7) and (22)bi-ai≤ε1-K1+Ki+Kib0-a0 for all i
∈
N
0.Proof.
If we define monotone increasing contraction mappingsφ
,
ψ
:
[
0
,
∞
)
→
[
0
,
∞
) by φ
(
x
)
≔
K
x and ψ
(
x
)
≔
φ
(
x
)
+
ε
=
K
x
+
ε, then, by Corollary 2, we have (23)bi-ai≤ε1-K+Ki1-Kψb0-a0-b0-a0=ε1-K+Ki1-Kε+Kib0-a0=ε1-K1+Ki+Kib0-a0 for all i
∈
N
0.Remark 4.
If we setε
n
=
ε, λ
n
=
K, L
=
K, and a
n
(
·
)
=
F
(
n
,
·
) in [13, Theorem 2], then we get K
<
q
<
1, n
(
q
)
=
2, and |
b
i
-
a
i
|
≤
M
ε, where M
≥
K
q
+
1
/
1
-
q, while we get |
b
i
-
a
i
|
≤
2
/
1
-
K
ε from Corollary 3 provided the initial condition a
0
=
b
0 is assumed. In general, the result of [13, Theorem 2] is better than that of Corollary 3.Example 5.
Given positive real constantsε and M with 0
<
M
<
e, let F
:
N
0
×
C
→
C be a function defined by (24)Fi,u≔Mi+1ue-u for all i
∈
N
0 and u
∈
C. Then we have (25)Fi,u-Fi,v=Mi+1ue-u-ve-v≤Mi+1supz∈Cddwwe-ww=zu-v=Mi+11eu-v≤Meu-v=Ku-vfor all i
∈
N
0 and u
,
v
∈
C, where we set K
=
M
/
e. Then we have 0
<
K
<
1, and hence inequality (21) holds. According to Corollary 3, if a complex-valued sequence {
a
i
}
i
∈
N
0 satisfies inequality (6) for all i
∈
N
0, then there exists a complex-valued sequence {
b
i
}
i
∈
N
0 satisfying (7) and (26)bi-ai≤ε1-K1+Ki+Kib0-a0=eεe-M1+Mei+Meib0-a0 for all i
∈
N
0.


## 3. Conditions for Nonstability

In this section, we introduce some conditions for the functionF, under which the first-order difference equation (1) is not stable in the sense of Hyers and Ulam.We replace inequality (5) in Theorem 1 with another one and prove the counterpart of Theorem 1 in the following theorem.Theorem 6.
Givenε
>
0, let F
:
N
0
×
C
→
C be a function satisfying the condition(27)Fi,u-Fi,v≥ξu-vfor all i
∈
N
0 and u
,
v
∈
C, where ξ
:
[
0
,
∞
)
→
[
0
,
∞
) is a monotone increasing function, and let ζ
:
[
0
,
∞
)
→
R be a monotone increasing function defined by (28)ζx≔ξx-ε for all x
≥
0. If a complex-valued sequence {
a
i
}
i
∈
N
0 satisfies inequality(29)ai+1-Fi,ai≤εfor all i
∈
N
0 and ζ
(
|
b
1
-
a
1
|
)
≥
0, then there exists a complex-valued sequence {
b
i
}
i
∈
N
0 satisfying(30)bi+1=Fi,bi,(31)bi+1-ai+1≥ζib1-a1for all i
∈
N
0.Proof.
We apply the induction oni to prove inequality (31). Trivially, (31) is true for i
=
0. For i
=
1, we use (27), (29), and (30) to show that (32)b2-a2≥F1,b1-F1,a1-a2-F1,a1≥ξb1-a1-ε=ζb1-a1; that is, inequality (31) holds for i
=
1. We now assume that inequality (33)bk+1-ak+1≥ζkb1-a1is true for all k
∈
{
0,1
,
…
,
i
-
1
}, where i is a positive integer. Then, it follows from (27), (29), and (30) that (34)bi+1-ai+1≥Fi,bi-Fi,ai-ai+1-Fi,ai≥ξbi-ai-ε=ζbi-ai≥ζζi-1b1-a1=ζib1-a1,which proves the validity of (31) for all i
∈
N
0.Lemma 7.
Letζ
:
[
C
,
∞
)
→
[
C
,
∞
) be an increasing continuous function for some C
∈
R. Assume that ζ
(
C
)
>
C and ζ has fixed points which are greater than C; that is, P
=
{
x
>
C
:
ζ
(
x
)
=
x
} is nonempty. If x
∗
≔
i
n
f
P, then ζ
i
(
x
)
≤
x
∗ for all x
∈
[
C
,
x
∗
] and for all i
∈
N
0. Moreover, (35)limi→∞ζix=x∗.Proof.
The fact thatζ is increasing and continuous implies that ζ
(
x
)
≤
ζ
(
x
∗
)
=
x
∗ for all x
∈
[
c
0
,
x
∗
], where c
0
∈
[
C
,
x
∗
]. Denoting c
i
=
ζ
i
(
c
0
) for each i
∈
N
0, assume that there exists m
∈
N
0 satisfying (36)cm≤x∗<cm+1for some c
0
∈
[
C
,
x
∗
]. Since ζ is an increasing continuous function, we have the following inequality: (37)x∗<cm+1=ζcm≤ζx∗=x∗,which is the contradiction. Consequently, x
∗ is an upper bound of the sequence {
ζ
i
(
x
)
}
i
∈
N
0 for each x
∈
[
C
,
x
∗
].
Next, we claim that{
ζ
i
(
x
)
}
i
∈
N
0 is an increasing sequence for each x
∈
[
C
,
x
∗
]. Suppose for contradiction that there exists m
∈
N
0 such that c
m
>
ζ
(
c
m
)
=
c
m
+
1 for some c
0
∈
[
C
,
x
∗
]. If we define g
(
x
)
≔
ζ
(
x
)
-
x, then the following are satisfied: (i)
g
(
x
) is continuous.
(ii)
g
(
C
)
=
ζ
(
C
)
-
C
>
0.
(iii)
g
(
c
m
)
=
ζ
(
c
m
)
-
c
m
=
c
m
+
1
-
c
m
<
0.
Thus, by the intermediate value theorem forg, the function ζ has a fixed point in (
C
,
c
m
) and c
m
≤
x
∗. It is contrary to the assumption that x
∗ is the infimum of P. Therefore, we conclude that {
ζ
i
(
x
)
}
i
∈
N
0 is an increasing sequence for each x
∈
[
C
,
x
∗
].
Since the sequence{
ζ
i
(
x
)
}
i
∈
N
0 is an increasing sequence bound above, it is convergent. Denote the limit lim
i
→
∞
⁡
c
i by c
∞ for each c
0
∈
[
C
,
x
∗
]. Then, we conclude that c
∞ is a fixed point of ζ by considering the relation (38)c∞=limi→∞ci=limi→∞ci+1=limi→∞ζci=ζc∞.Hence, we obtain x
∗
≤
c
∞ by the minimality of x
∗. On the other hand, x
∗ is an upper bound of c
m; that is, c
m
≤
x
∗ for all m
∈
N
0. Thus, we get (39)c∞=limi→∞ci≤x∗,which implies that lim
i
→
∞
⁡
ζ
i
(
c
0
)
=
c
∞
=
x
∗ for each c
0
∈
[
C
,
x
∗
].Lemma 8.
LetL, β, and ε be constants with L
>
0, 0
<
β
<
1, and 0
<
ε
<
(
L
/
2
)
1
/
1
-
β. Then, the function ζ
:
[
0
,
∞
)
→
R, defined by ζ
(
x
)
=
L
x
β
-
ε, has two distinct fixed points.Proof.
Sinceε
<
(
L
/
2
)
1
/
1
-
β, we have ζ
(
0
)
=
-
ε
<
0 and (40)ζε=Lεβ-ε=εLεβ-1-1>εL2L-1=ε.Since ζ
(
ε
)
-
ε
>
0, ζ has a fixed point x
1
∗ in (
0
,
ε
) by the intermediate value theorem for the continuous function ζ
(
x
)
-
x. Moreover, if x
≥
L
1
/
1
-
β, then we have x
1
-
β
≥
L and hence, x
≥
L
x
β
>
L
x
β
-
ε
=
ζ
(
x
); that is, ζ
(
x
)
-
x
<
0 for each x
≥
L
1
/
1
-
β. Hence, ζ has another fixed point x
2
∗ in (
ε
,
L
1
/
1
-
β
).Corollary 9.
LetL, β, and ε be constants with L
>
0, 0
<
β
<
1, and 0
<
ε
<
(
L
/
3
)
1
/
1
-
β. The function ζ
(
x
)
=
L
x
β
-
ε has a fixed point x
1
∗ in the interval (
0
,
ε
/
2
) and another one x
2
∗ in (
(
L
/
2
)
1
/
1
-
β
,
L
1
/
1
-
β
).Proof.
We see thatζ
(
0
)
=
-
ε
<
0 and(41)ζε2=Lεβ2β-ε>Lεβ2-ε=ε2Lεβ-1-2>ε2L3L-2=ε2.Hence, if we apply the intermediate value theorem with the continuous function η
(
x
)
=
ζ
(
x
)
-
x, then we have η
(
0
)
<
0 and η
(
ε
/
2
)
>
0, which implies that ζ has a fixed point in the interval (
0
,
ε
/
2
).
Similarly, we have(42)ζL1/1-β=L·Lβ/1-β-ε=L1/1-β-ε<L1/1-β and moreover, (43)ζL21/1-β=2·L2L2β/1-β-ε=2L21/1-β-ε>2L21/1-β-L21/1-β=L21/1-β.
We also apply the intermediate value theorem toη on the interval [
(
L
/
2
)
1
/
1
-
β
,
L
1
/
1
-
β
]. Then ζ has another fixed point x
2
∗ in (
(
L
/
2
)
1
/
1
-
β
,
L
1
/
1
-
β
).Remark 10.
The fixed pointx
2
∗ of ζ in Corollary 9 is contained in the interval (
(
L
/
2
)
1
/
1
-
β
,
L
1
/
1
-
β
) independent of ε which runs over the interval (
0
,
(
L
/
3
)
1
/
1
-
β
). Since L
1
/
1
-
β is a fixed point of the mapping ξ
(
x
)
=
L
x
β and ζ has only two fixed points, x
2
∗ converges to L
1
/
1
-
β as ε
→
0.In the following theorem, we introduce a condition for the functionF, under which the first-order difference equation (1) is not stable in the sense of Hyers and Ulam.Theorem 11.
LetL and β be constants with L
>
0 and 0
<
β
<
1. Given ε
>
0, let F
:
N
0
×
C
→
C be a function satisfying the condition (44)Fi,u-Fi,v≥Lu-vβ for all i
∈
N
0 and u
,
v
∈
C. If a complex-valued sequence {
a
i
}
i
∈
N
0 satisfies inequality(45)ai+1-Fi,ai≤εfor all i
∈
N
0, then for every sequence {
b
i
}
i
∈
N
0 satisfying b
i
+
1
=
F
(
i
,
b
i
) and |
a
0
-
b
0
|
≥
ε
/
2, it holds that (46)liminfi→∞bi-ai≥L31/1-β.Proof.
Letε be a constant with 0
<
ε
<
(
L
/
3
)
1
/
1
-
β. Denote ξ and ζ as follows: (47)ξx=Lxβ,ζx=Lxβ-ε.We set ε
i
=
|
a
i
+
1
-
F
(
i
,
a
i
)
| for all i
∈
N
0 and in view of (45), without loss of generality, we can choose ε
≔
sup
i
∈
N
0
⁡
ε
i. Then there exists k
∈
N
0 such that (48)ε2<εk≤ε.
Thus we may assume thatε
/
2
<
ε
0
≤
ε. By Lemma 8, ζ has two distinct fixed points x
1
∗ and x
2
∗ with x
1
∗
<
x
2
∗. Since ζ is an increasing continuous function and ε
>
0, there is the point x
0 satisfying ζ
(
x
0
)
=
0. Moreover, Corollary 9 implies that(49)0<x0<x1∗<ε2<ε<L31/1-β<x2∗<L1/1-β.
By the triangular inequality and Theorem6, we have(50)b1-a1≥F0,b0-F0,a0-a1-F0,a0≥ξb0-a0-ε0.Inequality (41) implies that (51)ξε2=ζε2+ε>32ε,(52)ξε2-ε0>32ε-ε0≥32ε-ε=ε2,(53)ζε2>ε2>0.Since the functions ξ
(
x
)
-
ε
0 and ζ
(
x
) are increasing functions, we obtain (54)ξb0-a0-ε0>ε2,ζb0-a0>0 for all |
b
0
-
a
0
|
∈
[
ε
/
2
,
∞
). Then, in view of (49), (50), and (52), |
b
1
-
a
1
| is greater than the fixed point x
1
∗; that is, (55)b1-a1≥ξb0-a0-ε0>ε2>x1∗.Moreover, since ζ is increasing, we have (56)ζb1-a1≥ζε2>ε2>0.
By Theorem6, inequality |
b
i
+
1
-
a
i
+
1
|
≥
ζ
i
(
|
b
1
-
a
1
|
) holds for all i
∈
N
0 and by putting C
=
ε
/
2 and x
∗
=
x
2
∗ in Lemma 7 and taking (41) into account, ζ
i
(
|
b
1
-
a
1
|
) converges to x
2
∗. Hence, by Theorem 6 and (49), we get (57)bi+1-ai+1≥ζib1-a1≥L31/1-β for all sufficiently large integers i.Remark 12.
Theorem11 can be regarded as a nonstability result because this theorem shows that the difference |
b
i
-
a
i
| can be quite large even in the case when ε is very small. An analogous nonstability result can be deduced from [14, Theorem 1] when β
=
1 and L
=
1. However, this is not the case when β
=
1 and L
>
1 because [13, Theorem 1] shows that the sequence has Hyers-Ulam stability.



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*Source: 6078298-2016-08-04.xml*