# A Linear Functional Equation of Third Order Associated with the Fibonacci Numbers

Authors: Soon-Mo Jung; Michael Th. Rassias
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

Given a vector spaceX, we investigate the solutions f
:
R
→
X of the linear functional equation of third order f
x
=
p
f
x
-
1
+
q
f
x
-
2
+
r
f
(
x
-
3
), which is strongly associated with a well-known identity for the Fibonacci numbers. Moreover, we prove the Hyers-Ulam stability of that equation.


---

## Body

## 1. Introduction

The problem of stability of functional equations was motivated by a question of Ulam [1] and a solution to it by Hyers [2]. Since then, numerous papers have been published on that subject and we refer to [3–6] for more details, some discussions, and further references; for examples of very recent results, see, for example, [7].In this paper, as usual,C, R, Z, and N stand for the sets of complex numbers, real numbers, integers, and positive integers, respectively. For a nonempty subset S of a vector space, let ξ
:
S
→
S be a function. Moreover, ξ
0
(
x
)
=
x, ξ
n
+
1
(
x
)
=
ξ
(
ξ
n
(
x
)
), and (only for bijective ξ) ξ
-
n
-
1
(
x
)
=
ξ
-
1
(
ξ
-
n
(
x
)
) for x
∈
S and n
∈
N
0
:
=
N
∪
{
0
}.Jung has proved in [3] (see also [8]) some results on solutions and Hyers-Ulam stability of the functional equation
(1)
f
(
x
)
=
p
f
(
ξ
(
x
)
)
-
q
f
(
ξ
2
(
x
)
)
,
in the case where S
=
R and ξ
(
x
)
=
x
-
1 for x
∈
R.IfS
:
=
N
0 and p
,
q
∈
Z, then solutions x
:
N
0
→
Z of the difference equation f
(
x
)
=
p
f
(
x
-
1
)
-
q
f
(
x
-
2
) are called the Lucas sequences (see, e.g., [9]). In some special cases they are called with specific names, for example, the Fibonacci numbers (p
=
1, q
=
-
1, x
(
0
)
=
0, and x
(
1
)
=
1), the Lucas numbers (p
=
1, q
=
-
1, x
(
0
)
=
2, and x
(
1
)
=
1), the Pell numbers (p
=
2, q
=
-
1, x
(
0
)
=
0, and x
(
1
)
=
1), the Pell-Lucas (or companion Lucas) numbers (p
=
2, q
=
-
1, x
(
0
)
=
2, and x
(
1
)
=
2), and the Jacobsthal numbers (p
=
1, q
=
-
2, x
(
0
)
=
0, and x
(
1
)
=
1).For some information and further references concerning the functional equations in a single variable, we refer to [10–12]. Let us mention yet that the problem of Hyers-Ulam stability of functional equations is connected to the notions of controlled chaos and shadowing (see [13]).We remark that ifξ
:
S
→
S is bijective, then (1) can be written in the following equivalent form:
(2)
f
(
η
2
(
x
)
)
=
p
f
(
η
(
x
)
)
-
q
f
(
x
)
,
where η
:
=
ξ
-
1.In view of the last remark, the following Hyers-Ulam stability result concerning (1) can be derived from [14, Theorem 2] (see also [15]).Theorem 1.
Letp
,
q
∈
R be given with q
≠
0 and let S be a nonempty subset of a vector space. Assume that a
1, a
2 are the complex roots of the quadratic equation x
2
-
p
x
+
q
=
0 with |
a
i
|
≠
1 for i
∈
{
1,2
}. Moreover, assume that X is either a real vector space if p
2
-
4
q
>
0 or a complex vector space if p
2
-
4
q
<
0. Let ξ
:
S
→
S be bijective. If a function f
:
S
→
X satisfies the inequality
(3)
∥
f
(
x
)
-
p
f
(
ξ
(
x
)
)
+
q
f
(
ξ
2
(
x
)
)
∥
≤
ɛ
for all x
∈
S and for some ɛ
≥
0, then there exists a unique solution F
:
S
→
X of (1) with
(4)
∥
f
(
x
)
-
F
(
x
)
∥
≤
ɛ
|
(
|
a
1
|
-
1
)
(
|
a
2
|
-
1
)
|
for all x
∈
S.In [16, Theorem 1.4], the method presented in [3] was modified so as to prove a theorem which is a complement of Theorem 1. Note that, for bijective ξ, the following theorem improves the estimation (4) in some cases (e.g., a
1
=
3
/
2, a
2
=
-
3
/
2, or a
1
=
1
/
2, a
2
=
-
1
/
2). However, in some other situations (e.g., a
1
=
3, a
2
=
-
3), the estimation (4) is better than (5). The following theorem also complements Theorem 1, because ξ can be quite arbitrary in the case of (
α
).Theorem 2.
Givenp
,
q
∈
R with q
≠
0, assume that the distinct complex roots a
1, a
2 of the quadratic equation x
2
-
p
x
+
q
=
0 satisfy one of the following two conditions:(
α
)
|
a
i
|
<
1 for i
∈
{
1,2
};
(
β
)
|
a
i
|
≠
1 for i
∈
{
1,2
} and ξ
:
S
→
S is bijective.Moreover, assume that X is either a real vector space if p
2
-
4
q
>
0 or a complex vector space if p
2
-
4
q
<
0. If a function f
:
S
→
X satisfies inequality (3), then there exists a solution F
:
S
→
X of (1) such that
(5)
∥
f
(
x
)
-
F
(
x
)
∥
≤
ɛ
|
a
1
-
a
2
|
(
|
a
1
|
|
|
a
1
|
-
1
|
+
|
a
2
|
|
|
a
2
|
-
1
|
)
for all x
∈
S. Moreover, if the condition (
β
) is true, then the F is the unique solution of (1) satisfying (5).In this paper, we investigate the solutions of the functional equation(6)
f
(
x
)
=
p
f
(
x
-
1
)
+
q
f
(
x
-
2
)
+
r
f
(
x
-
3
)
,
where p, q, r are real constants. Moreover, we also prove the Hyers-Ulam stability of that equation. Equation (6) is a kind of linear functional equations of third order because it is of the form
(7)
f
(
x
)
=
a
1
(
x
)
f
(
ξ
(
x
)
)
+
a
2
(
x
)
f
(
ξ
2
(
x
)
)
+
a
3
(
x
)
f
(
ξ
3
(
x
)
)
for the case of a
1
(
x
)
=
p, a
2
(
x
)
=
q, a
3
(
x
)
=
r, and ξ
(
x
)
=
x
-
1.


## 2. General Solution

In the following theorem, we apply [16, Theorem 1.1] for the investigation of general solutions of the functional equation (6).Theorem 3.
Letp, q, r be real constants such that the cubic equation
(8)
x
3
+
p
x
2
-
q
x
+
r
=
0
has the following properties:(i)
α
1 and α
2 are two distinct nonzero real roots of the cubic equation (8);
(ii)
it holds true that either(
α
i
+
p
)
2
+
4
r
/
α
i
>
0 for i
∈
{
1,2
} or (
α
i
+
p
)
2
+
4
r
/
α
i
<
0 for i
∈
{
1,2
}.Let X be either a real vector space if (
α
i
+
p
)
2
+
4
r
/
α
i
>
0 for i
∈
{
1,2
} or a complex vector space if (
α
i
+
p
)
2
+
4
r
/
α
i
<
0 for i
∈
{
1,2
}. Then, a function f
:
R
→
X is a solution of the functional equation (6) if and only if there exist functions h
1
,
h
2
:
[
-
1,1
)
→
X such that
(9)
f
(
x
)
=
α
1
α
1
-
α
2
V
[
x
]
+
1
h
2
(
x
-
[
x
]
)
+
α
1
r
α
2
(
α
1
-
α
2
)
V
[
x
]
h
2
(
x
-
[
x
]
-
1
)
-
α
2
α
1
-
α
2
U
[
x
]
+
1
h
1
(
x
-
[
x
]
)
-
α
2
r
α
1
(
α
1
-
α
2
)
U
[
x
]
h
1
(
x
-
[
x
]
-
1
)
,
where [
x
] denotes the largest integer not exceeding x, and U
n, V
n are defined in (13) and (23).Proof.
Assume thatf
:
R
→
X is a solution of (6). If we define an auxiliary function g
1
:
R
→
X by
(10)
g
1
(
x
)
:
=
f
(
x
)
+
α
1
f
(
x
-
1
)
,
then it follows from (6) that g
1 satisfies
(11)
g
1
(
x
)
=
(
α
1
+
p
)
g
1
(
x
-
1
)
+
r
α
1
g
1
(
x
-
2
)
for any x
∈
R. According to [16, Theorem 1.1] or [3, Theorem 2.1], there exists a function h
1
:
[
-
1,1
)
→
X such that
(12)
g
1
(
x
)
=
f
(
x
)
+
α
1
f
(
x
-
1
)
=
U
[
x
]
+
1
h
1
(
x
-
[
x
]
)
+
r
α
1
U
[
x
]
h
1
(
x
-
[
x
]
-
1
)
for all x
∈
R, where
(13)
U
n
=
a
n
-
b
n
a
-
b
(
n
∈
Z
)
and a, b are the distinct roots of the quadratic equation
(14)
x
2
-
(
α
1
+
p
)
x
-
r
α
1
=
0
,
that is,
(15)
a
=
α
1
+
p
2
+
(
α
1
+
p
2
)
2
+
r
α
1
,
b
=
α
1
+
p
2
-
(
α
1
+
p
2
)
2
+
r
α
1
.
Sincea is a root of the quadratic equation (14), we have
(16)
a
2
=
(
α
1
+
p
)
a
+
r
α
1
.
We multiply both sides of (16) with a and make use of (16) and (i) to get
(17)
a
3
=
p
a
2
+
α
1
a
2
+
r
α
1
a
=
p
a
2
+
α
1
(
(
α
1
+
p
)
a
+
r
α
1
)
+
r
α
1
a
=
p
a
2
+
a
α
1
(
α
1
3
+
p
α
1
2
+
r
)
+
r
=
p
a
2
+
q
a
+
r
.
Similarly, we also obtain
(18)
b
3
=
p
b
2
+
q
b
+
r
.
Using (13), (17), and (18), we have
(19)
p
U
n
-
1
+
q
U
n
-
2
+
r
U
n
-
3
=
(
p
a
2
+
q
a
+
r
)
a
n
-
3
-
(
p
b
2
+
q
b
+
r
)
b
n
-
3
a
-
b
=
a
n
-
b
n
a
-
b
=
U
n
for all n
∈
Z.
If we define an auxiliary functiong
2
:
R
→
X by
(20)
g
2
(
x
)
:
=
f
(
x
)
+
α
2
f
(
x
-
1
)
,
then it follows from (6) that g
2 satisfies
(21)
g
2
(
x
)
=
(
α
2
+
p
)
g
2
(
x
-
1
)
+
r
α
2
g
2
(
x
-
2
)
for any x
∈
R. According to [16, Theorem 1.1] or [3, Theorem 2.1], there exists a function h
2
:
[
-
1,1
)
→
X such that
(22)
g
2
(
x
)
=
f
(
x
)
+
α
2
f
(
x
-
1
)
=
V
[
x
]
+
1
h
2
(
x
-
[
x
]
)
+
r
α
2
V
[
x
]
h
2
(
x
-
[
x
]
-
1
)
for all x
∈
R, where
(23)
V
n
=
c
n
-
d
n
c
-
d
(
n
∈
Z
)
and c, d are the distinct roots of the quadratic equation
(24)
x
2
-
(
α
2
+
p
)
x
-
r
α
2
=
0
,
that is,
(25)
c
=
α
2
+
p
2
+
(
α
2
+
p
2
)
2
+
r
α
2
,
d
=
α
2
+
p
2
-
(
α
2
+
p
2
)
2
+
r
α
2
.
As in the first part, we verify that
(26)
V
n
=
p
V
n
-
1
+
q
V
n
-
2
+
r
V
n
-
3
for all n
∈
Z.
We now multiply (12) with α
2 and (22) with α
1, we subtract the former from the latter, and we then divide the resulting equation by (
α
1
-
α
2
) to get (9).
We assume that a functionf
:
R
→
X is given by (9), where h
1
,
h
2
:
[
-
1,1
)
→
X are arbitrarily given functions and U
n, V
n are given by (13) and (23), respectively. Then, by (9), (19), and (26), we have
(27)
p
f
(
x
-
1
)
+
q
f
(
x
-
2
)
+
r
f
(
x
-
3
)
=
α
1
α
1
-
α
2
(
p
V
[
x
]
+
q
V
[
x
]
-
1
+
r
V
[
x
]
-
2
)
h
2
(
x
-
[
x
]
)
+
α
1
r
α
2
(
α
1
-
α
2
)
(
p
V
[
x
]
-
1
+
q
V
[
x
]
-
2
+
r
V
[
x
]
-
3
)
×
h
2
(
x
-
[
x
]
-
1
)
-
α
2
α
1
-
α
2
(
p
U
[
x
]
+
q
U
[
x
]
-
1
+
r
U
[
x
]
-
2
)
h
1
(
x
-
[
x
]
)
-
α
2
r
α
1
(
α
1
-
α
2
)
(
p
U
[
x
]
-
1
+
q
U
[
x
]
-
2
+
r
U
[
x
]
-
3
)
×
h
1
(
x
-
[
x
]
-
1
)
=
α
1
α
1
-
α
2
V
[
x
]
+
1
h
2
(
x
-
[
x
]
)
+
α
1
r
α
2
(
α
1
-
α
2
)
V
[
x
]
h
2
(
x
-
[
x
]
-
1
)
-
α
2
α
1
-
α
2
U
[
x
]
+
1
h
1
(
x
-
[
x
]
)
-
α
2
r
α
1
(
α
1
-
α
2
)
U
[
x
]
h
1
(
x
-
[
x
]
-
1
)
=
f
(
x
)
for all x
∈
R, which implies that f is a solution of (6).According to [17, p. 92], the Fibonacci numbers F
n satisfy the identity
(28)
F
n
2
=
2
F
n
-
1
2
+
2
F
n
-
2
2
-
F
n
-
3
2
for all integers n
>
3. We can easily notice that the linear equation of third order
(29)
f
(
x
)
=
2
f
(
x
-
1
)
+
2
f
(
x
-
2
)
-
f
(
x
-
3
)
is strongly related to identity (28).Corollary 4.
LetX be a real vector space. A function f
:
R
→
X is a solution of the functional equation (29) if and only if there exist functions h
1
,
h
2
:
[
-
1,1
)
→
X such that
(30)
f
(
x
)
=
5
+
3
5
10
U
[
x
]
+
1
h
1
(
x
-
[
x
]
)
+
15
+
7
5
10
U
[
x
]
h
1
(
x
-
[
x
]
-
1
)
+
5
-
3
5
10
V
[
x
]
+
1
h
2
(
x
-
[
x
]
)
+
15
-
7
5
10
V
[
x
]
h
2
(
x
-
[
x
]
-
1
)
,
where U
n and V
n are defined in (33).Proof.
If we setp
=
2, q
=
2, and r
=
-
1 in (8), then the cubic equation
(31)
x
3
+
2
x
2
-
2
x
-
1
=
0
has three distinct nonzero roots including
(32)
α
1
=
-
3
2
+
5
2
,
α
2
=
-
3
2
-
5
2
.
Moreover, it holds that (
α
1
+
p
)
2
+
4
r
/
α
1
>
0 and (
α
2
+
p
)
2
+
4
r
/
α
2
>
0. By (13), (15), (23), and (25), we have
(33)
U
n
=
a
n
-
b
n
a
-
b
,
V
n
=
c
n
-
d
n
c
-
d
,
where we make use of (15) and (25) to calculate
(34)
a
=
3
+
5
2
,
b
=
-
1
,
c
=
3
-
5
2
,
d
=
-
1
.
Finally, in view of Theorem3, we conclude that the assertion of our corollary is true.Corollary 5.
If a functionf
:
R
→
R is a solution of functional equation (29), then there exist real constants μ
1, μ
2, ν
1, and ν
2 such that
(35)
f
(
n
)
=
5
+
3
5
10
μ
1
U
n
+
1
+
15
+
7
5
10
μ
2
U
n
+
5
-
3
5
10
ν
1
V
n
+
1
+
15
-
7
5
10
ν
2
V
n
for all n
∈
Z, where U
n and V
n are defined in (33).


## 3. Hyers-Ulam Stability

We apply the classical direct method to the proof of the following theorem. The classical direct method was first proposed by Hyers [2].Theorem 6.
Letp, q, r be real constants with r
≠
0, let α be a nonzero root of the cubic equation (8), and let a, b be the roots of the quadratic equation x
2
-
(
α
+
p
)
x
-
r
/
α
=
0 with |
a
|
>
1 and 0
<
|
b
|
<
1. Let X be either a real Banach space if (
α
+
p
)
2
+
4
r
/
α
>
0 or a complex Banach space if (
α
+
p
)
2
+
4
r
/
α
<
0. If a function f
:
R
→
X satisfies the inequality
(36)
∥
f
(
x
)
-
p
f
(
x
-
1
)
-
q
f
(
x
-
2
)
-
r
f
(
x
-
3
)
∥
≤
ɛ
for all x
∈
R and for some ɛ
≥
0, then there exists a solution G
:
R
→
X of (6) such that
(37)
∥
f
(
x
)
+
α
f
(
x
-
1
)
-
G
(
x
)
∥
≤
|
a
|
-
|
b
|
|
a
-
b
|
ɛ
(
|
a
|
-
1
)
(
1
-
|
b
|
)
for all x
∈
R.Proof.
If we define an auxiliary functiong
:
R
→
X by
(38)
g
(
x
)
:
=
f
(
x
)
+
α
f
(
x
-
1
)
,
then, as we did in (11), it follows from (36) that g satisfies the inequality
(39)
∥
g
(
x
)
-
(
α
+
p
)
g
(
x
-
1
)
-
r
α
g
(
x
-
2
)
∥
≤
ɛ
or
(40)
∥
g
(
x
)
-
a
g
(
x
-
1
)
-
b
[
g
(
x
-
1
)
-
a
g
(
x
-
2
)
]
∥
≤
ɛ
for any x
∈
R.
If we replacex with x
-
k in the last inequality, then we have
(41)
∥
[
g
(
x
-
k
-
1
)
-
a
g
(
x
-
k
-
2
)
]
g
(
x
-
k
)
-
a
g
(
x
-
k
-
1
)
-
b
[
g
(
x
-
k
-
1
)
-
a
g
(
x
-
k
-
2
)
]
∥
≤
ɛ
for all x
∈
R. Furthermore, we get
(42)
∥
b
k
[
g
(
x
-
k
)
-
a
g
(
x
-
k
-
1
)
]
-
b
k
+
1
[
g
(
x
-
k
-
1
)
-
a
g
(
x
-
k
-
2
)
]
b
k
[
g
(
x
-
k
)
-
a
g
(
x
-
k
-
1
)
]
∥
≤
|
b
|
k
ɛ
for all x
∈
R and k
∈
Z. By (42), we obviously have
(43)
∥
g
(
x
)
-
a
g
(
x
-
1
)
-
b
n
[
g
(
x
-
n
)
-
a
g
(
x
-
n
-
1
)
]
∥
≤
∑
k
=
0
n
-
1
‍
∥
b
k
[
g
(
x
-
k
)
-
a
g
(
x
-
k
-
1
)
]
h
-
b
k
+
1
[
g
(
x
-
k
-
1
)
-
a
g
(
x
-
k
-
2
)
]
∥
≤
∑
k
=
0
n
-
1
‍
|
b
|
k
ɛ
for x
∈
R and n
∈
N.
For anyx
∈
R, (42) implies that the sequence {
b
n
[
g
(
x
-
n
)
-
a
g
(
x
-
n
-
1
)
]
} is a Cauchy sequence (note that 0
<
|
b
|
<
1). Therefore, we can define a function G
1
:
R
→
X by
(44)
G
1
(
x
)
:
=
lim
⁡
n
→
∞
b
n
[
g
(
x
-
n
)
-
a
g
(
x
-
n
-
1
)
]
,
since X is complete. In view of the definition of G
1 and using the relations, a
+
b
=
α
+
p and a
b
=
-
r
/
α, we obtain
(45)
(
α
+
p
)
G
1
(
x
-
1
)
+
r
α
G
1
(
x
-
2
)
=
(
a
+
b
)
G
1
(
x
-
1
)
-
a
b
G
1
(
x
-
2
)
=
a
+
b
b
lim
⁡
n
→
∞
b
n
+
1
[
g
(
x
-
(
n
+
1
)
)
-
a
g
(
x
-
(
n
+
1
)
-
1
)
]
-
a
b
b
2
lim
⁡
n
→
∞
b
n
+
2
[
g
(
x
-
(
n
+
2
)
)
-
a
g
(
x
-
(
n
+
2
)
-
1
)
]
=
a
+
b
b
G
1
(
x
)
-
a
b
G
1
(
x
)
=
G
1
(
x
)
for all x
∈
R. Since α is a nonzero root of the cubic equation (8), it follows from (45) that
(46)
G
1
(
x
)
-
p
G
1
(
x
-
1
)
-
q
G
1
(
x
-
2
)
-
r
G
1
(
x
-
3
)
=
(
α
+
p
)
G
1
(
x
-
1
)
+
r
α
G
1
(
x
-
2
)
-
p
G
1
(
x
-
1
)
-
q
G
1
(
x
-
2
)
-
r
G
1
(
x
-
3
)
=
α
G
1
(
x
-
1
)
+
(
-
q
+
r
α
)
G
1
(
x
-
2
)
-
r
G
1
(
x
-
3
)
=
α
G
1
(
x
-
1
)
+
(
-
α
2
-
p
α
)
G
1
(
x
-
2
)
-
r
G
1
(
x
-
3
)
=
α
(
(
α
+
p
)
G
1
(
x
-
2
)
+
r
α
G
1
(
x
-
3
)
)
-
α
(
α
+
p
)
G
1
(
x
-
2
)
-
r
G
1
(
x
-
3
)
=
0
for all x
∈
R. Hence, we conclude that G
1 is a solution of (6).
Ifn tends to infinity, then (43) yields that
(47)
∥
g
(
x
)
-
a
g
(
x
-
1
)
-
G
1
(
x
)
∥
≤
ɛ
1
-
|
b
|
for every x
∈
R.
On the other hand, it also follows from (36) that
(48)
∥
g
(
x
)
-
b
g
(
x
-
1
)
-
a
[
g
(
x
-
1
)
-
b
g
(
x
-
2
)
]
∥
≤
ɛ
for all x
∈
R. Analogously to (42), replacing x by x
+
k in the last inequality and then dividing by |
a
|
k both sides of the resulting inequality, then we have
(49)
∥
a
-
k
[
g
(
x
+
k
)
-
b
g
(
x
+
k
-
1
)
]
-
a
-
k
+
1
[
g
(
x
+
k
-
1
)
-
b
g
(
x
+
k
-
2
)
]
∥
≤
|
a
|
-
k
ɛ
for all x
∈
R and k
∈
Z. By using (49), we further obtain
(50)
∥
a
-
n
[
g
(
x
+
n
)
-
b
g
(
x
+
n
-
1
)
]
-
[
g
(
x
)
-
b
g
(
x
-
1
)
]
∥
≤
∑
k
=
1
n
‍
∥
a
-
k
[
g
(
x
+
k
)
-
b
g
(
x
+
k
-
1
)
]
h
-
a
-
k
+
1
[
g
(
x
+
k
-
1
)
-
b
g
(
x
+
k
-
2
)
]
∥
≤
∑
k
=
1
n
‍
|
a
|
-
k
ɛ
for x
∈
R and n
∈
N.
On account of (49), we see that the sequence {
a
-
n
[
g
(
x
+
n
)
-
b
g
(
x
+
n
-
1
)
]
} is a Cauchy sequence for any fixed x
∈
R (note that |
a
|
>
1). Hence, we can define a function G
2
:
R
→
X by
(51)
G
2
(
x
)
:
=
lim
⁡
n
→
∞
a
-
n
[
g
(
x
+
n
)
-
b
g
(
x
+
n
-
1
)
]
.
Due to the definition of G
2 and the relations, a
+
b
=
α
+
p and a
b
=
-
r
/
α, we get
(52)
(
α
+
p
)
G
2
(
x
-
1
)
+
r
α
G
2
(
x
-
2
)
=
(
a
+
b
)
G
2
(
x
-
1
)
-
a
b
G
2
(
x
-
2
)
=
a
+
b
a
lim
⁡
n
→
∞
a
-
(
n
-
1
)
[
g
(
x
+
n
-
1
)
-
b
g
(
x
+
n
-
2
)
]
-
a
b
a
2
lim
⁡
n
→
∞
a
-
(
n
-
2
)
[
g
(
x
+
n
-
2
)
-
b
g
(
x
+
n
-
3
)
]
=
a
+
b
a
G
2
(
x
)
-
b
a
G
2
(
x
)
=
G
2
(
x
)
for any x
∈
R. Similarly as in the first part, we can show that G
2 is a solution of (6).
If we letn tend to infinity, then it follows from (50) that
(53)
∥
G
2
(
x
)
-
g
(
x
)
+
b
g
(
x
-
1
)
∥
≤
ɛ
|
a
|
-
1
for x
∈
R.
It follows from (47) and (53) that
(54)
∥
g
(
x
-
1
)
-
1
a
-
b
G
2
(
x
)
+
1
a
-
b
G
1
(
x
)
∥
≤
∥
1
a
-
b
G
1
(
x
)
-
1
a
-
b
g
(
x
)
+
a
a
-
b
g
(
x
-
1
)
∥
+
∥
1
a
-
b
g
(
x
)
-
b
a
-
b
g
(
x
-
1
)
-
1
a
-
b
G
2
(
x
)
∥
≤
|
a
|
-
|
b
|
|
a
-
b
|
ɛ
(
|
a
|
-
1
)
(
1
-
|
b
|
)
for any x
∈
R.
Finally, if we define a functionG
:
R
→
X by
(55)
G
(
x
)
:
=
1
a
-
b
G
2
(
x
+
1
)
-
1
a
-
b
G
1
(
x
+
1
)
for all x
∈
R, then G is also a solution of (6). Moreover, the validity of (37) follows from the last inequality.The following theorem is the main theorem of this paper.Theorem 7.
Given real constantsp, q, r with r
≠
0, let α
1 and α
2 be distinct nonzero roots of cubic equation (8) and let a
i, b
i be the roots of the quadratic equation x
2
-
(
α
i
+
p
)
x
-
r
/
α
i
=
0 with |
a
i
|
>
1 and 0
<
|
b
i
|
<
1 for i
∈
{
1,2
}. Assume that either (
α
i
+
p
)
2
+
4
r
/
α
i
>
0 for all i
∈
{
1,2
} or (
α
i
+
p
)
2
+
4
r
/
α
i
<
0 for all i
∈
{
1,2
}. Let X be either a real Banach space if (
α
i
+
p
)
2
+
4
r
/
α
i
>
0 or a complex Banach space if (
α
i
+
p
)
2
+
4
r
/
α
i
<
0. If a function f
:
R
→
X satisfies inequality (36) for all x
∈
R and for some ɛ
≥
0, then there exists a solution F
:
R
→
X of (6) such that
(56)
∥
f
(
x
)
-
F
(
x
)
∥
≤
|
a
1
|
-
|
b
1
|
|
a
1
-
b
1
|
|
α
2
|
|
α
1
-
α
2
|
ɛ
(
|
a
1
|
-
1
)
(
1
-
|
b
1
|
)
+
|
a
2
|
-
|
b
2
|
|
a
2
-
b
2
|
|
α
1
|
|
α
1
-
α
2
|
ɛ
(
|
a
2
|
-
1
)
(
1
-
|
b
2
|
)
for all x
∈
R.Proof.
According to Theorem6, there exists a solution F
i
:
R
→
X of (6) such that
(57)
∥
f
(
x
)
+
α
i
f
(
x
-
1
)
-
F
i
(
x
)
∥
≤
|
a
i
|
-
|
b
i
|
|
a
i
-
b
i
|
ɛ
(
|
a
i
|
-
1
)
(
1
-
|
b
i
|
)
for any x
∈
R and i
∈
{
1,2
}. In view of the last inequalities, we have
(58)
∥
f
(
x
)
-
α
1
α
1
-
α
2
F
2
(
x
)
+
α
2
α
1
-
α
2
F
1
(
x
)
∥
≤
∥
α
2
α
1
-
α
2
F
1
(
x
)
-
α
2
α
1
-
α
2
f
(
x
)
-
α
1
α
2
α
1
-
α
2
f
(
x
-
1
)
∥
+
∥
α
1
α
1
-
α
2
f
(
x
)
+
α
1
α
2
α
1
-
α
2
f
(
x
-
1
)
-
α
1
α
1
-
α
2
F
2
(
x
)
∥
≤
|
a
1
|
-
|
b
1
|
|
a
1
-
b
1
|
|
α
2
|
|
α
1
-
α
2
|
ɛ
(
|
a
1
|
-
1
)
(
1
-
|
b
1
|
)
+
|
a
2
|
-
|
b
2
|
|
a
2
-
b
2
|
|
α
1
|
|
α
1
-
α
2
|
ɛ
(
|
a
2
|
-
1
)
(
1
-
|
b
2
|
)
for all x
∈
R.
If we define a functionF
:
R
→
X by
(59)
F
(
x
)
:
=
α
1
α
1
-
α
2
F
2
(
x
)
-
α
2
α
1
-
α
2
F
1
(
x
)
for each x
∈
R, then F is also a solution of (6), and inequality (56) follows from the last inequality.



---

*Source: 137468-2014-06-30.xml*