On the Stability of a Functional Equation Associated with the Fibonacci Numbers
Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2014, Article ID 546046, 6 pages
http://dx.doi.org/10.1155/2014/546046
Research Article
On the Stability of a Functional Equation Associated with the
Fibonacci Numbers
Cristinel Mortici,1,2 Michael Th. Rassias,3 and Soon-Mo Jung4
1
Valahia University of Târgovişte, Bulevardul Unirii 18, 130082 Târgovişte, Romania
Academy of Romanian Scientists, Splaiul Independenţei 54, 050094 Bucharest, Romania
3
Department of Mathematics, ETH Zürich, Raemistrasse 101, 8092 Zürich, Switzerland
4
Mathematics Section, College of Science and Technology, Hongik University, Sejong 339-701, Republic of Korea
2
Correspondence should be addressed to Soon-Mo Jung;
Received 5 May 2014; Accepted 8 July 2014; Published 20 July 2014
Academic Editor: Chengjian Zhang
Copyright © 2014 Cristinel Mortici et al. 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.
We prove the Hyers-Ulam stability of the generalized Fibonacci functional equation 𝐹(𝑥) − 𝑔(𝑥)𝐹(ℎ(𝑥)) = 0, where 𝑔 and h are
given functions.
1. Introduction
In 1940, Ulam [1] gave a wide ranging talk before the
mathematics club of the University of Wisconsin in which he
discussed a number of important unsolved problems. Among
them was the question concerning the stability of group
homomorphisms.
Let 𝐺1 be a group and let 𝐺2 be a metric group with the
metric 𝑑(⋅, ⋅). Given 𝜀 > 0, does there exist a 𝛿 > 0 such
that if a function ℎ : 𝐺1 → 𝐺2 satisfies the inequality
𝑑(ℎ(𝑥𝑦), ℎ(𝑥)ℎ(𝑦)) < 𝛿, for all 𝑥, 𝑦 ∈ 𝐺1 , then there exists
a homomorphism 𝐻 : 𝐺1 → 𝐺2 with 𝑑(ℎ(𝑥), 𝐻(𝑥)) < 𝜀, for
all 𝑥 ∈ 𝐺1 ?
The case of approximately additive functions was solved
by Hyers [2] under the assumption that 𝐺1 and 𝐺2 are Banach
spaces. Indeed, he proved the following theorem.
Theorem 1. Let 𝑓 : 𝐺1 → 𝐺2 be a function between Banach
spaces such that
𝑓 (𝑥 + 𝑦) − 𝑓 (𝑥) − 𝑓 (𝑦) ≤ 𝜀,
(1)
for some 𝜀 > 0 and for all 𝑥, 𝑦 ∈ 𝐺1 . Then, the limit
𝐴 (𝑥) = lim 2−𝑛 𝑓 (2𝑛 𝑥)
𝑛→∞
(2)
exists for each 𝑥 ∈ 𝐺1 , and 𝐴 : 𝐺1 → 𝐺2 is the unique additive
function such that
𝑓 (𝑥) − 𝐴 (𝑥) ≤ 𝜀,
(3)
for any 𝑥 ∈ 𝐺1 . Moreover, if 𝑓(𝑡𝑥) is continuous in 𝑡, for each
fixed 𝑥 ∈ 𝐺1 , then the function 𝐴 is linear.
Hyers proved that each solution of the inequality ‖𝑓(𝑥 +
𝑦) − 𝑓(𝑥) − 𝑓(𝑦)‖ ≤ 𝜀 can be approximated by an exact
solution; say an additive function. In this case, the Cauchy
additive functional equation, 𝑓(𝑥 + 𝑦) = 𝑓(𝑥) + 𝑓(𝑦), is said
to have the Hyers-Ulam stability.
Since then, the stability problems of a large variety of
functional equations have been extensively investigated by
several mathematicians (cf. [3–14]).
In this paper, we investigate the Hyers-Ulam stability of
the functional equation
𝐹 (𝑥) − 𝑔 (𝑥) 𝐹 (ℎ (𝑥)) = 0,
(4)
where 𝑔 and ℎ are given functions.
In Section 2, we prove that the functional equation (4)
has a large class of nontrivial solutions. Section 3 is devoted
to the investigation of the Hyers-Ulam stability problems for
(4). In the last section, we prove the Hyers-Ulam stability of
(4) when 𝑔 is a constant function, which is a generalization
2
Abstract and Applied Analysis
of the papers [4, 7, 14]. More precisely, Jung [7] proved the
Hyers-Ulam stability of the generalized Fibonacci functional
equation
𝑓 (𝑥) = 𝑝𝑓 (𝑥 − 1) − 𝑞𝑓 (𝑥 − 2)
(5)
in the class of functions 𝑓 : R → 𝑋, where 𝑋 is a real (or
complex) Banach space.
Theorem 2 (see [7, Theorem 3.1]). Assume that the quadratic
equation 𝑥2 − 𝑝𝑥 + 𝑞 = 0 has real solutions 𝑎 and 𝑏 with 0 <
|𝑏| < 1 < |𝑎|. If a function 𝑓 : R → 𝑋 satisfies the inequality
(6)
𝑓 (𝑥) − 𝑝𝑓 (𝑥 − 1) + 𝑞𝑓 (𝑥 − 2) ≤ 𝜀
for all 𝑥 ∈ R and for some 𝜀 > 0, then there exists a unique
solution 𝐹 : R → 𝑋 of (5) such that
𝜀
|𝑎| − |𝑏|
⋅
𝑓 (𝑥) − 𝐹 (𝑥) ≤
|𝑎 − 𝑏| (|𝑎| − 1) (1 − |𝑏|)
(7)
for all 𝑥 ∈ R.
A similar case for 0 < |𝑏| < 1 < |𝑎| with |𝑏| ≠ 1/2 was
investigated by Brzdęk et al. [4] and Trif [14] who obtained
the estimate
4𝜀
.
𝑓 (𝑥) − 𝐹 (𝑥) ≤
|2 |𝑏| − 1| ⋅ (2 |𝑎| − 1)
(8)
If either 0 < |𝑏| < 1/2 and |𝑎| > 3/2 − |𝑏| or 1/2 < |𝑏| < 3/4
and |𝑎| > (5−6|𝑏|)/(6−8|𝑏|), then the inequality (7) is sharper
than that of (8).
In Section 4 of this paper, we improve the results of papers
[4, 7, 14] in the sense that we estimate ‖𝑓(𝑥) − 𝐹(𝑥)‖ even
when both |𝑎| and |𝑏| are larger or smaller than 1. Moreover,
we deal with a functional equation (4) that is regarded as a
more generalized form of the Fibonacci functional equation
(5).
In this paper, R, Z, and N stand for the sets of real
numbers, integers, and positive integers, respectively.
2. Solutions of (4)
Evidently, (4) admits the trivial solution 𝐹 = 0. In order to
avoid the trivial case, we search in this section for a class of
nontrivial solutions of (4).
Let 𝐷 be a subset of R. A function ℎ : 𝐷 → 𝐷 is said to
be of disjoint iterated images, shortly (DII)-function, if
(i) there exists a partition
𝑛≥1
(9)
(ii) ℎ maps bijectively 𝐷𝑛 onto 𝐷𝑛+1 for each integer 𝑛 ≥ 1.
As an example for a (DII)-function, we introduce a
function ℎ : (0, 1] → (0, 1] defined by
1
1
(𝑛𝑥 +
),
𝑛+2
𝑛+1
Theorem 3. Let ℎ : 𝐷 → 𝐷 be a (DII)-function and 𝑔 : 𝐷 →
R \ {0}. There is a one-to-one correspondence between the set
of all solutions 𝐹 : 𝐷 → R of the functional equation (4) and
the set of all functions 𝜑 : 𝐷1 → R.
Proof. Given a 𝜑 : 𝐷1 → R, we define a function 𝐹 on 𝐷1
as
𝐹 (𝑥) = 𝜑 (𝑥) ,
𝑥 ∈ 𝐷𝑛 := (
1 1
, ],
𝑛+1 𝑛
(10)
(11)
for all 𝑥 ∈ 𝐷1 . Assume that 𝐹 is defined on 𝐷𝑛−1 for some
𝑛 ≥ 2. If 𝑥 ∈ 𝐷𝑛 , then ℎ−1 (𝑥) ∈ 𝐷𝑛−1 and we put
𝐹 (𝑥) =
1
𝐹 (ℎ−1 (𝑥)) ,
𝑔 (ℎ−1 (𝑥))
(12)
for all 𝑥 ∈ 𝐷𝑛 . By this inductive procedure, 𝐹 is completely
defined.
We now show that 𝐹 is a solution of (4). Let 𝑧 be any point
of 𝐷 and let 𝑛 ≥ 2 be an integer such that ℎ(𝑧) ∈ 𝐷𝑛 . Put
𝑥 = ℎ(𝑧) in (12) to get
𝐹 (ℎ (𝑧)) =
1
𝐹 (𝑧) ,
𝑔 (𝑧)
(13)
which is (4).
Conversely, we associate to every solution 𝐹 of (4) the
function 𝜑 = 𝐹|𝐷1 .
We notice that a (DII)-function ℎ is injective as we see the
following: if 𝑥, 𝑦 ∈ 𝐷𝑛 for some 𝑛 ∈ N with 𝑥 ≠ 𝑦 but ℎ(𝑥) =
ℎ(𝑦), then ℎ(𝑥) = ℎ(𝑦) ∈ 𝐷𝑛+1 and, hence, 𝑥 = 𝑦 because
ℎ maps bijectively 𝐷𝑛 onto 𝐷𝑛+1 , a contradiction. If 𝑥 ∈ 𝐷𝑚
and 𝑦 ∈ 𝐷𝑛 for some 𝑚, 𝑛 ∈ N with 𝑚 ≠ 𝑛, it is then obvious
that ℎ(𝑥) ≠ ℎ(𝑦) because ℎ(𝑥) ∈ 𝐷𝑚+1 , ℎ(𝑦) ∈ 𝐷𝑛+1 , and
𝐷𝑚+1 ∩ 𝐷𝑛+1 = 0. But ℎ is not surjective, since I𝑚ℎ = 𝐷 \ 𝐷1 .
We now study the set of solutions of (4) under the
assumption that ℎ : 𝐷 → 𝐷 is a bijection. For any pair of
points 𝑥, 𝑦 ∈ 𝐷, we use the notation 𝑥 ≍ 𝑦 if there exists a
𝑘 ∈ Z with 𝑦 = ℎ𝑘 (𝑥). Since “≍” is an equivalence relation in
𝐷, let
𝐷 = ∐Δ 𝑖
𝑖∈𝐼
𝐷 = ∐𝐷𝑛 ;
ℎ (𝑥) =
for all 𝑛 ∈ N. For every 𝑛 ∈ N, this function is linear on 𝐷𝑛
and it transforms each 𝐷𝑛 onto 𝐷𝑛+1 .
We are now in a position to prove that the set of all
solutions of (4) is not empty but it is an infinite set.
(14)
be the corresponding p (...truncated)
