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 Jung
4
1
Valahia University of Tˆ argovis ¸te, Bulevardul Unirii 18, 130082 Tˆ argovis ¸te, Romania
2
Academy of Romanian Scientists, Splaiul Independent ¸ei 54, 050094 Bucharest, Romania
3
Department of Mathematics, ETH Z¨ urich, Raemistrasse 101, 8092 Z¨ urich, Switzerland
4
Mathematics Section, College of Science and Technology, Hongik University, Sejong 339-701, Republic of Korea
Correspondence should be addressed to Soon-Mo Jung; smjung@hongik.ac.kr
Received 5 May 2014; Accepted 8 July 2014; Published 20 July 2014
Academic Editor: Chengjian Zhang
Copyright © 2014 Cristinel Mortici et al. his 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 () − ()(ℎ()) = 0, where 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
1
be a group and let
2
be a metric group with the
metric (⋅,⋅). Given >0, does there exist a >0 such
that if a function ℎ:
1
→
2
satisies the inequality
(ℎ(),ℎ()ℎ()) < , for all , ∈
1
, then there exists
a homomorphism :
1
→
2
with (ℎ(),()) < , for
all ∈
1
?
he case of approximately additive functions was solved
by Hyers [2] under the assumption that
1
and
2
are Banach
spaces. Indeed, he proved the following theorem.
heorem 1. Let :
1
→
2
be a function between Banach
spaces such that
(+)− ()− ()
≤, (1)
for some >0 and for all ,∈
1
. hen, the limit
()= lim
→∞
2
−
(2
)
(2)
exists for each ∈
1
, and :
1
→
2
is the unique additive
function such that
()−()
≤, (3)
for any ∈
1
. Moreover, if () is continuous in , for each
ixed ∈
1
, then the function is linear.
Hyers proved that each solution of the inequality ‖( +
) − () − ()‖ ≤ can be approximated by an exact
solution; say an additive function. In this case, the Cauchy
additive functional equation, (+)=()+(), 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
()−()(ℎ())=0, (4)
where 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 is a constant function, which is a generalization
Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2014, Article ID 546046, 6 pages
http://dx.doi.org/10.1155/2014/546046