# On the Stability of a Functional Equation Associated with the Fibonacci Numbers

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

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## Abstract

We prove the Hyers-Ulam stability of the generalized Fibonacci functional equationF(x)-g(x)F(h(x))=0, where g and h are given functions.


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## Body

## 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
G
1
be a group and let
G
2
be a metric group with the metric
d
(
·
,
·
)
. Given
ε
>
0
, does there exist a
δ
>
0
such that if a function
h
:
G
1
→
G
2
satisfies the inequality
d
(
h
(
x
y
)
,
h
(
x
)
h
(
y
)
)
<
δ
, for all
x
,
y
∈
G
1
, then there exists a homomorphism
H
:
G
1
→
G
2
with
d
(
h
(
x
)
,
H
(
x
)
)
<
ε
, for all
x
∈
G
1
?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.
Letf:G1→G2 be a function between Banach spaces such that
(1)∥f(x+y)-f(x)-f(y)∥≤ε,
for some ε>0 and for all x,y∈G1. Then, the limit
(2)A(x)=lim⁡n→∞2-nf(2nx)
exists for each x∈G1, and A:G1→G2 is the unique additive function such that
(3)∥f(x)-A(x)∥≤ε,
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(4)F(x)-g(x)F(h(x))=0,
where g and h are given functions.In Section2, 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 of the papers [4, 7, 14]. More precisely, Jung [7] proved the Hyers-Ulam stability of the generalized Fibonacci functional equation
(5)f(x)=pf(x-1)-qf(x-2)
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 equationx2-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
(7)∥f(x)-F(x)∥≤|a|-|b||a-b|·ε(|a|-1)(1-|b|)
for all x∈R.A similar case for0<|b|<1<|a| with |b|≠1/2 was investigated by Brzdęk et al. [4] and Trif [14] who obtained the estimate
(8)∥f(x)-F(x)∥≤4ε|2|b|-1|·(2|a|-1).
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 Section4 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).LetD be a subset of R. A function h:D→D is said to beof disjoint iterated images, shortly (DII)-function, if(i)
there exists a partition(9)D=∐n≥1Dn;(ii)
h maps bijectively Dn onto Dn+1 for each integer n≥1.As an example for a (DII)-function, we introduce a functionh:(0,1]→(0,1] defined by
(10)h(x)=1n+2(nx+1n+1),x∈Dn≔(1n+1,1n],
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.Theorem 3.
Leth: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 aφ:D1→R, we define a function F on D1 as
(11)F(x)=φ(x),
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
(12)F(x)=1g(h-1(x))F(h-1(x)),
for all x∈Dn. By this inductive procedure, F is completely defined.
We now show thatF 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
(13)F(h(z))=1g(z)F(z),
which is (4).
Conversely, we associate to every solutionF of (4) the function φ=F|D1.We notice that a (DII)-functionh 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=∅. 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
(14)D=∐i∈I‍Δi
be the corresponding partition in “≍-equivalence classes” Δi=xi^(i∈I); that is,
(15)Δi={hk(xi)∣k∈Z}.Theorem 4.
Given a subsetD of R, let h:D→D be a bijective function and g:D→R∖{0}. Assume that
(16)D=∐i∈I‍Δi
is a partition of D corresponding to the equivalence relation ≍ with the property (15). Then, there exists a one-to-one correspondence between the set of all solutions F:D→R of the functional equation (4) and the set of all real sequences {yi}i∈I.Proof.
For any real sequence{yi}i∈I, we define F(xi)=yi for all i∈I, where I is the index set for the partition corresponding to the equivalence relation ≍ with the property (15). We further define the function F:D→R by
(17)F(h(xi))=1g(xi)F(xi)=1g(xi)yi,F(h-1(xi))=g(h-1(xi))F(xi)=g(h-1(xi))yi.
In general, if F is defined at hk(xi) and h-k(xi), then F is defined at hk+1(xi) and h-k-1(xi) by
(18)F(hk+1(xi))=1g(hk(xi))F(hk(xi)),F(h-k-1(xi))=g(h-k-1(xi))F(h-k(xi)).
For each i∈I, we can use such an inductive procedure to define the function F on Δi and we see that F|Δi is uniquely determined by the value of yi.
Conversely, every solutionF:D→R of (4) can be associated to the real sequence {F(xi)}i∈I.Corollary 5.
Given a subsetD of R, let h:D→D be a bijective function and α∈R∖{0}. Assume that
(19)D=∐i∈I‍Δi
is a partition of D corresponding to the equivalence relation ≍ with the property (15). Then there exists a one-to-one correspondence between the set of all solutions F:D→R of the functional equation
(20)F(x)-αF(h(x))=0
and the set of all real numbers {yi}i∈I.


## 3. Hyers-Ulam Stability of (4)

The above conditions imposed on the functionh were necessary for showing that the functional equations (4) and (20) have large classes of nontrivial solutions. The stability results presented in the sequel are valid also under weaker conditions as we shall see in the following theorems.Theorem 6.
Given real numbersa and b with a<b, let h:(a,b)→(a,b) and g:(a,b)→I be given functions, where I⊂(0,1) is an interval of length l. Assume that a bounded function f:(a,b)→R satisfies the inequality
(21)|f(x)-g(x)f(h(x))|≤ε
for all x∈(a,b) and for some ε>0. Then, for every α∈I, there exists a solution F:(a,b)→R of (20) such that
(22)|f(x)-F(x)|≤ε+l|f|∞1-α,
for any x∈(a,b), where |f|∞=sup⁡x∈(a,b)|f(x)|.Proof.
First, we prove that(23)|f(x)-αf(h(x))|≤λ,
for all x∈(a,b), where we set λ=ε+l|f|∞. Indeed, it follows from (21) that
(24)|f(x)-αf(h(x))|≤|f(x)-g(x)f(h(x))|+|g(x)f(h(x))-αf(h(x))|≤ε+|g(x)-α|·|f(h(x))|≤ε+l|f|∞=λ,
for every x∈(a,b).
By replacingx with hk(x) and then multiplying with αk both sides of (23), we get
(25)|αkf(hk(x))-αk+1f(hk+1(x))|≤λαk,
for all x∈(a,b) and k∈N. Since
(26)|f(x)-αnf(hn(x))|≤∑k=0n-1‍|αkf(hk(x))-αk+1f(hk+1(x))|≤∑k=0n-1‍λαk=λ1-αn1-α,
we have
(27)|f(x)-αnf(hn(x))|≤λ1-αn1-α,
for any x∈(a,b) and n∈N.
The inequality (25) shows that the sequence {αnf(hn(x))}n∈N is a Cauchy sequence for every x∈(a,b). Thus, we can define a function F:(a,b)→R by
(28)F(x)=lim⁡n→∞αnf(hn(x)),
for all x∈(a,b). Hence, it follows from (23) that
(29)|F(x)-αF(h(x))|=|lim⁡n→∞αnf(hn(x))-lim⁡n→∞αn+1f(hn+1(x))|=lim⁡n→∞|αn(f(hn(x))-αf(hn+1(x)))|≤lim⁡n→∞αnλ=0,
for each x∈(a,b), which implies that F is a solution of (20).
Finally, inequality (22) is an immediate consequence of (27) if we take the limit as n→∞.


## 4. Wheng Is Constant

In the case ofg(x)=s∉{-1,+1} for all x∈R, we investigate the Hyers-Ulam stability of the functional equation
(30)F(x)-sF(h(x))=0,
where h:R→R and F:R→X are functions and X is a real Banach space.Theorem 7.
LetX be a real Banach space and let s be a real number with |s|<1. If a function f:R→X satisfies the inequality
(31)∥f(x)-sf(h(x))∥≤ε,
for all x∈R and for some ε>0, then there exists a solution F:R→X of (30) such that
(32)∥f(x)-F(x)∥≤ε1-|s|,
for all x∈R.Proof.
By replacingx with hk(x) and multiplying with sk both sides of (31), we get
(33)∥skf(hk(x))-sk+1f(hk+1(x))∥≤ε|s|k,
for all x∈R and k∈{0,1,2,…}. By (33), we have
(34)∥f(x)-snf(hn(x))∥≤∑k=0n-1‍∥skf(hk(x))-sk+1f(hk+1(x))∥≤∑k=0n-1‍ε|s|k=ε1-|s|n1-|s|,
for all x∈R and n∈N. Hence, we get
(35)∥f(x)-snf(hn(x))∥≤ε1-|s|n1-|s|,
for any x∈R and n∈N.
The inequality (33) shows that the sequence {snf(hn(x))}n∈N is a Cauchy sequence for any fixed x∈R. Thus, since X is a complete space, we can define a function F:R→X by
(36)F(x)=lim⁡n→∞snf(hn(x)),
for all x∈R.
It follows from (31) that
(37)∥F(x)-sF(h(x))∥=∥lim⁡n→∞snf(hn(x))-lim⁡n→∞sn+1f(hn+1(x))∥=lim⁡n→∞sn∥f(hn(x))-sf(hn+1(x))∥≤lim⁡n→∞snε=0,
which implies that F is a solution of (30).
Finally, the inequality (32) immediately follows from (35) provided that we take the limit as n→∞.Assume now thath:R→R is bijective. A similar theorem can be proved when |s|>1.Theorem 8.
LetX be a real Banach space, let h:R→R be a bijective function, and let s∈R be given with |s|>1. If a function f:R→X satisfies the inequality
(38)∥f(x)-sf(h(x))∥≤ε,
for all x∈R and for some ε>0, then there exists a solution F:R→X of (30) such that
(39)∥f(x)-F(x)∥≤ε|s|-1,
for all x∈R.Proof.
By replacingx with h-1(x) and dividing by s both sides of (38), we get
(40)∥f(x)-1sf(h-1(x))∥≤ε|s|,
for any x∈R. Since the constant 1/|s| is less than 1, our assertion follows from Theorem 7. In particular, we have
(41)F(x)=lim⁡n→∞s-nf(h-n(x)),
for each x∈R.Corollary 9.
LetX be a real Banach space, let h:R→R be a bijective function, and let s∈R be given with |s|≠1. If a function f:R→X satisfies the inequality
(42)∥f(x)-sf(h(x))∥≤ε,
for any x∈R and for some ε>0, then there exists a solution F:R→X of (30) such that
(43)∥f(x)-F(x)∥≤ε||s|-1|,
for all x∈R.By combining the results of Theorems7 and 8, we can present a stability result of the following functional equation
(44)F(x)-pF(h(x))+qF(h(h(x)))=0,
where h:R→R is bijective and the range space of the function F:R→X is a real Banach space.Theorem 10.
Letp and q be given real numbers such that the quadratic equation x2-px+q=0 has distinct real solutions a and b with |a|≠1 and |b|≠1. Assume that a bijective function h:R→R is given and X is a real Banach space. If a function f:R→X satisfies
(45)∥f(x)-pf(h(x))+qf(h(h(x)))∥≤ε,
for all x∈R and for some ε>0, then there exists a solution F:R→X of (44) such that
(46)∥f(x)-F(x)∥≤1|a-b|(1||a|-1-1|+1||b|-1-1|)ε,
for all x∈R.As we mentioned in the Introduction, our result extends Jung’s result in [7], since
(47)1|a-b|(1||a|-1-1|+1||b|-1-1|)ε=|a|-|b||a-b|·ε(|a|-1)(1-|b|)
when 0<|b|<1<|a|. Moreover, Jung’s result is a particular case of Theorem 10 when we set h(x)=x-1 in (45).Proof of Theorem10.
If we set(48)u(x)=f(x)-af(h(x)),
then the inequality (45) yields
(49)∥u(x)-bu(h(x))∥≤ε,
for all x∈R. According to Corollary 9, there exists a solution U:R→X of
(50)U(x)-bU(h(x))=0
with
(51)∥u(x)-U(x)∥≤ε||b|-1|,
for any x∈R.
If we set(52)v(x)=f(x)-bf(h(x)),
then the inequality (45) yields
(53)∥v(x)-av(h(x))∥≤ε,
for any x∈R. In view of Corollary 9 again, there exists a solution V:R→X of
(54)V(x)-aV(h(x))=0
with
(55)∥v(x)-V(x)∥≤ε||a|-1|,
for all x∈R.
We now define a functionF:R→X by
(56)Fλ(x)=(1-λ)U(x)+λV(x),
for any x∈R, where λ is a real number. Then, it follows from (50) and (54) that
(57)Fλ(x)-pFλ(h(x))+qFλ(h(h(x)))=Fλ(x)-bFλ(h(x))-a(Fλ(h(x))-bFλ(h(h(x))))=(1-λ)((U(h(x))-bU(h(h(x))))U(x)-bU(h(x))hhhhhhhhhhh-a(U(h(x))-bU(h(h(x)))))+λ(V(x)-bV(h(x))-a(V(h(x))-bV(h(h(x)))))=λ(V(x)-bV(h(x))-a(V(h(x))-bV(h(h(x)))))=λ((1-ba)V(x)-a(1-ba)V(h(x)))=λ(1-ba)(V(x)-aV(h(x)))=0,
for all x∈R, which implies that Fλ is a solution of (44) for every fixed real number λ.
We now set(58)λ=-ab-a
and assert that the function
(59)F(x)=bb-aU(x)-ab-aV(x)
satisfies the requirements of this theorem. Indeed, it follows from (51) and (55) that
(60)∥f(x)-(bb-aU(x)-ab-aV(x))∥=1|b-a|∥(b-a)f(x)-(bU(x)-aV(x))∥≤|b||b-a|∥U(x)-(f(x)-af(h(x)))∥+|a||b-a|∥V(x)-(f(x)-bf(h(x)))∥=|b||b-a|∥U(x)-u(x)∥+|a||b-a|∥V(x)-v(x)∥≤|b||b-a|·ε||b|-1|+|a||b-a|·ε||a|-1|=1|a-b|(1||a|-1-1|+1||b|-1-1|)ε,
for any x∈R.



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