# The Stability of a Quadratic Functional Equation with the Fixed Point Alternative

Authors: Choonkil Park; Ji-Hye Kim
Journal: Abstract and Applied Analysis
 (2009)

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## Abstract

Lee, An and Park introduced the quadratic
functional equationf(2x+y)+f(2x−y)=8f(x)+2f(y) and proved the stability of the quadratic functional equation in the spirit of Hyers, Ulam and Th. M. Rassias. Using the fixed point method, we prove the generalized Hyers-Ulam stability of the quadratic
functional equation in Banach spaces.


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## Body

## 1. Introduction

The stability problem of functional equations originated from a question of Ulam [1] concerning the stability of group homomorphisms. Hyers [2] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki [3] for additive mappings and by Th. M. Rassias [4] for linear mappings by considering an unbounded Cauchy difference.Theorem 1.1 (Th. M. Rassias).
Letf:E→E′ be a mapping from a normed vector space E into a Banach space E′ subject to the inequality
(1.1)∥f(x+y)-f(x)-f(y)∥≤ϵ(∥x∥p+∥y∥p)
for all x,y∈E, where ϵ and p are constants with ϵ>0 and p<1. Then the limit
(1.2)L(x)=limn→∞f(2nx)2n
exists for all x∈E, and L:E→E′ is the unique additive mapping which satisfies
(1.3)∥f(x)-L(x)∥≤2ϵ2-2p∥x∥p
for all x∈E. Also, if for each x∈E the function f(tx) is continuous in t∈ℝ, then L is ℝ-linear.The above inequality  (1.1) has provided a lot of influence in the development of what is now known as a generalized Hyers-Ulam stability of functional equations. Beginning around the year 1980, the topic of approximate homomorphisms, or the stability of the equation of homomorphism, was studied by a number of mathematicians. Găvruta [5] generalized the Rassias’ result.Theorem 1.2 (see [6–8]).
LetX be a real normed linear space and Y a real complete normed linear space. Assume that f:X→Y is an approximately additive mapping for which there exist constants θ≥0 and p∈ℝ-{1} such that f satisfies inequality
(1.4)∥f(x+y)-f(x)-f(y)∥≤θ·∥x∥p/2·∥y∥p/2
for all x,y∈X. Then there exists a unique additive mapping L:X→Y satisfying
(1.5)∥f(x)-L(x)∥≤θ|2p-2|∥x∥p
for all x∈X. If, in addition, f:X→Y is a mapping such that the transformation t→f(tx) is continuous in t∈ℝ for each fixed x∈X, then L is an ℝ-linear mapping.The functional equation(1.6)f(x+y)+f(x-y)=2f(x)+2f(y)
is called a quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic function. A generalized Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [9] for mappings f:X→Y, where X is a normed space and Y is a Banach space. Cholewa [10] noticed that the theorem of Skof is still true if the relevant domain X is replaced by an Abelian group. Czerwik [11] proved the generalized Hyers-Ulam stability of the quadratic functional equation. Several functional equations have been investigated in [12–25].LetX be a set. A function d:X×X→[0,∞] is called a generalized metric on X if d satisfies(1)
d(x,y)=0 if and only if x=y;(2)
d(x,y)=d(y,x) for all x,y∈X;(3)
d(x,z)≤d(x,y)+d(y,z) for all x,y,z∈X.We recall a fundamental result in fixed point theory.Theorem 1.3 (see [26–28]).
Let(X,d) be a complete generalized metric space and let J:X→X be a strictly contractive mapping with Lipschitz constant L<1. Then for each given element x∈X, either
(1.7)d(Jnx,Jn+1x)=∞
for all nonnegative integers n or there exists a positive integer n0 such that(1)
d(Jnx,Jn+1x)<∞,  for all n≥n0;(2)
the sequence{Jnx} converges to a fixed point y* of J;(3)
y* is the unique fixed point of J in the set Y={y∈X∣d(Jn0x,y)<∞};(4)
d(y,y*)≤(1/(1-L))d(y,Jy) for all y∈Y.Lee et al. [29] proved that   a mapping f:X→Y satisfies(1.8)f(2x+y)+f(2x-y)=8f(x)+2f(y)
for all x,y∈X if and only if the mapping f:X→Y satisfies(1.9)f(x+y)+f(x-y)=2f(x)+2f(y)
for all x,y∈X.Using the fixed point method,   Park [14] proved the generalized Hyers-Ulam stability of the quadratic functional equation(1.10)f(2x+y)=4f(x)+f(y)+f(x+y)-f(x-y)
in Banach spaces.In this paper, using the fixed point method, we prove the generalized Hyers-Ulam stability of the quadratic functional equation  (1.8) in Banach spaces.Throughout this paper, assume thatX is a normed vector space with norm ||·|| and that Y is a Banach space with norm ∥·∥.


## 2. Fixed Points and Generalized Hyers-Ulam Stability of a Quadratic Functional Equation

For a given mappingf:X→Y, we define(2.1)Cf(x,y):=f(2x+y)+f(2x-y)-8f(x)-2f(y)
for all x,y∈X.Using the fixed point method, we prove the generalized Hyers-Ulam stability of the quadratic functional equationCf(x,y)=0.Theorem 2.1.
Letf:X→Y be a mapping for which there exists a function φ:X2→[0,∞) with f(0)=0 such that
(2.2)∥Df(x,y)∥≤φ(x,y)
for all x,y∈X. If there exists an L<1 such that φ(x,y)≤4Lφ(x/2,y/2) for all x,y∈X, then there exists a unique quadratic mapping Q:X→Y satisfying  (1.8) and
(2.3)∥f(x)-Q(x)∥≤18-8Lφ(x,0)
for all x∈X.Proof.
Consider the set(2.4)S:={g:X→Y},
and introduce thegeneralized metric  on S:
(2.5)d(g,h)=inf{K∈ℝ+:∥g(x)-h(x)∥≤Kφ(x,0),∀x∈X}.
It is easy to show that (S,d) is complete.
Now we consider the linear mappingJ:S→S such that
(2.6)Jg(x):=14g(2x)
for all x∈X.
By  [30, Theorem 3.1],
(2.7)d(Jg,Jh)≤Ld(g,h)
for all g,h∈S.
Lettingy=0 in  (2.2), we get
(2.8)∥2f(2x)-8f(x)∥≤φ(x,0)
for all x∈X. So
(2.9)∥f(x)-14f(2x)∥≤18φ(x,0)
for all x∈X. Hence  d(f,Jf)≤1/8.
By Theorem1.3, there exists a mapping Q:X→Y such that
(1)Q is a fixed point of J, that is,
(2.10)Q(2x)=4Q(x)
for all x∈X. The mapping Q is a unique fixed point of J in the set
(2.11)M={g∈S:d(f,g)<∞}.
This implies that Q is a unique mapping satisfying  (2.10) such that there exists K∈(0,∞) satisfying
(2.12)∥f(x)-Q(x)∥≤Kφ(x,0)
for all x∈X.
(2)d(Jnf,Q)→0 as n→∞. This implies the equality
(2.13)limn→∞f(2nx)4n=Q(x)
for all x∈X.
(3)d(f,Q)≤(1/(1-L))d(f,Jf), which implies the inequality
(2.14)d(f,Q)≤18-8L.
This implies that the inequality  (2.3) holds.
It follows from  (2.2) and  (2.13) that
(2.15)∥CQ(x,y)∥=limn→∞14n∥Cf(2nx,2ny)∥≤limn→∞14nφ(2nx,2ny)≤limn→∞Lnφ(x,y)=0
for all x,y∈X. So CQ(x,y)=0 for all x,y∈X.
By   [29, Proposition 2.1], the mapping Q:X→Y is quadratic, as desired.Corollary 2.2.
Let0<p<2 and θ be positive real numbers, and let f:X→Y be a mapping such that
(2.16)∥Cf(x,y)∥≤θ(∥x∥p+∥y∥p)
for all x,y∈X. Then there exists a unique quadratic mapping Q:X→Y satisfying  (1.8) and
(2.17)∥f(x)-Q(x)∥≤θ8-2p+1∥x∥p
for all x∈X.Proof.
The proof follows from Theorem2.1 by taking
(2.18)φ(x,y):=θ(∥x∥p+∥y∥p)
for all x,y∈X. Then L=2p-2, and we get the desired result.Theorem 2.3.
Letf:X→Y be a mapping for which there exists a function φ:X2→[0,∞) satisfying  (2.2) and f(0)=0. If there exists an L<1 such that φ(x,y)≤(L/4)φ(2x,2y) for all x,y∈X, then there exists a unique quadratic mapping Q:X→Y satisfying  (1.8) and
(2.19)∥f(x)-Q(x)∥≤L8-8Lφ(x,0)
for all x∈X.Proof.
We consider the linear mappingJ:S→S such that
(2.20)Jg(x):=4g(x2)
for all x∈X.
It follows from  (2.8) that
(2.21)∥f(x)-4f(x2)∥≤12φ(x2,0)≤L8φ(x,0)
for all x∈X. Hence  d(f,Jf)≤L/8.
By Theorem1.3, there exists a mapping Q:X→Y such that
(1)Q is a fixed point of J, that is,
(2.22)Q(2x)=4Q(x)
for all x∈X. The mapping Q is a unique fixed point of J in the set
(2.23)M={g∈S:d(f,g)<∞}.
This implies that Q is a unique mapping satisfying  (2.22) such that there exists K∈(0,∞) satisfying
(2.24)∥f(x)-Q(x)∥≤Kφ(x,0)
for all x∈X.
(2)d(Jnf,Q)→0 as n→∞. This implies the equality
(2.25)limn→∞4nf(x2n)=Q(x)
for all x∈X.
(3)d(f,Q)≤(1/(1-L))d(f,Jf), which implies the inequality
(2.26)d(f,Q)≤L8-8L,
which implies that the inequality  (2.19) holds.
The rest of the proof is similar to the proof of Theorem2.1.Corollary 2.4.
Letp>2 and θ be positive real numbers, and let f:X→Y be a mapping satisfying  (2.16). Then there exists a unique quadratic mapping Q:X→Y satisfying  (1.8) and
(2.27)∥f(x)-Q(x)∥≤θ2p+1-8∥x∥p
for all x∈X.Proof.
The proof follows from Theorem2.3 by taking
(2.28)φ(x,y):=θ(∥x∥p+∥y∥p)
for all x,y∈X. Then L=22-p and, we get the desired result.Theorem 2.5.
Letf:X→Y be a mapping for which there exists a function φ:X2→[0,∞) satisfying  (2.2). If there exists an L<1 such that φ(x,y)≤9Lφ(x/3,y/3) for all x,y∈X, then there exists a unique quadratic mapping Q:X→Y satisfying  (1.8) and
(2.29)∥f(x)-Q(x)∥≤19-9Lφ(x,x)
for all x∈X.Proof.
Consider the set(2.30)S:={g:X→Y},
and introduce the generalized metric on S:
(2.31)d(g,h)=inf{K∈ℝ+:∥g(x)-h(x)∥≤Kφ(x,x),∀x∈X}.
It is easy to show that (S,d) is complete.
Now we consider the linear mappingJ:S→S such that
(2.32)Jg(x):=19g(3x)
for all x∈X.
By   [30, Theorem 3.1],
(2.33)d(Jg,Jh)≤Ld(g,h)
for all g,h∈S.
Lettingy=x in  (2.2), we get
(2.34)∥f(3x)-9f(x)∥≤φ(x,x)
for all x∈X. So
(2.35)∥f(x)-19f(3x)∥≤19φ(x,x)
for all x∈X.  Hence  d(f,Jf)≤1/9.
By Theorem1.3, there exists a mapping Q:X→Y such that
(1)Q is a fixed point of J, that is,
(2.36)Q(3x)=9Q(x)
for all x∈X. The mapping Q is a unique fixed point of J in the set
(2.37)M={g∈S:d(f,g)<∞}.
This implies that Q is a unique mapping satisfying  (2.36) such that there exists K∈(0,∞) satisfying
(2.38)∥f(x)-Q(x)∥≤Kφ(x,x)
for all x∈X.
(2)d(Jnf,Q)→0 as n→∞. This implies the equality
(2.39)limn→∞f(3nx)9n=Q(x)
for all x∈X.
(3)d(f,Q)≤(1/(1-L))d(f,Jf), which implies the inequality
(2.40)d(f,Q)≤19-9L.
This implies that the inequality  (2.29) holds.
It follows from  (2.2) and  (2.39) that
(2.41)∥CQ(x,y)∥=limn→∞19n∥Cf(3nx,3ny)∥≤limn→∞19nφ(3nx,3ny)≤limn→∞Lnφ(x,y)=0
for all x,y∈X. So CQ(x,y)=0 for all x,y∈X.
By   [29, Proposition 2.1], the mapping Q:X→Y is quadratic, as desired.Corollary 2.6.
Let0<p<2 and θ be positive real numbers, and let f:X→Y be a mapping satisfying  (2.16). Then there exists a unique quadratic mapping Q:X→Y satisfying  (1.8) and
(2.42)∥f(x)-Q(x)∥≤2θ9-3p∥x∥p
for all x∈X.Proof.
The proof follows from Theorem2.5 by taking
(2.43)φ(x,y):=θ(∥x∥p+∥y∥p)
for all x,y∈X. Then L=3p-2 and, we get the desired result.Corollary 2.7.
Let0<p<1 and θ be positive real numbers, and let f:X→Y be a mapping such that
(2.44)∥Df(x,y)∥≤θ·∥x∥p·∥y∥p
for all x,y∈X. Then there exists a unique quadratic mapping Q:X→Y satisfying  (1.8) and
(2.45)∥f(x)-Q(x)∥≤θ9-9p∥x∥2p
for all x∈X.Proof.
The proof follows from Theorem2.5 by taking
(2.46)φ(x,y):=θ·∥x∥p·∥y∥p
for all x,y∈X. Then L=9p-1 and, we get the desired result.Theorem 2.8.
Letf:X→Y be a mapping for which there exists a function φ:X2→[0,∞) satisfying  (2.2). If there exists an L<1 such that φ(x,y)≤(L/9)φ(3x,3y) for all x,y∈X, then there exists a unique quadratic mapping Q:X→Y satisfying  (1.8) and
(2.47)∥f(x)-Q(x)∥≤L9-9Lφ(x,x)
for all x∈X.Proof.
We consider the linear mappingJ:S→S such that
(2.48)Jg(x):=9g(x3)
for all x∈X.
The rest of the proof is similar to the proof of Theorem2.1.Corollary 2.9.
Letp>2 and θ be positive real numbers, and let f:X→Y be a mapping satisfying  (2.16). Then there exists a unique quadratic mapping Q:X→Y satisfying  (1.8) and
(2.49)∥f(x)-Q(x)∥≤2θ3p-9∥x∥p
for all x∈X.Proof.
The proof follows from Theorem2.8 by taking
(2.50)φ(x,y):=θ(∥x∥p+∥y∥p)
for all x,y∈X. Then L=32-p, and we get the desired result.Corollary 2.10.
Letp>1 and θ be positive real numbers, and let f:X→Y be a mapping satisfying  (2.44). Then there exists a unique quadratic mapping Q:X→Y satisfying  (1.8) and
(2.51)∥f(x)-Q(x)∥≤θ9p-9∥x∥2p
for all x∈X.Proof.
The proof follows from Theorem2.8 by taking
(2.52)φ(x,y):=θ·∥x∥p·∥y∥p
for all x,y∈X. Then L=91-p, and we get the desired result.



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