AAAAbstract and Applied Analysis1687-04091085-3375Hindawi Publishing CorporationS108533750862817810.1155/2008/628178628178Research ArticleOn the Stability of Quadratic Functional EquationsLeeJung Rye^{1}AnJong Su^{2}ParkChoonkil^{3}EloePaul^{1}Department of MathematicsDaejin UniversityKyeonggi 487-711South Korea^{2}Department of Mathematics EducationPusan National UniversityPusan 609-735South Korea^{3}Department of MathematicsHanyang UniversitySeoul 133-791South Korea20081501200820080510200727112007040120082008Copyright _ 2008 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.

Let X,Y be vector spaces and k a fixed positive integer. It is shown that a mapping f(kx+y)+f(kx-y)=2k2f(x)+2f(y) for all x,y∈X if and only if the mapping f:X→Y satisfies f(x+y)+f(x-y)=2f(x)+2f(y) for all
x,y∈X. Furthermore, the Hyers-Ulam-Rassias stability of the above functional
equation in Banach spaces is proven.

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 answer to the question of Ulam for Banach spaces. Hyers' theorem
was generalized by Aoki [3]
for additive mapping and by Th. M. Rassias [4] for linear mappings by considering an unbounded Cauchy difference. The paper of
Th. M. Rassias [4] has
provided a lot of influence in the development of what we now call Hyers-Ulam-Rassias stability of functional
equations. Th. M. Rassias [5] during the 27th International Symposium on Functional
Equations asked the question whether such a theorem can also be proved for p≥1. Gajda [6], following the same approach as in
[4], gave an
affirmative solution to this question for p>1. It was shown by Gajda [6] as well as by Rassias
and Šemrl [7] that one cannot prove a Th.M. Rassias' type theorem
when p=1. J. M. Rassias [8], following the spirit of the innovative approach of
Th. M. Rassias [4] for
the unbounded Cauchy difference, proved a similar stability theorem in which he
replaced the factor ∥x∥p+∥y∥p by ∥x∥p⋅∥y∥q for p,q∈ℝ with p+q≠1.

The functional equationf(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
Hyers-Ulam-Rassias 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. In [11],
Czerwik proved the Hyers-Ulam-Rassias stability of the quadratic functional
equation. Several functional equations have been investigated in [12–17].

Throughout this paper, assume that k is a fixed
positive integer.

In this paper, we solve the functional
equationf(kx+y)+f(kx−y)=2k2f(x)+2f(y)and prove the Hyers-Ulam-Rassias
stability of the functional equation
(1.2) in Banach spaces.

2. Hyers-Ulam-Rassias Stability of the Quadratic Functional Equation

Proposition 2.1.

Let X and Y be vector
spaces. A mapping f:X→Y satisfiesf(kx+y)+f(kx−y)=2k2f(x)+2f(y)for all x,y∈X if and only if
the mapping f:X→Y satisfiesf(x+y)+f(x−y)=2f(x)+2f(y)for all x,y∈X.

Proof.

Assume that f:X→Y satisfies (2.1).

Letting x=y=0 in (2.1), we get f(0)=0.

Letting y=0 in (2.1), we get f(kx)=k2f(x) for all x∈X.

Letting x=0 in (2.1), we get f(−y)=f(y) for all y∈X.

It follows from
(2.1) thatf(kx+y)+f(kx−y)=2k2f(x)+2f(y)=2f(kx)+2f(y)for all x,y∈X. So the mapping f:X→Y satisfiesf(x+y)+f(x−y)=2f(x)+2f(y)for all x,y∈X.

Assume that f:X→Y satisfies f(x+y)+f(x−y)=2f(x)+2f(y) for all x,y∈X.

We prove (2.1)
for k=j by induction on j.

For the case j=1, (2.1) holds
by the assumption.

For the case j=2, sincef(2x+y)+f(2x−y)=f(x+y+x)+f(x−y+x)=2f(x+y)+2f(x)−f(y)+2f(x−y)+2f(x)−f(−y)=2f(x+y)+2f(x−y)+4f(x)−2f(y)=4f(x)+4f(y)+4f(x)−2f(y)=8f(x)+2f(y)for all x,y∈X, then (2.1)
holds.

Assume that
(2.1) holds for j=n−2 and j=n−1 (2<n≤k). By the
assumption,f(nx+y)+f(nx−y)=f((n−1)x+y+x)+f((n−1)x−y+x)=2f((n−1)x+y)+2f(x)−f((n−2)x+y)+2f((n−1)x−y)+2f(x)−f((n−2)x−y)=4(n−1)2f(x)+4f(y)+4f(x)−2(n−2)2f(x)−2f(y)=2n2f(x)+2f(y)for all x,y∈X, (2.1) holds
for j=n. Hence the mapping f:X→Y satisfies (2.1) for j=k.

From now on, assume that X is a normed
vector space with norm ∥⋅∥ and that Y is a Banach
space with norm ∥⋅∥.

For a given mapping f:X→Y, we defineDf(x,y):=f(kx+y)+f(kx−y)−2k2f(x)−2f(y)for all x,y∈X.

Now we prove the Hyers-Ulam-Rassias stability of the
quadratic functional equation Df(x,y)=0.

Theorem 2.2.

Let f:X→Y be a mapping
with f(0)=0 for which there
exists a function φ:X2→[0,∞) such
thatφ˜(x,y):=∑j=0∞1k2jφ(kjx,kjy)<∞,∥Df(x,y)∥≤φ(x,y)for all x,y∈X. Then there exists a unique quadratic mapping Q:X→Y such
that∥f(x)−Q(x)∥≤12k2φ˜(x,0)for all x∈X.

Proof.

Letting y=0 in (2.9), we get∥2f(kx)−2k2f(x)∥≤φ(x,0)for all x∈X. So∥f(x)−1k2f(kx)∥≤12k2φ(x,0)for all x∈X. Hence∥1k2lf(klx)−1k2mf(kmx)∥≤∑j=lm−112k2j+2φ(kjx,0)for all nonnegative integers m and l with m>l and all x∈X. It follows from
(2.13) that the sequence {(1/k2n)f(knx)} is Cauchy for
all x∈X. Since Y is complete,
the sequence {(1/k2n)f(knx)} converges. So
one can define the mapping Q:X→Y byQ(x):=limn→∞1k2nf(knx)for all x∈X.

By
(2.8),∥DQ(x,y)∥=limn→∞1k2n∥Df(knx,kny)∥≤limn→∞1k2nφ(knx,kny)=0for all x,y∈X. So DQ(x,y)=0. By Proposition 2.1, the mapping Q:X→Y is quadratic.
Moreover, letting l=0 and passing the
limit m→∞ in (2.13), we get
(2.10).

Now, let T:X→Y be another
quadratic mapping satisfying (2.1)
and (2.10). Then we have∥Q(x)−T(x)∥=1k2n∥Q(knx)−T(knx)∥≤1k2n(∥Q(knx)−f(knx)∥+∥T(knx)−f(knx)∥)≤1k2n+2φ˜(knx,0),which tends to zero as n→∞ for all x∈X. So we can conclude that Q(x)=T(x) for all x∈X. This proves the uniqueness of Q. So there exists a unique quadratic mapping Q:X→Y satisfying (2.10).

Corollary 2.3.

Let p<2 and θ be positive
real numbers, and let f:X→Y be a mapping
such that∥Df(x,y)∥≤θ(||x||p+||y||p)for all x,y∈X. Then there exists a unique quadratic mapping Q:X→Y such
that∥f(x)−Q(x)∥≤θ8−2p+1||x||pfor all x∈X.

Proof.

The
proof follows from Theorem 2.2 by takingφ(x,y):=θ(||x||p+||y||p)for all x,y∈A.

Theorem 2.4.

Let f:X→Y be a mapping
with f(0)=0 for which there
exists a function φ:X2→[0,∞) satisfying (2.9) such thatφ˜(x,y):=∑j=0∞k2jφ(xkj,ykj)<∞for all x,y∈X. Then there exists a unique quadratic mapping Q:X→Y such
that∥f(x)−Q(x)∥≤12φ˜(xk,0)for all x∈X.

Proof.

It
follows from (2.11) that∥f(x)−k2f(xk)∥≤12φ(xk,0)for all x∈X. Hence∥k2lf(xkl)−k2mf(xkm)∥≤∑j=lm−1k2j2φ(xkj+1,0)for all nonnegative integers m and l with m>l and all x∈X. It follows from
(2.23) that the sequence {k2nf(x/kn)} is Cauchy for
all x∈X. Since Y is complete,
the sequence {k2nf(x/kn)} converges. So
one can define the mapping Q:X→Y byQ(x):=limn→∞k2nf(xkn)for all x∈X.

By
(2.20),∥DQ(x,y)∥=limn→∞k2n∥Df(xkn,ykn)∥≤limn→∞k2nφ(xkn,ykn)=0for all x,y∈X. So DQ(x,y)=0. By Proposition 2.1, the mapping Q:X→Y is quadratic.
Moreover, letting l=0 and passing the
limit m→∞ in (2.23), we get (2.21).

The rest of the proof is similar to the proof of
Theorem 2.2.

Corollary 2.5.

Let p>2 and θ be positive
real numbers, and let f:X→Y be a mapping
satisfying (2.17). Then there exists a
unique quadratic mapping Q:X→Y such
that∥f(x)−Q(x)∥≤θ2p+1−8||x||pfor all x∈X.

Proof.

The
proof follows from Theorem 2.4 by takingφ(x,y):=θ(||x||p+||y||p)for all x,y∈A.

From now on, assume that k=2.

Theorem 2.6.

Let f:X→Y be a mapping
with f(0)=0 for which there
exists a function φ:X2→[0,∞) satisfying (2.9) such thatφ˜(x,y):=∑j=0∞19jφ(3jx,3jy)<∞for all x,y∈X. Then there exists a unique quadratic mapping Q:X→Y such
that∥f(x)−Q(x)∥≤19φ˜(x,x)for all x∈X.

Proof.

Letting y=x in (2.9), we get∥f(3x)−9f(x)∥≤φ(x,x)for all x∈X. So∥f(x)−19f(3x)∥≤19φ(x,x)for all x∈X. Hence∥19lf(3lx)−19mf(3mx)∥≤∑j=lm−119j+1φ(3jx,3jx)for all nonnegative integers m and l with m>l and all x∈X. It follows from
(2.32) that the sequence {(1/9n)f(3nx)} is Cauchy for
all x∈X. Since Y is complete,
the sequence {(1/9n)f(3nx)} converges. So
one can define the mapping Q:X→Y byQ(x):=limn→∞19nf(3nx)for all x∈X.

By
(2.28),∥DQ(x,y)∥=limn→∞19n∥Df(3nx,3ny)∥≤limn→∞19nφ(3nx,3ny)=0for all x,y∈X. So DQ(x,y)=0. By Proposition 2.1, the mapping Q:X→Y is quadratic.
Moreover, letting l=0 and passing the
limit m→∞ in (2.32), we get (2.29).

The rest of the proof is similar to the proof of
Theorem 2.2.

Corollary 2.7.

Let p<1 and θ be positive
real numbers, and let f:X→Y be a mapping
such that∥Df(x,y)∥≤θ⋅||x||p⋅||y||pfor all x,y∈X. Then there exists a unique quadratic mapping Q:X→Y such
that∥f(x)−Q(x)∥≤θ9−9p||x||2pfor all x∈X.

Proof.

The
proof follows from Theorem 2.6 by takingφ(x,y):=θ⋅||x||p⋅||y||pfor all x,y∈A.

Theorem 2.8.

Let f:X→Y be a mapping
with f(0)=0 for which there
exists a function φ:X2→[0,∞) satisfying (2.9) such thatφ˜(x,y):=∑j=0∞9jφ(x3j,y3j)<∞for all x,y∈X. Then there exists a unique quadratic mapping Q:X→Y such
that∥f(x)−Q(x)∥≤φ˜(x3,x3)for all x∈X.

Proof.

It
follows from (2.30) that∥f(x)−9f(x3)∥≤φ(x3,x3)for all x∈X. Hence∥9lf(x3l)−9mf(x3m)∥≤∑j=lm−19jφ(x3j+1,x3j+1)for all nonnegative integers m and l with m>l and all x∈X. It follows from
(2.41) that the sequence {9nf(x/3n)} is Cauchy for
all x∈X. Since Y is complete,
the sequence {9nf(x/3n)} converges. So
one can define the mapping Q:X→Y byQ(x):=limn→∞9nf(x3n)for all x∈X.

By
(2.38),∥DQ(x,y)∥=limn→∞19n∥Df(3nx,3ny)∥≤limn→∞19nφ(3nx,3ny)=0for all x,y∈X. So DQ(x,y)=0. By Proposition 2.1, the mapping Q:X→Y is quadratic.
Moreover, letting l=0 and passing the
limit m→∞ in (2.41), we get (2.39).

The rest of the proof is similar to the proof of
Theorem 2.2.

Corollary 2.9.

Let p>1 and θ be positive
real numbers, and let f:X→Y be a mapping
satisfying (2.35). Then there exists a
unique quadratic mapping Q:X→Y such
that∥f(x)−Q(x)∥≤θ9p−9||x||2pfor all x∈X.

Proof.

The
proof follows from Theorem 2.8 by takingφ(x,y):=θ⋅||x||p⋅||y||pfor all x,y∈A.

Acknowledgments

Jung Rye Lee was supported by Daejin University grants in 2007. The authors
would like to thank the referees for a number of valuable suggestions regarding a
previous version of this paper.

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