Abstract and Applied Analysis

VolumeΒ 2008Β (2008), Article IDΒ 628178, 8 pages

http://dx.doi.org/10.1155/2008/628178

## On the Stability of Quadratic Functional Equations

^{1}Department of Mathematics, Daejin University, Kyeonggi 487-711, South Korea^{2}Department of Mathematics Education, Pusan National University, Pusan 609-735, South Korea^{3}Department of Mathematics, Hanyang University, Seoul 133-791, South Korea

Received 5 October 2007; Revised 27 November 2007; Accepted 4 January 2008

Academic Editor: PaulΒ Eloe

Copyright Β© 2008 Jung Rye Lee 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.

#### Abstract

Let be vector spaces and a fixed positive integer. It is shown that a mapping for all if and only if the mapping satisfies for all . 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 . Gajda [6], following the same approach as in
[4], gave an
affirmative solution to this question for . It was shown by Gajda [6] as well as by Rassias
and Ε emrl [7] that one cannot prove a Th.M. Rassias' type theorem
when . 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 by for with .

The functional equationis 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 , where is a normed
space and is a Banach
space. Cholewa [10]
noticed that the theorem of Skof is still true if the relevant domain 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 [12β17].

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

In this paper, we solve the functional equationand 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 and be vector
spaces. A mapping satisfies **for all if and only if
the mapping satisfies **for all .*

*Proof. *Assume that satisfies (2.1).

Letting in (2.1), we get .

Letting in (2.1), we get for all .

Letting in (2.1), we get for all .

It follows from
(2.1) thatfor all . So the mapping satisfiesfor all .

Assume that satisfies for all .

We prove (2.1)
for by induction on .

For the case , (2.1) holds
by the assumption.

For the case , sincefor all , then (2.1)
holds.

Assume that
(2.1) holds for and (). By the
assumption,for all , (2.1) holds
for . Hence the mapping satisfies (2.1) for .

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

For a given mapping , we definefor all .

Now we prove the Hyers-Ulam-Rassias stability of the quadratic functional equation .

Theorem 2.2. *Let be a mapping
with for which there
exists a function such
that **for all . Then there exists a unique quadratic mapping such
that **for all .*

*Proof. *Letting in (2.9), we getfor all . Sofor all . Hencefor all nonnegative integers and with and all . It follows from
(2.13) that the sequence is Cauchy for
all . Since is complete,
the sequence converges. So
one can define the mapping byfor all .

By
(2.8),for all . So . By Proposition 2.1, the mapping is quadratic.
Moreover, letting and passing the
limit in (2.13), we get
(2.10).

Now, let be another
quadratic mapping satisfying (2.1)
and (2.10). Then we havewhich tends to zero as for all . So we can conclude that for all . This proves the uniqueness of . So there exists a unique quadratic mapping satisfying (2.10).

Corollary 2.3. *Let and be positive
real numbers, and let be a mapping
such that **for all . Then there exists a unique quadratic mapping such
that **for all .*

*Proof. *The
proof follows from Theorem 2.2 by takingfor all .

Theorem 2.4. *Let be a mapping
with for which there
exists a function satisfying (2.9) such that **for all . Then there exists a unique quadratic mapping such
that **for all .*

*Proof. *It
follows from (2.11) thatfor all . Hencefor all nonnegative integers and with and all . It follows from
(2.23) that the sequence is Cauchy for
all . Since is complete,
the sequence converges. So
one can define the mapping byfor all .

By
(2.20),for all . So . By Proposition 2.1, the mapping is quadratic.
Moreover, letting and passing the
limit 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 and be positive
real numbers, and let be a mapping
satisfying (2.17). Then there exists a
unique quadratic mapping such
that **for all .*

*Proof. *The
proof follows from Theorem 2.4 by takingfor all .

From now on, assume that .

Theorem 2.6. *Let be a mapping
with for which there
exists a function satisfying (2.9) such that **for all . Then there exists a unique quadratic mapping such
that **for all .*

*Proof. *Letting in (2.9), we getfor all . Sofor all . Hencefor all nonnegative integers and with and all . It follows from
(2.32) that the sequence is Cauchy for
all . Since is complete,
the sequence converges. So
one can define the mapping byfor all .

By
(2.28),for all . So . By Proposition 2.1, the mapping is quadratic.
Moreover, letting and passing the
limit 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 and be positive
real numbers, and let be a mapping
such that **for all . Then there exists a unique quadratic mapping such
that **for all .*

*Proof. *The
proof follows from Theorem 2.6 by takingfor all .

Theorem 2.8. *Let be a mapping
with for which there
exists a function satisfying (2.9) such that **for all . Then there exists a unique quadratic mapping such
that **for all .*

*Proof. *It
follows from (2.30) thatfor all . Hencefor all nonnegative integers and with and all . It follows from
(2.41) that the sequence is Cauchy for
all . Since is complete,
the sequence converges. So
one can define the mapping byfor all .

By
(2.38),for all . So . By Proposition 2.1, the mapping is quadratic.
Moreover, letting and passing the
limit 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 and be positive
real numbers, and let be a mapping
satisfying (2.35). Then there exists a
unique quadratic mapping such
that **for all .*

*Proof. *The
proof follows from Theorem 2.8 by takingfor all .

#### 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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