# Superstability of Generalized Multiplicative Functionals

- Takeshi Miura
^{1}Email author, - Hiroyuki Takagi
^{2}, - Makoto Tsukada
^{3}and - Sin-Ei Takahasi
^{1}

**2009**:486375

https://doi.org/10.1155/2009/486375

© Takeshi Miura et al. 2009

**Received: **2 March 2009

**Accepted: **20 May 2009

**Published: **16 June 2009

## Abstract

## 1. Introduction

It seems that the stability problem of functional equations had been first raised by S. M. Ulam (cf. [1, Chapter VI]). "For what metric groups
is it true that an
-automorphism of
is necessarily near to a strict automorphism? (An
-automorphism of
means a transformation
of
into itself such that
for all
.)" D. H. Hyers [2] gave an affirmative answer to the problem: if
and
is a mapping between two real Banach spaces
and
satisfying
for all
, then there exists a unique additive mapping
such that
for all
. If, in addition, the mapping
is continuous for each fixed
, then
is linear. This result is called *Hyers-Ulam stability* of the *additive* Cauchy equation
. J. A. Baker [3, Theorem
] considered stability of the multiplicative Cauchy equation
: if
and
is a complex valued function on a semigroup
such that
for all
, then
is multiplicative, or
for all
. This result is called superstability of the functional equation
. Recently, A. Najdecki [4, Theorem
] proved the superstability of the functional equation
: if
,
is a (real or complex valued) functional from a commutative semigroup
and
is a mapping from
into itself such that
for all
, then
holds for all
, or
is bounded.

In this paper, we show that superstability of the functional equation holds for a set with a binary operation under an additional assumption.

## 2. Main Result

Theorem 2.1.

then for all , or for all . In the latter case, the constant is the best possible.

Proof.

Thus, . Now it is easy to see that . Consequently, if is bounded, then for all . The constant is the best possible since for satisfies for each . It should be mentioned that the above proof is essentially due to P. Šemrl [5, Proof of Theorem and Proposition ] (cf. [6, Proposition ]).

Suppose that is an unbounded functional satisfying the inequality (2.2). Since is unbounded, there exists a sequence such that . Take arbitrarily. Set

Next we consider the case when is infinite. By a quite similar argument as in the case when is infinite, we see that there exists a subsequence such that for every . Then

Consequently, if is unbounded, then for all .

Remark 2.2.

for all . Therefore, Theorem 2.1 is a generalization of Najdecki [4, Theorem ] and Baker [3, Theorem ].

Remark 2.3.

If satisfies (2.2) for some , then by quite similar arguments to the proof of Theorem 2.1, we can prove that for all , or for all . Thus, Theorem 2.1 is still true under the weaker condition (2.16) instead of (2.2). This was pointed out by the referee of this paper. The condition (2.16) is related to that introduced by Kannappan [7].

Example 2.4.

Let and be mappings from a semigroup into itself with the following properties.

as claimed.

Let be a ring homomorphism from into itself, that is, and for each . It is well known that there exist infinitely many such homomorphisms on (cf. [8, 9]). If is not identically , then we see that for every , the field of all rational real numbers. Thus, if we consider the case when , a nonzero ring homomorphism, and , then satisfies the conditions (a), (b), and (c).

If we define for each , then holds for every . In fact,

Example 2.5.

for each . Then satisfies the condition (2.1). In fact, let .

Thus, in general. In the same way, we see that if , then need not to be true.

## Declarations

### Acknowledgments

The authors would like to thank the referees for valuable suggestions and comments to improve the manuscript. The first and fourth authors were partly supported by the Grant-in-Aid for Scientific Research.

## Authors’ Affiliations

## References

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## Copyright

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