Superstability of Generalized Multiplicative Functionals

Abstract

Let be a set with a binary operation such that, for each , either , or . We show the superstability of the functional equation . More explicitly, if and satisfies for each , then for all , or for all . In the latter case, the constant is the best possible.

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.

Let and a set with a binary operation such that, for each , either

(2.1)

If satisfies

(2.2)

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

Proof.

Let be a functional satisfying (2.2). Suppose that is bounded. There exists a constant such that for all . Set . By (2.2), we have, for each , , and therefore

(2.3)

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

(2.4)

By (2.1), . Thus either or is an infinite subset of . First we consider the case when is infinite. Take arbitrarily. Choose with . Since is assumed to be infinite, for each there exists such that . Then is a subsequence of with for every . By the choice of , we have

(2.5)

Thus we may and do assume that for every . By (2.2) we have, for each and , . According to (2.5), we have

(2.6)

Consequently, we have, for each

(2.7)

Since , we have for every . Applying (2.7), we have

(2.8)

By (2.2) and (2.5), we have

(2.9)

Consequently, we have by (2.8) and (2.7)

(2.10)

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

(2.11)

In the same way as in the proof of (2.7), we have

(2.12)

for every . According to (2.2) and (2.11), we have

(2.13)

Since for every , (2.11) and (2.12) show that

(2.14)

Consequently, if is unbounded, then for all .

Remark 2.2.

Let be a mapping from a commutative semigroup into itself. We define the binary operation by for each . Then satisfies (2.1) since

(2.15)

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

Remark 2.3.

Let be a set, and . Suppose that has a binary operation such that, for each , either

(2.16)

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.

(a) for every .

(b).

(c) for every .

If we define for each , then we have for every . In fact, if , then we have

(2.17)

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,

(2.18)

Example 2.5.

Let and, let . We define the binary operation by

(2.19)

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

(a)If , then we have

(2.20)

(b)If , then

(2.21)

(c)If and , then

(2.22)

(d)If and , then

(2.23)

(e)If , then we have

(2.24)

Therefore, satisfies the condition (2.1). On the other hand, if , then

(2.25)

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