Article
KYUNGPOOK Math. J. 2019; 59(4): 689-702
Published online December 23, 2019
Copyright © Kyungpook Mathematical Journal.
On the Stability of a Higher Functional Equation in Banach Algebras
Yong-Soo Jung
Department of Mathematics, Sun Moon University, Asan, Chungnam 336-708, Korea
e-mail : ysjung@sunmoon.ac.kr
Received: July 25, 2018; Accepted: May 10, 2019
Let and ℬ be real (or complex) algebras. We investigate the stability of a sequence
for each
Keywords: higher functional equation, stability.
1. Introduction and Preliminaries
The study of stability problems originated from a famous talk given by S.M. Ulam [18] in 1940 :
In 1991, Z. Gajda [6] answered the question for the case
In 1992, a generalization of the Rassias theorem was obtained by P. Gǎvruţǎ [7]:
Throughout this note, let ℕ be the set of natural numbers and we assume that and ℬ are algebras over the real or complex field . An additive mapping is said to be a
M. Brešar and J. Vukman [5, Proposition 1.6] showed that every ring left derivation on a semiprime ring is a ring derivation which maps the ring into its center.
Definition 1.1
A sequence
holds for each
Let . The sequence
Remark 1.2
K.-H. Park [14] proved that every strong higher ring left derivation
Consider a sequence
holds for each
For convenience’ sake, we will say that the relation (
Remark 1.3
Let
Example 1.4
Given any ring left derivation
In 1949, D.G. Bourgin [4] proved the following stability result, which is sometimes called the superstability of ring homomorphisms:
R. Badora [2] gave a generalization of the Bourgin’s result. Badora [3] also obtained the following results for the stability in the sense of Hyers and Ulam and for the superstability of ring derivations:
On the other hand, T. Miura
In [11] and [12], we dealt with the stability of higher ring derivations and higher ring left derivations, respectively.
Here it is natural to ask that there exists an approximate sequence of mappings which is not an exactly sequence of mappings satisfying the functional
Example 1.5
Let
for all
for each
We claim that
for all
Observe that for all with
Indeed, fix ,
This yields, for each
for all
Now, it follows that
for each
Our objective is to investigate the stability of the higher functional
Theorem 2.1
Putting
which yields
i.e.,
for each
Using Hyers’ direct method on inequality (
for each
for each
for each
Next, we need to show that the sequence
for each
which implies that
for each
for each
for each
That is, we obtain that
for each
Hence we get
for each
for each
for each
Remark 2.2
Let be an algebra and a function such that
for all . If we replace (
then (
is not valid since lim
Corollary 2.3
Let
it follows from (
for each
Assume that
it follows from (
for each
By setting
Corollary 2.4
As a consequence of Corollary
Theorem 2.5
As in (
for each
for each
for each
We continue the next result.
Theorem 2.6
For all , we have, by (
and so
From the similar way as in the proof of Theorem 2.5 using the induction and the relation (
for all
for all
The Singer-Wermer theorem [16], which is a fundamental result in a Banach algebra theory, states that every continuous linear derivation (or linear left derivation) on a commutative Banach algebra maps into the Jacobson radical. They also conjectured that the assumption of continuity is unnecessary. M.P. Thomas [17] proved the conjecture. According to the Thomas’ result, it is easy to see that every linear derivation (or linear left derivation) on a commutative semisimple Banach algebra is identically zero which is the result of B.E. Johnson [9].
The following is similar to B.E. Johnson’ result [9] in the sense of Hyers-Ulam [8].
Theorem 2.7
Put in (
for each
for each
as
for all and all .
Clearly,
Therefore, we have
for each
Since
it follows from the hypothesis that
for all . This implies that
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