Article
Kyungpook Mathematical Journal 2018; 58(2): 291-305
Published online June 23, 2018
Copyright © Kyungpook Mathematical Journal.
A General Uniqueness Theorem concerning the Stability of AQCQ Type Functional Equations
Yang-Hi Lee and Soon-Mo Jung*
Department of Mathematics Education, Gongju National University of Education, 32553 Gongju, Republic of Korea, e-mail : yanghi2@hanmail.net, Mathematics Section, College of Science and Technology, Hongik University, 30016 Sejong, Republic of Korea, e-mail: smjung@hongik.ac.kr
Received: August 24, 2017; Accepted: April 12, 2018
In this paper, we prove a general uniqueness theorem which is useful for proving the uniqueness of the relevant additive mapping, quadratic mapping, cubic mapping, quartic mapping, or the additive-quadratic-cubic-quartic mapping when we investigate the (generalized) Hyers-Ulam stability.
Keywords: uniqueness, stability, Hyers-Ulam stability, generalized Hyers-Ulam stability, AQCQ type functional equation, AQCQ mapping
1. Introduction
From now on, we assume that
for
If a mapping
We call a mapping
Whenever we investigate the stability problems for additive-quadratic-cubic-quartic type functional equations or others, we encounter some uniqueness problems. To our best of knowledge, however, no author has succeeded in proving even a uniqueness theorem for these cases ([1, 2, 7, 12, 14, 15, 17]) except our papers ([8, 9, 10, 11]). The ideas of the present paper are strongly based on the previous papers [8, 9, 10, 11]. But this paper includes more general results than the previous three papers.
In this paper, a general uniqueness theorem will be proved which is useful for proving the uniqueness of the relevant additive mapping, quadratic mapping, cubic mapping, quartic mapping, or the additive-quadratic-cubic-quartic mapping when we investigate the (generalized) Hyers-Ulam stability. In Section 3, we make use of our uniqueness theorem to improve the stability theorems presented in [5, 13], where the uniqueness of exact solutions have not been proved.
We assume that
Theorem 2.1
Suppose a mapping
(i) We consider
for all
When Φ :
for all
(ii) We consider the mapping
for all
When Φ :
for all
for all
When Φ :
for all
(iv) Finally, we consider
for all
Now, we deal with the case when Φ :
for all
Consequently, since
3. Applications
Theorem 2.1 seems to be impractical for applications in general cases. Thus, it is necessary to introduce some corollaries which are easily applicable to the uniqueness problems for the generalized Hyers-Ulam stability. For the exact definition of the generalized Hyers-Ulam stability, we refer the reader to [3, 6].
Corollary 3.1
When
Corollary 3.2
We set Φ(
for all
for any
for all
On the other hand, we make use of the above equality to prove
for all
for all
Similarly, it holds that
for all
for any
for any
for
Theorem 2.1 implies that our conclusion for this corollary is true.
Corollary 3.3
We set Φ(
for all
for any
for all
On the other hand, it follows from the above equality that
for each
for all
Similarly, we have
for all
for any
for every
for each
Corollary 3.4
Let us set Φ(
for all
for any
for all
Similarly, we obtain
for all
for any
for each
The following corollary states that if, for any given mapping
Corollary 3.5
The following corollaries are immediate consequences of Corollaries 3.2, 3.3, and 3.4, respectively, because each of additive, quadratic, cubic, quartic, and additive-quadratic-cubic-quartic mapping satisfies the conditions in (
Corollary 3.6
Corollary 3.7
Corollary 3.8
If we set Φ(
Hence, by Theorem 2.1, we have the following corollary concerning the Hyers-Ulam-Rassias stability. (For the exact definition of the Hyers-Ulam-Rassias stability, we refer to [4, 16, 18].)
Corollary 3.9
Since each of additive, quadratic, cubic, quartic, or additive-quadratic-cubic-quartic mappings satisfies the conditions in (
Corollary 3.10
Acknowledgements
Soon-Mo Jung was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. 2016R1D1A1B03931061).
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