- Research Article
- Open Access
Superstability and Stability of the Pexiderized Multiplicative Functional Equation
Journal of Inequalities and Applications volume 2010, Article number: 486325 (2010)
We obtain the superstability of the Pexiderized multiplicative functional equation and investigate the stability of this equation in the following form: .
The superstability of the functional equation was studied by Baker et al. . They proved that if is a functional on a real vector space satisfying for some fixed and all , then is either bounded or else for all . This result was genealized with a simplified proof by Baker  as follows.
Theorem 1.1 (Baker ).
Let , be a semigroup and satisfying
for all . Put Then for all or else for all .
A different generalization of the result of Baker et al. was given by Székelyhidi . It involves an interesting generalization of the class of bounded functions on a group or semigroup and may be stated as follows.
Theorem 1.2 (Székelyhidi ).
Let be a commutative group with identity and let be functions such that there exist functions with
for all . Then is bounded or is an exponential and .
In this paper, we prove the superstability of the Pexiderized multiplicative functional equation (PMFE)
That is, we prove that if are functional on a semigroup with identity satisfying and
for all and for a function with some coditions, then is bounded or else is an exponential and . This is a generalization of the result of Székelyhidi. Also we investigate the stability of the Pexiderized multipicative functional equation (1.3) in the sense of Ger .
2. Superstability of the PMFE
In this section, let be a semigroup with identity and a function with
for all and
The following functions satisfy conditions (2.1) and (2.2) above.
(a) for every and
(b), for every and is a functional on .
(c), for every .
(d), for every .
Let and also ,
for all and for some . Let . Then satisfy the conditions (1.4), (2.1), (2.2) and
In particular, we know that , and
Let be a semigroup with identity . If are functions with for some satisfying and condition (1.4), that is,
for all and .
If we replace by and also by in (1.4), we get
Also we replace by in (1.4), then we have
for all . An induction argument implies that for all ,
Indeed, if inequality (2.9) holds, using inequality (1.4) and (2.8) we have
for all . By (2.9), we have
Thus we can easily show that from as and thus as . By (1.4),
and thus we have
for all . Then, by (2.2),
for all . Thus we have as . Since
as , we can define by
for all . Then
for all .
Let be a semigroup with identity and functions satisfying the inequality
for all . If , then is bounded or else is exponential and
Let be a semigroup with identity and functions satisfying condition (1.4), that is,
If satisfies that for some and for some then
for all and .
Let and for every and . Then
for all , and where . By Theorem 2.3, we complete the proof.
Let be a semigroup with identity . If are nonzero functions satisfying condition (1.4), that is,
then either is bounded, or else
for all and .
Let for some . If is unbounded, then there exists such that . By Theorem 2.5, we complete the proof.
3. Stability of the PMFE
In 1940, Ulam gave a wide-ranging talk in the Mathematical Club of the University of Wisconsin in which he discussed a number of important unsolved problems . One of those was the question concerning the stability of homomorphisms.
Let be a group and let be a metric group with a metric . Given , does there exist a such that if a mapping satisfies the inequality for all , then there exists a homomorphism with for all ?
In the next year, Hyers  answered the Ulam's question for the case of the additive mapping on the Banach spaces . Thereafter, the result of Hyers has been generalized by Rassias . Since then, the stability problems of various functional equations have been investigated by many authors (see [6, 8–18]).
Ger  suggested another type of stability for the exponential equation in the following type:
In this section, the stability problem for the Pexiderized multiplicative functional equation in the following form:
will be investigated.
Throughout this section, we denote by a commutative semigroup and by a function such that
for all . Also we let
for all Inequality (3.3) implies that
(d)for all for
The following functions satisfy condition (3.3) above.
(a) for every and
(b), for every .
Let and also ,
for all and for some . Let . Then satisfy condition (3.3) and
In particular, we know that if we let then
If are functions such that
for all , then there exists a function and there exists a constant such that for all and
for all . Moreover, if is bounded, then
for all and for some constant .
If we define functions by
for all , then equality (3.13) may be transformed into
for all . For the case of , the above inequality implies
for all . Putting instead of and instead of in (3.20), respectively, we get
Letting by and by in (3.20), we have
From (3.21), (3.22) and (3.23),
for all . Now replacing by and by , respectively, we have
for all . Replacing by and by in (3.21), (3.22), and (3.23), respectively, one obtains
for all . Also from (3.22) and (3.23), we have
for all . Thus we have
for all . For arbitrary positive integer , putting instead of in (3.24) and instead of in (3.28), respectively, we see that
for all . By (3.29) with ,
for all . By (3.30), for every positive integer with , we have
as . This proves that is a Cauchy sequence in . Thus we can define a function by
for all . Then, by (3.20) and (3.31), we have
for all . Thus
for all . Now replacing by and then by in (3.20), respectively, we obtain
for all . By (3.22), (3.23), and (3.36), we have
for all and for fixed . By (3.3) and (3.30),
for all . By (3.20), (3.38) and (3.39), for all with , there exits a constant such that
Now we define a function by
for all . Then
for all . By (3.40), we have
and thus for all
If is bounded, there exist constants such that
and by the same method above, we have
for all . Therefore, we have
for all .
Baker JA, Lawrence J, Zorzitto F: The stability of the equation . Proceedings of the American Mathematical Society 1979, 74(2):242–246.
Baker JA: The stability of the cosine equation. Proceedings of the American Mathematical Society 1980, 80(3):411–416. 10.1090/S0002-9939-1980-0580995-3
Székelyhidi L: On a theorem of Baker, Lawrence and Zorzitto. Proceedings of the American Mathematical Society 1982, 84(1):95–96.
Ger R: Superstability is not natural. Rocznik Naukowo-Dydaktyczny WSP Krakkowie 1993, 159(13):109–123.
Ulam SM: Problems in Modern Mathematics. John Wiley & Sons, New York, NY, USA; 1964:xvii+150.
Hyers DH: On the stability of the linear functional equation. Proceedings of the National Academy of Sciences of the United States of America 1941, 27: 222–224. 10.1073/pnas.27.4.222
Rassias ThM: On the stability of the linear mapping in Banach spaces. Proceedings of the American Mathematical Society 1978, 72(2):297–300. 10.1090/S0002-9939-1978-0507327-1
Forti GL: Hyers-Ulam stability of functional equations in several variables. Aequationes Mathematicae 1995, 50(1–2):143–190. 10.1007/BF01831117
Hyers DH, Rassias ThM: Approximate homomorphisms. Aequationes Mathematicae 1992, 44(2–3):125–153. 10.1007/BF01830975
Hyers DH, Isac G, Rassias ThM: Stability of Functional Equations in Several Variables, Progress in Nonlinear Differential Equations and Their Applications, 34. Birkhäuser, Boston, Mass, USA; 1998:vi+313.
Jun KW, Kim GH, Lee YW: Stability of generalized gamma and beta functional equations. Aequationes Mathematicae 2000, 60(1–2):15–24. 10.1007/s000100050132
Jung S-M: On a general Hyers-Ulam stability of gamma functional equation. Bulletin of the Korean Mathematical Society 1997, 34(3):437–446.
Jung S-M: On the stability of gamma functional equation. Results in Mathematics 1998, 33(3–4):306–309.
Kim GH, Lee YW: The stability of the beta functional equation. Babeş-Bolyai. Mathematica 2000, 45(1):89–96.
Lee YW: On the stability of a quadratic Jensen type functional equation. Journal of Mathematical Analysis and Applications 2002, 270(2):590–601. 10.1016/S0022-247X(02)00093-8
Lee YW: The stability of derivations on Banach algebras. Bulletin of the Institute of Mathematics. Academia Sinica 2000, 28(2):113–116.
Lee YW, Choi BM: The stability of Cauchy's gamma-beta functional equation. Journal of Mathematical Analysis and Applications 2004, 299(2):305–313. 10.1016/j.jmaa.2003.12.050
Rassias ThM: The problem of S. M. Ulam for approximately multiplicative mappings. Journal of Mathematical Analysis and Applications 2000, 246(2):352–378. 10.1006/jmaa.2000.6788
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Lee, Y. Superstability and Stability of the Pexiderized Multiplicative Functional Equation. J Inequal Appl 2010, 486325 (2010). https://doi.org/10.1155/2010/486325
- Banach Space
- Positive Integer
- Vector Space
- Commutative Group
- Functional Equation