- Research Article
- Open access
- Published:
Stability Problems of Quintic Mappings in Quasi-
-Normed Spaces
Journal of Inequalities and Applications volume 2010, Article number: 368981 (2010)
Abstract
We investigate the generalized Hyers-Ulam-Rassias stability problem in quasi--normed spaces and then the stability by using a subadditive function for the quintic function
such that
, for all
.
1. Introduction
In 1940 Ulam [1] proposed the problem concerning the stability of group homomorphisms as follows.Letbe a group and let
be a metric group with the metric
. Given that
, does there exist a
such that if a function
satisfies the inequality
for all
then there is a homomorphism
with
for all
? In other words, when is it true that a mapping satisfying a functional equation approximately must be close to the solution of the given functional equation ? In 1941, Hyers [2] considered the case of approximately additive mappings under the assumption that
and
are Banach spaces.
The famous Hyers stability result that appeared in [2] was generalized by Aoki [3] for the stability of the additive mapping involving a sum of powers of -norms. In 1978, Th. M. Rassias [4] provided a generalization of Hyers' Theorem for the stability of the linear mapping, which allows the Cauchy difference to be unbounded. This result of Th. M. Rassias lead mathematicians working in stability of functional equations to establish what is known today as Hyers-Ulam-Rassias stability or Cauchy-Rassias stability as well as to introduce new definitions of stability concepts. During the last three decades, several stability problems of a large variety of functional equations have been extensively studied and generalized by a number of authors [5–14]. In particular, J. M. Rassias [15] introduced the quartic functional equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F368981/MediaObjects/13660_2010_Article_2133_Equ1_HTML.gif)
It is easy to see that is a solution of (1.1) by virtue of the identity
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F368981/MediaObjects/13660_2010_Article_2133_Equ2_HTML.gif)
For this reason, (1.1) is called a quartic functional equation. Also Chung and Sahoo [16] determined the general solution of (1.1) without assuming any regularity conditions on the unknown function. In fact, they proved that the function is a solution of (1.1) if and only if
where the function
is symmetric and additive in each variable.
Similar to the quartic functional equation, we may define quintic functional equation as follows.
Definition 1.1.
Let be a linear space and let
be a real complete linear space. Then a mapping
is called quintic if the quintic functional equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F368981/MediaObjects/13660_2010_Article_2133_Equ3_HTML.gif)
holds for all
Note that the mapping is called quintic because the following algebraic identity
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F368981/MediaObjects/13660_2010_Article_2133_Equ4_HTML.gif)
holds for all .
Let be a real number with
and let
be either
or
We will consider the definition and some preliminary results of a quasi-
-norm on a linear space.
Definition 1.2.
Let be a linear space over a field
A
-
-norm
is a real-valued function on
satisfying the followings.
(1) for all
and
if and only if
.
(2) for all
and all
.
(3)There is a constant such that
for all
.
The pair is called aquasi-
-normed space if
is a quasi-
-norm on
The smallest possible
is called the modulus of concavity of
A quasi-Banach space is a complete quasi-
-normed space.
A quasi--norm
is called a
-norm (
if
for all
In this case, a quasi-
-Banach space is called a
-Banach space; see [17–19].
In this paper, we consider the following quintic functional equation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F368981/MediaObjects/13660_2010_Article_2133_Equ5_HTML.gif)
for all We investigate the generalized Hyers-Ulam-Rassias stability problem in quasi-
-normed spaces and then the stability by using a subadditive function for the quintic function
satisfying (1.5).
2. Quintic Functional Equations
Lemma 2.1.
Let be a quintic mapping satisfying (1.3). Then
(1), for all
and
,
(2)
(3) is an odd mapping,
Proof.
-
(1)
Letting
in (1.3), we have
(2.1)
that is, , for all
. Now inductively replacing
by
, we have the desired result. (2) Putting
in (1.3),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F368981/MediaObjects/13660_2010_Article_2133_Equ7_HTML.gif)
Hence (3) Letting
in (1.3), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F368981/MediaObjects/13660_2010_Article_2133_Equ8_HTML.gif)
for all By (1) and (2), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F368981/MediaObjects/13660_2010_Article_2133_Equ9_HTML.gif)
for all Thus it is an odd mapping.
Note that for all
and
.
3. Stabilities
Throughout this section, let be a quasi-
-normed space and let
be a quasi-
-Banach space with a quasi-
-norm
. Let
be the modulus of concavity of
. We will investigate the generalized Hyers-Ulam-Rassias stability problem for the functional equation (1.5). After that we will study the stability by using a subadditive function. For a given mapping
, let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F368981/MediaObjects/13660_2010_Article_2133_Equ10_HTML.gif)
for all .
Theorem 3.1.
Suppose that there exists a mapping for which a mapping
satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F368981/MediaObjects/13660_2010_Article_2133_Equ11_HTML.gif)
and the series converges for all
Then there exists a unique quintic mapping
which satisfies (1.3) and the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F368981/MediaObjects/13660_2010_Article_2133_Equ12_HTML.gif)
for all
Proof.
By letting in the inequality (3.2), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F368981/MediaObjects/13660_2010_Article_2133_Equ13_HTML.gif)
that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F368981/MediaObjects/13660_2010_Article_2133_Equ14_HTML.gif)
for all Now, replacing
by
and multiplying
in the inequality (3.5), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F368981/MediaObjects/13660_2010_Article_2133_Equ15_HTML.gif)
for all Combining the two equations (3.5) and (3.6), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F368981/MediaObjects/13660_2010_Article_2133_Equ16_HTML.gif)
for all Inductively, since
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F368981/MediaObjects/13660_2010_Article_2133_Equ17_HTML.gif)
for all For all
and
with
and inductively switching
and
and multiplying
in the inequality (3.5), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F368981/MediaObjects/13660_2010_Article_2133_Equ18_HTML.gif)
for all Since the right-hand side of the previous inequality tends to 0 as
hence
is a Cauchy sequence in the quasi-
-Banach space
Thus we may define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F368981/MediaObjects/13660_2010_Article_2133_Equ19_HTML.gif)
for all Since
, replacing
and
by
and
respectively, and dividing by
in the inequality (3.2), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F368981/MediaObjects/13660_2010_Article_2133_Equ20_HTML.gif)
for all By taking
the definition of
implies that
satisfies (1.3) for all
that is,
is the quintic mapping. Also, the inequality (3.8) implies the inequality (3.3). Now, it remains to show the uniqueness. Assume that there exists
satisfying (1.3) and (3.3). Lemma 2.1 implies that
and
for all
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F368981/MediaObjects/13660_2010_Article_2133_Equ21_HTML.gif)
for all By letting
we immediately have the uniqueness of
Theorem 3.2.
Suppose that there exists a mapping for which a mapping
satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F368981/MediaObjects/13660_2010_Article_2133_Equ22_HTML.gif)
and the series converges for all
Then there exists a unique quintic mapping
which satisfies (1.3) and the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F368981/MediaObjects/13660_2010_Article_2133_Equ23_HTML.gif)
for all
Proof.
If is replaced by
in the inequality (3.5), then the proof follows from the proof of Theorem 3.1.
Now we will recall a subadditive function and then investigate the stability under the condition that the space is a
-Banach space. The basic definitions of subadditive functions follow from [19].
A function having a domain
and a codomain
that are both closed under addition is called
(1)a subadditive function if ,
(2)a contractively subadditive function if there exists a constant with
such that
,
(3)an expansively superadditive function if there exists a constant with
such that
,
for all .
Theorem 3.3.
Suppose that there exists a mapping for which a mapping
satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F368981/MediaObjects/13660_2010_Article_2133_Equ24_HTML.gif)
for all and the map
is contractively subadditive with a constant
such that
Then there exists a unique quintic mapping
which satisfies (1.3) and the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F368981/MediaObjects/13660_2010_Article_2133_Equ25_HTML.gif)
for all
Proof.
By the inequalities (3.5) and (3.9) of the proof of Theorem 3.1, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F368981/MediaObjects/13660_2010_Article_2133_Equ26_HTML.gif)
that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F368981/MediaObjects/13660_2010_Article_2133_Equ27_HTML.gif)
for all and for all
and
with
Hence
is a Cauchy sequence in the space
Thus we may define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F368981/MediaObjects/13660_2010_Article_2133_Equ28_HTML.gif)
for all Now, we will show that the map
is a generalized quintic mapping. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F368981/MediaObjects/13660_2010_Article_2133_Equ29_HTML.gif)
for all Hence the mapping
is a quintic mapping. Note that the inequality (3.18) implies the inequality (3.16) by letting
and taking
Assume that there exists
satisfying (1.5) and (3.16). We know that
for all
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F368981/MediaObjects/13660_2010_Article_2133_Equ30_HTML.gif)
that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F368981/MediaObjects/13660_2010_Article_2133_Equ31_HTML.gif)
for all By letting
, we immediately have the uniqueness of
.
Theorem 3.4.
Suppose that there exists a mapping for which a mapping
satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F368981/MediaObjects/13660_2010_Article_2133_Equ32_HTML.gif)
for all and the map
is expansively superadditive with a constant
such that
Then there exists a unique quintic mapping
which satisfies (1.3) and the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F368981/MediaObjects/13660_2010_Article_2133_Equ33_HTML.gif)
for all
Proof.
By letting in (3.23), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F368981/MediaObjects/13660_2010_Article_2133_Equ34_HTML.gif)
and then replacing by
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F368981/MediaObjects/13660_2010_Article_2133_Equ35_HTML.gif)
for all . For all
and
with
, inductively we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F368981/MediaObjects/13660_2010_Article_2133_Equ36_HTML.gif)
for all . The remains follow from the proof of Theorem 3.3.
References
Ulam SM: Problems in Modern Mathematics. Wiley, New York, NY, USA; 1960.
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
Aoki T: On the stability of the linear transformation in Banach spaces. Journal of the Mathematical Society of Japan 1950, 2: 64–66. 10.2969/jmsj/00210064
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
Bae J-H, Park W-G: On the generalized Hyers-Ulam-Rassias stability in Banach modules over a -algebra. Journal of Mathematical Analysis and Applications 2004, 294(1):196–205. 10.1016/j.jmaa.2004.02.009
Czerwik S: On the stability of the quadratic mapping in normed spaces. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 1992, 62: 59–64. 10.1007/BF02941618
Czerwik S: Functional Equations and Inequalities in Several Variables. World Scientific, River Edge, NJ, USA; 2002:x+410.
Gajda Z: On stability of additive mappings. International Journal of Mathematics and Mathematical Sciences 1991, 14(3):431–434. 10.1155/S016117129100056X
Hyers DH, Rassias ThM: Approximate homomorphisms. Aequationes Mathematicae 1992, 44(2–3):125–153. 10.1007/BF01830975
Rassias ThM: On the stability of functional equations and a problem of Ulam. Acta Applicandae Mathematicae 2000, 62(1):23–130. 10.1023/A:1006499223572
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.
Rassias ThM: On the stability of functional equations in Banach spaces. Journal of Mathematical Analysis and Applications 2000, 251(1):264–284. 10.1006/jmaa.2000.7046
Rassias ThM, Šemrl P: On the Hyers-Ulam stability of linear mappings. Journal of Mathematical Analysis and Applications 1993, 173(2):325–338. 10.1006/jmaa.1993.1070
Rassias ThM, Shibata K: Variational problem of some quadratic functionals in complex analysis. Journal of Mathematical Analysis and Applications 1998, 228(1):234–253. 10.1006/jmaa.1998.6129
Rassias JM: Solution of the Ulam stability problem for quartic mappings. Glasnik Matematički. Serija III 1999, 34(2):243–252.
Chung JK, Sahoo PK: On the general solution of a quartic functional equation. Bulletin of the Korean Mathematical Society 2003, 40(4):565–576.
Benyamini Y, Lindenstrauss J: Geometric Nonlinear Functional Analysis. Vol. 1, American Mathematical Society Colloquium Publications. Volume 48. American Mathematical Society, Providence, RI, USA; 2000:xii+488.
Rolewicz S: Metric Linear Spaces. 2nd edition. PWN-Polish Scientific Publishers, Warsaw, Poland; 1984:xi+459.
Rassias JM, Kim H-M: Generalized Hyers-Ulam stability for general additive functional equations in quasi--normed spaces. Journal of Mathematical Analysis and Applications 2009, 356(1):302–309. 10.1016/j.jmaa.2009.03.005
Author information
Authors and Affiliations
Corresponding author
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
Cho, I., Kang, D. & Koh, H. Stability Problems of Quintic Mappings in Quasi--Normed Spaces.
J Inequal Appl 2010, 368981 (2010). https://doi.org/10.1155/2010/368981
Received:
Accepted:
Published:
DOI: https://doi.org/10.1155/2010/368981