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Stability Problems of Quintic Mappings in QuasiNormed Spaces
Journal of Inequalities and Applications volume 2010, Article number: 368981 (2010)
Abstract
We investigate the generalized HyersUlamRassias stability problem in quasinormed 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 letbe a metric group with the metric. Given that, does there exist asuch that if a functionsatisfies the inequalityfor allthen there is a homomorphismwithfor 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 HyersUlamRassias stability or CauchyRassias 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
It is easy to see that is a solution of (1.1) by virtue of the identity
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
holds for all
Note that the mapping is called quintic because the following algebraic identity
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 quasinorm on a linear space.
Definition 1.2.
Let be a linear space over a field A norm is a realvalued 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 aquasinormed space if is a quasinorm on The smallest possible is called the modulus of concavity of A quasiBanach space is a complete quasinormed space.
A quasinorm is called a norm ( if for all In this case, a quasiBanach space is called a Banach space; see [17–19].
In this paper, we consider the following quintic functional equation:
for all We investigate the generalized HyersUlamRassias stability problem in quasinormed 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),
Hence (3) Letting in (1.3), we get
for all By (1) and (2), we have
for all Thus it is an odd mapping.
Note that for all and .
3. Stabilities
Throughout this section, let be a quasinormed space and let be a quasiBanach space with a quasinorm . Let be the modulus of concavity of . We will investigate the generalized HyersUlamRassias 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
for all .
Theorem 3.1.
Suppose that there exists a mapping for which a mapping satisfies
and the series converges for all Then there exists a unique quintic mapping which satisfies (1.3) and the inequality
for all
Proof.
By letting in the inequality (3.2), we have
that is,
for all Now, replacing by and multiplying in the inequality (3.5), we get
for all Combining the two equations (3.5) and (3.6), we have
for all Inductively, since we have
for all For all and with and inductively switching and and multiplying in the inequality (3.5), we have
for all Since the righthand side of the previous inequality tends to 0 as hence is a Cauchy sequence in the quasiBanach space Thus we may define
for all Since , replacing and by and respectively, and dividing by in the inequality (3.2), we have
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
for all By letting we immediately have the uniqueness of
Theorem 3.2.
Suppose that there exists a mapping for which a mapping satisfies
and the series converges for all Then there exists a unique quintic mapping which satisfies (1.3) and the inequality
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
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
for all
Proof.
By the inequalities (3.5) and (3.9) of the proof of Theorem 3.1, we have
that is,
for all and for all and with Hence is a Cauchy sequence in the space Thus we may define
for all Now, we will show that the map is a generalized quintic mapping. Then
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
that is,
for all By letting , we immediately have the uniqueness of .
Theorem 3.4.
Suppose that there exists a mapping for which a mapping satisfies
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
for all
Proof.
By letting in (3.23), we have
and then replacing by ,
for all . For all and with , inductively we have
for all . The remains follow from the proof of Theorem 3.3.
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Cho, I., Kang, D. & Koh, H. Stability Problems of Quintic Mappings in QuasiNormed Spaces. J Inequal Appl 2010, 368981 (2010). https://doi.org/10.1155/2010/368981
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DOI: https://doi.org/10.1155/2010/368981