On the generalized Hyers-Ulam-Rassias stability problem of radical functional equations
© Kim et al.; licensee Springer 2012
Received: 1 December 2011
Accepted: 9 August 2012
Published: 31 August 2012
In this paper, the generalized Hyers-Ulam-Rassias stability problem of radical quadratic and radical quartic functional equations in quasi-β-Banach spaces and then the stability by using subadditive and subquadratic functions for radical functional equations in -Banach spaces are given.
MSC:39B82, 39B52, 46H25.
In 1960, the stability problem of functional equations originated from the question of Ulam [1, 2] concerning the stability of group homomorphisms. The famous Ulam stability problem was partially solved by Hyers  in Banach spaces. Later, Aoki  and Bourgin  considered the stability problem with unbounded Cauchy differences. Rassias [6–9] provided a generalization of Hyers’ theorem by proving the existence of unique linear mappings near approximate additive mappings. On the other hand, Rassias [10, 11] considered the Cauchy difference controlled by a product of different powers of norm. The above results have been generalized by Forti  and Gǎvruta , who permitted the Cauchy difference to become arbitrary unbounded. Gajda and Ger  showed that one can get analogous stability results for subadditive multifunctions. Gruber  remarked that Ulam’s problem is of particular interest in probability theory and in the case of functional equations of different types. Recently, Baktash et al. , Cho et al. [17–20], Gordji et al. [21–24], Lee et al. [25, 26], Najati et al. [27, 28], Park et al. , Saadati et al.  and Savadkouhi et al.  have studied and generalized several stability problems of a large variety of functional equations.
Thus it is natural that each equation is said to be a quadratic functional equation. In particular, every solution of the quadratic equation (1.1) is said to be a quadratic function.
The functional equation (1.2) clearly has as a solution when f is a real valued function of a real variable. So, it is said to be a quartic functional equation.
for all and if and only if ;
There is a constant such that for all .
Then is called a quasi-β-normed space if is a quasi-β-norm on X. The smallest possible K is called the module of concavity of . A quasi-β-Banach space ia a complete quasi-β-normed space.
A quasi-β-norm is called a -norm if for all . In this case, a quasi-β-Banach space is called a -Banach space.
and discuss the generalized Hyers-Ulam-Rassias stability problem in quasi-β-normed spaces and then the stability by using subadditive and subquadratic functions for the functional equations (1.3) and (1.4) in -Banach spaces.
2 Stability of radical functional equations
Using an idea of Gǎvruta , we prove the generalized stability of (1.3) and (1.4) in the spirit of Ulam, Hyers and Rassias.
In , Khodaei et al. proved the following result:
If f satisfies the functional equation (1.3), then f is quadratic.
If f satisfies the functional equation (1.4), then f is quartic.
for all . So it follows from (2.5) and (2.7) that f satisfies (1.2). Therefore, f is quartic. This completes the proof. □
for all .
for all .
for all . Then and, by Lemma 2.1, is a quadratic function. Taking the limit in (2.15) with , we find that a function satisfies (2.10) near the approximate function f of the functional equation (1.3).
for all . Letting , we establish for all . This completes the proof. □
for all .
Proof If x is replaced by in the inequality (2.13), then the proof follows from the proof of Theorem 2.2. □
for all .
for all .
Now, we give an example to illustrate that the functional equation (1.3) is not stable for in a quasi-1-Banach space with .
which contradicts (2.18). Therefore, the quadratic equation (1.3) is not stable for in Corollary 2.5.
for all .
for all . Then, we have , and by Lemma 2.1, is quartic. Taking the limit in (2.27) with , we obtain that a function satisfies (2.19) near the approximate function f of the functional equation (1.4). The remaining assertion is similar to the corresponding part of Theorem 2.2. This completes the proof. □
for all .
for all .
for all and .
for all and .
From now on, we establish the modified Hyers-Ulam-Rassias stability of the equations (1.3) and (1.4) in a -Banach space .
for all . Letting , we have the uniqueness of . This completes the proof. □
for all .
The remaining part follows as the proof of Theorem 2.9. This completes the proof. □
for all .
for all . Thus the function is quartic. Taking the limit in (2.33) with , satisfies (2.31) near the approximate function f of the functional equation (1.4). The remaining proof is similar to that of Theorem 2.9. This completes the proof. □
for all .
This research was supported by Dongeui University (2011AA088) and the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant Number: 2011-0021821).
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