On the Stability of a Generalized Quadratic and Quartic Type Functional Equation in Quasi-Banach Spaces
© M. Eshaghi Gordji et al. 2009
Received: 31 May 2009
Accepted: 9 September 2009
Published: 12 October 2009
for all Moreover, if is continuous in for each fixed then is -linear. In 1978, Th. M. Rassias  provided a generalization of Hyers' theorem which allows the Cauchy difference to be unbounded. The functional equation
The generalized Hyers-Ulam stability problem for the quadratic functional equation (1.3) was proved by Skof for mappings , where is a normed space and is a Banach space (see ). Cholewa  noticed that the theorem of Skof is still true if relevant domain is replaced by an abelian group. In  , Czerwik proved the generalized Hyers-Ulam stability of the functional equation (1.3). Grabiec  has generalized these results mentioned above.
In , Park and Bae considered the following quartic functional equation:
In fact, they proved that a mapping between two real vector spaces and is a solution of (1.5) if and only if there exists a unique symmetric multiadditive mapping such that for all . It is easy to show that satisfies the functional equation (1.5), which is called a quartic functional equation (see also ).
In addition, Kim  has obtained the generalized Hyers-Ulam stability for a mixed type of quartic and quadratic functional equation between two real linear Banach spaces. Najati and Zamani Eskandani  have established the general solution and the generalized Hyers-Ulam stability for a mixed type of cubic and additive functional equation, whenever is a mapping between two quasi-Banach spaces (see also [16, 17]).
in quasi-Banach spaces. It is easy to see that the function is a solution of the functional equation (1.6). In the present paper we investigate the general solution of the functional equation (1.6) when is a mapping between vector spaces, and we establish the generalized Hyers-Ulam stability of this functional equation whenever is a mapping between two quasi-Banach spaces.
We recall some basic facts concerning quasi-Banach space and some preliminary results.
Given a -norm, the formula gives us a translation invariant metric on X. By the Aoki-Rolewicz theorem  (see also ), each quasi-norm is equivalent to some -norm. Since it is much easier to work with -norms, henceforth we restrict our attention mainly to -norms. In , Tabor has investigated a version of Hyers-Rassias-Gajda theorem (see [3, 21]) in quasi-Banach spaces.
2. General Solution
If a mapping satisfies the functional equation (1.6), then is a quadratic and quartic mapping.
for all where the mapping is symmetric multi-additive and is bi-additive. By a simple computation, one can show that the mappings and satisfy the functional equation (1.6), so the mapping f satisfies (1.6).
3. Generalized Hyers-Ulam Stability of (1.6)
From now on, let and be a quasi-Banach space with quasi-norm and a -Banach space with -norm , respectively. Let be the modulus of concavity of . In this section, using an idea of G vruta , we prove the stability of (1.6) in the spirit of Hyers, Ulam, and Rassias. For convenience we use the following abbreviation for a given mapping :
Lemma 3.1 (see ).
The proof is similar to the proof of Theorem 3.2.
The proof is similar to the proof of Theorem 3.6.
The third and corresponding author was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2009-0070788). Also, the second author would like to thank the office of gifted students at Semnan University for its financial support.
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