Quadratic-Quartic Functional Equations in RN-Spaces
© M. Eshaghi Gordji et al. 2009
Received: 20 July 2009
Accepted: 3 November 2009
Published: 1 December 2009
The stability problem of functional equations originated from a question of Ulam  in concerning the stability of group homomorphisms. Let be a group and let be a metric group with the 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 other words, under what condition does there exists a homomorphism near an approximate homomorphism? The concept of stability for functional equation arises when we replace the functional equation by an inequality which acts as a perturbation of the equation. Hyers  gave a first affirmative answer to the question of Ulam for Banach spaces. Let be a mapping between Banach spaces such that
for all Moreover, if is continuous in for each fixed then is -linear. In 1978, Rassias  provided a generalization of Hyers' theorem which allows the Cauchy difference to be unbounded. In Gajda  answered the question for the case , which was raised by Rassias. This new concept is known as Hyers-Ulam-Rassias stability of functional equations (see [5–12]). The functional equation
is related to a symmetric biadditive mapping. It is natural that this equation is called a quadratic functional equation. In particular, every solution of the quadratic functional equation (1.3) is said to be a quadratic mapping. It is well known that a mapping between real vector spaces is quadratic if and only if there exits a unique symmetric biadditive mapping such that for all (see [5, 13]). The biadditive mapping is given by
The Hyers-Ulam-Rassias 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 an abelian group. In , Czerwik proved the Hyers-Ulam-Rassias stability of the functional equation (1.3). Grabiec  has generalized the 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 the function satisfies the functional equation (1.5), which is called a quartic functional equation (see also ). In addition, Kim  has obtained the Hyers-Ulam-Rassias stability for a mixed type of quartic and quadratic functional equation.
The Hyers-Ulam-Rassias stability of different functional equations in random normed and fuzzy normed spaces has been recently studied in [21–26]. It should be noticed that in all these papers the triangle inequality is expressed by using the strongest triangular norm .
In this sequel, we adopt the usual terminology, notations, and conventions of the theory of random normed spaces, as in [22, 23, 27–29]. Throughout this paper, is the space of distribution functions, that is, the space of all mappings such that is left-continuous and nondecreasing on and . Also, is a subset of consisting of all functions for which , where denotes the left limit of the function at the point , that is, . The space is partially ordered by the usual point-wise ordering of functions, that is, if and only if for all in . The maximal element for in this order is the distribution function given by
Definition 1.1 (see ).
Typical examples of continuous -norms are , and (the Lukasiewicz -norm). Recall (see [30, 31]) that if is a -norm and is a given sequence of numbers in , then is defined recurrently by and for . is defined as It is known  that for the Lukasiewicz -norm, the following implication holds:
Definition 1.2 (see ).
Theorem 1.4 (see ).
In this paper, we deal with the following functional equation:
In Section 2, we investigate the general solution of the functional equation (1.9) when is a mapping between vector spaces and in Section 3, we establish the stability of the functional equation (1.9) in RN-spaces.
2. General Solution
Finally, to prove the uniqueness of the quadratic mapping subject to (3.3), let us assume that there exists another quadratic mapping which satisfies (3.3). Since for all and all from (3.3), it follows that
C. Park was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2009-0070788).
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