Stability of Mixed Type Cubic and Quartic Functional Equations in Random Normed Spaces
© M. Eshaghi Gordji and M. B. Savadkouhi 2009
Received: 22 June 2009
Accepted: 5 August 2009
Published: 31 August 2009
for all Moreover if is continuous in for each fixed then is linear. In 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]).
and they established the general solution and the generalized Hyers-Ulam-Rassias stability for the functional equation (1.3) The function satisfies the functional equation (1.3) which is thus called a cubic functional equation. Every solution of the cubic functional equation is said to be a cubic function. Jun and Kim proved that a function between real vector spaces X and Y is a solution of (1.3) if and only if there exits a unique function such that for all and is symmetric for each fixed one variable and is additive for fixed two variables.
In fact they proved that a function between real vector spaces and is a solution of (1.4) if and only if there exists a unique symmetric multiadditive function such that for all (see also [15–18]). It is easy to show that the function satisfies the functional equation (1.4) which is called a quartic functional equation and every solution of the quartic functional equation is said to be a quartic function.
In the sequel we adopt the usual terminology, notations, and conventions of the theory of random normed spaces, as in [19–21]. Throughout this paper, is the space of distribution functions that is, the space of all mappings , such that is leftcontinuous and nondecreasing on and 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 pointwise 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 ).
Definition 1.2 (see ).
Theorem 1.4 (see ).
on random normed spaces. It is easy to see that the function is a solution of the functional equation (1.8) In the present paper we establish the stability of the functional equation (1.8) in random normed spaces.
2. Main Results
To prove (2.3) take the limit as in (2.9) Finally, to prove the uniqueness of the cubic function subject to (2.3) let us assume that there exists a cubic function which satisfies (2.3) Since and for all and from (2.3) it follows that
To prove (2.14) take the limit as in (2.20) Finally, to prove the uniqueness property of subject to (2.14) let us assume that there exists a quartic function which satisfies (2.14) Since and for all and from (2.14) it follows that
Obviously, (2.25) follows from (2.28) and (2.31).
The second author would like to thank the Office of Gifted Students at Semnan University for its financial support.
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