Stability of Cubic Functional Equation in Random Normed Space

In this paper, we present the Hyers-Ulam stability of Cubic functional equation. 
 
where n is greater than or equal to 4, in Random Normed Space.


Introduction
The theory of random normed spaces (briefly, RN-Spaces) is important as a generalization of deterministic result of normed spaces and also in the study of random operator equations. The notion of an RN-Space corresponds to the situations when we do not know exactly the norm of the point and we know only probabilities of possible values of this norm.
Random Theory is a setting in which uncertainty arising from problems in various fields of science, can be modelled. It is a practical tool for handling situations where classical theories fail to explain. Random Theory has many applications in several fields, for example, population dynamics, computer programming, nonlinear dynamical system, nonlinear operators, statistical convergence and so forth. Jun and Kim [5] introduced the following cubic functional equation  were dealt by Seong Sik Kim et al., [17], S, Murthy et al., [9]. Some of the non-cubic functional equations discussed in various spaces of papers are used to develop this paper which are [1,3,7,8,10,11,12,13,14,16,18,19]. In this paper, the authors investigate the general solution and generalized Hyers-Ulam stability of a new type of n-dimensional cubic functional equation where n is greater than or equal to 4, in Random Normed Space by using direct and fixed-point method.

Preliminaries
In this part, we evoke some notations and basic definitions used in this article. Tx =+ is defined recurrently by ' It is known that, for the Lukasiewicz t-norm, the following implication holds: XT  , where X is a vector space. T is a continuous t-norm and  is a mapping from X into D + satisfies the following conditions: 3) XT  is said to be complete, if every Cauchy sequence in X is convergent to a point in X.

General Solution of the n-Dimensional Cubic Functional Equation (1.4):
In this part, the authors discuss the general solution of the functional equation (1.4) by considering X and Y are real vector space. , , ,..., xX  . Hence f is an odd function. Again replacing ( ) for all xX  . In general, for any positive integer a , we obtain x y X  . Adding (3.5) and (3.6), We achieve our required result (1.1).
All over this paper we use the following notation for a given mapping

Random Stability Results: Direct Method
In this part, the generalized Ulam-Hyers Stability of the cubic functional equation (1.4) in RN-Space is provided.
All through this part, let us consider X be a linear space ( ) ,, YT  is a complete RN-Space.
This completes the proof.
The following corollary is an immediate consequence of Theorem 4.1, concerning the stability of (1.4).
for all xX  and all 0 t  .

Random Stability Results: Fixed Point Method.
In this part, the author presents the generalized Ulam-Hyers Stability of the functional equation (1.4), in Random Normed Space using fixed point method. for all xX  and 0 t  .

Proof.
Let d be a general metric on ,, x