INTRODUCTION
The functional equation:
f(2x+y)+f(2xy) = 2f(x+y) +2f(xy)+12f(x)

is said to be the cubic functional equation. Skof (1983)
by proving that if f is a mapping from a normed space X into a Banach space
Y satisfying:
f(x+y)+f(xy)2f(x)2f(y)≤ε 
for some ε>0, then there is a unique quadratic function g:X→Y such that:
Cholewa (1984) extended Skofs theorem by replacing X
by an abelian group G. Skof’s result was later generalized by Czerwik
(1992) in the spirit of HyersUlamRassias (Ulam, 1964).
The stability problem of the quadratic equation has been extensively investigated
by a number of mathematicians and references therein. In addition, Alsina
(1987) and Mihet and Radu (2008) investigated the
stability in the settings of fuzzy, probabilistic and random normed spaces.
In the sequel, the usual terminology, notations and conventions of the theory
of random normed spaces shall be adopted, as by Schweizer and
Sklar (1983). Throughout present study, the space of all probability distribution
functions (briefly, d.f.’s) is denoted by:
where, F is left continuous and non decreasing on R. Also the subset is the set:
D^{+} = {FεΔ^{+}: l^{–}F(+∞)
= 1}

where, l^{–}f (x) denotes the left limit of the function f at the point x, l^{–}f(x) = lim_{t→x} f(t). The space Δ^{+} is partially ordered by the usual pointwise ordering of functions, i.e., D≤G if and only if F(t)≤G(t) for all t in R. The maximal element for Δ^{+} in this order is the d.f. given by:
Definition 1: A mapping T:[0, 1]x[0, 1]→[0, 1] is a continuous
tnorm if T satisfies the following conditions:
• 
T is commutative and associative 
• 
T is continuous 
• 
T (a, 1) = a for all a ε [0, 1] 
• 
T (a, b)≤T(c, d) whenever a≤c and b≤d and a, b, c, d ε [0,
1] 
Two typical examples of continuous tnorm are T(a,b) = ab and T(a,b) = min
(a, b)
Now tnorms are recursively defined by T^{1} = T and
for n≤2 and x_{i} ε[0,1], for all I ε{1,2,...,n+1}
The tnorm T is Hadzic type if for given ε ε (0,1) there is δ ε (0,1) such that:
A typical example of such tnorms is T (a, b) = min (a, b).
Recall that if T is a tnorm and {X_{n}} is a given sequence of numbers
in is
defined recursively
by:
and
is defined as
Definition 2: A Menger Probabilistic normed space (briefly, Menger PN
space) is a triple (X, μ,T), where X is a nonempty set, T is a continuous
tnorm and μ is a mapping from X into D^{+} such that, the following
conditions hold:
• 
(PN1) μ_{x} for all t>0 if and only if x = 0 
• 
(PN2) μ_{xy} (t) = ε_{0} for all x, y in X
and t≥0 
• 

• 
(PN4) μ_{x+y} (t+s)≥T(μ_{x}(t), μ_{y}(s))
for all x, y, z ε X and t, s≥0 
Clearly every Menger PNspace is a probabilistic metric space having a metrizable
uniformity on X if sup_{a<1}T(a, a) = 1.
Definition 3: Let (X, μ, T) be a Menger PNspace:
• 
A sequence {x_{n}} in X is said to be convergent to x in X if,
for every t>0 and ε>0, there exists positive integer N such that
μx_{n}x (t)>1ε whenever n≥N 
• 
A sequence {x_{n}} in X is called Cauchy sequence if, for every
t>0 and ε>0, there exists positive integer N such that μx_{n},
xm (t)>1ε whenever ≥m≥N 
• 
A Menger PNspace (X, μ, T) is said to be complete if and only if
every Cauchy sequence in X is convergent to a point in X 
Theorem 1: If (X, μ, T) is a Menger PNspace and {x_{n}}
is a sequence such that
In this study, the stability of the quadratic functional equation in the setting
of Menger probabilistic normed spaces is established.
MAIN RESULTS
Definition 4: Let X,Y be vector spaces. The functional equation f: X→Y
defined by:
is called cubic functional equation.
Theorem 2: Let (X, v, R) be Menger PN space and (Y, μ, T) be a
complete Menger PN space. If f: X→Y be a mapping such that:
for t>0 in which ξ: X^{2} →D^{+} and
Then there exists a unique quadratic mapping Q: X→Y such that:
Proof: Putting y = 0 in Eq. 2, then:
Replacing x by 2x in Eq. 5, then:
Triangular inequality implies that:
Thus
Replacing x by 4x in Eq. 5 and triangular inequality implies
that:
Using the induction on n, is obtained that:
In order to prove convergence of the sequence replace x with 2^{m}x
in Eq. 10 to find that for m, n>0:
Since the right hand side of the inequality tend to 1 as m tends to infinity, the sequence
is a Cauchy sequence. Therefore, define:
for all x ∈ X.
Now, it is showed that Q is a quadratic map. Replacing x, y with 2^{n}x
and 2^{n}y, respectively in Eq. 2. Then it follows
that:
Taking the limit as m→∞, can be found that Q satisfies Eq.
2 for all x, y ε X.
To prove Eq. 4, take the limit as n→∞ in Eq.
10.
To prove the uniqueness of the quadratic function Q subject to
Eq. 4, assume that there exists a quadratic function Q’ which satisfies
Eq. 4. Obviously;
Hence, it follows from Eq. 4 that:
for all x ε X. By letting n→∞ in Eq. 4,
implies that the uniqueness of Q.