Intuitionistic random almost additive-quadratic mappings
© Park et al.; licensee Springer 2012
Received: 31 January 2012
Accepted: 23 August 2012
Published: 4 September 2012
In this paper, we investigate the Hyers-Ulam stability of the additive-quadratic functional equation () in intuitionistic random normed spaces.
MSC:39B52, 34K36, 46S50, 47S50, 34Fxx.
The concept of stability of a functional equation arises when one replaces a functional equation by an inequality which acts as a perturbation of the equation. The first stability problem concerning group homomorphisms was raised by Ulam  in 1940 and affirmatively solved by Hyers . The result of Hyers was generalized by Aoki  for approximate additive mappings and by Rassias  for approximate linear mappings by allowing the difference Cauchy equation to be controlled by . In 1994, a generalization of the Th.M. Rassias’ theorem was obtained by Gǎvruta , who replaced by a general control function . For more details about the results concerning such problems, the reader is referred to [6–16].
is related to a symmetric bi-additive mapping [17, 18]. It is natural that this equation is called a quadratic functional equation. In particular, every solution of the quadratic equation (1.1) is said to be a quadratic mapping. It is well known that a mapping f between real vector spaces is quadratic if and only if there exists a unique symmetric bi-additive mapping such that for all x. The bi-additive mapping is given by . The Hyers-Ulam stability problem for the quadratic functional equation was solved by Skof . In , Czerwik proved the Hyers-Ulam stability of the function equation (1.1).
in quasi-Banach spaces, where k is a nonzero integer with . Obviously, the function is a solution of the functional equation (1.2). Interesting new results concerning mixed functional equations have recently been obtained by Najati et al. [22–24], Jun and Kim [25, 26] as well as for the fuzzy stability of a mixed-type functional equation by Park et al. [27–29].
was investigated by Najati and Rassias .
The theory of random normed spaces (RN-spaces) is important as a generalization of the deterministic result of linear normed spaces and also in the study of random operator equations. The RN-spaces may also provide us with the appropriate tools to study the geometry of nuclear physics and have important application in quantum particle physics. The Hyers-Ulam stability of different functional equations in random normed spaces and RN-spaces has been recently studied in Alsina , Eshaghi Gordji et al. [31, 32], Miheţ and Radu [33–35], Miheţ, Saadati and Vaezpour [36, 37], and Saadati et al. . Recently, Zhang et al.  investigated the intuitionistic random stability problems for the cubic functional equation.
In this paper, we prove the Hyers-Ulam stability of the additive and quadratic functional equation (1.3) in intuitionistic random spaces.
We start our work with the following notion of intuitionistic random normed spaces. In the sequel, we adopt the usual terminology, notations and conventions of the theory of intuitionistic Menger probabilistic normed spaces as in  and [40–44].
If X is a nonempty set, then is called a probabilistic measure on X and is denoted by .
If X is a nonempty set, then is called a probabilistic non-measure on X and is denoted by .
Then is a complete lattice.
We denote the units by and . Classically, for all , a triangular norm on [0,1] is defined as an increasing, commutative, associative mapping satisfying , and a triangular conorm is defined as an increasing, commutative, associative mapping satisfying .
By use of the lattice , these definitions can be straightforwardly extended.
Definition 2.2 
, (boundary condition);
, , (monotonicity).
If is an Abelian topological monoid with unit , then ϒ is said to be a continuous t-norm.
Definition 2.3 
Typical examples of continuous t-representable are and for all , .
for all and .
for all . is defined as .
A negator on is any decreasing mapping satisfying and . If for all , then ℵ is called an involutive negator. A negator on [0,1] is a decreasing mapping satisfying and . denotes the standard negator on defined by for all .
Definition 2.4 . Let μ and ν be measure and non-measure distribution functions from to such that for all and all . The triple is said to be an intuitionistic random normed space (briefly IRN-space) if X is a vector space, ϒ is a continuous t-representable, and is a mapping such that the following conditions hold for all and all :
() if and only if ;
() for all ;
In this case, is called an intuitionistic random norm. Here, .
Every normed space defines an IRN-space , where for all and for all , . This space is called the induced IRN-space.
A sequence in X is said to be convergent to x in X if, as for every .
An IRN-space is said to be complete if and only if every Cauchy sequence in X is convergent to a point in X.
From now on, let X be a linear space and be a complete IRN-space.
for all , where is a fixed integer.
3 Results in intuitionistic random spaces
In the following theorem, we prove the Hyers-Ulam stability of the functional equation (1.3) in IRN-spaces for quadratic mappings.
for all and all . By letting in (3.22), we find that for all , which implies . Thus Q satisfies (1.3). Hence the mapping is quadratic.
To prove (3.4), take the limit as in (3.20).
for all and all . By letting in (3.23), we find that . □
Proof Let . Then the corollary follows immediately from Theorem 3.1. □
Now, we prove the Hyers-Ulam stability of the functional equation (1.3) in IRN-spaces for additive mappings.
for all and all . By letting in (3.38), we find that for all , which implies . Thus A satisfies (1.3). Hence the mapping is additive. To prove (3.26), take the limit as in (3.36).
The rest of the proof is similar to the proof of Theorem 3.1. □
The main result of this paper is the following:
for all and all .
for all and all . Hence (3.39) follows from (3.40) and (3.41). □
for all and all .
Now, we give an example to validate the result of quadratic mappings as follows:
Therefore, all the conditions of Theorem 3.1 hold, and so there exists a unique quadratic mapping such that .
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