Th. M. Rassias (1984) proved that the norm defined over a real vector space

Studies on fuzzy normed linear spaces are relatively recent in the field of fuzzy functional analysis. In 1984, Katsaras [

Nowadays, fixed point and operator theory play an important role in different areas of mathematics, and its applications, particularly in mathematics, physics, differential equation, game theory and dynamic programming. Since fuzzy mathematics and fuzzy physics along with the classical ones are constantly developing, the fuzzy type of the fixed point and operator theory can also play an important role in the new fuzzy area and fuzzy mathematical physics. Many authors [

One of the most interesting questions in the theory of functional analysis concerning the Ulam stability problem of functional equations is as follows: when is it true that a mapping satisfying a functional equation approximately must be close to an exact solution of the given functional equation?

The first stability problem concerning group homomorphisms was raised by Ulam [

On the other hand, J. M. Rassias [

Let

Very recently, K. Ravi [

For more details about the results concerning such problems and mixed product-sum stability (J. M. Rassias Stability), the reader is referred to [

Quadratic functional equations were used to characterize inner product spaces [

It was shown by Th. M. Rassias [

The main objective of this paper is to prove the the generalized Hyers-Ulam stability of the following functional equation associated with inner product spaces

The stability of different functional equations in fuzzy normed spaces and random normed spaces has been studied in [

We start our work with the following notion of fixed point theory. For the proof, refer to [

Let

We recall the following theorem by Margolis and Diaz.

Suppose that one is given a complete generalized metric space

the sequence

Next, we define the notion of a fuzzy normed linear space.

Let

In the following we will suppose that

A fuzzy normed linear space is a pair

Let

Let

A sequence

From now on, let

In the following theorem, we prove the fuzzy stability of the functional equation (

Let

Letting

Without loss of generality, we consider

It follows from (

Due to Theorem

Also,

According to the fixed point alterative, since

In the following theorem, we prove the fuzzy stability of the functional equation (

Let

Letting

It follows from (

Due to Theorem

Also,

The rest of the proof is similar to the proof of Theorem

The main result of the paper is the following.

Let

Let

The second author would like to thank the Office of Gifted Students at Semnan University for its financial support.