ON THE STABILITY PROBLEM OF QUADRATIC FUNCTIONAL EQUATIONS IN 2-BANACH SPACES

In 1940, Ulam [17] suggested the stability problem of functional equations concerning the stability of group homomorphisms as follows: When is it true that a mapping satisfying a stability problem of functional equation approximately must be close to an exact solution of the given functional equation? The famous Ulam stability problem was partially solved by Hyers [10] of Banach spaces in 1941. Th.M. Rassias [14] given a generalization of Hyers' theorem by proving the existence of unique linear mappings near approximate additive mappings. The paper of Rassias has provided a lot of influence in the development of what we call the generalized Hyers-Ulam-Rassias stability of functional equations. In 1982, J.M. Rassias [13] provided a generalizations of the Hyer-Ulam stability theorem which allows the Cauchy difference controlled by a product of different powers of norm. And then, the result of Th.M. Rassias theorem has been generalized by G vruta [8] by replacing the unbounded Cauchy difference by a generalized control function.


INTRODUCTION
In 1940, Ulam [17] suggested the stability problem of functional equations concerning the stability of group homomorphisms as follows: When is it true that a mapping satisfying a stability problem of functional equation approximately must be close to an exact solution of the given functional equation? The famous Ulam stability problem was partially solved by Hyers [10] of Banach spaces in 1941. Th.M. Rassias [14] given a generalization of Hyers' theorem by proving the existence of unique linear mappings near approximate additive mappings. The paper of Rassias has provided a lot of influence in the development of what we call the generalized Hyers-Ulam-Rassias stability of functional equations. In 1982, J.M. Rassias [13] provided a generalizations of the Hyer-Ulam stability theorem which allows the Cauchy difference controlled by a product of different powers of norm. And then, the result of Th.M. Rassias theorem has been generalized by G vruta [8] by replacing the unbounded Cauchy difference by a generalized control function.
The quadratic function the mapping satisfies the following functional equation (1.1) and therefore the functional equation (1.1) is called the quadratic functional equation. The generalized Hyers-Ulam-Rassias stability theorem for the quadratic functional equation (1.1) was proved by [4,5,9,16] and the references therein.
Recently, many mathematicians came out with results in linear 2-normed spaces, analogous with that in classical normed spaces and Banach spaces. In particular, the stability problems of the functional equations in 2-normed spaces have investigated by [1,3,11,12]. Before we present our results, we introduce some basic facts concerning linear 2-normed spaces. The theory of linear 2-normed spaces was first developed by G hler [7] in the mid 1960's, while that of 2-Banach spaces was studied later by White [18]. For further details on linear 2-normed spaces, refer to the books [2] and [6].  Geometrically, a 2-norm function generalizes the concept of area function of parallelogram due to the fact that it represents the area of the usual parallelogram spanned by the two associated elements. For example, take being equipped with the 2-norm = the area of the parallelogram spanned by the vectors and , which may be given explicitly by the formula where Then, is a linear 2-normed space.
(2) If and are linearly dependent, then or .
( (1) A sequence in is called a 2-convergent sequence if there is an such that for all .
If the sequence 2-converges to then we write and is called the limit of Throughout this paper, let be a real normed space. We consider that there is a 2-norm on which makes a 2-Banach space (in shortly, we substitute for ). For any in , let denote the subspace of generated by . Whenever the notation is used, the vectors are linearly independent. For a given mapping we define a difference operator by for all .

G vrute stability of (1.2)
In this section, we obtain stability in the spirit of G vrute [8] for the quadratic functional equation (1.2). This results generalized and modifies the Hyers-Ulam stability in 2-Banach spaces.

ACKNOWLEDGMENTS
The first author was supported by Dong-eui University (201702480001).