Fuzzy Stability of Jensen-Type Quadratic Functional Equations

and Applied Analysis 3 a vector space from various points of view 28–30 . In particular, Bag and Samanta 31 , following Cheng and Mordeson 32 , gave an idea of fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Michálek type 33 . They established a decomposition theorem of a fuzzy norm into a family of crisp norms and investigated some properties of fuzzy normed spaces 34 . We use the definition of fuzzy normed spaces given in 31 and 35–38 to investigate a fuzzy version of the generalized Hyers-Ulam stability for the quadratic functional equations 2f ( x y 2 ) 2f ( x − y 2 ) f x f ( y ) , 1.6 f ( ax ay ) f ( ax − ay 2a2f x 2a2fy 1.7 in the fuzzy normed vector space setting. Definition 1.1 see 31, 35–38 . Let X be a real vector space. A function N : X × R → 0, 1 is called a fuzzy norm on X if for all x, y ∈ X and all s, t ∈ R, N1 N x, t 0 for t ≤ 0; N2 x 0 if and only ifN x, t 1 for all t > 0; N3 N cx, t N x, t/|c| if c / 0; N4 N x y, s t ≥ min{N x, s ,N y, t }; N5 N x, · is a non-decreasing function of R and limt→∞N x, t 1; N6 for x / 0, N x, · is continuous on R. The pair X,N is called a fuzzy normed vector space. The properties of fuzzy normed vector spaces and examples of fuzzy norms are given in 35–38 . Definition 1.2 see 31, 35–38 . Let X,N be a fuzzy normed vector space. A sequence {xn} in X is said to be convergent or converge if there exists an x ∈ X such that limn→∞N xn − x, t 1 for all t > 0. In this case, x is called the limit of the sequence {xn} and we denote it by Nlimn→∞xn x. Definition 1.3 see 31, 35–38 . Let X,N be a fuzzy normed vector space. A sequence {xn} in X is called Cauchy if for each ε > 0 and each t > 0 there exists an n0 ∈ N such that for all n ≥ n0 and all p > 0, we have N xn p − xn, t > 1 − ε. It is well known that every convergent sequence in a fuzzy normed vector space is Cauchy. If each Cauchy sequence is convergent, then the fuzzy norm is said to be complete and the fuzzy normed vector space is called a fuzzy Banach space. We say that a mapping f : X → Y between fuzzy normed vector spaces X and Y is continuous at a point x0 ∈ X if for each sequence {xn} converging to x0 in X, then the sequence {f xn } converges to f x0 . If f : X → Y is continuous at each x ∈ X, then f : X → Y is said to be continuous on X see 34 . In this paper, we prove the generalized Hyers-Ulam stability of the quadratic functional equations 1.6 and 1.7 in fuzzy Banach spaces. Throughout this paper, assume that X is a vector space and that Y,N is a fuzzy Banach space. Let a be a nonzero real number with a/ ±1/2 . 4 Abstract and Applied Analysis 2. Fuzzy Stability of Quadratic Functional Equations We prove the fuzzy stability of the quadratic functional equation 1.6 . Theorem 2.1. Let f : X → Y be an even mapping with f 0 0. Suppose that φ is a mapping from X to a fuzzy normed space Z,N ′ such that N ( 2f ( x y 2 ) 2f ( x − y 2 ) − f x − fy, t s ) ≥ min { N ′( φ x , t ) ,N ′( φ ( y ) , s )}