Fuzzy Stability of a Functional Equation Deriving from Quadratic and Additive Mappings

and Applied Analysis 3 is Cauchy. If each Cauchy sequence is convergent, then the fuzzy norm is said to be complete, and the fuzzy normed space is called a fuzzy Banach space. Let X,N be a fuzzy normed space and Y,N ′ a fuzzy Banach space. For a given mapping f : X → Y , we use the abbreviation Df ( x, y ) : f ( 2x y ) f ( 2x − y 2f x − fx y − fx − y − 2f 2x , 2.1 for all x, y ∈ X. Recall Df ≡ 0 means that f is a general quadratic mapping. For given q > 0, the mapping f is called a fuzzy q-almost general quadratic mapping if N ′ ( Df ( x, y ) , t s ) ≥ minN x, s ,Ny, tq, 2.2 for all x, y ∈ X \ {0} and all s, t ∈ 0,∞ . Now, we get the general stability result in the fuzzy normed linear setting. Theorem 2.2. Let q be a positive real number with q / 1/2, 1. And let f be a fuzzy q-almost general quadratic mapping from a fuzzy normed space X,N into a fuzzy Banach space Y,N ′ . Then, there is a unique general quadratic mapping F : X → Y such that N ′ ( F x − f x , t ≥ sup 0<t′<t N ( x, t′q 7 2p 3p 4p / |4−2p|3p 5 2 · 2p 3p / 2|2−2p| q ) ,