Solution and Hyers-Ulam-Rassias Stability of Generalized Mixed Type Additive-Quadratic Functional Equations in Fuzzy Banach Spaces

and Applied Analysis 3 with f 0 0 in a non-Archimedean space. It is easy to see that the function f x ax bx2 is a solution of the functional equation 1.8 , which explains why it is called additive-quadratic functional equation. For more detailed definitions of mixed type functional equations, we can refer to 26–47 . Definition 1.1 see 48 . 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 nondecreasing 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. Example 1.2. Let X, ‖ · ‖ be a normed linear space and α, β > 0. Then