Orthogonal Stability of an Additive-quartic Functional Equation in Non-Archimedean Spaces

Using fixed point method, we prove the Hyers-Ulam stability of the orthogonally additive-quartic functional equation f(2x+y)+ f(2x-y)=4 f(x+y)+ 4 f(x-y) + 10 f(x) + 14f(-x) - 3 f(y)-3f(-y) for all $x, y$ with $xperp y$, in non-Archimedean Banach spaces. Here $perp$ is the orthogonality in the sense of Rätz

Fixed Points and Fuzzy Stability of Functional Equations Related to Inner Product

In , Th.M. Rassias introduced the following equality sum_{i,j=1}^m |x_i - x_j |^2 = 2m sum_{i=1}^m|x_i|^2, qquad sum_{i=1}^m x_i =0 for a fixed integer $m ge 3$. Let $V, W$ be real vector spaces. It is shown that if a mapping $f : V ightarrow W$ satisfies sum_{i,j=1}^m f(x_i - x_j ) = 2m sum_{i=1}^m f(x_i) for all $x_1, ldots, x_{m} in V$ with $sum_{i=1}^m x_i =0$, then the mapping $f : V ightarrow W$ is realized as the sum of an additive mapping and a quadratic mapping. From the above equality we can define the functional equation f(x-y) +f(2x+y) + f(x+2y)= 3f(x)+ 3f(y) + 3f(x+y) , which is called a {it quadratic functional equation}. Every solution of the quadratic functional equation is said to be a {it quadratic mapping}. Using fixed point theorem we prove the Hyers-Ulam stability of the functional equation () in fuzzy Banach spaces