In this paper, we prove the stability of the following functional equation ${\sum }_{i=0}^n{}_n{C}_i{(-1)}^{n-i}f(ix+y)-n!f\left(x\right)=0$ on a restricted domain by employing the direct method in the sense of Hyers.

In this paper, we study the generalized Hyers-Ulam stability of the quartic functional equation \[ f(x+3y) - 5f(x+2y) + 10f(x+y) - 10f(x) + 5f(x-y) - f(x-2y) = 0, \] by applying the fixed point method.

The general quintic functional equation and the general sextic functional equation are generalizations of many functional equations such as the additive function equation and the quadratic function equation. In this paper, we investigate Hyers–Ulam–Rassias stability of the general quintic functional equation and the general sextic functional equation.

We establish the hyperstability of -Jordan homomorphisms from a normed algebra to a Banach algebra, and also we show that an -Jordan homomorphism between two commutative Banach algebras is an -ring homomorphism.

In this paper, we investigate the stability of an additive-quadratic-quartic functional equation f ( x + 2 y ) + f ( x − 2 y ) − 2 f ( x + y ) − 2 f ( − x − y ) − 2 f ( x − y ) − 2 f ( y − x ) + 4 f ( − x ) + 2 f ( x ) − f ( 2 y ) − f ( − 2 y ) + 4 f ( y ) + 4 f ( − y ) = 0 $$\begin{align*}f(x+2y)& +f(x-2y)-2f(x+y)-2f(-x- y)-2f(x-y)-2f(y-x)\nonumber \\ &+4f(-x)+ 2f(x)-f(2y)-f(-2y)+4f(y)+4f(-y)=0 \end{align*}$$by the direct method in the sense of Găvruta.