It is easy to show that the function satisfies the functional equation (1.4) which is called a quartic functional equation and every solution of the quartic functional equation is said to be a quartic function.
It is easy to see that the function f(x) = d x4 is a solution of the functional equation (1.3), which is called a quartic functional equation.
It is easy to show that the function satisfies the functional equation (1.2), which is called a quartic functional equation.
It is easy to show that the function satisfies the functional equation (1.5), which is called a quartic functional equation (see also [19]).
It is easy to show that satisfies the functional equation (1.5), which is called a quartic functional equation (see also [13]).
It is easy to show that the function f(x) = x4 satisfies the functional equation (1.3), which is called a quartic functional equation (see also [33]).
For this reason, (1.1) is called a quartic functional equation.
The bi-quadratic mapping B 2 is given by B 2 ( x, y ) = 1 12 ( f ( x + y ) + f ( x − y ) − 2 f ( x ) − 2 f ( y ) ). Obviously, the function f ( x ) = a x 4 satisfies functional equation (1.4), which is called a quartic functional equation.
The bi-quadratic mapping B 2 is given by B 2 ( x, y ) = 1 12 ( f ( x + y ) + f ( x − y ) − 2 f ( x ) − 2 f ( y ) ). Obviously, the function f ( x ) = a x 4 satisfies the functional equation (1.4), which is called a quartic functional equation.
It is easy to show that the function f ( x ) = x 4 satisfies the functional equation (1.3), which is called a quartic functional equation, and every solution of the quartic functional equation is said to be a quartic mapping (for the stability of the ACQ and quartic functional equations, see [26, 31] and others).
