Solution and stability of system of quartic functional equations

K. Balamurugana,* M. Arunkumarb and P. Ravindiranc

Solution and stability of system of quartic functional equations

Authors :

K. Balamurugan^a,* M. Arunkumar^b and P. Ravindiran^c

Author Address :

^a,bDepartment of Mathematics, Government Arts College, Tiruvannamalai - 606 603, Tamil Nadu, India.
^cDepartment of Mathematics, Arignar Anna Government Arts College, Villupuram - 605 602, Tamil Nadu, India.

*Corresponding author.

Abstract :

In this paper, the authors introduced and investigated the general solution of system of quartic functional equations
\begin{align*}
f(x+y&+z) + f(x+y-z) + f(x-y+z) + f(x-y-z)\\ &=2[f(x+y) + f(x-y) + f(x+z) + f(x-z) + f(y+z) + f(y-z)]\\
&\qquad \qquad -4[f(x) + f(y) + f(z)],\\
f(3x+2y&+z) + f(3x+2y-z) + f(3x-2y+z) + f(3x-2y-z)\\ &= 72[f(x+y) + f(x-y)] + 18[f(x+z) + f(x-z)] + 8[f(y+z) + f(y-z)]\\
&\qquad \qquad + 144f(x) - 96f(y) - 48f(z),\\
f(x+2y&+3z) + f(x+2y-3z) + f(x-2y+3z) + f(x-2y-3z) \\ &= 8[f(x+y) + f(x-y)] + 18[f(x+z) + f(x-z)] + 72[f(y+z) + f(y-z)]\\
&\qquad \qquad - 48f(x) - 96f(y) + 144f(z).
\end{align*}
Its generalized Hyers-Ulam stability using Hyers direct method and fixed point method are discussed. Counter examples for non stable cases are also given.