A Fixed point approach to the stability of a mixed Additive-Quadratic-Cubic-Quartic(AQCQ) functional equation in quasi$-\beta-$normed spaces

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

A Fixed point approach to the stability of a mixed Additive-Quadratic-Cubic-Quartic(AQCQ) functional equation in quasi$-\beta-$normed spaces

Authors :

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

Author Address :

^a,bDepartment of Mathematics, Government 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 we establish the general solution and investigate the generalized Hyers-Ulam stability of the following additive-quadratic-cubic-quartic functional equation
\begin{align}\label{a4}
&f(3x+2y+z) + f(3x+2y-z) + f(3x-2y+z) + f(3x-2y-z)+24\tilde{f}(z) \notag\\ &=48[f(x+y) + f(x-y)] +24[f(-x+y) + f(-x-y)] + 12f(x+z) \notag\\& \quad+ 12f(x-z)+6[f(-x+z) + f(-x-z)] +4[\tilde{f}(y+z) +\tilde{f}(y-z)] \notag \\&\quad+ 20f(2x)+4f(-2x)- 160f(x)-80f(-x)+ 2\tilde{f}(2y)-80\tilde{f}(y)
\end{align}
in the quasi$-\beta-$normed spaces via fixed point method where $\tilde{f}(x)=f(x)+f(-x)$. Counterexamples for non-stability cases are also discussed.

Keywords :

Hyers-Ulam stability, additive-quadratic-cubic-quartic mapping, mixed type functional equation, fixed point.

DOI :

Article Info :

Received : April 10, 2015; Accepted : June 12, 2015.