Stability of $n$-Dimensional Additive  Functional Equation: Direct and Fixed Point Methods

M. Arunkumara,*, A. Vijayakumarb , S. Karthikeyanc and S. Ramamoorthid

Stability of $n$-Dimensional Additive Functional Equation: Direct and Fixed Point Methods

Authors :

M. Arunkumar^a,*, A. Vijayakumar^b , S. Karthikeyan^c and S. Ramamoorthi^d

Author Address :

^aDepartment of Mathematics, Government Arts College, Tiruvannamalai - 606 603, TamilNadu, India.
^b,cDepartment of Mathematics, R.M.K. Engineering College, Kavaraipettai - 601 206, Tamil Nadu, India.
^dDepartment of Mathematics, Arunai Engineering College, Tiruvannamalai - 606 604, TamilNadu, India.

*Corresponding author.

Abstract :

In this paper, the authors established the generalized Ulam - Hyers stability of $n$-dimensional additive functional equation
\begin{align*}
\sum\limits^{n}_{i=1}f\left(\sum\limits^{n}_{j=1}x_{ij}\right)=(n-2)\sum\limits^{n}_{j=1}f\left(x_{j}\right) \ \ \ \ \
\textnormal{where} ~~~~~~~~~ x_{ij}=\left\{
\begin{array}{rll}
-x_j & if & i=j\\
x_j & if & i\neq j
\end{array}\right.
\end{align*}
and
$n$ is a positive integer with $n\neq 2$ using direct and fixed point methods.

Keywords :

Additive functional equation, Generalized Ulam-Hyers stability, Fixed point method.

DOI :

Article Info :

Received : April 10, 2015; Accepted : May 23, 2015.