$a_i$ Type $n-$ Variable Multi $n-$ Dimensional Additive Functional  Equation

Matina J. Rassias, a ∗ M. Arunkumar, b and E. Sathya c

$a_i$ Type $n-$ Variable Multi $n-$ Dimensional Additive Functional Equation

Authors :

Matina J. Rassias, ^{a ∗} M. Arunkumar, ^b and E. Sathya ^c

Author Address :

^a Department of Statistical Science , University College London, 1-19 Torrington Place, #140, London, WC1E 7HB, UK.
^b,cDepartment of Mathematics, Government Arts College, Tiruvannamalai - 606 603, TamilNadu, India.

*Corresponding author

Abstract :

In this paper, the authors investigated the general solution and generalized Ulam - Hyers stability of $a_i$ type $n-$ variable multi $n-$ dimensional additive functional equation
\begin{align*}
& 2h\left(\sum_{i=1}^n a_i~x_{1i}, \sum_{i=1}^n a_i~x_{2i}, \dots \dots,\sum_{i=1}^n a_i~x_{ni}\right)\\
& \qquad = \left(\sum_{i=1}^n a_i\right) h\left(\sum_{i=1}^n x_{1i}, \sum_{i=1}^n x_{2i}, \dots \dots,\sum_{i=1}^n x_{ni}\right)\\
& \qquad \qquad \quad + \left(a_1 - \sum_{i=2}^n a_i\right) h\left(x_{11}-\sum_{i=2}^n x_{1i}, x_{21}-\sum_{i=2}^n x_{2i}, \dots \dots,x_{n1}-\sum_{i=2}^n x_{ni}\right)
\end{align*}
where $a_i(i=1,2,\dots n)$ are different integers greater than $1$, using two different technique.

Keywords :

Additive functional equations, Ulam - Hyers stability, Ulam - Hyers - Rassias stability, Ulam - Gavruta - Rassias stability, Ulam - JRassias stability.

DOI :

Article Info :

Received : April 13, 2016; Accepted : November 12, 2016.