Stability of $n-$ Dimensional Quartic Functional Equation In Generalized 2 Normed Spaces Using Two Different Methods

M. Arunkumara,* , S. Murthyb , T. Namachivayamc G. Ganapathyd

Stability of $n-$ Dimensional Quartic Functional Equation In Generalized 2 Normed Spaces Using Two Different Methods

Authors :

M. Arunkumar^a,* , S. Murthy^b , T. Namachivayam^c G. Ganapathy^d

Author Address :

^a,cDepartment of Mathematics, Government Arts College, Tiruvannamalai - 606 603, Tamil Nadu, India.
^bDepartment of Mathematics, Government Arts College for Men, Krishnagiri-635 001, Tamil Nadu, India.
^dDepartment of Mathematics, R.M.D. Engineering College, Kavaraipettai - 601 206, Tamil Nadu, India.

*Corresponding author.

Abstract :

In this paper, we have proved generalized Ulam-Hyers stability of a $n-$ dimensional quartic functional equation
\begin{align*}
&\sum\limits_{i=1}^nf\left(\sum\limits_{j=1}^ix_j\right)\!=\!\sum\limits_{i=1}^n\sum\limits_{1\leq j <k<l <m\leq i}^{}f(x_j+x_k+x_l+x_m)
-\sum\limits_{i=1}^n(i-4)\!\sum\limits_{1\leq j < k<l \leq i}^{}\!f(x_j+x_k+x_l)
\notag\\
&\qquad\qquad
\!+\!\sum\limits_{i=1}^n\!\left(\frac{(i-3)(i-4)}{2}\right)\!\sum\limits_{1\leq j <k\leq i}^{}\!f(x_j+x_k)
-\frac{1}{16}\sum\limits_{i=1}^n\left(\frac{(i-4)(i-3)(i-2)}{6}\right)\sum\limits_{ j= 1}^{i}f(2x_j)
\end{align*}
with $n>5$ in generalized $2-$ normed spaces via two different methods.

Keywords :

Quartic functional equations, Generalized Ulam - Hyers stability, Felbin’s type Fuzzy Cone normed space, Fixed point.

DOI :

Article Info :

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