General Solution and Two Methods of Generalized Ulam - Hyers Stability of $n-$ Dimensional AQCQ  Functional Equation

Sandra Pinelas, a∗ M. Arunkumar, b T. Namachivayam c and E. Sathya d

General Solution and Two Methods of Generalized Ulam - Hyers Stability of $n-$ Dimensional AQCQ Functional Equation

Authors :

Sandra Pinelas,^a∗ M. Arunkumar, ^b T. Namachivayam ^c and E. Sathya^d

Author Address :

^aPedagogical Department E.E., Section of Mathematics and Informatics, National and Capodistrian University of Athens, Athens 15342, Greece.
^b,c,d Department of Mathematics, Government Arts College, Tiruvannamalai - 606 603, TamilNadu, India.

*Corresponding author

Abstract :

In this paper, we achieve the general solution and generalized Ulam - Hyers stability of a $n-$ dimensional additive-quadratic-cubic-quartic (AQCQ) functional equation
\begin{align*}
f\left(\sum\limits_{i=1}^{n-1}{v_i}+2v_n\right)
+f\left(\sum\limits_{i=1}^{n-1}{v_i}-2v_n\right)
&= 4f\left(\sum\limits_{i=1}^n{v_i }\right)
+ 4f\left(\sum\limits_{i=1}^{n-1}{v_i}-v_n \right)
- 6f\left(\sum\limits_{i=1}^{n-1}{v_i}\right) \\
&\qquad + f\left(2v_n\right) + f\left(- 2v_n\right)
- 4f\left(v_n\right) - 4f\left(- v_n\right)
\end{align*}
where $n$ is a positive integer with $n\ge 3$ in Banach Space (\textbf{BS}) via direct and fixed point methods. The stability results are discussed in two different ways by assuming $n$ is an odd positive integer and $n$ is an even positive integer.

Keywords :

AQCQ functional equation, generalized Ulam - Hyers stability, Banach space, fixed point.

DOI :

Article Info :

Received : July 22, 2016; Accepted : December 10, 2016.