General Solution and Generalized Ulam - Hyers Stability  of A Additive Functional Equation Originating From $N$ Observations of An Arithmetic Mean In Banach Spaces Using Various Substitutions In Tw

M. Arunkumar a,∗ , E. Sathya b and S. Ramamoorthi c

General Solution and Generalized Ulam - Hyers Stability of A Additive Functional Equation Originating From $N$ Observations of An Arithmetic Mean In Banach Spaces Using Various Substitutions In Tw

Authors :

M. Arunkumar ^a,∗ , E. Sathya^b and S. Ramamoorthi ^c

Author Address :

^a,b Department of Mathematics, Government Arts College, Tiruvannamalai - 606 603, TamilNadu, India.
^cDepartment of Mathematics, Arunai Engineering College, Tiruvannamalai - 606 604, TamilNadu, India.

*Corresponding author

Abstract :

In this paper, we introduce and investigate the general solution and generalized Ulam- Hyers stability of a additive functional equation

\begin{align*}
%f\left(\frac{x_1+x_2+x_3+\cdots\cdots+x_N}{N}\right)
%&= \frac{1}{N} \left(f(x_1)+f(x_2)+\cdots\cdots+x_N\right)\\
f\left(\displaystyle{\frac{\sum\limits_{k=1}^N x_k}{N}}\right)
&= \frac{1}{N} \sum_{k=1}^N f(x_k)
\end{align*}
originating from $N$ observations of an arithmetic mean in Banach spaces using various substitutions in two different approaches with $N \geq 2$.

Keywords :

Arithmetic mean, additive functional equation, Generalized Hyers-Ulam stability, fixed point.

DOI :

Article Info :

Received : August 12, 2016; Accepted : December 22, 2017.