Stability Results of Additive-Quadratic \(n\)-Dimensional Functional Equation: Fixed Point Approach

Abstract

In this paper, the authors discussed the Ulam-Hyers stability results of \(n\)-dimensional mixed type additive and quadratic functional equation:
\begin{align*}
\sum\limits^{n}_{i=1}f\!\bigg(\sum\limits^{n}_{j=1}x_{ij}\bigg)\!& = \bigg(\frac{-n^2+7n-6}{2}\bigg)\!\!
\sum_{i=1}^n f(x_i)+\!\bigg(\frac{-n^2+5n-2}{2}\bigg)\!\!
\sum_{i=1}^n f(-x_i) \\
&\quad +\left(\frac{n-4}{2}\right)
\sum_{1\le i<j\le n}(f(x_i+x_j)
+f(-x_i-x_j))\,,
\end{align*}
where
\begin{align*}
x_{ij}=
\begin{cases}
-x_j&\text{if} \ i=j, \\
x_j &\text{if} \ i\neq j,
\end{cases}
\end{align*} in Banach spaces using fixed point method.