General solution and generalized Ulam - Hyers stability of $r_i$-type $n$ dimensional quadratic-cubic functional equation in random normed spaces: Direct and fixed point methods

Abstract

In this paper, the authors introduce and establish the general solution and generalized Ulam- Hyers stability of a $r_i$ type $n$-dimensional Quadratic-Cubic functional equation
$$
\begin{aligned}
& \sum_{i=0}^n\left[f\left(r_{2 i} x_{2 i}+r_{2 i+1} x_{2 i+1}\right)\right] \\
& =\sum_{i=0}^n\left(\sum_{u=0}^1\left[\sum_{v=0}^1\left(\frac{r_{2 i} r_{2 i+1}(-1)^{u+v}+r_{2 i} r_{2 i+1}^2(-1)^u+r_{2 i}^2 r_{2 i+1}(-1)^v}{4}\right)\left(f\left((-1)^u x_{2 i}+(-1)^v x_{2 i+1}\right)\right)\right]\right. \\
& +\left(\frac{r_{2 i}^3+r_{2 i}^2-r_{2 i} r_{2 i+1}^2}{4}\right) f\left(x_{2 i}\right)+\left(\frac{r_{2 i}^2-r_{2 i}^3+r_{2 i} r_{2 i+1}^2}{4}\right) f\left(-x_{2 i}\right) \\
& \left.+\left(\frac{r_{2 i+1}^3+r_{2 i+1}^2-r_{2 i}^2 r_{2 i+1}}{4}\right) f\left(x_{2 i+1}\right)+\left(\frac{r_{2 i+1}^2-r_{2 i+1}^3+r_{2 i}^2 r_{2 i+1}}{4}\right) f\left(-x_{2 i+1}\right)\right)
\end{aligned}
$$
where $r_{2 i}, r_{2 i+1} \in R-\{0\},(i=0,1,2 \cdots n)$ and $n$ is a positive integer in Random normed spaces.