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In this paper, the authors investigated the intuitionistic random stability of a quadratic reciprocal functional equation$$f(x+2 y)+f(2 x+y)=\frac{f(x) f(y)[5 f(x)+5 f(y)+8 \sqrt{f(x) f(y)}]}{[2 f(x)+2 f(y)+5 \sqrt{f(x)+f(y)}]^2}$$using direct and fixed point methods.

In this paper, we achieve the general solution and generalized Ulam - Hyers stability of a $n$-dimensional additive-quadratic-cubic-quartic (AQCQ) functional equation$$\begin{aligned}f\left(\sum_{i=1}^{n-1} v_i+2 v_n\right)+f\left(\sum_{i=1}^{n-1} v_i-2 v_n\right)= & 4 f\left(\sum_{i=1}^n v_i\right)+4 f\left(\sum_{i=1}^{n-1} v_i-v_n\right)-6 f\left(\sum_{i=1}^{n-1} v_i\right) \\& +f\left(2 v_n\right)+f\left(-2 v_n\right)-4 f\left(v_n\right)-4 f\left(-v_n\right)\end{aligned}$$where $n$ is a positive integer with $n \geq 3$ in Banach Space (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.

In this paper, the authors proved the generalized Ulam-Hyers stability of 2-variable Additive-QuadraticCubic-Quartic functional equation$$\begin{aligned}f(x+2 y, u+2 v)+f(x-2 y, u-2 v)= & 4 f(x+y, u+v)-4 f(x-y, u-v)-6 f(x, u)+f(2 y, 2 v) \\& +f(-2 y,-2 v)-4 f(y, v)-4 f(-y,-v)\end{aligned}$$using fixed point method.

In this paper, the authors investigated the general solution and generalized Ulam - Hyers stability of $a_i$ type $n$ - variable multi $n$ - dimensional additive functional equation$$\begin{aligned}2 h\left(\sum_{i=1}^n a_i x_{1 i}\right. & \left., \sum_{i=1}^n a_i x_{2 i}, \ldots \ldots, \sum_{i=1}^n a_i x_{n i}\right) \\= & \left(\sum_{i=1}^n a_i\right) h\left(\sum_{i=1}^n x_{1 i}, \sum_{i=1}^n x_{2 i}, \ldots \ldots, \sum_{i=1}^n x_{n i}\right) \\& +\left(a_1-\sum_{i=2}^n a_i\right) h\left(x_{11}-\sum_{i=2}^n x_{1 i}, x_{21}-\sum_{i=2}^n x_{2 i}, \ldots \ldots, x_{n 1}-\sum_{i=2}^n x_{n i}\right)\end{aligned}$$where $a_i(i=1,2, \ldots n)$ are different integers greater than 1 , using two different technique.

In this paper, the authors established the general solution and generalized Ulam - Hyers stability of an 3 dimensional additive quadratic functional equation$$\begin{aligned}& h(x+2 y+3 z)+h(x+2 y-3 z)+h(x-2 y+3 z)+h(-x+2 y+3 z) \\&=h(x+y+z)+h(x+y-z)+h(x-y+z)+h(-x+y+z) \\&+2 h(y)+4 h(z)+5[h(y)+h(-y)]+14[h(z)+h(-z)]\end{aligned}$$via Banach space and non-Archimedean fuzzy Banach Space using direct and fixed point methods.

In this paper, the authors obtain the general solution and generalized Ulam - Hyers stability of an (AQQ): additive - quadratic - quartic functional equation of the form$$\begin{aligned}f(x+y+z) & +f(x+y-z)+f(x-y+z)+f(x-y-z) \\=2[f(x+y) & +f(x-y)+f(y+z)+f(y-z)+f(x+z)+f(x-z)] \\& -4 f(x)-4 f(y)-2[f(z)+f(-z)]\end{aligned}$$by using the classical Hyers' direct method. Counter examples for non stability are discussed also.

In this paper, we achieve the general solution and generalized Ulam-Hyers stability of a generalized 3dimensional AQCQ functional equation$$\begin{aligned}f(x+r(y+z))+f(x- & r(y+z))=r^2[f(x+y+z)+f(x-y-z)]+2\left(1-r^2\right) f(x) \\+ & \frac{\left(r^4-r^2\right)}{12}[f(2(y+z))+f(-2(y+z))-4 f(y+z)-4 f(-(y+z))]\end{aligned}$$for all positive integers $r$ with $r \geq 2$ in Banach Space using two different methods.

In this paper, we introduce and investigate the general solution and generalized Ulam- Hyers stability of a additive functional equation$$f\left(\frac{\sum_{k=1}^N x_k}{N}\right)=\frac{1}{N} \sum_{k=1}^N f\left(x_k\right)$$originating from $N$ observations of an arithmetic mean in Banach spaces using various substitutions in two different approaches with $N \geq 2$.

In this paper, the authors introduced and investigated the general solution of system of quartic functional equations$$\begin{aligned}& f(x+y+z)+f(x+y-z)+f(x-y+z)+f(x-y-z) \\& =2[f(x+y)+f(x-y)+f(x+z)+f(x-z)+f(y+z)+f(y-z)] \\& -4[f(x)+f(y)+f(z)] \\& f(3 x+2 y+z)+f(3 x+2 y-z)+f(3 x-2 y+z)+f(3 x-2 y-z) \\& =72[f(x+y)+f(x-y)]+18[f(x+z)+f(x-z)]+8[f(y+z)+f(y-z)] \\& +144 f(x)-96 f(y)-48 f(z) \\& f(x+2 y+3 z)+f(x+2 y-3 z)+f(x-2 y+3 z)+f(x-2 y-3 z) \\& =8[f(x+y)+f(x-y)]+18[f(x+z)+f(x-z)]+72[f(y+z)+f(y-z)] \\& -48 f(x)-96 f(y)+144 f(z) \text {. } \\&\end{aligned}$$Its generalized Hyers-Ulam stability using Hyers direct method and fixed point method are discussed. Counter examples for non stable cases are also given.

In this paper, we obtain the generalized Ulam - Hyers stability of a 2 - variable AC - mixed type functional equation$$f(2 x+y, 2 z+w)-f(2 x-y, 2 z-w)=4[f(x+y, z+w)-f(x-y, z-w)]-6 f(y, w)$$in Quasi - Beta normed space using direct and fixed point methods.

In this paper, the authors has proved the solution of a new type of functional equation$$f\left(\sum_{j=1}^k j^p x_j\right)=\sum_{j=1}^k\left(j^p f\left(x_j\right)\right), \quad k, p \geq 1$$which is originating from sum of higher powers of an arithmetic progression. Its generalized Ulam - Hyers stability in Banach space using direct and fixed point methods are investigated. An application of this functional equation is also studied.

In this paper, the authors established the solution of the additive functional equation and inequality$$f(x)+f(y+z)-f(x+y)=f(z)$$and$$\|f(x)+f(y+z)-f(x+y)\| \leq\|f(z)\| .$$We also prove that the above functional equation and inequality are stable in Banach space in the sense of Ulam, Hyers, Rassias. An application of this functional equation is also studied.