# Modular Stability Analysis of a Nonlinear Stochastic Fractional Volterra IDE
## 2 Preliminaries
: _The arithmetic mean function \(AM:\mathbb{R}^{k}\longrightarrow\mathbb{R}\) is defined by_  
\[AM(p)=\frac{1}{k}\sum_{j=1}^{k}p_{j}.\]  
**Example 2**. : _The geometric mean function \(GM:\mathbb{R}^{k}\longrightarrow\mathbb{R}\) is defined by_  
\[GM(p)=(\prod_{j=1}^{k}p_{j})^{\frac{1}{k}}.\]  
**Example 3**. : _The projection function \(\mathcal{P}_{\Im}:\mathbb{R}^{k}\longrightarrow\mathbb{R}\) for \(\Im\in[k]\) and \(\Im\)th argument is defined by_  
\[\mathcal{P}_{\Im}(p)=p_{\Im},\]  
_where \(p_{(\Im)}\) is the \(\Im\)th lowest coordinate of \(p\), i.e., \(p_{(1)}\leq\cdots\leq p_{(\Im)}\leq\cdots p_{(k)}\). Furthermore, the following functions show the \(\mathcal{PF}\) in the first and last coordinates_  
\[\mathcal{P}_{F}(p) =\mathcal{P}_{1}(p)=p_{1}, \tag{10}\] \[\mathcal{P}_{L}(p) =\mathcal{P}_{k}(p)=p_{p}.\]  
**Example 4**. : _The order statistic function \(\mathcal{OS}_{\Im}:\mathbb{R}^{k}\longrightarrow\mathbb{R}\) with the \(\Im\)th argument and \(\Im\)th lowest coordinate is defined by_  
\[\mathcal{OS}_{\Im}(p)=p_{(\Im)},\]  
_for any \(\Im\in[k]\)._  
**Example 5**. : _The minimum function and maximum function are defined as follows, respectively,_  
\[MIN(p) =\bigwedge_{j=1}^{k}p_{j}, \tag{11}\] \[MAX(p) =\bigvee_{j=1}^{k}p_{j}.\]  
**Example 6**. : _The median function is defined for odd and even values of \((p_{1},\cdots,p_{23-1})\) and \((p_{1},\cdots,p_{23})\), respectively,_  
\[MED(p_{1},\cdots,p_{23-1})=p_{(\Im)}, \tag{12}\] \[AM(p)=\frac{1}{k}\sum_{j=1}^{k}p_{j}.\]0  
According to the above functions, we consider the following set:  
\[AM(p)=\frac{1}{k}\sum_{j=1}^{k}p_{j}.\]1  
and the necessary calculations were performed on the considered set, and the results are shown in the table below. From the calculations in Table 1, we consider the minimum function as the control function and define it as follows  
\[AM(p)=\frac{1}{k}\sum_{j=1}^{k}p_{j}.\]2  
In the following, after the table, in Figure 1, we provide a graphical representation of some aggregation functions  
**Definition 9**. : _Consider the continuously differentiable random operator \(\hbar(\varrho,\theta)\) and let \(\varphi_{\tau}(\theta)\) be a modular control function satisfying \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(\theta\) & \(AM\) (\(\Delta\)) & \(GM\) (\(\Delta\)) & \(MAX\) (\(\Delta\)) & \(MIN\) (\(\Delta\)) & \(MED\) (\(\Delta\)) \\ \hline
0.2 & 0.8080213850 & 0.1309431397 & 1.124151808 & 0.00005753566833 & 1.105170918 \\ \hline
0.3 & 0.8643139212 & 0.1490999010 & 1.54862650 & 0.00008260506014 & 1.161834243 \\ \hline
0.4 & 0.9247680550 & 0.1663238147 & 1.668199962 & 0.0001071434790 & 1.221402758 \\ \hline
0.5 & 0.9898018722 & 0.1835845210 & 1.794667420 & 0.0001318755004 & 1.284025417 \\ \hline
0.6 & 1.059899323 & 0.2013893746 & 1.928408155 & 0.0001573709088 & 1.349858808 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Calculation of aggregation function according to special functions for different values. Figure 1: Graph of aggregate functions AM and GM for, \(\tau=2\) and different values \(\theta\). (**a**) The aggregation arithmetic mean function for \(\theta\in(\frac{3}{2},\frac{10}{3})\). (**b**) The aggregation arithmetic mean function for \(\theta\in(1.5,10.5)\). (**c**) The aggregation geometric mean function for \(\theta\in(0,10)\). \[v_{\tau}\bigg{(}{{}^{H}\!\!\!\!\!D}_{0+}^{\ell,x,\aleph}h(\varrho,\theta)-f(\varrho, \theta,\hbar(\varrho,\theta))-\int_{0}^{\theta}k(\varrho,\theta,\vartheta,\hbar (\varrho,\theta))d\theta\bigg{)}\leq\varphi_{\tau}(\theta),\]  
_for each \(\theta\in\Xi_{1}\), \(\tau\in\Xi_{2}\) and \(\varrho\in\mathrm{Y}\). If there exists a solution \(\hbar_{0}(\varrho,\theta)\) of the VIDE (1) and a fixed number \(C>0\) with_  
\[v_{\tau}(\hbar(\varrho,\theta)-\hbar_{0}(\varrho,\theta))\leq\varphi_{\frac{ \tau}{\ell}}(\theta),\]  
_for all \(\theta\in\Xi_{1}\), \(\tau\in\Xi_{2}\) and \(\varrho\in\mathrm{Y}\), in which \(C\) is autonomous of \(\hbar(\varrho,\theta)\) and \(\hbar_{0}(\varrho,\theta)\), then we say that (1) has Hyers-Ulam-Rassias stability._