# Modular Stability Analysis of a Nonlinear Stochastic Fractional Volterra IDE
## 2 Preliminaries
: _The generalized exponential function, which is called Mittag-Leffler function, is defined as_\[E_{t}(\theta)=\sum_{\varsigma=0}^{\infty}\frac{\theta\varsigma}{\theta(1+t\varsigma)} \ \ p\in C,\ \ R(p)>0,\ \ \theta\in C, \tag{4}\]  
Definition 5: The Gauss Hypergeometric series, which is called the Hypergeometric function, is defined as  
\[E_{t}^{s_{1},s_{2}}(\theta)=\frac{\Gamma(t)}{\Gamma(s_{1})\Gamma(s_{2})}\sum_{ \varsigma=0}^{\infty}\frac{\Gamma(s_{1}+\varsigma)\Gamma(s_{2}+\varsigma)}{ \Gamma(t+\varsigma)}\frac{\theta\varsigma}{\varsigma!}\ \ s_{1},s_{2},t\in C,\ \ R(s_{1}),R(s_{2}),R(t)>0. \tag{5}\]  
We can rewrite the above series via the Mellin-Barnes integral as  
\[E_{t}^{s_{1},s_{2}}(\theta)=\frac{\Gamma(t)}{\Gamma(s_{1})\Gamma(s_{2})}\frac{ 1}{2\pi i}\int\frac{\Gamma(x)\Gamma(s_{1}-x)\Gamma(s_{2}-x)}{\Gamma(t-x)}(- \theta)^{-x}dx. \tag{6}\]  
Definition 6: The Maitland function, which is called the Wright function, is defined as  
\[E_{t}^{s_{1}}(\theta)=\sum_{\varsigma=0}^{\infty}\frac{\theta\varsigma}{ \varsigma!\Gamma(s_{1}\varsigma+t)},\ \ s_{1}\in(-1,\infty),\ \ t,\theta\in C. \tag{7}\]  
The generalized Wright function, which is called the Fox-Wright function, is defined as  
\[E_{\left[(t_{1},\tau_{1}),(s_{2},s_{2}),\ldots,(s_{\varsigma},\tau_{\xi}) \right]}^{\left[(s_{1},\tau_{1}),(s_{2},\tau_{2}),\ldots,(t_{\xi},\tau_{\xi}) \right]}(\theta)=\sum_{\varsigma=0}^{\infty}\frac{\prod_{j=1}^{\xi}\Gamma(S_{j }\varsigma+s_{j})}{\prod_{j=1}^{\xi}\Gamma(T_{j}\varsigma+t_{j})}\frac{\theta \varsigma}{\varsigma!}. \tag{8}\]  
Definition 7: For \(0\leq\gamma_{1}\leq\gamma_{2}\), \(1\leq\gamma_{3}\leq\gamma_{4}\), \(\{x_{\ell},y_{\ell}\}\in\mathbb{C}\), \(\{u_{\ell},v_{\ell}\}\in\mathbb{R}^{+}\), we define the following functions  
* \(\psi_{1}(f)=\prod_{\ell=1}^{\gamma_{1}}\Gamma(y_{\ell}-v_{\ell}f)\),
* \(\psi_{2}(f)=\prod_{\ell=1}^{\gamma_{3}}\Gamma(1-x_{\ell}+u_{\ell}f)\),
* \(\psi_{3}(f)=\prod_{\ell=\gamma_{3}+1}^{\gamma_{3}}\Gamma(1-y_{\ell}+v_{\ell}f)\),
* \(\psi_{4}(f)=\prod_{\ell=\gamma_{1}+1}^{\gamma_{2}}\Gamma(x_{\ell}-u_{\ell}f)\). In the above functions, \(\gamma_{1}=0\) if and only if \(\psi_{2}(f)=1\), \(\gamma_{3}=\gamma_{4}\) if and only if \(\psi_{3}(z)=1\) and \(\gamma_{1}=\gamma_{2}\) if and only if \(\gamma_{4}(f)=1\). According to the above functions, we consider \(\mathcal{H}_{\gamma_{2}\gamma_{4}}^{\gamma_{3}\gamma_{1}}(f)=\frac{\psi_{1}(f )\psi_{2}(f)}{\psi_{3}(f)\psi_{4}(f)}\). The Mellin-Barnes integral (M-BI) representation of the H-Fox function (I-FF) is  
\[H_{\gamma_{2}\gamma_{4}}^{\gamma_{3}\gamma_{1}}(u)=\frac{1}{2\pi i}\int_{ \mathcal{A}}\mathcal{H}_{\gamma_{2}\gamma_{4}}^{\gamma_{2}\gamma_{1}}(f)u^{f }df, \tag{9}\]  
where \(u^{f}=\exp\{f(\log|u|+i\arg u)\}\) and \(\mathcal{A}\in\mathbb{C}\) is a path. Furthermore, the symbol \(H_{\gamma_{2}\gamma_{4}}^{\gamma_{3}\gamma_{1}}(u)=H_{\gamma_{2}\gamma_{4}}^{ \gamma_{3}\gamma_{1}}\bigg{[}u\bigg{|} \begin{matrix}(x_{\ell},\epsilon_{\ell}) _{\ell=1,\cdots,\gamma_{2}}\\ (y_{\ell},\rho_{\ell})_{\ell=1,\cdots,\gamma_{2}}\end{matrix} \bigg{]}\) is considered for this integral. Now we introduce the aggregation function because, in this paper, we use this function as a control function. Definition 8: For a natural and fixed number k and \(\mathfrak{J}\in\mathbb{R}\), an aggregation function is a function \(\mathfrak{A}^{k}:\mathfrak{J}^{k}\rightarrow\mathfrak{J}\), which is nondecreasing, that is, for all \(j\in[1,\ldots,k]\)  
\[p_{j}\leq q_{j}\Longrightarrow\mathfrak{A}^{k}(p_{1},\ldots,p_{k})\leq \mathfrak{A}^{k}(q_{1},\ldots,q_{k}),\]  
hold for the desired k-tuples \((p_{1},\ldots,p_{k})\in\mathfrak{J}^{k},(q_{1},\ldots,q_{k})\in\mathfrak{J}^{k}\). The natural k represents the arity of the aggregation function when no confusion arises, and the aggregation function can be given as \(\mathfrak{A}\). Now we consider some examples of aggregation functions. The arithmetic, the geometric, the projection, the order statistic, the minimum and maximum, the median are aggregation functions. **Example 1**.