Throughout this work, to get our main results for the given ISRNNMD (1), we suggest the following standard postulations:
- \((P_{1})\):
For every \(j\in \mathbb{D}\) the functions \(f_{j}(\cdot)\), \(g _{j}(\cdot)\) and \(h_{j}(\cdot)\) are Lipschitz continuous with Lipschitz constants \(L_{j}(\cdot)\), \(M_{j}(\cdot)\) and \(N_{j}(\cdot)\) individually.
- \((P_{2})\):
For any positive constant ν such that \(V(t_{k},z+P_{ik}(z))\leq \nu _{i}V(t_{k}^{-},z)\) for \(t=t_{k}\).
- \((P_{3})\):
There exist non-negative constants \(\mu _{i}\), \(\phi _{i}\), \(\psi _{i} (i\in \mathbb{D})\), such that
$$ \operatorname{tr} \bigl(\sigma _{ij}^{T}(t,u_{j},v_{j},w_{j}) \sigma _{ij}(t,u_{j},v_{j},w _{j}) \bigr) \leq \mu _{i} \vert u_{j} \vert ^{2}+\phi _{i} \vert v_{j} \vert ^{2}+\psi _{i} \vert w_{j} \vert ^{2}. $$
- \((P_{4})\):
There exist a delay kernel function \(\kappa _{ij}\) and a non-negative constant \(\chi _{ij}\) and \(\mathfrak{K}_{ij}\) such that
$$ \bigl\vert \kappa _{ij}(t) \bigr\vert \leq \chi _{ij},\quad t\in [0,\infty )\ (i,j\in \mathbb{D}). $$
- \((P_{5})\):
-
$$ \int _{0}^{\infty }\kappa _{ij}(t)\,dt=1 \quad \mbox{and}\quad \int _{0}^{\infty }e^{rt}\kappa _{ij}(t)\,dt=\mathfrak{K}_{ij}< \infty . $$
- \((P_{6})\):
\(f_{j}(0)=g_{j}(0)=h_{j}(0)=0\) and \(\sigma _{ij}(0,0,0)=0\).
- \((P_{7})\):
For every \(k\in \mathbb{L}\) there exist scalars \(\upsilon _{i}\) such that
$$ \bigl\vert P_{ik}\bigl(z_{i}\bigl(t_{k}^{-} \bigr)\bigr) \bigr\vert \leq \upsilon _{i} \bigl\vert z_{i}\bigl(t_{k}^{-}\bigr) \bigr\vert . $$
Theorem 5.1
Let
\(\zeta =\inf_{k\in \mathbb{N}}\{t_{k}-t_{k-1}\}\)be finite and let
\((\mathcal{G},\mathcal{A})\)be strongly connected, there exist constantsσ, λ, ν, η, \(1<\nu <e^{(\sigma -\lambda \nu )\zeta }\)and suppose that the system (1) allows the vertex-Lyapunov function
\(v_{i}(t,z_{i}(t))\)and the assumption
\((P_{2})\)holds, then the trivial solution of the system (1) is exponentially stable in thepth moment.
Proof
There exists a positive constant \(\delta (\varepsilon )>0\), for any \(\varepsilon >0\) such that \(b_{i}\delta <\nu _{i}a_{i}\varepsilon \). We assign \(z(t)=z(t,t_{0},\xi )\) to be the solution of (1) by means of \((t_{0},\xi )\), for some \(t_{0}\geq 0\) and \(z(t_{0})=z_{t_{0}}= \xi \) which is in \(PC_{\mathcal{F}_{0}(\delta )}^{b}\). We will show that
$$ \mathbb{E} \bigl\vert z_{i}(t) \bigr\vert ^{p} < \varepsilon , \quad \forall t\geq t_{0}. $$
Let us consider the global Lyapunov function
$$ V\bigl(t,z(t)\bigr)=\sum_{i=1}^{n} c_{i} v_{i}\bigl(t,z_{i}(t)\bigr). $$
Here \(c_{i}\) indicates the cofactor element of \(L_{p}\) of the digraph \((\mathcal{G},\mathcal{A})\), since the digraph is strongly connected, by Lemma 4.4 we have \(c_{i}>0\) for any \(i\in \mathbb{D}\). Choose an arbitrary constant θ and we set
$$ W\bigl(t,z(t)\bigr)=e^{\theta t} V\bigl(t,z(t)\bigr). $$
When \(t\neq t_{k}\), we use Itô’s formula,
$$\begin{aligned} dW\bigl(t,z(t)\bigr) =&\mathcal{L}W\bigl(t,z(t)\bigr)\,dt \\ &{}+\frac{\partial W(t,z(t))}{\partial z(t)} \sigma \biggl(t,z(t),z\bigl(t-\tau (t)\bigr), \int _{-\infty }^{t}\kappa (t-s)h\bigl(z(s)\bigr)\,ds \biggr)\,d\omega (t), \end{aligned}$$
where \(t\in [t_{k-1},t_{k})\), \(k\in \mathbb{N}\). By integrating the above expression from \(t_{k}\) to \(t+\Delta t\) for small enough \(\Delta t>0\) and taking the mathematical expectation, we obtain, for \(t\geq 0\), \(t+ \Delta t\in [t_{k},t_{k+1})\),
$$ \mathbb{E}W(t+\Delta t)= \mathbb{E}W(t_{k})+\mathbb{E} \int _{t _{k}}^{t}\mathcal{L}W\bigl(t,z(t)\bigr)\,dt, $$
which implies that
$$ \mathbb{E}D^{+}W\bigl(t,z(t)\bigr)=\mathbb{E}\mathcal{L}W \bigl(t,z(t)\bigr), \quad t\in [t_{k-1},t_{k}), k\in \mathbb{D}. $$
Here, \(D^{+}W(t,z(t))\) denotes the upper-right Dini derivative of \(W(t,z(t))\) defined by
$$ D^{+}W\bigl(t,z(t)\bigr)= \limsup_{h\rightarrow 0^{+}} \frac{\mathbb{E}W(t+h,z(t+h))-\mathbb{E}W(t,z(t))}{h} . $$
Also
$$ \mathbb{E}\mathcal{L}W\bigl(t,z(t)\bigr)=\theta e^{\theta t}\mathbb{E}V \bigl(t,z(t)\bigr)+e ^{\theta t}\mathbb{E}\mathcal{L}V\bigl(t,z(t)\bigr) $$
and we can choose \(\delta >0\) for any given \(\varepsilon > 0\), such that \(b_{i}\delta <\nu _{i}a_{i}\varepsilon \). Currently, we assign \(\mathbb{E}\|\xi \|^{p}<\delta \). By (3) we obtain
$$ \mathbb{E}W\bigl(t,z(t)\bigr)\leq b_{i}\mathbb{E} \Vert \xi \Vert ^{p}< b_{i}\delta < \nu _{i}a_{i} \varepsilon ,\quad t\in [t_{0}-\tau ,t_{0}]. $$
We mainly prove that
$$ \mathbb{E}W\bigl(t,z(t)\bigr)\leq a_{i}\varepsilon ,\quad t\in (t_{0},t_{1}). $$
(8)
Suppose, on the contrary, that we have \(s\in (t_{0},t_{1})\) such that
$$ \mathbb{E}W\bigl(t,z(t)\bigr)> a_{i}\varepsilon . $$
Set
$$\begin{aligned}& s_{1}= \inf \bigl\{ t\in (t_{0},t_{1}): \mathbb{E}W\bigl(t,z(t)\bigr)>a_{i}\varepsilon \bigr\} ,\quad \mbox{then } s_{1}\in (t_{0},t_{1}) \\& s_{2}= \sup \biggl\{ t\in [t_{0},s_{1}): \mathbb{E}W\bigl(t,z(t)\bigr)< \frac{1}{\nu _{i}} a_{i}\varepsilon \biggr\} . \end{aligned}$$
In such a way it is obvious for \(t\in [s_{2},s_{1}]\), for \(z_{i}(t- \tau _{i}(t))= (z_{1}(t-\tau _{1}(t)),z_{2}(t-\tau _{2}(t)),\ldots,z_{n}(t- \tau _{n}(t)) )\), where \(\tau =\max_{1\leq j\leq n}\{\tau _{j}\}\), for \(t\in [s_{2},s_{1}]\) that
$$ \mathbb{E}W\bigl(t,z\bigl(t-\tau (t)\bigr)\bigr)\leq a_{i} \varepsilon = \frac{1}{\nu _{i}}\nu _{i} a_{i} \varepsilon \leq \nu _{i} \mathbb{E}W\bigl(t,z(t)\bigr), \quad \forall - \tau \leq \theta \leq 0. $$
Hence,
$$ \mathbb{E}V\bigl(t,z\bigl(t-\tau (t)\bigr)\bigr)\leq \nu _{i} e^{\theta \tau }\mathbb{E}V\bigl(t,z(t)\bigr), \quad \forall t\in [s_{2},s_{1}]. $$
Therefore from (6), \(t\in [s_{2},s_{1}]\), we induce that
$$\begin{aligned} D^{+}\mathbb{E}W\bigl(t,z(t)\bigr)\leq{} & \mathcal{L}\mathbb{E}W \bigl(t,z(t)\bigr) \\ \leq{} &\sum_{i=1}^{n}c_{i}e^{\theta t} \mathbb{E} \Biggl[-\sigma _{i} v_{i} \bigl(t,z_{i}(t)\bigr)+\lambda _{i} v_{i} \bigl(t,z_{i}\bigl(t-\tau (t)\bigr)\bigr) \\ &{}+\sum_{j=1}^{n}a_{ij}F_{ij} \bigl(t,z_{i}(t),z_{j}(t)\bigr) \\ &{}+\eta _{i} \int _{-\infty }^{t}\kappa _{ij}(t-s)v_{i} \bigl(s,z_{i}(s)\bigr)\,ds \Biggr]+ \sum_{i=1}^{n} \theta c_{i} e^{\theta t}\mathbb{E}v_{i}\bigl(t,z _{i}(t)\bigr) \\ \leq{} & \bigl[-\sigma _{i} +\lambda _{i} \nu _{i} e^{\theta \tau }+\eta _{i}+ \theta \bigr] \mathbb{E}W\bigl(t,z(t)\bigr). \end{aligned}$$
Integrating the above inequality, for \(t\in [s_{1},s_{2}]\), we get
$$\begin{aligned}& \int _{s_{2}}^{s_{1}} \frac{D^{+}\mathbb{E}W(t,z(t))}{\mathbb{E}W(t,z(t))}\leq \mathbb{E} \int _{s_{2}}^{s_{1}}\bigl[-\sigma _{i} + \lambda _{i} \nu _{i} e^{ \theta \tau }+\eta _{i}+\theta \bigr]\,dt, \\& \begin{aligned} \mathbb{E}W\bigl(s_{1},z(s_{1})\bigr)&\leq \mathbb{E}W \bigl(s_{2},z(s_{2})\bigr)e^{(- \sigma _{i} +\theta +\eta _{i}+\lambda _{i} \nu _{i} e^{\theta \tau }) \zeta } \\ &\leq \nu _{i} a_{i}\varepsilon e^{(-\sigma _{i} +\theta +\eta _{i}+ \lambda _{i} \nu _{i} e^{\theta \tau })\zeta } \\ &< a_{i}\varepsilon . \end{aligned} \end{aligned}$$
Hence,
$$ \mathbb{E}W\bigl(s_{1},z(s_{1})\bigr)< a_{i} \varepsilon $$
which is a contradiction with
$$ \mathbb{E}W\bigl(t,z(t)\bigr)\leq a_{i}\varepsilon . $$
Next, we consider, for \(m=1,2,3,\ldots,k\) and \(k\in \mathbb{N}\),
$$ \mathbb{E}W\bigl(t,z(t)\bigr)\leq a_{i}\varepsilon , \quad \forall t\in [t_{m-1},t_{m}) $$
(9)
We want to prove that
$$ \mathbb{E}W\bigl(t,z(t)\bigr)\leq a_{i}\varepsilon ,\quad \forall t\in [t_{k},t_{k+1}). $$
(10)
On the contrary, there exist some \(t\in [t_{k},t_{k+1})\) such that
$$ \mathbb{E}W\bigl(t,z(t)\bigr)>a_{i}\varepsilon . $$
By using assumption \((P_{2})\) and (9), we obtain
$$\begin{aligned} \mathbb{E}W\bigl(t_{k},z(t_{k})\bigr) = &\mathbb{E} \bigl(e^{\theta t_{k}}V\bigl(t_{k},z(t _{k})\bigr) \bigr) \\ \leq &\mathbb{E}\bigl(e^{\theta t_{k}}V\bigl(t_{k},z+P_{ik}(z) \bigr)\bigr) \\ \leq &\nu _{i} \mathbb{E}W\bigl(t_{k}^{-},z \bigl(t_{k}^{-}\bigr)\bigr) \\ \leq &a_{i}\nu _{i}\varepsilon . \end{aligned}$$
Now, set
$$ s_{1}=\inf \bigl\{ t\in (t_{k},t_{k+1}): \mathbb{E}W\bigl(t,z(t)\bigr)>a_{i}\varepsilon \bigr\} ,\quad \mbox{then } s_{1}\in (t_{k},t_{k+1}). $$
Let
$$ s_{2}=\sup \bigl\{ t\in [t_{k},s_{1}): \mathbb{E}W\bigl(t,z(t)\bigr)< \nu _{i}a_{i} \varepsilon \bigr\} . $$
For \(t\in (s_{2},s_{1})\), we get
$$ \mathbb{E}W\bigl(t,z\bigl(t-\tau (t)\bigr)\bigr)\leq a_{i} \varepsilon \leq \frac{1}{\nu _{i}}\nu _{i}a_{i} \varepsilon \leq \frac{1}{\nu _{i}}EW\bigl(t,z(t)\bigr),\quad - \tau \leq \theta \leq 0. $$
Hence,
$$ \mathbb{E}W\bigl(t,z\bigl(t-\tau (t)\bigr)\bigr)\leq \frac{e^{a_{i}\tau }}{\nu _{i}} \mathbb{E}V\bigl(t,z(t)\bigr). $$
Similarly, we can derive \(\mathbb{E}W(s_{2},z(s_{2}))< a_{i}\varepsilon \) which is a contradiction to \(\mathbb{E}W(t,z(t))\leq a_{i}\varepsilon \) for \(t\epsilon [t_{k},t_{k+1})\). Hence the proof of the theorem is completed.
By mathematical induction \(\mathbb{E}W(t,z(t))\leq a_{i} \varepsilon \) for \(t\geq t_{0}\). Hence,
$$ \mathbb{E} \bigl\vert z(t) \bigr\vert ^{p}\leq e^{-\theta t} \varepsilon ,\quad t\geq t_{0}. $$
□
Remark 5.2
In the study of the stability of ISNNMD, to construct a Lyapunov function is a formidable task. However, Theorem 5.1 offers a technique to construct systematically a Lyapunov function for (1) by using the Lyapunov function \(v_{i}(t,z_{i}(t))=v_{i1}(t,z_{i}(t))+v_{i2}(t,z_{i}(t))+v_{i3}(t,z_{i}(t))\) of each vertex system, which avoids the difficulty of finding a Lyapunov function directly for ISNNMD. In the final section, an example is presented to show the validity of the technique.
Remark 5.3
It should be noticed that Theorem 5.1 holds if \(c_{i}>0\), that is, the graph \((\mathcal{G},\mathcal{A})\) is strongly connected, which means that exponential stability of RNNs has a close relationship with the topology property of the network. Therefore, we can get some better results in the following.
Theorem 5.4
Assume that the assumptions
\((P_{1})\)–\((P_{7})\)hold; then the considered system (1) is exponentially stable.
Proof
Let us define the subsequent Lyapunov–Krasovskii functional for (1) as follows:
$$\begin{aligned} v_{i}\bigl(t,z_{i}(t)\bigr)={} &e^{rt} \bigl\vert z_{i}(t) \bigr\vert ^{p}+\sum _{j=1}^{n}e^{r \tau } \int _{t-\tau _{i}(t)}^{t} e^{rs} \bigl\vert z_{i}(s) \bigr\vert ^{p-2}g_{j}^{2} \bigl(z_{j}(s)\bigr)\,ds \\ &{}+\sum_{j=1}^{n}\sum _{l=1}^{n}m_{l} \int _{0}^{ \infty }\kappa _{lj}(\theta ) \\ &{}\times \int _{t-\theta }^{t} e^{r(s+\theta )} \bigl\vert z_{i}(s) \bigr\vert ^{p-2}h _{j}^{2} \bigl(z_{j}(s)\bigr)\,ds\,d\theta . \end{aligned}$$
(11)
Now, we can calculate the Lie derivative of \(v_{i}(t,z_{1}(t))\) for \(t\neq t_{k}\). By using Itô’s formula along the trajectories of the model (1), we obtain
$$ \mathcal{L}v_{i}\bigl(t,z_{i}(t)\bigr)= \mathcal{L}v_{i1}\bigl(t,z_{i}(t)\bigr)+ \mathcal{L}v_{i2}\bigl(t,z_{i}(t)\bigr)+ \mathcal{L}v_{i3}\bigl(t,z_{i}(t)\bigr), $$
(12)
where
$$\begin{aligned}& \mathcal{L}v_{i1}\bigl(t,z_{i}(t)\bigr) \\& \quad \leq r e^{rt} \bigl\vert z_{i}(t) \bigr\vert ^{p}+pe^{rt} \bigl\vert z_{i}(t) \bigr\vert ^{p-2}z_{i}(t) \Biggl[-d _{i}z_{i}(t)+ \sum_{j=1}^{n}\alpha _{ij}f_{j} \bigl(z_{j}(t)\bigr)+\sum_{j=1}^{n} \beta _{ij} \\& \qquad {} \times g_{j}\bigl(z_{j}\bigl(t-\tau _{i}(t)\bigr)\bigr)+\sum_{j=1}^{n} \gamma _{ij} \int _{-\infty }^{t}\kappa _{ij}(t-s)h_{j} \bigl(z_{j}(s)\bigr)\,ds \Biggr]+ \frac{p(p-1)}{2}e^{rt} \\& \qquad {} \times \bigl\vert z_{i}(t) \bigr\vert ^{p-2}\sum _{j=1}^{n}\sigma _{ij}^{2} \biggl(t,z_{i}(t),z_{i}\bigl(t-\tau _{i}(t)\bigr), \int _{-\infty }^{t}\kappa _{ij}(t-s)h _{j}\bigl(z_{j}(s)\bigr)\,ds \biggr) \\& \quad \leq re^{rt} \bigl\vert z_{i}(t) \bigr\vert ^{p}-d_{i}pe^{rt} \bigl\vert z_{i}(t) \bigr\vert ^{p-2}z^{2}_{i}(t)+pe ^{rt}\sum_{j=1}^{n}\alpha _{ij} \bigl\vert z_{i}(t) \bigr\vert ^{p-2}z_{i}(t)f_{j}\bigl(z _{j}(t)\bigr) \\& \qquad {} +pe^{rt}\sum_{j=1}^{n} \beta _{ij} \bigl\vert z_{i}(t) \bigr\vert ^{p-2}z_{i}(t)g _{j}\bigl(z_{j} \bigl(t-\tau _{i}(t)\bigr)\bigr) \\& \qquad {} +\sum_{j=1}^{n}\gamma _{ij}pe^{rt} \bigl\vert z_{i}(t) \bigr\vert ^{p-2}z_{i}(t) \int _{-\infty }^{t}\kappa _{ij}(t-s)h_{j} \bigl(z_{j}(s)\bigr)\,ds \\& \qquad {} +\frac{p(p-1)}{2}e^{rt} \bigl\vert z_{i}(t) \bigr\vert ^{p-2} \\& \qquad {} \times \sigma _{ij}^{2} \biggl(t,z_{i}(t),z_{i} \bigl(t-\tau _{i}(t)\bigr), \int _{-\infty }^{t}\kappa _{ij}(t-s)h_{j} \bigl(z_{j}(s)\bigr)\,ds \biggr). \end{aligned}$$
(13)
By using Lemma 4.3, we obtain
$$\begin{aligned}& pe^{rt}\sum_{j=1}^{n} \alpha _{ij} \bigl\vert z_{i}(t) \bigr\vert ^{p-2}z_{i}(t)f_{j}\bigl(z _{j}(t)\bigr) \\& \quad \leq \sum_{j=1}^{n} \vert \alpha _{ij} \vert pe^{rt}L_{j} \bigl[ \bigl\vert z_{i}(t) \bigr\vert ^{p-2}z _{i}(t) \bigl\vert z_{j}(t) \bigr\vert \bigr] \\& \quad \leq \sum_{j=1}^{n} \vert \alpha _{ij} \vert pe^{rt}L_{j} \biggl[ \frac{p-1}{p} \bigl\vert z _{i}(t) \bigr\vert ^{p}+\frac{1}{p} \bigl\vert z_{j}(t) \bigr\vert ^{p} \biggr] \\& \quad = \sum_{j=1}^{n} \vert \alpha _{ij} \vert e^{rt}L_{j} \bigl[(p-1) \bigl\vert z_{i}(t) \bigr\vert ^{p}+ \bigl\vert z _{j}(t) \bigr\vert ^{p} \bigr]. \end{aligned}$$
(14)
Similarly,
$$\begin{aligned}& pe^{rt}\sum_{j=1}^{n}\beta _{ij} \bigl\vert z_{i}(t) \bigr\vert ^{p-2}z_{i}(t)g_{j}\bigl(z _{j} \bigl(t-\tau _{i}(t)\bigr)\bigr) \\& \quad \leq pe^{rt}\sum_{j=1}^{n} \vert \beta _{ij} \vert \bigl\vert z_{i}(t) \bigr\vert ^{p-1}M_{j} \bigl\vert z _{j} \bigl(t-\tau _{i}(t)\bigr) \bigr\vert \\& \quad \leq pe^{rt}\sum_{j=1}^{n} \vert \beta _{ij} \vert M_{j} \biggl[ \frac{p-1}{p} \bigl\vert z _{i}(t) \bigr\vert ^{p}+\frac{1}{p} \bigl\vert z_{j}\bigl(t- \tau _{i}(t)\bigr) \bigr\vert ^{p} \biggr] \\& \quad \leq \sum_{j=1}^{n} \vert \beta _{ij} \vert e^{rt}M_{j} \bigl[(p-1) \bigl\vert z_{i}(t) \bigr\vert ^{p}+ \bigl\vert z _{j}\bigl(t-\tau _{i}(t)\bigr) \bigr\vert ^{p} \bigr] \end{aligned}$$
(15)
and
$$\begin{aligned}& \sum_{j=1}^{n}\gamma _{ij}pe^{rt} \bigl\vert z_{i}(t) \bigr\vert ^{p-2}z_{i}(t) \int _{-\infty }^{t}\kappa _{ij}(t-s)h_{j} \bigl(z_{j}(s)\bigr)\,ds \\& \quad \leq \sum_{j=1}^{n} \vert \gamma _{ij} \vert pe^{rt} \bigl\vert z_{i}(t) \bigr\vert ^{p-1} \int _{-\infty }^{t}\kappa _{ij}(t-s)N_{j} \bigl\vert z_{j}(s) \bigr\vert \,ds \\& \quad \leq \sum_{j=1}^{n} \vert \gamma _{ij} \vert pe^{rt}N_{j} \int _{-\infty } ^{t}\kappa _{ij}(t-s) \bigl\vert z_{i}(t) \bigr\vert ^{p-1} \bigl\vert z_{j}(s) \bigr\vert \,ds \\& \quad \leq \sum_{j=1}^{n}(p-1) \vert \gamma _{ij} \vert e^{rt}N_{j} \bigl\vert z_{i}(t) \bigr\vert ^{p} \int _{-\infty }^{t}\kappa _{ij}(t-s)\,ds \\& \qquad {} +\sum_{j=1}^{n} \vert \gamma _{ij} \vert e^{rt}N_{j} \int _{-\infty } ^{t}\kappa _{ij}(t-s) \bigl\vert z_{j}(s) \bigr\vert ^{p}\,ds \\& \quad \leq \sum_{j=1}^{n} \vert \gamma _{ij} \vert e^{rt}N_{j} [(p-1) \bigl\vert z_{i}(t) \bigr\vert ^{p}+ \sum _{j=1}^{n} \vert \gamma _{ij} \vert \\& \qquad {} \times e^{rt}N_{j} \int _{-\infty }^{t}\kappa _{ij}(t-s) \bigl\vert z_{j}(s) \bigr\vert ^{p}\,ds. \end{aligned}$$
(16)
We use the assumption and the well-known Cauchy–Schwartz inequality
$$\begin{aligned}& \frac{p(p-1)}{2}e^{rt} \bigl\vert z_{i}(t) \bigr\vert ^{p-2}\sigma _{ij}^{2} \biggl(t,z_{i}(t),z _{i}\bigl(t-\tau _{i}(t)\bigr), \int _{-\infty }^{t}\kappa _{ij}(t-s)h_{j} \bigl(z_{j}(s)\bigr)\,ds \biggr) \\& \quad \leq \frac{p(p-1)}{2}e^{rt} \bigl\vert z_{i}(t) \bigr\vert ^{p-2} \biggl[\mu _{i}\bigl(z _{i}(t)\bigr)^{2}+\phi _{i} \biggl[ \frac{p-2}{p} \bigl\vert z_{i}(t) \bigr\vert ^{p}+\frac{2}{p} \bigl\vert z _{i}\bigl(t- \tau _{i}(t)\bigr) \bigr\vert ^{p} \biggr] \\& \qquad {} +\psi _{i}\biggl( \int _{-\infty }^{t}\kappa _{ij}(t-s)h_{j} \bigl(z_{j}(s)\bigr)\,ds\biggr)^{2} \biggr]. \end{aligned}$$
(17)
Substituting (13)–(17) in (12) we get
$$\begin{aligned}& \mathcal{L}v_{1i}\bigl(t,z_{i}(t)\bigr) \\& \quad \leq re^{rt} \bigl\vert z_{i}(t) \bigr\vert ^{p}-d_{i}pe^{rt} \bigl\vert z_{i}(t) \bigr\vert ^{p}+\sum _{j=1}^{n} \vert \alpha _{ij} \vert e^{rt}L_{j}(p-1) \bigl\vert z_{i}(t) \bigr\vert ^{p}+\sum _{j=1} ^{n} \vert \alpha _{ij} \vert e^{rt}L_{j} \bigl\vert z_{j}(t) \bigr\vert ^{p} \\& \qquad {} +\sum_{j=1}^{n} \vert \beta _{ij} \vert (p-1)e^{rt}M_{j} \bigl\vert z_{i}(t) \bigr\vert ^{p}+ \sum _{j=1}^{n} \vert \beta _{ij} \vert e^{rt}M_{j} \bigl\vert z_{j}\bigl(t-\tau _{i}(t)\bigr) \bigr\vert ^{p}+ \sum _{j=1}^{n}(p-1) \vert \gamma _{ij} \vert \\& \qquad {} \times e^{rt}N_{j} \bigl\vert z_{i}(t) \bigr\vert ^{p}+\sum_{j=1}^{n} \vert \gamma _{ij} \vert e^{rt}N_{j} \int _{-\infty }^{t}\kappa _{ij}(t-s) \bigl\vert z_{j}(s) \bigr\vert ^{p}\,ds+ \frac{p(p-1)}{2}e^{rt} \bigl\vert z_{i}(t) \bigr\vert ^{p-2} \\& \qquad {} \times \biggl[\mu _{i}\bigl(z_{i}(t) \bigr)^{2}+\phi _{i} \biggl[\frac{p-2}{p}z _{i}(t) \vert ^{p}+\frac{2}{p} \bigl\vert z_{i}\bigl(t-\tau _{i}(t)\bigr) \bigr\vert ^{p} \biggr]+\psi _{i}\biggl( \int _{-\infty }^{t}\kappa _{ij}(t-s) \\& \qquad {} \times h_{j}\bigl(z_{j}(s)\bigr)\,ds \biggr)^{2} \biggr]. \end{aligned}$$
(18)
Next,
$$\begin{aligned}& \mathcal{L}v_{2i}\bigl(t,z_{i}(t)\bigr) \\& \quad \leq \sum _{j=1}^{n}e^{r\tau } \bigl[ e^{rt} \bigl\vert z_{i}(t) \bigr\vert ^{p-2}g_{j}^{2}\bigl(z_{j}(t) \bigr)-e^{r(t-\tau _{i}(t))} \bigl\vert z _{i}\bigl(t-\tau _{ij}(t)\bigr) \bigr\vert ^{p-2}g_{j}^{2} \bigl(z_{j}\bigl(t-\tau _{i}(t)\bigr)\bigr) \bigr] \\& \quad \leq \sum_{j=1}^{n}M_{j}^{2}e^{r(t+\tau )} \biggl[\frac{p-2}{p} \bigl\vert z _{i}(t) \bigr\vert ^{p}+\frac{2}{p} \bigl\vert z_{j}(t) \bigr\vert ^{p} \biggr]-\sum_{j=1}^{n}M _{j}^{2}e^{rt} \biggl[\frac{p-2}{p} \bigl\vert z_{i}\bigl(t-\tau _{i}(t)\bigr) \bigr\vert ^{p} \\& \qquad {} +\frac{2}{p} \bigl\vert z_{j}\bigl(t-\tau _{i}(t)\bigr) \bigr\vert ^{p} \vert \biggr]. \end{aligned}$$
(19)
And
$$\begin{aligned}& \mathcal{L}v_{3i}\bigl(t,z_{i}(t)\bigr) \\& \quad = \sum _{j=1}^{n}\sum _{l=1} ^{n}m_{l} \int _{0}^{\infty }\kappa _{lj}(\theta )e^{r(t+\theta )} \bigl\vert z_{i}(t) \bigr\vert ^{p-2}h _{j}^{2}\bigl(z_{j}(t) \bigr)\,d\theta -\sum_{j=1}^{n}\sum _{l=1}^{n}m _{l} \int _{0}^{\infty }\kappa _{lj}(\theta )e^{rt} \\& \qquad {} \times \bigl\vert z_{i}(t-\theta ) \bigr\vert ^{p-2}h_{j}^{2}\bigl(z_{j}(t- \theta )\bigr)d \theta \\& \quad \leq \sum_{j=1}^{n}\sum _{l=1}^{n}m_{l}N_{j}^{2} \biggl[\frac{p-2}{p} \bigl\vert z _{i}(t) \bigr\vert ^{p}+\frac{2}{p} \bigl\vert z_{j}(t) \bigr\vert ^{p} \biggr] \int _{0}^{\infty } \kappa _{lj}(\theta )e^{r\theta }\,d\theta -\sum_{j=1}^{n} \sum_{l=1}^{n}m_{l}e^{rt} \\& \qquad {} \times \biggl( \int ^{t}_{-\infty }\kappa _{lj}(t-s) \bigl\vert z_{i}(s) \bigr\vert ^{p-2}h _{j}^{2} \bigl(z_{j}(s)\bigr)\,ds \biggr)^{2}. \end{aligned}$$
(20)
Substituting (18)–(20) in (11) we get
$$\begin{aligned}& \mathcal{L}v_{i}\bigl(t,z_{i}(t)\bigr) \\& \quad \leq re^{rt} \bigl\vert z_{i}(t) \bigr\vert ^{p}-d_{i}pe^{rt} \bigl\vert z _{i}(t) \bigr\vert ^{p}+\sum _{j=1}^{n} \vert \alpha _{ij} \vert e^{rt}L_{j}(p-1) \bigl\vert z_{i}(t) \bigr\vert ^{p}+ \sum _{j=1}^{n} \vert \alpha _{ij} \vert e^{rt}L_{j} \bigl\vert z_{j}(t) \bigr\vert ^{p} \\& \qquad {}+\sum_{j=1}^{n} \vert \beta _{ij} \vert (p-1)e^{rt}M_{j} \bigl\vert z_{i}(t) \bigr\vert ^{p}+ \sum _{j=1}^{n} \vert \beta _{ij} \vert e^{rt}M_{j} \bigl\vert z_{j}\bigl(t-\tau _{ij}(t)\bigr) \bigr\vert ^{p}+ \sum _{j=1}^{n}(p-1) \vert \gamma _{ij} \vert \\& \qquad {} \times e^{rt}N_{j} \bigl\vert z_{i}(t) \bigr\vert ^{p}+\sum_{j=1}^{n} \vert \gamma _{ij} \vert e^{rt}N_{j} \int _{-\infty }^{t}\kappa _{ij}(t-s) \bigl\vert z_{j}(s) \bigr\vert ^{p}\,ds+ \frac{p(p-1)}{2}e^{rt} \bigl\vert z_{i}(t) \bigr\vert ^{p-2} \\& \qquad {} \times \biggl[\mu _{i}\bigl(z_{i}(t) \bigr)^{2}+\phi _{i} \biggl[\frac{p-2}{p}z _{i}(t) \vert ^{p}+\frac{2}{p} \bigl\vert z_{i}\bigl(t-\tau _{j}(t)\bigr) \bigr\vert ^{p} \biggr]+\psi _{i}\biggl( \int _{-\infty }^{t}\kappa _{ij}(t-s) \\& \qquad {} \times h_{i}\bigl(z_{j}(s)\bigr)\,ds \biggr)^{2} \biggr]+\sum_{j=1}^{n}M _{j}^{2}e^{r(t+\tau )} \biggl[\frac{p-2}{p} \bigl\vert z_{i}(t) \bigr\vert ^{p}+ \frac{2}{p} \bigl\vert z _{j}(t) \bigr\vert ^{p} \biggr]-\sum_{j=1}^{n}M_{j}^{2}e^{rt} \\& \qquad {} \times \biggl[\frac{p-2}{p} \bigl\vert z_{i}\bigl(t-\tau _{ij}(t)\bigr) \bigr\vert ^{p}+ \frac{2}{p} \bigl\vert z_{j}\bigl(t-\tau _{ij}(t)\bigr) \bigr\vert ^{p} \vert \biggr]+\sum_{j=1}^{n} \sum_{l=1}^{n}m_{l}N_{j}^{2} \biggl[\frac{p-2}{p} \bigl\vert z_{i}(t) \bigr\vert ^{p} \\& \qquad {} +\frac{2}{p} \bigl\vert z_{j}(t) \bigr\vert ^{p} \biggr] \int _{0}^{\infty }\kappa _{lj}( \theta )e^{r\theta }\,d\theta -\sum_{j=1}^{n} \sum_{l=1} ^{n}m_{l}e^{rt} \biggl( \int ^{t}_{-\infty }\kappa _{lj}(t-s) \bigl\vert z_{i}(s) \bigr\vert ^{p-2} \\& \qquad {} \times h_{j}^{2}\bigl(z_{j}(s)\bigr)\,ds \biggr)^{2} \end{aligned}$$
(21)
$$\begin{aligned}& \quad \leq e^{rt} \bigl\vert z_{i}(t) \bigr\vert ^{p} [r+ \vert d_{i} \vert p+\sum _{j=1}^{n} \vert \alpha _{ij} \vert L_{j}(p-1)+\sum_{j=1}^{n} \vert \beta _{ij} \vert (p-1)M_{j}+ \sum _{j=1}^{n}(p-1) \vert \gamma _{ij} \vert N_{j} \\& \qquad {} +\frac{p(p-1)}{2} \vert \mu _{i} \vert + \frac{(p-1)(p-2)}{2}\phi _{i}+ \sum_{j=1}^{n} \frac{p-2}{p}e^{r\tau }M_{j}^{2}+ \frac{p-2}{p} \sum_{j=1}^{n}\sum _{l=1}^{n}m_{l}N_{j}^{2} \mathcal{K} \\& \qquad {} +e^{rt} \bigl\vert z_{j}(t) \bigr\vert ^{p} \Biggl[\sum_{j=1}^{n} \vert \alpha _{ij} \vert L _{j}+e^{r\tau } \sum_{j=1}^{n}\frac{2}{p}M_{j}^{2}+ \sum_{j=1}^{n}\sum _{l=1}^{n}m_{l}N_{j}^{2} \frac{2}{p}\mathcal{k} \Biggr]+ \bigl\vert z_{j}\bigl(t- \tau _{ij}(t)\bigr) \bigr\vert ^{p}e^{rt} \\& \qquad {} \times \Biggl[\sum_{j=1}^{n} \vert \beta _{ij} \vert M_{j}-\frac{2n}{p}\Biggr]+ \Biggl[e ^{rt}\sum_{j=1}^{n} \vert \gamma _{ij} \vert e^{rt}N_{j} \Biggr] \int _{-\infty } ^{t}\kappa _{ij}(t-s) \bigl\vert z_{j}(s) \bigr\vert ^{p}\,ds \\& \quad \leq -T_{i1}v_{i}\bigl(t,z_{i}(t)\bigr)+ \sum_{j=1}^{n}a_{ij}F_{ij} \bigl(z_{i}(t),z _{j}(t),t\bigr)+T_{i2}v_{i}(t,z_{j} \bigl(t-\tau _{i}(t)\bigr)+T_{i3} \int _{-\infty } ^{t}\kappa _{ij}(t-s) \\& \qquad {} \times \bigl\vert z_{j}(s) \bigr\vert ^{p}\,ds, \quad \mbox{for } t\neq t_{k}. \end{aligned}$$
(22)
On the other hand, for \(t=t_{k}\),
$$\begin{aligned}& v_{i}\bigl(t_{k},z_{i} \bigl(t_{k}^{-}+P_{ik}\bigl(z_{i} \bigl(t_{k}^{-}\bigr)\bigr)\bigr)\bigr) \\& \quad = e^{rt_{k}} \bigl\vert z _{i}\bigl(t_{k}^{-} \bigr)+P_{ik}\bigl(z_{i}\bigl(t_{k}^{-} \bigr)\bigr) \bigr\vert ^{p}+\sum_{j=1}^{n}e ^{r\tau } \int _{t_{k}-\tau _{i}(t_{k})}^{t_{k}} e^{rs} \bigl\vert z_{i}(s) \bigr\vert ^{p-2}g _{j}^{2} \bigl(z_{j}(s)\bigr)\,ds \\& \qquad {} +\sum_{j=1}^{n}\sum _{l=1}^{n}m_{l} \int _{0}^{ \infty }\kappa _{lj}(\theta ) \int _{t_{k}-\theta }^{t_{k}} e^{r(s+ \theta )} \bigl\vert z_{i}(s) \bigr\vert ^{p-2} h_{j}^{2} \bigl(z_{j}(s)\bigr)\,ds\,d\theta \\& \quad \leq e^{rt_{k}} \bigl\vert z_{i}\bigl(t_{k}^{-} \bigr) \bigr\vert ^{p}+\sum_{j=1}^{n}e^{r \tau } \int _{t_{k}-\tau _{i}(t_{k})}^{t_{k}} e^{rs} \bigl\vert z_{i}(s) \bigr\vert ^{p-2}g _{j}^{2} \bigl(z_{j}(s)\bigr)\,ds+\sum_{j=1}^{n} \sum_{l=1}^{n}m_{l} \\& \qquad {} \times \int _{0}^{\infty }\kappa _{lj}(\theta ) \int _{t_{k}- \theta }^{t_{k}} e^{r(s+\theta )} \bigl\vert z_{i}(s) \bigr\vert ^{p-2}h_{j}^{2} \bigl(z_{j}(s)\bigr)\,ds\,d\theta + \bigl\vert P_{ik} \bigl(z_{i}\bigl(t_{k}^{-}\bigr)\bigr) \bigr\vert ^{p} \\& \quad \leq V_{i}\bigl(t_{k}^{-},z_{i} \bigl(t_{k}^{-}\bigr)\bigr)+\upsilon _{i} \bigl\vert z_{i}\bigl(t_{k}^{-}\bigr) \bigr\vert ^{p} \\& \quad \leq \varUpsilon _{i}V_{i}\bigl(t_{k}^{-},z_{i} \bigl(t_{k}^{-}\bigr)\bigr). \end{aligned}$$
Now, we define \(v_{i}(t,z_{i}(t))=e^{rt}|z_{i}(t)|^{p}\) and the suitable constants \(T_{i1}\), \(T_{i2}\), \(T_{i3}\) such that the condition (4) is satisfied, then by using Theorem 5.1 we ensure the pth moment’s exponential stability of (1). Here
$$\begin{aligned}& \begin{aligned} T_{i1}= {}& r+ \vert d_{i} \vert p+\sum _{j=1}^{n} \vert \alpha _{ij} \vert L_{j}(p-1)+ \sum_{j=1}^{n} \vert \beta _{ij} \vert (p-1)M_{j}+\sum _{j=1}^{n}(p-1) \vert \gamma _{ij} \vert N_{j} \\ &{} +\frac{p(p-1)}{2} \vert \mu _{i} \vert + \frac{(p-1)(p-2)}{2}\phi _{i}+ \sum_{j=1}^{n} \frac{p-2}{p}e^{r\tau }M_{j}^{2}+ \frac{p-2}{p} \sum_{j=1}^{n}\sum _{l=1}^{n}m_{l} \\ &{} \times N_{j}^{2}\mathcal{K}+\sum _{i=1}^{n} \vert \alpha _{ij} \vert L _{i}+e^{r\tau }\sum_{i=1}^{n} \frac{2}{p}M_{i}^{2}+\sum _{i=1}^{n}\sum_{l=1}^{n}m_{l}N_{i}^{2} \frac{2}{p}\mathcal{k}, \end{aligned} \\& T_{i2}= \sum_{j=1}^{n} \vert \beta _{ij} \vert M_{j}-\frac{2n}{p}, \\& T_{i3}= e^{rt}\sum_{j=1}^{n} \vert \gamma _{ij} \vert e^{rt}N_{j}, \\& \sum_{j=1}^{n}F_{ij} \bigl(z_{i}(t),z_{j}(t),t\bigr) \\& \quad = e^{rt} \Biggl[ \bigl\vert z_{j}(t) \bigr\vert ^{p} \Biggl[ \sum_{j=1}^{n} \vert \alpha _{ij} \vert L_{j}+e^{r\tau }\sum _{j=1} ^{n}\frac{2}{p}M_{j}^{2}+ \sum_{j=1}^{n}\sum _{l=1}^{n}m_{l}N_{j}^{2} \frac{2}{p}\mathcal{k} \Biggr] \Biggr] \\& \qquad {} -e^{rt} \Biggl[ \bigl\vert z_{i}(t) \bigr\vert ^{p} \Biggl[\sum_{i=1}^{n} \vert \alpha _{ij} \vert L_{i}+e^{r\tau }\sum _{i=1}^{n}\frac{2}{p}M_{i}^{2}+ \sum_{i=1}^{n}\sum _{l=1}^{n}m_{l}N_{i}^{2} \frac{2}{p}\mathcal{K} \Biggr] \Biggr]. \end{aligned}$$
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