Appendix
1.1 A Proof of Theorem 1
According to system (3), we have:
$$\begin{aligned} x^*_i&= c_i^{-1}\bigg [\displaystyle \sum _{j=1}^{n}a_{ij}f_j(x_j^*)+\displaystyle \sum _{j=1}^{n}b_{ij}g_j(x_j^*)+\displaystyle \sum _{j=1}^{n}d_{ij}h_j(x_j^*)\nonumber \\&\qquad + \displaystyle \sum _{j=1}^{n}\sum _{k=1}^{n}T_{ijk}g_k(x^*_k)g_j(x^*_j)+\overline{I}_i\bigg ] \end{aligned}$$
(11)
Define the mapping \({\varPsi }:\mathbb {R}^n\rightarrow \mathbb {R}^n\) as:
$$\begin{aligned} {\varPsi }(x_1,\ldots ,x_n)=\left( \begin{array}{ccc} c_1^{-1}\bigg [\displaystyle \sum _{j=1}^{n}a_{1j}f_j(x_j)+\displaystyle \sum _{j=1}^{n}b_{1j}g_j(x_j) \\ +\displaystyle \sum _{j=1}^{n}d_{1j}h_j(x_j)\\ +\displaystyle \sum _{j=1}^{n}\sum _{k=1}^{n}T_{1jk}g_k(x_k)g_j(x_j)+\overline{I}_1\bigg ] \\ \vdots \\ c_n^{-1}\bigg [\displaystyle \sum _{j=1}^{n}a_{nj}f_j(x_j)+\displaystyle \sum _{j=1}^{n}b_{nj}g_j(x_j)\\ +\displaystyle \sum _{j=1}^{n}d_{nj}h_j(x_j)\\ +\displaystyle \sum _{j=1}^{n}\sum _{k=1}^{n}T_{njk}g_k(x_k)g_j(x_j)+\overline{I}_n\bigg ] \end{array}\right) \end{aligned}$$
We will show that \({\varPsi }:\mathbb {R}^n\rightarrow \mathbb {R}^n\) is a global contraction map on \(\mathbb {R}^n\).
In fact, from Assumptions (H1) and (H2), for \(\overline{x}=(\overline{x}_1,\ldots ,\overline{x}_n)^T, x=(x_1,\ldots ,x_n)^T\in \mathbb {R}^n\), we get:
$$\begin{aligned}&\big |{\varPsi }_i(\overline{x}_1,\ldots ,\overline{x}_n))-{\varPsi }_i(x_1,\ldots ,x_n)\big | \\&\quad= \bigg |c^{-1}_i\bigg \{\displaystyle \sum _{j=1}^{n}a_{ij}\bigg [f_j(\overline{x}_j)-f_j(x_j)\bigg ]+\displaystyle \sum _{j=1}^{n}b_{ij}\bigg [g_j(\overline{x}_j)-g_j(x_j)\bigg ]\\&\quad +\displaystyle \sum _{j=1}^{n}d_{ij}\bigg [h_j(\overline{x}_j)-h_j(x_j)\bigg ]\\&\quad +\displaystyle \sum _{j=1}^{n}\sum _{k=1}^{n}T_{ijk}\bigg [g_k(\overline{x}_k)g_j(\overline{x}_j)-g_k(x_k)g_j(x_j)\bigg ]\bigg \}\bigg |\\&\le \bigg |c^{-1}_i\bigg \{\displaystyle \sum _{j=1}^{n}a_{ij}\bigg [f_j(\overline{x}_j)-f_j(x_j)\bigg ]+\displaystyle \sum _{j=1}^{n}b_{ij}\bigg [g_j(\overline{x}_j)-g_j(x_j)\bigg ]\\&\quad +\displaystyle \sum _{j=1}^{n}d_{ij}\bigg [h_j(\overline{x}_j)-h_j(x_j)\bigg ]\\&\quad +\displaystyle \sum _{j=1}^{n}\sum _{k=1}^{n}T_{ijk}\bigg [g_k(\overline{x}_k)g_j(\overline{x}_j)-g_k(\overline{x}_k)g_j(x_j)\\&\quad +g_k(\overline{x}_k)g_j(x_j)-g_k(x_k)g_j(x_j)\bigg ]\\ &\le \displaystyle \bigg |c^{-1}_i\bigg \{\displaystyle \sum _{j=1}^{n}a_{ij}\bigg [f_j(\overline{x}_j)-f_j(x_j)\bigg ]+\displaystyle \sum _{j=1}^{n}b_{ij}\bigg [g_j(\overline{x}_j)-g_j(x_j)\bigg ]\\&\quad +\displaystyle \sum _{j=1}^{n}d_{ij}\bigg [h_j(\overline{x}_j)-h_j(x_j)\bigg ]+\displaystyle \sum _{j=1}^{n}\sum _{k=1}^{n}T_{ijk}\qquad \bigg [g_k(\overline{x}_k)\bigg (g_j(\overline{x}_j)-g_j(x_j)\bigg )\\&\quad +g_j(x_j)\bigg (g_k(\overline{x}_k)-g_k(x_k)\bigg )\bigg ] \end{aligned}$$
$$\begin{aligned}\le & c_i^{-1}\displaystyle \sum _{j=1}^{n}|a_{ij}|L_j\bigg |\overline{x}_j-x_j\bigg |+\displaystyle \sum _{i=1}^{n}c_i^{-1}\displaystyle \sum _{j=1}^{n}|b_{ij}|M_j\bigg |\overline{x}_j-x_j\bigg |\nonumber \\&\quad +\displaystyle \sum _{i=1}^{n}c_i^{-1}\displaystyle \sum _{j=1}^{n}|d_{ij}|N_j\bigg |\overline{x}_j-x_j\bigg |\nonumber \\&\quad +c_i^{-1}\displaystyle \sum _{j=1}^{n}\sum _{k=1}^{n}|T_{ijk}|\bigg [\big |g_k(\overline{x}_k)\big |\bigg |g_j(\overline{x}_j)-g_j(x_j)\bigg |\nonumber \\&\quad +\big |g_j(x_j)\big |\bigg |g_k(\overline{x}_k)-g_k(x_k)\bigg |\bigg ]\nonumber \\\le & c_i^{-1}\displaystyle \sum _{j=1}^{n}\bigg [L_j|a_{ij}|+M_j|b_{ij}|+\tau N_j|d_{ij}|\bigg ]\bigg |\overline{x}_j-x_j\bigg |\nonumber \\&\quad +c_i^{-1}\displaystyle \sum _{j=1}^{n}\sum _{k=1}^{n}|T_{ijk}|\bigg [\chi _kM_j\bigg |\overline{x}_j-x_j\bigg |+\chi _jM_k\bigg |\overline{x}_k-x_k\bigg |\bigg ]\nonumber \\\le & c_i^{-1}\displaystyle \sum _{j=1}^{n}\bigg [L_j|a_{ij}|+M_j|b_{ij}|+ N_j|d_{ij}|+\sum _{k=1}^{n}|T_{ijk}|\chi _kM_j\bigg ]\nonumber \\\times & \bigg |\overline{x}_j-x_j\bigg |+c_i^{-1}\displaystyle \sum _{j=1}^{n}\sum _{k=1}^{n}\bigg [|T_{ikj}|\chi _kM_j\bigg ]\bigg |\overline{x}_j-x_j\bigg |\nonumber \\\le & c_i^{-1}\displaystyle \sum _{j=1}^{n}\bigg [L_j|a_{ij}|+M_j|b_{ij}|+ N_j|d_{ij}|+\sum _{k=1}^{n}|T_{ijk}|\chi _kM_j\nonumber \\&\quad +\displaystyle \sum _{k=1}^{n}|T_{ikj}|\chi _kM_j\bigg ]\bigg |\overline{x}_j-x_j\bigg | \end{aligned}$$
(12)
which implies that
$$\begin{aligned} \Vert {\varPsi }(\overline{x})-{\varPsi }(x)\Vert ^2= & \displaystyle \sum _{i=1}^{n}\big |\psi _i(\overline{x})-\psi _i(x)\big |^2 \\\le & \displaystyle \sum _{i=1}^{n}(c_i^{-1})^2\bigg (\displaystyle \sum _{j=1}^{n}\bigg [L_j|a_{ij}|+M_j|b_{ij}|+N_j|d_{ij}| \\&\quad +\sum _{k=1}^{n}|T_{ijk}|\chi _kM_j+\displaystyle \sum _{k=1}^{n}|T_{ikj}|\chi _kM_j\bigg ]\bigg |\overline{x}_j-x_j\bigg |\bigg )^2 \\\le & \displaystyle \sum _{i=1}^{n}(c_i^{-1})^2\displaystyle \sum _{j=1}^{n}\bigg [L_j|a_{ij}|+M_j|b_{ij}|+N_j|d_{ij}| \\&\quad +\sum _{k=1}^{n}|T_{ijk}|\chi _kM_j+\displaystyle \sum _{k=1}^{n}|T_{ikj}|\chi _kM_j\bigg ]^2\displaystyle \sum _{j=1}^{n}\bigg |\overline{x}_j-x_j\bigg |^2 \\\le & \gamma \Vert \overline{x}-x\Vert ^2 \end{aligned}$$
where
$$\begin{aligned} \gamma= & \displaystyle \sum _{i=1}^{n}(c_i^{-1})^2\displaystyle \sum _{j=1}^{n}\bigg [L_j|a_{ij}|+M_j|b_{ij}|+ N_j|d_{ij}|+\sum _{k=1}^{n}|T_{ijk}|\chi _kM_j \\&\quad +\displaystyle \sum _{k=1}^{n}|T_{ikj}|\chi _kM_j\bigg ]^2<1 \end{aligned}$$
From (4) we obtain \(\Vert {\varPsi }(\overline{x})-\psi (x)\Vert \le \sqrt{\gamma }\Vert \overline{x}-x\Vert \) which implies that the mapping \({\varPsi }:\mathbb {R}^n\rightarrow \mathbb {R}^n\) is a contraction on \(\mathbb {R}^n\) endowed with the Euclidean vector norm \(\Vert .\Vert \), and thus, there exists a unique fixed point \(x^*\in \mathbb {R}^n\) such that \({\varPsi }(x^*)=x^*\) which defines the unique solution of the system (1). \(\blacksquare \)
1.2 B Proof of Theorem 2
We only need to prove the zero solution of system (5) is globally exponentially stable.
Construct a Lyapunov–Krasovskii functional as follows: \(V(t)=V_1(t)+V_2(t)+V_3(t)+V_4(t)\), where:
$$\begin{aligned}&V_1(t)=e^{\overline{\varepsilon }t}y^T(t)P_1y(t) \\&\quad V_2(t)=\frac{1}{1-\rho }\displaystyle \int _{t-\tau (t)}^{t}e^{\overline{\varepsilon }s}G^T(y(s))[Q_2+R_2+2T^{*T}T^*]G(y(s))\mathrm{d}s \\&\quad V_3(t)=\displaystyle \sum _{j=1}^{n}\left[ q^{(3)}_j+r^{(3)}_j\right] \displaystyle \int _{0}^{+\infty }K_j(\theta )\displaystyle \int _{t-\theta }^{t}e^{\overline{\varepsilon }(s+\theta )}H_j^2(y_j(s))\mathrm{d}s\mathrm{d}\theta \end{aligned}$$
where \(Q_3=\mathrm{diag}(q^{(3)}_1,q^{(3)}_2,\ldots ,q^{(3)}_n)\), \(R_3=\mathrm{diag}(r^{(3)}_1,r^{(3)}_2,\ldots ,r^{(3)}_n)\)
$$\begin{aligned} V_4(t)=\sigma \displaystyle \int _{-\sigma }^{0}\int _{t+\theta }^{t}e^{\overline{\varepsilon }s}\dot{y}^T(s)C[Q_4+R_4]C\dot{y}(s)\mathrm{d}s\mathrm{d}\theta \end{aligned}$$
From the definition of V(t), we know
$$\begin{aligned}&\lambda _\mathrm{min}(P_1)e^{\overline{\varepsilon }t}\Vert y(t)\Vert ^2\le V(t)\le \bigg [\lambda _\mathrm{max}(P_1)e^{\overline{\varepsilon }t}\Vert y(t)\Vert ^2\nonumber \\&\quad +\frac{\lambda _\mathrm{max}(Q_2+R_2+2T^{*T}T^*))}{\overline{\varepsilon }(1-\rho )}M\bigg (1-e^{-\overline{\varepsilon }\tau }\bigg )\bigg )\nonumber \\&\quad +\displaystyle \sum _{j=1}^{n}\left[ q^{(3)}_j+r^{(3)}_j\right] \max _{1\le j\le n}N_j^2\displaystyle \int _{0}^{+\infty }K_j(\theta )\frac{1}{\overline{\varepsilon }}(e^{\overline{\varepsilon }\theta }-1)\mathrm{d}\theta \nonumber \\&\quad +\lambda _\mathrm{max}\left( C(Q_4+R_4)C\right) \sigma \left( \frac{1}{\overline{\varepsilon }^2}(\sigma \overline{\varepsilon }+e^{-\overline{\varepsilon }\sigma }-1)\right) \bigg ]e^{\overline{\varepsilon }t}\Vert y(t)\Vert ^2_{\sigma }. \end{aligned}$$
(13)
In addition we have from system (5), Lemmas 2 and 3 for \(t\in [t_k,t_{k+1}[\), \(k=1,2,\ldots ,\):
$$\begin{aligned}D^+V_1(t)&\le \overline{\varepsilon }e^{\overline{\varepsilon }t}y^T(t)P_1y(t)+2e^{\overline{\varepsilon }t}y^T(t)P_1\dot{y}(t)\nonumber \\&\quad +2e^{\overline{\varepsilon }t}\dot{y}^T(t)P_2[-\dot{y}(t)+\dot{y}(t)]\nonumber \\ & \le\, \overline{\varepsilon }e^{\overline{\varepsilon }t}y^T(t)P_1y(t)+2e^{\overline{\varepsilon }t}y(t)^TP_1\bigg [-Cy(t)+C\displaystyle \int _{t-\sigma (t)}^{t}\dot{y}(s)\mathrm{d}s\nonumber \\&\quad +AF(y(t))+BG(y(t-\tau (t)))+{\varGamma }^TT^*G(y(t-\tau (t)))\nonumber \\&\quad +D\displaystyle \int _{-\infty }^{t}K(t-s)H(y(s))\mathrm{d}s\bigg ]+2e^{\overline{\varepsilon }t}\dot{y}^T(t)\\&\quad P_2\bigg [-\dot{y}(t)-Cy(t)\nonumber \quad +C\displaystyle \int _{t-\sigma (t)}^{t}\dot{y}(s)\mathrm{d}s+AF(y(t))\\&\quad+BG(y(t-\tau (t)))\nonumber \\&\quad +{\varGamma }^TT^*G(y(t-\tau (t)))+D\displaystyle \int _{-\infty }^{t}K(t-s)H(y(s))\mathrm{d}s\bigg ]\nonumber \\\le & \overline{\varepsilon }e^{\overline{\varepsilon }t}y^T(t)P_1y(t)+e^{\overline{\varepsilon }t}\bigg [-2y(t)^TP_1Cy(t)\nonumber \\&\quad +2y(t)^TP_1AF(y(t))+2y(t)^TP_1BG(y(t-\tau (t))\nonumber \\&\quad +2y(t)^TP_1{\varGamma }^TT^*G(y(t-\tau (t)))\nonumber \\&\quad +2y(t)^TP_1D\int _{-\infty }^{t}K(t-s)H(y(s))\mathrm{d}s+2y(t)^TP_1C\displaystyle \int _{t-\sigma (t)}^{t}\dot{y}(s)\mathrm{d}s-2\dot{y}^T(t)P_2\dot{y}(t)\nonumber \\&\quad -2\dot{y}^T(t)P_2Cy(t)+2\dot{y}^T(t)P_2AF(y(t))+2\dot{y}^T(t)P_2BG(y(t-\tau (t))\nonumber \\&\quad +2\dot{y}(t)^TP_2{\varGamma }^TT^*G(y(t-\tau (t)))\nonumber \\&\quad +2\dot{y}^T(t)P_2D\int _{-\infty }^{t}K(t-s)H(y(s))\mathrm{d}s+2\dot{y}^T(t)P_2C\displaystyle \int _{t-\sigma (t)}^{t}\dot{y}(s)\mathrm{d}s\bigg ] \end{aligned}$$
(14)
$$\begin{aligned}D^+V_2(t)&\le \frac{e^{\overline{\varepsilon }t}}{1-\rho }G^T(y(t))[Q_2+R_2+2T^{*T}T^*]G(y(t))\nonumber \\&\quad -e^{\overline{\varepsilon }(t-\tau (t))}G^T(y(t-\tau (t)))[Q_2+R_2+2T^{*T}T^*]G(y(t-\tau (t)))\nonumber \\&\le \frac{e^{\overline{\varepsilon }t}}{1-\rho }\lambda _\mathrm{max}(Q_2+R_2+2T^{*T}T^*)My^T(t)y(t)\nonumber \\&\quad -e^{\overline{\varepsilon }(t-\tau (t))}G^T(y(t-\tau (t)))[Q_2+R_2+2T^{*T}T^*]G(y(t-\tau (t))) \end{aligned}$$
(15)
$$\begin{aligned}D^+V_3(t)&\le \displaystyle \sum _{j=1}^{n}[q^{(3)}_j+r^{(3)}_j]\displaystyle \int _{0}^{+\infty }K_j(\theta )e^{\overline{\varepsilon }(t+\theta )}H^2_j(y_j(t))\mathrm{d}\theta \nonumber \\&\quad -\displaystyle \sum _{j=1}^{n}[q^{(3)}_j+r^{(3)}_j]\displaystyle \int _{0}^{+\infty }K_j(\theta )e^{\overline{\varepsilon }t}H^2_j(y_j(t-\theta ))\mathrm{d}\theta \nonumber \\ &\le e^{\overline{\varepsilon }t}\displaystyle \sum _{j=1}^{n}[q^{(3)}_j+r^{(3)}_j]H^2_j(y_j(t))\displaystyle \int _{0}^{+\infty }K_j(\theta )e^{\overline{\varepsilon }\theta }\mathrm{d}\theta \nonumber \\&\quad -e^{\overline{\varepsilon }t}\displaystyle \sum _{j=1}^{n}[q^{(3)}_j+r^{(3)}_j]\displaystyle \int _{0}^{+\infty }K_j(\theta )H^2_j(y_j(t-\theta ))\mathrm{d}\theta \nonumber \\&\le \beta _1e^{\overline{\varepsilon }t}\displaystyle \sum _{j=1}^{n}[q^{(3)}_j+r^{(3)}_j]H^2_j(y_j(t))-e^{\overline{\varepsilon }t}\displaystyle \sum _{j=1}^{n}[q^{(3)}_j+r^{(3)}_j]\displaystyle \int _{0}^{+\infty }K_j(\theta )\mathrm{d}\theta \nonumber \\ &\quad\times \displaystyle \int _{0}^{+\infty }K_j(\theta )H^2_j(y_j(t-\theta ))\mathrm{d}\theta \nonumber \\ &\le \beta _1\lambda _\mathrm{max}(Q_3+R_3)Ny^T(t)y(t)-e^{\overline{\varepsilon }t}\bigg (\displaystyle \int _{-\infty }^{t}K(t-s)H(y(s))\mathrm{d}s\bigg )^T\nonumber \\&\quad\times [Q_3+R_3]\bigg (\displaystyle \int _{-\infty }^{t}K(t-s)H(y(s))\mathrm{d}s\bigg ) \end{aligned}$$
(16)
$$\begin{aligned}D^+V_4(t)&\le \sigma ^2e^{\overline{\varepsilon }t}\dot{y}^T(t)C[Q_4+R_4]C\dot{y}(t)\nonumber \\&\quad -\sigma \displaystyle \int _{t-\sigma }^{t}e^{\overline{\varepsilon }s}\dot{y}^T(s)C[Q_4+R_4]C\dot{y}(s)\mathrm{d}s\nonumber \\&\le \sigma ^2e^{\overline{\varepsilon }t}\dot{y}^T(t)C[Q_4+R_4]C\dot{y}(t)\nonumber \\&\quad -\sigma (t)\displaystyle \int _{t-\sigma (t)}^{t}e^{\overline{\varepsilon }s}\dot{y}^T(s)C[Q_4+R_4]C\dot{y}(s)\mathrm{d}s\nonumber \\&\le \sigma ^2e^{\overline{\varepsilon }t}\dot{y}^T(t)C[Q_4+R_4]C\dot{y}(t)\nonumber \\&\quad -e^{-\overline{\varepsilon }\sigma }e^{\overline{\varepsilon }t}\bigg (\displaystyle \int _{t-\sigma (t)}^{t}\dot{y}(s)\mathrm{d}s\bigg )^TC[Q_4+R_4]C\bigg (\displaystyle \int _{t-\sigma (t)}^{t}\dot{y}(s)\mathrm{d}s\bigg )\nonumber \\&\le \overline{\sigma }^2e^{\overline{\varepsilon }t}\dot{y}^T(t)C[Q_4+R_4]C\dot{y}(t)\nonumber \\&\quad -e^{-\overline{\varepsilon }\sigma }e^{\overline{\varepsilon }t}\bigg (C\displaystyle \int _{t-\sigma (t)}^{t}\dot{y}(s)\mathrm{d}s\bigg )^T[Q_4+R_4]\bigg (C\displaystyle \int _{t-\sigma (t)}^{t}\dot{y}(s)\mathrm{d}s\bigg ) \end{aligned}$$
(17)
Moreover, by Lemmas 1, 2 and 3 we get:
$$\begin{aligned}&2y^T(t)P_1AF(y(t))=2F^T(y(t))A^TP_1y(t)\nonumber \\&\le F^T(y(t))Q_1F(y(t))+y^T(t)P_1AQ_1^{-1}A^TP_1y(t)\nonumber \\&\le \lambda _\mathrm{max}(Q_1)F^T(y(t))F(y(t))+y^T(t)P_1AQ_1^{-1}A^TP_1y(t)\nonumber \\&\le y^T(t)\big [P_1AQ_1^{-1}A^TP_1+\lambda _\mathrm{max}(Q_1)LI\big ]y(t) \end{aligned}$$
(18)
$$\begin{aligned}2y^T(t)P_1BG(t-\tau (t))&=2G^T(y(t-\tau (t))B^TP_1y(t)\nonumber \\&= \,2\bigg [G(y(t-\tau (t))\sqrt{e^{-\tau \overline{\varepsilon }}}\bigg ]^T\bigg [B^TP_1y(t)\frac{1}{\sqrt{e^{-\tau \overline{\varepsilon }}}}\bigg ]\nonumber \\&\le e^{-\tau \overline{\varepsilon }}G^T(y(t-\tau (t))Q_2G(y(t-\tau (t))\nonumber \\&\quad +e^{\tau \overline{\varepsilon }}y^T(t)P_1BQ_2^{-1}B^TP_1y(t) \end{aligned}$$
(19)
$$\begin{aligned}&2y^T(t)P_1D\displaystyle \int _{-\infty }^{t}K(t-s)H(y(s))\mathrm{d}s\nonumber \\&= 2\bigg [\displaystyle \int _{-\infty }^{t}K(t-s)H(y(s))\mathrm{d}s\bigg ]^TD^TP_1y(t)\nonumber \\&= 2\bigg [\displaystyle \int _{-\infty }^{t}K(t-s)H(y(s))\mathrm{d}s\sqrt{e^{-\tau \overline{\varepsilon }}}\bigg ]^T\bigg [D^TP_1y(t)\frac{1}{\sqrt{e^{-\tau \overline{\varepsilon }}}}\bigg ]\nonumber \\&\le e^{-\tau \overline{\varepsilon }}\bigg [\displaystyle \int _{-\infty }^{t}K(t-s)H(y(s))\mathrm{d}s\bigg ]^TQ_3\bigg [\displaystyle \int _{-\infty }^{t}K(t-s)H(y(s))\mathrm{d}s\bigg ]\nonumber \\&\quad +e^{\tau \overline{\varepsilon }}y^T(t)P_1DQ_3^{-1}D^TP_1y(t) \end{aligned}$$
(20)
$$\begin{aligned}2y(t)^TP_1C\displaystyle \int _{t-\sigma (t)}^{t}\dot{y}(s)\mathrm{d}s&=2\bigg [C\displaystyle \int _{t-\sigma (t)}^{t}\dot{y}(s)\mathrm{d}s\bigg ]^TP_1y(t)\nonumber \\&= 2\bigg [C\displaystyle \int _{t-\sigma (t)}^{t}\dot{y}(s)\mathrm{d}s\sqrt{e^{-\sigma \overline{\varepsilon }}}\bigg ]^T\bigg [P_1y(t)\frac{1}{\sqrt{e^{-\sigma \overline{\varepsilon }}}}\bigg ]\nonumber \\&\le e^{-\sigma \overline{\varepsilon }}\bigg [C\displaystyle \int _{t-\sigma (t)}^{t}\dot{y}(s)\mathrm{d}s\bigg ]^TQ_4\bigg [C\displaystyle \int _{t-\sigma (t)}^{t}\dot{y}(s)\mathrm{d}s\bigg ]\nonumber \\&\quad +e^{\sigma \overline{\varepsilon }}y^T(t)P_1Q_4^{-1}P_1y(t) \end{aligned}$$
(21)
In addition, since \({\varGamma }^T{\varGamma }=\Vert \xi \Vert ^2I\) and \(\Vert \xi \Vert \le \Vert \chi \Vert \), then we have \(y^T(t)P_1{\varGamma }^T{\varGamma }P_1y(t)\le \Vert \chi \Vert ^2y^T(t)P^2_1y(t)\) where \(\chi =(\chi _1,\chi _2,\ldots ,\chi _n)^T\), Thus, we obtain
$$\begin{aligned}&2y^T(t)P_1{\varGamma }^TT^*G(y(t-\tau (t)))=2G^T(y(t-\tau (t)))T^{*T}{\varGamma }P_1y(t)\nonumber \\&= 2\bigg (T^*G(y(t-\tau (t)))\sqrt{e^{-\tau \overline{\varepsilon }}}\bigg )^T\bigg ({\varGamma }P_1y(t)\frac{1}{e^{-\tau \overline{\varepsilon }}}\bigg )\nonumber \\&\le e^{-\tau \overline{\varepsilon }} G^T(y(t-\tau (t))T^{*T}T^*G(y(t-\tau (t))\nonumber \\&\quad +\frac{1}{e^{-\tau \overline{\varepsilon }}}y^T(t)P_1{\varGamma }^T{\varGamma }P_1y(t)\nonumber \\&\le e^{-\tau \overline{\varepsilon }} G^T(y(t-\tau (t))T^{*T}T^*G(y(t-\tau (t))\nonumber \\&\quad +e^{\tau \overline{\varepsilon }}\Vert \chi \Vert ^2y^T(t)P^2_1y(t) \end{aligned}$$
(22)
Similarly, to (18)–(22) we obtain easily
$$\begin{aligned}2\dot{y}^T(t)P_2AF(y(t))&\le F^T(y(t))R_1F(y(t))\nonumber \\&\quad +\dot{y}^T(t)P_2AR_1^{-1}A^TP_2\dot{y}(t)\nonumber \\&\le \lambda _\mathrm{max}(R_1)F^T(y(t))F(y(t))+\dot{y}^T(t)P_2AR_1^{-1}A^TP_2\dot{y}(t)\nonumber \\&\le y^T(t)\lambda _\mathrm{max}(R_1)LIy(t)+\dot{y}^T(t)P_2AR_1^{-1}A^TP_2\dot{y}(t) \end{aligned}$$
(23)
$$\begin{aligned} 2\dot{y}^T(t)P_2BG(t-\tau (t))\le & e^{-\tau \overline{\varepsilon }}G^T(y(t-\tau (t))R_2G(y(t-\tau (t))\nonumber \\&\quad + e^{\tau \overline{\varepsilon }}\dot{y}^T(t)P_2BR_2^{-1}B^TP_2\dot{y}(t) \end{aligned}$$
(24)
$$\begin{aligned}&2\dot{y}^T(t)P_2{\varGamma }^TT^*G(y(t-\tau (t)))\nonumber \\&\le e^{-\tau \overline{\varepsilon }} G^T(y(t-\tau (t))T^{*T}T^*G(y(t-\tau (t))\\&\quad+\,e^{\tau \overline{\varepsilon }}\Vert \chi \Vert ^2\dot{y}^T(t)P^2_2\dot{y}(t) \end{aligned}$$
(25)
$$\begin{aligned}&2\dot{y}^T(t)P_2D\displaystyle \int _{-\infty }^{t}K(t-s)H(y(s))\mathrm{d}s\nonumber \\&\le e^{-\tau \overline{\varepsilon }}\bigg [\displaystyle \int _{-\infty }^{t}K(t-s)H(y(s))\mathrm{d}s\bigg ]^TR_3\bigg [\displaystyle \int _{-\infty }^{t}K(t-s)H(y(s))\mathrm{d}s\bigg ]\nonumber \\&\quad +e^{\tau \overline{\varepsilon }}\dot{y}^T(t)P_2DR_3^{-1}D^TP_2\dot{y}(t)\end{aligned}$$
(26)
$$\begin{aligned} 2\dot{y}(t)^TP_2C\displaystyle \int _{t-\sigma (t)}^{t}\dot{y}(s)\mathrm{d}s\le & e^{-\sigma \overline{\varepsilon }}\bigg [C\displaystyle \int _{t-\sigma (t)}^{t}\dot{y}(s)\mathrm{d}s\bigg ]^TR_4\nonumber \\\times & \bigg [C\displaystyle \int _{t-\sigma (t)}^{t}\dot{y}(s)\mathrm{d}s\bigg ]+e^{\sigma \overline{\varepsilon }}\dot{y}^T(t)P_2R_4^{-1}P_2\dot{y}(t) \end{aligned}$$
(27)
$$\begin{aligned} -2\dot{y}^T(t)P_2Cy(t)= & -2(Cy)^T(t)P_2\dot{y}^T(t)\le y^T(t)CR_5Cy(t)\nonumber \\&\quad +\dot{y}^T(t)P_2R_5^{-1}P_2\dot{y}^T(t) \end{aligned}$$
(28)
Then, by (18)–(28) we get
$$\begin{aligned}&D^+V_1(t)\le \overline{\varepsilon }e^{\overline{\varepsilon }t}y^T(t)P_1y(t)+e^{\overline{\varepsilon }t}\bigg \{-2y(t)^TP_1Cy(t)\\&\quad +y^T(t)\big [P_1AQ_1^{-1}A^TP_1+\lambda _\mathrm{max}(Q_1)LI\big ]y(t)\\&\quad +e^{-\tau \overline{\varepsilon }}G^T(y(t-\tau (t))Q_2G(y(t-\tau (t))\\&\quad +e^{\tau \overline{\varepsilon }}y^T(t)P_1BQ_2^{-1}B^TP_1y(t)+e^{-\tau \overline{\varepsilon }} G^T(y(t-\tau (t))T^{*T}T^*G(y(t-\tau (t))\\&\quad +e^{\tau \overline{\varepsilon }}\Vert \chi \Vert ^2y^T(t)P^2_1y(t)+e^{-\tau \overline{\varepsilon }}\bigg [\displaystyle \int _{-\infty }^{t}K(t-s)H(y(s))\mathrm{d}s\bigg ]^TQ_3\\&\quad \times \bigg [\displaystyle \int _{-\infty }^{t}K(t-s)H(y(s))\mathrm{d}s\bigg ]+e^{\tau \overline{\varepsilon }}y^T(t)P_1DQ_3^{-1}D^TP_1y(t)\\&\quad +e^{-\sigma \overline{\varepsilon }}\bigg [C\displaystyle \int _{t-\sigma (t)}^{t}\dot{y}(s)\mathrm{d}s\bigg ]^TQ_4\bigg [C\displaystyle \int _{t-\sigma (t)}^{t}\dot{y}(s)\mathrm{d}s\bigg ]\\&\quad +e^{\sigma \overline{\varepsilon }}y^T(t)P_1Q_4^{-1}P_1y(t)-2\dot{y}^T(t)P_2\dot{y}(t)+y^T(t)CR_5Cy(t)\\&\quad +\dot{y}^T(t)P_2R_5^{-1}P_2\dot{y}^T(t)+y^T(t)\lambda _\mathrm{max}(R_1)LIy(t)\\&\quad +\dot{y}^T(t)P_2AR_1^{-1}A^TP_2\dot{y}(t)+e^{-\tau \overline{\varepsilon }}G^T(y(t-\tau (t))R_2\\&\quad \times G(y(t-\tau (t))+e^{\tau \overline{\varepsilon }}\dot{y}^T(t)P_2BR_2^{-1}B^TP_2\dot{y}(t)\\&\quad +e^{-\tau \overline{\varepsilon }}G^T(y(t-\tau (t))T^{*T}T^*G(y(t-\tau (t))\\&\quad +e^{\tau \overline{\varepsilon }}\Vert \chi \Vert ^2\dot{y}^T(t)P^2_2\dot{y}(t)+e^{-\tau \overline{\varepsilon }}\bigg [\displaystyle \int _{-\infty }^{t}K(t-s)H(y(s))\mathrm{d}s\bigg ]^TR_3\bigg [\displaystyle \int _{-\infty }^{t}K(t-s)H(y(s))\mathrm{d}s\bigg ]\\&\quad +e^{\tau \overline{\varepsilon }}\dot{y}^T(t)P_2DR_3^{-1}D^TP_2\dot{y}(t)\\&\quad +e^{-\sigma \overline{\varepsilon }}\bigg [C\displaystyle \int _{t-\sigma (t)}^{t}\dot{y}(s)\mathrm{d}s\bigg ]^TR_4\bigg [C\displaystyle \int _{t-\sigma (t)}^{t}\dot{y}(s)\mathrm{d}s\bigg ]\\&\quad +e^{\sigma \overline{\varepsilon }}\dot{y}^T(t)P_2QR_4^{-1}P_2\dot{y}(t)\bigg \} \end{aligned}$$
then
$$\begin{aligned}D^+V_1(t)&\le e^{\overline{\varepsilon }t}\bigg \{y(t)^T\big [\overline{\varepsilon }P_1-2P_1C+P_1AQ_1^{-1}A^TP_1\nonumber \\&\quad +\lambda _\mathrm{max}(Q_1)LI+e^{\tau \overline{\varepsilon }}P_1BQ_2^{-1}B^TP_1+e^{\tau \overline{\varepsilon }}\Vert \chi \Vert ^2P^2_1\nonumber \\&\quad +e^{\tau \overline{\varepsilon }}P_1DQ_3^{-1}D^TP_1+e^{\sigma \overline{\varepsilon }}P_1Q_4^{-1}P_1+CR_5C\nonumber \\&\quad +\lambda _\mathrm{max}(R_1)LI\big ]y(t)\nonumber \\&\quad +e^{-\tau \overline{\varepsilon }}G^T(y(t-\tau (t))\big [Q_2+R_2+2T^{*T}T^*\big ]G(y(t-\tau (t))\nonumber \\&\quad +e^{-\tau \overline{\varepsilon }}\bigg [\displaystyle \int_{-\infty }^{t}K(t-s)H(y(s))\mathrm{d}s\bigg ]^T\big [Q_3+R_3\big ]\left[\displaystyle \int_{-\infty }^{t}K(t-s)H(y(s))\mathrm{d}s\right]\nonumber \\&\quad +e^{-\sigma \overline{\varepsilon }}\bigg [C\displaystyle \int _{t-\sigma (t)}^{t}\dot{y}(s)\mathrm{d}s\bigg ]^T\big [Q_4+R_4\big ]\bigg [C\displaystyle \int _{t-\sigma (t)}^{t}\dot{y}(s)\mathrm {d}s\bigg ]\nonumber \\&\quad +\dot{y}^T(t)\big [-2P_2+P_2R_5^{-1}P_2+P_2AR_1^{-1}A^TP_2+e^{\tau \overline{\varepsilon }}P_2BR_2^{-1}B^TP_2\nonumber \\&\quad +e^{\tau \overline{\varepsilon }}\Vert \chi \Vert ^2P^2_2+e^{\tau \overline{\varepsilon }}P_2DR_3^{-1}D^TP_2+e^{\sigma \overline{\varepsilon }}P_2R_4^{-1}P_2\big ]\dot{y}(t)\bigg \} \end{aligned}$$
(29)
by (29) and derivate of \(V_i,\; i=2,3,4\) we get
$$\begin{aligned}D^+V(t)&\le e^{\overline{\varepsilon }t}y^T(t)\bigg [\overline{\varepsilon }P_1-2P_1C+P_1AQ_1^{-1}A^TP_1+\lambda _\mathrm{max}(Q_1)LI\nonumber \\&\quad +e^{\tau \overline{\varepsilon }}P_1BQ_2^{-1}B^TP_1+e^{\tau \overline{\varepsilon }}\Vert \chi \Vert ^2P^2_1+e^{\tau \overline{\varepsilon }}P_1DQ_3^{-1}D^TP_1\nonumber \\&\quad +e^{\tau \overline{\varepsilon }}P_1Q_4^{-1}P_1+CR_5C+\lambda _\mathrm{max}(R_1)LI\nonumber \\&\quad +\frac{1}{1-\rho }\lambda _\mathrm{max}(Q_2+R_2+2T^{*T}T^*)M\nonumber \\&\quad +\beta _1\lambda _\mathrm{max}(Q_3+R_3)N\bigg ]y(t)+e^{\overline{\varepsilon }t}\dot{y}^T(t)\bigg [-2P_2+P_2R_5^{-1}P_2\nonumber \\&\quad +P_2AR_1^{-1}A^TP_2+e^{\tau \overline{\varepsilon }}P_2BR_2^{-1}B^TP_2+e^{\tau \overline{\varepsilon }}\Vert \chi \Vert ^2P^2_2\nonumber \\&\quad +e^{\tau \overline{\varepsilon }}P_2DR_3^{-1}D^TP_2+e^{\tau \overline{\varepsilon }}P_2R_4^{-1}P_2+\sigma ^2 C[Q_4+R_4]C\bigg ]\dot{y}(t)\nonumber \\&\quad +\big [e^{\overline{\varepsilon }(t-\tau )}-e^{\overline{\varepsilon }(t-\tau )}\big ]G^T(y(t-\tau (t))[Q_2+R_2+2T^{*T}T^*]\nonumber \\&\quad \times G(y(t-\tau (t))\nonumber \\&\quad +e^{\overline{\varepsilon }t}\big [e^{-\overline{\varepsilon }\tau }-1\big ]\bigg [\displaystyle \int _{-\infty }^{t}K(t-s)H(y(s))\mathrm {d}s\bigg ]^T[Q_3+R_3]\nonumber \\&\quad \times \bigg [\displaystyle \int _{-\infty }^{t}K(t-s)H(y(s))\mathrm {d}s\bigg ]\nonumber \\&\quad +e^{\overline{\varepsilon }t}\big [e^{-\overline{\varepsilon }\sigma }-e^{-\overline{\varepsilon }\sigma }\big ]\bigg [C\displaystyle \int _{t-\sigma (t)}^{t}\dot{y}(s)\mathrm {d}s\bigg ]^T[Q_4+R_4]\bigg [C\displaystyle \int _{t-\sigma (t)}^{t}\dot{y}(s)\mathrm {d}s\bigg ] \end{aligned}$$
(30)
then we have
$$\begin{aligned}&D^+V(t)\le e^{\overline{\varepsilon }t}y^T(t)\bigg [\overline{\varepsilon }P_1-2P_1C+P_1AQ_1^{-1}A^TP_1\nonumber \\&\quad +\lambda _\mathrm{max}(Q_1)LI+e^{\tau \overline{\varepsilon }}P_1BQ_2^{-1}B^TP_1+e^{\tau \overline{\varepsilon }}\Vert \chi \Vert ^2P^2_1\nonumber \\&\quad +e^{\tau \overline{\varepsilon }}P_1DQ_3^{-1}D^TP_1+e^{\sigma \overline{\varepsilon }}P_1Q_4^{-1}P_1+CR_5C+\lambda _\mathrm{max}(R_1)LI\nonumber \\&\quad +\frac{1}{1-\rho }\lambda _\mathrm{max}(Q_2+R_2+2T^{*T}T^*)M+\beta _1\lambda _\mathrm{max}(Q_3+R_3)N\bigg ]y(t)\nonumber \\&\quad +e^{\overline{\varepsilon }t}\dot{y}^T(t)\bigg [-2P_2+P_2R_5^{-1}P_2+P_2AR_1^{-1}A^TP_2+e^{\tau \overline{\varepsilon }}P_2BR_2^{-1}B^TP_2\nonumber \\&\quad +e^{\tau \overline{\varepsilon }}\Vert \chi \Vert ^2P^2_2+e^{\tau \overline{\varepsilon }}P_2DR_3^{-1}D^TP_2+e^{\sigma \overline{\varepsilon }}P_2R_4^{-1}P_2\nonumber \\&\quad +\sigma ^2 C[Q_4+R_4]C\bigg ]\dot{y}(t) \end{aligned}$$
(31)
By using (i) and (ii), we obtain
$$\begin{aligned} D^+V(t)\le 0 \end{aligned}$$
(32)
In addition, we note by system (5) and Lemma 1
$$\begin{aligned}V(t_k)&=e^{\overline{\varepsilon }t_k}y^T(t_k)P_1y(t_k)+\frac{1}{1-\rho }\displaystyle \int _{t_k-\tau (t_k)}^{t_k}e^{\overline{\varepsilon }s}G^T(y(s))\nonumber \\&\quad \times [Q_2+R_2+2T^{*T}T^*]G(y(s))\mathrm{d}s\nonumber \\&\quad +\displaystyle \sum _{j=1}^{n}[q^{(3)}_j+r^{(3)}_j]\displaystyle \int _{0}^{+\infty }K_j(\theta )\displaystyle \int _{t_k-\theta }^{t_k}e^{\overline{\varepsilon }(s+\theta )}H_j^2(y_j(s))\mathrm {d}s\mathrm{d}\theta \nonumber \\&\quad +\sigma \displaystyle \int _{-\sigma }^{0}\int _{t_k+\theta }^{t_k}e^{\overline{\varepsilon }s}\dot{y}^T(s)C[Q_4+R_4]C\dot{y}(s)\mathrm {d}s\mathrm{d}\theta \nonumber \\&= e^{\overline{\varepsilon }t^{-}_k}y^T(t^{-}_k)E_kP_1E_ky(t^{-}_k)+\frac{1}{1-\rho }\displaystyle \int _{t^{-}_k-\tau (t^{-}_k)}^{t^{-}_k}e^{\overline{\varepsilon }s}G^T(y(s))\nonumber \\&\quad \times [Q_2+R_2+2T^{*T}T^*]G(y(s))\mathrm {d}s\nonumber \\&\quad +\displaystyle \sum _{j=1}^{n}[q^{(3)}_j+r^{(3)}_j]\displaystyle \int _{0}^{+\infty }K_j(\theta )\displaystyle \int _{t^{-}_k-\theta }^{t^{-}_k}e^{\overline{\varepsilon }(s+\theta )}H_j^2(y_j(s))\mathrm {d}s\mathrm{d}\theta \nonumber \\&\quad +\sigma \displaystyle \int _{-\sigma }^{0}\int _{t^{-}_k+\theta }^{t^{-}_k}e^{\overline{\varepsilon }s}\dot{y}^T(s)C[Q_4+R_4]C\dot{y}(s)\mathrm {d}s\mathrm{d}\theta \nonumber \\&\le e^{\overline{\varepsilon }t^{-}_k}\xi _ky^T(t^{-}_k)P_1y(t^{-}_k)+\frac{1}{1-\rho }\displaystyle \int _{t^{-}_k-\tau (t^{-}_k)}^{t^{-}_k}e^{\overline{\varepsilon }s}G^T(y(s))\nonumber \\&\quad \times [Q_2+R_2+2T^{*T}T^*]G(y(s))\mathrm {d}s\nonumber \\&\quad +\displaystyle \sum _{j=1}^{n}[q^{(3)}_j+r^{(3)}_j]\displaystyle \int _{0}^{+\infty }K_j(\theta )\displaystyle \int _{t^{-}_k-\theta }^{t^{-}_k}e^{\overline{\varepsilon }(s+\theta )}H_j^2(y_j(s))\mathrm {d}s\mathrm{d}\theta \nonumber \\&\quad +\sigma \displaystyle \int _{-\sigma }^{0}\int _{t^{-}_k+\theta }^{t^{-}_k}e^{\overline{\varepsilon }s}\dot{y}^T(s)C[Q_4+R_4]C\dot{y}(s)\mathrm {d}s\mathrm{d}\theta \nonumber \\&\le \max \big \{\xi _k,1\big \}V(t^{-}_k) \end{aligned}$$
(33)
By (32) and (33) we obtain for \(k\ge 1\)
$$\begin{aligned} \lambda
_\mathrm{min}(P_1)e^{\overline{\varepsilon }t}\Vert y(t)\Vert ^2\le & V(t)\le V(t_0)\displaystyle \prod _{t_0<t_k\le t}\max \big \{\xi _k,1\big \} \end{aligned}$$
(34)
In addition, utilizing (13), for \(t=t_0\) we get
$$\begin{aligned} V(t_0)\le & \bigg [\lambda _\mathrm{max}(P_1)+\frac{\lambda _\mathrm{max}(Q_2+R_2+2T^{*T}T^*)}{\overline{\varepsilon }(1-\rho )}M\bigg (1-e^{-\overline{\varepsilon }\tau }\bigg )\bigg )\nonumber \\&\quad +\displaystyle \sum _{j=1}^{n}[q^{(3)}_j+r^{(3)}_j]\max _{1\le j\le n}N_j^2\displaystyle \int _{0}^{+\infty }K_j(\theta )\frac{1}{\overline{\varepsilon }}(e^{\overline{\varepsilon }\theta }-1)\mathrm{d}\theta \nonumber \\&\quad +\lambda _\mathrm{max}\big (C(Q_4+R_4)C\big )\sigma \big (\frac{1}{\overline{\varepsilon }^2}(\sigma \overline{\varepsilon }+e^{-\overline{\varepsilon }\sigma }-1)\big )\bigg ]e^{\overline{\varepsilon }t_0}\Vert \varphi \Vert ^2_\sigma \end{aligned}$$
(35)
Hence, we have from (34) and (35),
$$\begin{aligned} \Vert y(t)\Vert ^2\le & \frac{1}{\lambda _\mathrm{min}(P_1)}\bigg [\lambda _\mathrm{max}(P_1)+\frac{\lambda _\mathrm{max}(Q_2+R_2+2T^{*T}T^*)}{\overline{\varepsilon }(1-\rho )} \\&\quad \times M\bigg (1-e^{-\overline{\varepsilon }\tau }\bigg )\bigg ) \\&\quad +\displaystyle \sum _{j=1}^{n}[q^{(3)}_j+r^{(3)}_j]\max _{1\le j\le n}N_j^2\displaystyle \int _{0}^{+\infty }K_j(\theta )\frac{1}{\overline{\varepsilon }}(e^{\overline{\varepsilon }\theta }-1)\mathrm{d}\theta \\&\quad +\lambda _\mathrm{max}\big (C(Q_4+R_4)C\big )\sigma \big (\frac{1}{\overline{\varepsilon }^2}(\sigma \overline{\varepsilon }+e^{-\overline{\varepsilon }\sigma }-1)\big )\bigg ] \\&\quad \times e^{-\overline{\varepsilon }(t-t_0)}\Vert \varphi \Vert ^2_\sigma \displaystyle \prod _{t_0<t_k\le t}\max \big \{\xi _k,1\big \} \end{aligned}$$
Using condition (iii), we furthermore obtain
$$\begin{aligned} \Vert y(t)\Vert ^2\le & \frac{1}{\lambda _\mathrm{min}(P_1)}\bigg [\lambda _\mathrm{max}(P_1)+\frac{\lambda _\mathrm{max}(Q_2+R_2+2T^{*T}T^*)}{\overline{\varepsilon }(1-\rho )} \\&\quad \times M\bigg (1-e^{-\overline{\varepsilon }\tau }\bigg )\bigg ) \\&\quad +\displaystyle \sum _{j=1}^{n}[q^{(3)}_j+r^{(3)}_j]\max _{1\le j\le n}N_j^2\displaystyle \int _{0}^{+\infty }K_j(\theta )\frac{1}{\overline{\varepsilon }}(e^{\overline{\varepsilon }\theta }-1)\mathrm{d}\theta \\&\quad +\lambda _\mathrm{max}\big (C(Q_4+R_4)C\big )\sigma \big (\frac{1}{\overline{\varepsilon }^2}(\sigma \overline{\varepsilon }+e^{-\overline{\varepsilon }\sigma }-1)\big )\bigg ] \\&\quad \times e^{-(\overline{\varepsilon }-\overline{\alpha })(t-t_0)}e^{\lambda }\Vert \varphi \Vert ^2_\sigma \end{aligned}$$
Then
$$\begin{aligned} \Vert y(t)\Vert\le & M^*e^{-\frac{1}{2}(\overline{\varepsilon }-\overline{\alpha })(t-t_0)}e^{\lambda }\Vert \varphi \Vert ,\;\;\; \forall t\ge t_0 \end{aligned}$$
where
$$\begin{aligned} M^{*2}= & \frac{1}{\lambda _\mathrm{min}(P_1)}\bigg [\lambda _\mathrm{max}(P_1)+\frac{\lambda _\mathrm{max}(Q_2+R_2+2T^{*T}T^*)}{\overline{\varepsilon }(1-\rho )} \\&\quad \times M\bigg (1-e^{-\overline{\varepsilon }\tau }\bigg )\bigg ) \\&\quad +\displaystyle \sum _{j=1}^{n}[q^{(3)}_j+r^{(3)}_j]\max _{1\le j\le n}N_j^2\displaystyle \int _{0}^{+\infty }K_j(\theta )\frac{1}{\overline{\varepsilon }}(e^{\overline{\varepsilon }\theta }-1)\mathrm{d}\theta \\&\quad +\lambda _\mathrm{max}\big (C(Q_4+R_4)C\big )\sigma \big (\frac{1}{\overline{\varepsilon }^2}(\sigma \overline{\varepsilon }+e^{-\overline{\varepsilon }\sigma }-1)\big )\bigg ]e^{\lambda }\ge 1 \end{aligned}$$
by which we can conclude the exponential stability of the zero solution of system (5), i.e., the equilibrium point of system (1) is globally exponentially stable and the approximate exponential convergent rate is \(\frac{1}{2}(\overline{\varepsilon }-\overline{\alpha })\).\(\blacksquare \)
1.3 C Proof of Theorem 3
We first prove the zero of system (5) is uniformly stable.
Construct a Lyapunov–Krasovskii functional in the form \(V(t)=V_1(t)+V_2(t)+V_3(t)+V_4(t)\), where
$$\begin{aligned}V_1(t)&=[1+\varepsilon^*(t-t_0)^2]y^T(t)P_1y(t) \\V_2(t)&=\frac{1}{1-\rho }\displaystyle \int _{t-\tau (t)}^{t}(1+(s-t_0)^2) G^T(y(s))[Q_2+R_2+2T^{*T}T^*] \\&\quad \times G(y(s))\mathrm {d}s \\ V_3(t)&=\displaystyle \sum _{j=1}^{n}[q^{(3)}_j+r^{(3)}_j]\displaystyle \int _{0}^{+\infty }K_j(\theta )\displaystyle \int _{t-\theta }^{t}(1+(s+\theta -t_0)^2) \\&\quad \times H_j^2(y_j(s))\mathrm {d}s\mathrm{d}\theta \\ V_4(t)&=\sigma \displaystyle \int _{-\sigma }^{0}\int _{t+\theta }^{t}(1+(s-t_0)^2)\dot{y}^T(s)C[Q_4+R_4]C\dot{y}(s)\mathrm {d}s\mathrm{d}\theta \end{aligned}$$
\(\forall t_0\ge 0\), let y(t) be an arbitrary solution of system (5). So, for any \(\varepsilon >0\), we choose \(\delta \) as:
$$\begin{aligned} \delta =\frac{\varepsilon }{\sqrt{\prod }} \end{aligned}$$
where
$$\begin{aligned} \prod= & \bigg [\lambda _\mathrm{max}(P_1)+\bigg (\tau +\frac{\tau ^3}{3}\bigg )\frac{\lambda _\mathrm{max}(Q_2+R_2+2T^{*T}T^*)}{1-\rho }M \\&+\lambda _\mathrm{max}(Q_3+R_3)\max _{1\le j\le n}N_j^2\displaystyle \int _{0}^{+\infty }K_j(\theta )\left(\theta +\frac{\theta ^3}{3}\right)\mathrm{d}\theta \\&+\sigma \bigg (\frac{\sigma ^2}{2}+\frac{\sigma ^4}{12}\bigg )\lambda _\mathrm{max}(C(Q_4+R_4)C)\bigg ]\frac{\overline{M}}{\lambda _\mathrm{min}(P_1)} \end{aligned}$$
Now we consider the derivation of \(V_i(t)\), \(i=1,\ldots ,4\) for any \(t\in [t_k,t_{k+1}[,\;\;k=1,2,\ldots \) along the solution of (5)
$$\begin{aligned} D^+V_1(t)\le & 2\varepsilon ^*(t-t_0)y^T(t)P_1y(t)+2[1+\varepsilon ^*(t-t_0)^2]y^T(t)P_1\dot{y}(t)\nonumber \\&\quad +2[1+\varepsilon ^*(t-t_0)^2]\dot{y}^T(t)P_2[-\dot{y}(t)+\dot{y}(t)]\nonumber \\\le & 2\varepsilon ^*(t-t_0)y^T(t)P_1y(t)\nonumber \\&\quad +2[1+\varepsilon ^*(t-t_0)^2]y(t)^TP_1\bigg [-Cy(t)+C\displaystyle \int _{t-\sigma (t)}^{t}\dot{y}(s)\mathrm {d}s\nonumber \\&\quad +AF(y(t))+BG(y(t-\tau (t)))\\&\quad+{\varGamma }^TT^*G(y(t-\tau (t))) \qquad\qquad\qquad \qquad (36)\\ &\quad +D\displaystyle \int _{-\infty }^{t}K(t-s)H(y(s))\mathrm {d}s\bigg ]+2[1+\varepsilon ^*(t-t_0)^2]\dot{y}^T(t)P_2\nonumber \\&\quad \times \bigg [-\dot{y}(t)-Cy(t)\nonumber \\&\quad +C\displaystyle \int _{t-\sigma (t)}^{t}\dot{y}(s)\mathrm {d}s\nonumber \\&\quad +AF(y(t))+BG(y(t-\tau (t)))\nonumber \\&\quad +{\varGamma }^TT^*G(y(t-\tau (t)))\nonumber \\&\quad +D\displaystyle \int _{-\infty }^{t}K(t-s)H(y(s))\mathrm {d}s\bigg ]\nonumber \\\le & 2\varepsilon ^*(t-t_0)y^T(t)P_1y(t)\nonumber \\&\quad +[1+\varepsilon ^*(t-t_0)^2]\bigg [-2y(t)^TP_1Cy(t)\nonumber \\&\quad +2y(t)^TP_1AF(y(t))+2y(t)^TP_1BG(y(t-\tau (t))\nonumber \\&\quad +2y(t)^TP_1{\varGamma }^TT^*G(y(t-\tau (t)))\nonumber \\&\quad +2y(t)^TP_1D\int _{-\infty }^{t}K(t-s)H(y(s))\mathrm {d}s\nonumber \\&\quad +2y(t)^TP_1C\displaystyle \int _{t-\sigma (t)}^{t}\dot{y}(s)\mathrm {d}s\nonumber \\&\quad -2\dot{y}^T(t)P_2\dot{y}(t)-2\dot{y}^T(t)P_2Cy(t)\nonumber \\&\quad +2\dot{y}^T(t)P_2AF(y(t))\nonumber \\&\quad +2\dot{y}^T(t)P_2BG(y(t-\tau (t))\nonumber \\&\quad +2\dot{y}(t)^TP_2{\varGamma }^TT^*G(y(t-\tau (t)))\nonumber \\&\quad +2\dot{y}^T(t)P_2D\int _{t-\tau (t)}^{t}H(y(s))\mathrm {d}s\nonumber \\&\quad +2\dot{y}^T(t)P_2C\displaystyle \int _{t-\sigma (t)}^{t}\dot{y}(s)\mathrm {d}s\bigg ] \end{aligned}$$
(37)
$$\begin{aligned} D^+V_2(t)\le & \frac{1+(t-t_0)^2}{1-\rho }G^T(y(t))[Q_2+R_2+2T^{*T}T^*]\nonumber \\&\quad \times G(y(t))\nonumber \\&\quad (1+(t-\tau (t)-t_0)^2)G^T(y(t-\tau (t)))\nonumber \\&\quad \times [Q_2+R_2+2T^{*T}T^*]G(y(t-\tau (t)))\nonumber \\&\le \frac{1+(t-t_0)^2}{1-\rho }\lambda _\mathrm{max}(Q_2+R_2+2T^{*T}T^*)My^T(t)y(t)\nonumber \\&\quad (1+(t-\tau (t)-t_0)^2)G^T(y(t-\tau (t)))\nonumber \\&\quad \times (Q_2+R_2+2T^{*T}T^*)G(y(t-\tau (t))) \end{aligned}$$
(38)
$$\begin{aligned}D^+V_3(t)&=\displaystyle \sum _{j=1}^{n}[q^{(3)}_j+r^{(3)}_j]\displaystyle \int _{0}^{+\infty }K_j(\theta )(1+(t+\theta -t_0)^2)\nonumber \\&\quad \times H^2_j(y_j(t))\mathrm{d}\theta \nonumber \\&\quad \displaystyle \sum _{j=1}^{n}[q^{(3)}_j+r^{(3)}_j]\displaystyle \int _{0}^{+\infty }K_j(\theta )(1+(t-t_0)^2))H^2_j(y_j(t-\theta ))\mathrm{d}\theta \nonumber \\&\le \displaystyle \sum _{j=1}^{n}[q^{(3)}_j+r^{(3)}_j]H^2_j(y_j(t))\displaystyle \int _{0}^{+\infty }K_j(\theta )(1+(t+\theta -t_0)^2)\mathrm{d}\theta \nonumber \\&\quad (1+(t-t_0)^2))\displaystyle \sum _{j=1}^{n}[q^{(3)}_j+r^{(3)}_j]\displaystyle \int _{0}^{+\infty }K_j(\theta )\mathrm{d}\theta \displaystyle \int _{0}^{+\infty }K_j(\theta )\nonumber \\&\quad \times H^2_j(y_j(t-\theta ))\mathrm{d}\theta \nonumber \\&\le (1+(t-t_0)^2))\displaystyle \sum _{j=1}^{n}[q^{(3)}_j+r^{(3)}_j]H^2_j(y_j(t))\nonumber \\&\quad \times \frac{\displaystyle \int _{0}^{+\infty }K_j(\theta )(1+(t+\theta -t_0)^2)\mathrm{d}\theta }{(1+(t-t_0)^2))}\nonumber \\&\quad (1+(t-t_0)^2))\displaystyle \sum _{j=1}^{n}[q^{(3)}_j+r^{(3)}_j]\bigg (\displaystyle \int _{0}^{+\infty }K_j(\theta )H_j(y_j(t-\theta ))\mathrm{d}\theta \bigg )^2\nonumber \\&\le (1+(t-t_0)^2))\lambda _\mathrm{max}(Q_3+R_3)\beta _1Ny^T(t)y(t)\nonumber \\&\quad (1+(t-t_0)^2))\bigg (\displaystyle \int _{-\infty }^{t}K(t-s)H(y(s))\mathrm {d}s\bigg )^T[Q_3+R_3]\nonumber \\&\quad \times \bigg (\displaystyle \int _{-\infty }^{t}K(t-s)H(y(s))\mathrm {d}s\bigg ) \end{aligned}$$
(39)
$$\begin{aligned}D^+V_4(t)&\le \sigma ^2(1+(t-t_0)^2)\dot{y}^T(t)C[Q_4+R_4]C\dot{y}(t)\nonumber \\&\quad \sigma \displaystyle \int _{t-\sigma }^{t}(1+(s-t_0)^2)\dot{y}^T(s)C[Q_4+R_4]C\dot{y}(s)\mathrm {d}s\nonumber \\&\le \sigma ^2(1+(t-t_0)^2)\dot{y}^T(t)C[Q_4+R_4]C\dot{y}(t)\nonumber \\&\quad \sigma (t)\displaystyle \int _{t-\sigma (t)}^{t}(1+(s-t_0)^2)\dot{y}^T(s)C[Q_4+R_4]C\dot{y}(s)\mathrm {d}s\nonumber \\&\le \sigma ^2(1+(t-t_0)^2)\dot{y}^T(t)C[Q_4+R_4]C\dot{y}(t)\nonumber \\&\quad \beta _3(1+(t-t_0)^2)\bigg (\displaystyle \int _{t-\sigma (t)}^{t}\dot{y}(s)\mathrm {d}s\bigg )^TC[Q_4+R_4]C\bigg (\displaystyle \int _{t-\sigma (t)}^{t}\dot{y}(s)\mathrm {d}s\bigg )\nonumber \\&\le \sigma ^2(1+(t-t_0)^2)\dot{y}^T(t)C[Q_4+R_4]C\dot{y}(t)\nonumber \\&\quad \beta _3(1+(t-t_0)^2)\bigg (C\displaystyle \int _{t-\sigma (t)}^{t}\dot{y}(s)\mathrm {d}s\bigg )^T[Q_4+R_4]\bigg (C\displaystyle \int _{t-\sigma (t)}^{t}\dot{y}(s)\mathrm {d}s\bigg ) \end{aligned}$$
(40)
Moreover, by Lemmas 1, 2 and 3 we get:
$$\begin{aligned}2y^T(t)P_1AF(y(t))&=2F^T(y(t))A^TP_1y(t)\nonumber \\\le & F^T(y(t))Q_1F(y(t))+y^T(t)P_1AQ_1^{-1}A^TP_1y(t)\nonumber \\\le & \lambda _\mathrm{max}(Q_1)F^T(y(t))F(y(t))+y^T(t)P_1AQ_1^{-1}A^TP_1y(t)\nonumber \\\le & y^T(t)\big [P_1AQ_1^{-1}A^TP_1+\lambda _\mathrm{max}(Q_1)LI\big ]y(t) \end{aligned}$$
(41)
$$\begin{aligned}2y^T(t)P_1BG(t-\tau (t))&=2G^T(y(t-\tau (t))B^TP_1y(t)\nonumber \\= & 2\bigg [G(y(t-\tau (t))\sqrt{\alpha _1}\bigg ]^T\bigg [B^TP_1y(t)\frac{1}{\sqrt{\alpha _1}}\bigg ]\nonumber \\\le & \alpha _1 G^T(y(t-\tau (t))Q_2G(y(t-\tau (t))\nonumber \\&\quad +\frac{1}{\alpha _1}y^T(t)P_1BQ_2^{-1}B^TP_1y(t) \end{aligned}$$
(42)
$$\begin{aligned}&2y^T(t)P_1D\displaystyle \int _{-\infty }^{t}K(t-s)H(y(s))\mathrm {d}s\nonumber \\= & 2\bigg [\displaystyle \int _{t-\tau (t)}^{t}H(y(s))\mathrm {d}s\bigg ]^TD^TP_1y(t)\nonumber \\\le & \bigg [\displaystyle \int _{-\infty }^{t}K(t-s)H(y(s))\mathrm {d}s\bigg ]^TQ_3\bigg [\displaystyle \int _{-\infty }^{t}K(t-s)H(y(s))\mathrm {d}s\bigg ]\nonumber \\&\quad +y^T(t)P_1DQ_3^{-1}D^TP_1y(t) \end{aligned}$$
(43)
$$\begin{aligned}&2y(t)^TP_1C\displaystyle \int _{t-\sigma (t)}^{t}\dot{y}(s)\mathrm {d}s=2\bigg [C\displaystyle \int _{t-\sigma (t)}^{t}\dot{y}(s)\mathrm {d}s\bigg ]^TP_1y(t)\nonumber \\= & 2\bigg [C\displaystyle \int _{t-\sigma (t)}^{t}\dot{y}(s)\mathrm {d}s\sqrt{\alpha _2}\bigg ]^T\bigg [P_1y(t)\frac{1}{\sqrt{\alpha _2}}\bigg ]\nonumber \\\le & \alpha _2\bigg [C\displaystyle \int _{t-\sigma (t)}^{t}\dot{y}(s)\mathrm {d}s\bigg ]^TQ_4\bigg [C\displaystyle \int _{t-\sigma (t)}^{t}\dot{y}(s)\mathrm {d}s\bigg ]\nonumber \\&\quad +\frac{1}{\alpha _2}y^T(t)P_1Q_4^{-1}P_1y(t) \end{aligned}$$
(44)
Furthermore, since \({\varGamma }^T{\varGamma }=\Vert \xi \Vert ^2I\), \({\varUpsilon }^T{\varUpsilon }=\Vert \zeta \Vert I\) then we have \(y^T(t)P_1{\varGamma }^T{\varGamma }P_1y(t)\le \Vert \chi \Vert ^2y^T(t)P^2_1y(t)\)
where \(\chi =(\chi _1,\chi _2,\ldots ,\chi _n)^T\).
Thus, we obtain
$$\begin{aligned}&2y^T(t)P_1{\varGamma }^TT^*G(y(t-\tau (t)))\nonumber \\= & 2G^T(y(t-\tau (t)))T^{*T}{\varGamma }P_1y(t)\nonumber \\= & 2\bigg (T^*G(y(t-\tau (t)))\sqrt{\alpha _1}\bigg )^T\bigg ({\varGamma }P_1y(t)\frac{1}{\alpha _1}\bigg )\nonumber \\\le & \alpha _1 G^T(y(t-\tau (t))T^{*T}T^*G(y(t-\tau (t))\nonumber \\&\quad +\frac{1}{\alpha _1}y^T(t)P_1{\varGamma }^T{\varGamma }P_1y(t)\nonumber \\\le & \alpha _1 G^T(y(t-\tau (t))T^{*T}T^*G(y(t-\tau (t))\nonumber \\&\quad +\frac{1}{\alpha _1 }\Vert \chi \Vert ^2y^T(t)P^2_1y(t) \end{aligned}$$
(45)
Similarly, to (41)–(45), it follows that
$$\begin{aligned}&2\dot{y}^T(t)P_2AF(y(t))\nonumber \\ & \le F^T(y(t))R_1F(y(t))+\dot{y}^T(t)P_2AR_1^{-1}A^TP_2\dot{y}(t) \\ &\le \lambda _\mathrm{max}(R_1)F^T(y(t))F(y(t))+\dot{y}^T(t)P_2AR_1^{-1}A^TP_2\dot{y}(t)\\ & \le y^T(t)\lambda _\mathrm{max}(R_1)LIy(t)+\dot{y}^T(t)P_2AR_1^{-1}A^TP_2\dot{y}(t) \end{aligned}$$
(46)
$$\begin{aligned} 2\dot{y}^T(t)P_2BG(t-\tau (t))\le & \alpha _1 G^T(y(t-\tau (t))R_2G(y(t-\tau (t))\nonumber \\&\quad +\frac{1}{\alpha _1}\dot{y}^T(t)P_2BR_2^{-1}B^TP_2\dot{y}(t)\;\;\;\;\;\; \end{aligned}$$
(47)
$$\begin{aligned} 2\dot{y}^T(t)P_2{\varGamma }^TT^*G(y(t-\tau (t)))\le & \alpha _1 G^T(y(t-\tau (t))T^{*T}T^*\nonumber \\\times & G(y(t-\tau (t))\nonumber \\&\quad +\frac{1}{\alpha _1}\Vert \chi \Vert ^2\dot{y}^T(t)P^2_2\dot{y}(t) \end{aligned}$$
(48)
$$\begin{aligned}&2\dot{y}^T(t)P_2D\displaystyle \int _{-\infty }^{t}K(t-s)H(y(s))\mathrm {d}s\nonumber \\\le & \bigg [\displaystyle \int _{-\infty }^{t}K(t-s)H(y(s))\mathrm {d}s\bigg ]^TR_3\bigg [\displaystyle \int _{-\infty }^{t}K(t-s)H(y(s))\mathrm {d}s\bigg ]\nonumber \\&\quad +\dot{y}^T(t)P_2DR_3^{-1}D^TP_2\dot{y}(t) \end{aligned}$$
(49)
$$\begin{aligned} 2\dot{y}(t)^TP_2C\displaystyle \int _{t-\sigma (t)}^{t}\dot{y}(s)\mathrm {d}s\le & \alpha _2\bigg [C\displaystyle \int _{t-\sigma (t)}^{t}\dot{y}(s)\mathrm {d}s\bigg ]^TR_4\nonumber \\\times & \bigg [C\displaystyle \int _{t-\sigma (t)}^{t}\dot{y}(s)\mathrm {d}s\bigg ]\nonumber \\&\quad +\frac{1}{\alpha _2}\dot{y}^T(t)P_2R_4^{-1}P_2\dot{y}(t) \end{aligned}$$
(50)
$$\begin{aligned} -2\dot{y}^T(t)P_2Cy(t)= & -2(Cy)^T(t)P_2\dot{y}^T(t)\nonumber \\\le & y^T(t)CR_5Cy(t)+\dot{y}^T(t)P_2R_5^{-1}P_2\dot{y}^T(t)\;\;\; \end{aligned}$$
(51)
Then, from (41) and (51), we obtain
$$\begin{aligned}&D^+V_1(t)\le [1+(t-t_0)^2]\bigg \{y(t)^T\big [2\frac{\varepsilon ^*(t-t_0)}{1+(t-t_0)^2}P_1-2P_1C\nonumber \\&\quad +P_1AQ_1^{-1}A^TP_1+\lambda _\mathrm{max}(Q_1)LI+\frac{1}{\alpha _1}P_1BQ_2^{-1}B^TP_1+\frac{1}{\alpha _1}\Vert \chi \Vert ^2P^2_1\nonumber \\&\quad +P_1DQ_3^{-1}D^TP_1+\frac{1}{\alpha _2}P_1Q_4^{-1}P_1+CR_5C+\lambda _\mathrm{max}(R_1)LI\big ]y(t)\nonumber \\&\quad +\dot{y}^T(t)\big [-2P_2+P_2R_5^{-1}P_2+P_2AR_1^{-1}A^TP_2+\frac{1}{\alpha _1}P_2BR_2^{-1}B^TP_2\nonumber \\&\quad +\frac{1}{\alpha _1}\Vert \chi \Vert ^2P^2_2+P_2DR_3^{-1}D^TP_2+\frac{1}{\alpha _2}P_2Q_4^{-1}P_2\big ]\dot{y}(t)\nonumber \\&\quad +\alpha _1 G^T(y(t-\tau (t))\big [Q_2+R_2+2T^{*T}T^*\big ]G(y(t-\tau (t))\nonumber \\&\quad +\bigg [\displaystyle \int _{-\infty }^{t}K(t-s)H(y(s))\mathrm {d}s\bigg ]^T\big [Q_3+R_3\big ]\nonumber \\\times & \bigg [\displaystyle \int _{-\infty }^{t}K(t-s)H(y(s))\mathrm {d}s\bigg ]\nonumber \\&\quad +\alpha _2\bigg [C\displaystyle \int _{t-\sigma (t)}^{t}\dot{y}(s)\mathrm {d}s\bigg ]^T\big [Q_4+R_4\big ]\bigg [C\displaystyle \int _{t-\sigma (t)}^{t}\dot{y}(s)\mathrm {d}s\bigg ]\bigg \} \end{aligned}$$
(52)
By \(D^+V_1,D^+V_2,D^+V_3,D^+V_4\), we obtain
$$\begin{aligned}&D^+V(t)\le [1+(t-t_0)^2]y^T(t)\bigg [\varepsilon ^*P_1-2P_1C\nonumber \\&\quad +P_1AQ_1^{-1}A^TP_1+\lambda _\mathrm{max}(Q_1)LI+\frac{1}{\alpha _1}P_1BQ_2^{-1}B^TP_1\nonumber \\&\quad +\frac{1}{\alpha _1}\Vert \chi \Vert ^2P^2_1+P_1DQ_3^{-1}D^TP_1+\frac{1}{\alpha _2}P_1Q_4^{-1}P_1+CR_5C\nonumber \\&\quad +\lambda _\mathrm{max}(R_1)LI+\frac{1}{1-\rho }\lambda _\mathrm{max}(Q_2+R_2+2T^{*T}T^*)M\nonumber \\&\quad +\beta _1\lambda _\mathrm{max}(Q_3+R_3+2\overline{T}^{*T}\overline{T}^*)N\bigg ]y(t)\nonumber \\&\quad +\dot{y}^T(t)\bigg [-2P_2+P_2R_5^{-1}P_2+P_2AR_1^{-1}A^TP_2\nonumber \\&\quad +\frac{1}{\alpha _1}P_2BR_2^{-1}B^TP_2+\frac{1}{\alpha _1}\Vert \chi \Vert ^2P^2_2+\frac{1}{P}_2DR_3^{-1}D^TP_2\nonumber \\&\quad +\frac{1}{\alpha _2}P_2R_4^{-1}P_2+\sigma ^2 C[Q_4+R_4]C\bigg ]\dot{y}(t)\nonumber \\&\quad +[1+(t-t_0)^2]\bigg [\alpha _1-\frac{(1+(t-\tau (t)-t_0)^2))}{1+(t-t_0)^2}\bigg ]\nonumber \\&\quad \times G^T(y(t-\tau (t))[Q_2+R_2+2T^{*T}T^*]G(y(t-\tau (t))\nonumber \\&\quad +[1+(t-t_0)^2]\big [1-1\big ]\bigg [\displaystyle \int _{-\infty }^{t}K(t-s)H(y(s))\mathrm {d}s\bigg ]^T\nonumber \\&\quad \times [Q_3+R_3]\bigg [\displaystyle \int _{-\infty }^{t}K(t-s)H(y(s))\mathrm {d}s\bigg ]\nonumber \\&\quad +[1+(t-t_0)^2]\big [\alpha _2-\beta _3\big ]\bigg [C\displaystyle \int _{t-\sigma (t)}^{t}\dot{y}(s)\mathrm {d}s\bigg ]^T[Q_4+R_4]\nonumber \\&\quad \times \bigg [C\displaystyle \int _{t-\sigma (t)}^{t}\dot{y}(s)\mathrm {d}s\bigg ] \end{aligned}$$
(53)
According to the hypothesis of Theorem 3, \(\alpha _1\le \beta _2\), \(\alpha _2\le \beta _3\) and \(-\frac{(1+(t-\tau (t)-t_0)^2)}{(1+(t-t_0)^2)}\le -\beta _2\) we have
$$\begin{aligned}D^+V(t)&\le [1+(t-t_0)^2]y^T(t)\bigg [\varepsilon ^*P_1-2P_1C\nonumber \\&\quad +P_1AQ_1^{-1}A^TP_1+\lambda _\mathrm{max}(Q_1)LI+\frac{1}{\alpha _1}P_1BQ_2^{-1}B^TP_1\nonumber \\&\quad +\frac{1}{\alpha _1}\Vert \chi \Vert ^2P^2_1+P_1DQ_3^{-1}D^TP_1+\frac{1}{\alpha _2}P_1Q_4^{-1}P_1+CR_5C\nonumber \\&\quad +\lambda _\mathrm{max}(R_1)LI+\frac{1}{1-\rho }\lambda _\mathrm{max}(Q_2+R_2+2T^{*T}T^*)M\nonumber \\&\quad +\beta _1\lambda _\mathrm{max}(Q_3+R_3+2\overline{T}^{*T}\overline{T}^*)N\bigg ]y(t)\nonumber \\&\quad +[1+(t-t_0)^2]\dot{y}^T(t)\bigg [-2P_2+P_2R_5^{-1}P_2+P_2AR_1^{-1}A^TP_2\nonumber \\&\quad +\frac{1}{\alpha _1}P_2BR_2^{-1}B^TP_2+\frac{1}{\alpha _1}\Vert \chi \Vert ^2P^2_2+P_2DR_3^{-1}D^TP_2\nonumber \\&\quad +\frac{1}{\alpha _2}P_2R_4^{-1}P_2+\sigma ^2 C[Q_4+R_4]C\bigg ]\dot{y}(t)\nonumber \\&\le 0 \end{aligned}$$
(54)
Moreover, we note by system (5)
$$\begin{aligned}V(t_k)&=[1+(t_k-t_0)^2]y^T(t_k)P_1y(t_k)\nonumber \\&\quad +\frac{1}{1-\rho }\displaystyle \int _{t_k-\tau (t_k)}^{t_k}(1+(s-t_0)^2)\nonumber \\&\quad \times G^T(y(s))[Q_2+R_2+2T^{*T}T^*]G(y(s))\mathrm {d}s\nonumber \\&\quad +\displaystyle \sum _{j=1}^{n}[q^{(3)}_j+r^{(3)}_j]\displaystyle \int _{0}^{+\infty }K_j(\theta )\displaystyle \int _{t_k-\theta }^{t_k}(1+(s+\theta -t_0)^2)\nonumber \\&\quad \times H_j^2(y_j(s))\mathrm {d}s\mathrm{d}\theta \nonumber \\&\quad +\sigma \displaystyle \int _{-\sigma }^{0}\int _{t_k+\theta }^{t_k}(1+(s-t_0)^2)\dot{y}^T(s)C[Q_4+R_4]C\dot{y}(s)\mathrm {d}s\mathrm{d}\theta \nonumber \\&= [1+(t^{-}_k-t_0)^2]y^T(t^{-}_k)E_kP_1E_ky(t^{-}_k)\nonumber \\&\quad +\frac{1}{1-\rho }\displaystyle \int _{t^{-}_k-\tau (t^{-}_k)}^{t^{-}_k}(1+(s-t_0)^2)\nonumber \\&\quad \times G^T(y(s))[Q_2+R_2+2T^{*T}T^*]G(y(s))\mathrm {d}s\nonumber \\&\quad +\displaystyle \sum _{j=1}^{n}[q^{(3)}_j+r^{(3)}_j]\displaystyle \int _{0}^{+\infty }K_j(\theta )\displaystyle \int _{t^{-}_k-\theta }^{t^{-}_k}(1+(s+\theta -t_0)^2)\nonumber \\&\quad \times H_j^2(y_j(s))\mathrm {d}s\mathrm{d}\theta \nonumber \\&\quad +\sigma \displaystyle \int _{-\sigma }^{0}\int _{t^{-}_k+\theta }^{t^{-}_k}(1+(s-t_0)^2)\dot{y}^T(s)C[Q_4+R_4]C\dot{y}(s)\mathrm {d}s\mathrm{d}\theta \nonumber \\&\le [1+(t^{-}_k-t_0)^2]\xi _ky^T(t^{-}_k)P_1y(t^{-}_k)\nonumber \\&\quad +\frac{1}{1-\rho }\displaystyle \int _{t^{-}_k-\tau (t^{-}_k)}^{t^{-}_k}(1+(s-t_0)^2)\nonumber \\&\quad \times G^T(y(s))[Q_2+R_2+2T^{*T}T^*]G(y(s))\mathrm {d}s\nonumber \\&\quad +\displaystyle \sum _{j=1}^{n}[q^{(3)}_j+r^{(3)}_j]\displaystyle \int _{0}^{+\infty }K_j(\theta )\displaystyle \int _{t^{-}_k-\theta }^{t^{-}_k}(1+(s+\theta -t_0)^2)\nonumber \\&\quad \times H_j^2(y_j(s))\mathrm {d}s\mathrm{d}\theta \nonumber \\&\quad +\sigma \displaystyle \int _{-\sigma }^{0}\int _{t^{-}_k+\theta }^{t^{-}_k}(1+(s-t_0)^2)\dot{y}^T(s)C[Q_4+R_4]C\dot{y}(s)\mathrm {d}s\mathrm{d}\theta \nonumber \\&\le \max \big \{\xi _k,1\big \}V(t^{-}_k) \end{aligned}$$
(55)
By (54) and (55) we obtain for \(k\ge 1\)
$$\begin{aligned}&\lambda _\mathrm{min}(P_1)[1+\varepsilon ^*(t-t_0)^2]\Vert y(t)\Vert ^2\le V(t)\nonumber \\&\le V(t_0)\displaystyle \prod _{t_0<t_k\le t}\max \big \{\xi _k,1\big \} \end{aligned}$$
(56)
On other hand, we have
$$\begin{aligned}&V(t)=[1+\varepsilon ^*(t-t_0)^2]y^T(t)P_1y(t)\nonumber \\&\quad +\frac{1}{1-\rho }\displaystyle \int _{t-\tau (t)}^{t}(1+(s-t_0)^2)\nonumber \\&\quad\times G^T(y(s))[Q_2+R_2+2T^{*T}T^*]G(y(s))\mathrm {d}s\nonumber \\&\quad +\displaystyle \sum _{j=1}^{n}[q^{(3)}_j+r^{(3)}_j]\displaystyle \int _{0}^{+\infty }K_j(\theta )\displaystyle \int _{t-\theta }^{t}(1+(s+\theta -t_0)^2)\nonumber \\ &\quad\times H_j^2(y_j(s))\mathrm {d}s\mathrm{d}\theta \nonumber \\&\quad +\sigma \displaystyle \int _{-\sigma }^{0}\int _{t+\theta }^{t}(1+(s-t_0)^2)\dot{y}^T(s)C[Q_4+R_4]C\dot{y}(s)\mathrm {d}s\mathrm{d}\theta \nonumber \\&\le \bigg \{[1+\varepsilon ^*(t-t_0)^2]\lambda _\mathrm{max}(P_1)\nonumber \\&\quad +\bigg (\tau (t)+\frac{(t-t_0)^3-(t-t_0-\tau (t))^3}{3}\bigg )\\nonumber \\&\quad \times \frac{\lambda _\mathrm{max}(Q_2+R_2+2T^{*T}T^*)}{1-\rho }M+\lambda _\mathrm{max}(Q_3+R_3)\max _{1\le j\le n}N_j^2\nonumber \\&\quad \times \displaystyle \int _{0}^{+\infty }K_j(\theta )\left(\theta +\frac{1}{3}(t+\theta -t_0)^3\right)\mathrm{d}\theta \nonumber \\&\quad +\sigma \bigg (-\frac{1}{12}(t-t_0)^4+\frac{1}{3}(t-t_0)^3\sigma +\frac{\sigma ^2}{2}\nonumber \\&\quad +\frac{1}{12}(t-\sigma -t_0)^4\bigg )\lambda _\mathrm{max}(C(Q_4+R_4)C)\bigg \}\Vert y(t)\Vert ^2_\sigma \end{aligned}$$
(57)
For \(t=t_0\) in (57) we get
$$\begin{aligned}&V(t_0)\le \bigg \{\lambda _\mathrm{max}(P_1)+\bigg (\tau (t_0)+\frac{\tau (t_0)^3}{3}\bigg )\nonumber \\\&\quad times \frac{\lambda _\mathrm{max}(Q_2+R_2+2T^{*T}T^*)}{1-\rho }M\nonumber \\&\quad +\lambda _\mathrm{max}(Q_3+R_3)\max _{1\le j\le n}N_j^2\displaystyle \int _{0}^{+\infty }K_j(\theta )\left(\theta +\frac{\theta ^3}{3}\right)\mathrm{d}\theta \nonumber \\&\quad +\sigma \bigg (\frac{\sigma ^2}{2}+\frac{\sigma ^4}{12}\bigg )\lambda _\mathrm{max}(C(Q_4+R_4)C)\bigg \}\Vert \varphi \Vert ^2_\sigma \nonumber \\&\le \bigg \{\lambda _\mathrm{max}(P_1)+\bigg (\tau +\frac{\tau ^3}{3}\bigg )\frac{\lambda _\mathrm{max}(Q_2+R_2+2T^{*T}T^*)}{1-\rho }M\nonumber \\&\quad +\lambda _\mathrm{max}(Q_3+R_3)\max _{1\le j\le n}N_j^2\displaystyle \int _{0}^{+\infty
}K_j(\theta )\left(\theta +\frac{\theta ^3}{3}\right)\mathrm{d}\theta \nonumber \\&\quad +\sigma \bigg (\frac{\sigma ^2}{2}+\frac{\sigma ^4}{12}\bigg )\lambda _\mathrm{max}(C(Q_4+R_4)C)\bigg \}\Vert \varphi \Vert ^2_\sigma \end{aligned}$$
(58)
From (56) and (58), we obtain
$$\begin{aligned} \Vert y(t)\Vert ^2&\le \bigg [\lambda _\mathrm{max}(P_1)+\bigg (\tau +\frac{\tau ^3}{3}\bigg ) \\&\quad \times \frac{\lambda _\mathrm{max}(Q_2+R_2+2T^{*T}T^*)}{1-\rho }M \\&\quad +\lambda _\mathrm{max}(Q_3+R_3)\max _{1\le j\le n}N_j^2\displaystyle \int _{0}^{+\infty }K_j(\theta )\left(\theta +\frac{\theta ^3}{3}\right)\mathrm{d}\theta \\&\quad +\sigma \left(\frac{\sigma ^2}{2}+\frac{\sigma ^4}{12}\right)\lambda _\mathrm{max}(C(Q_4+R_4)C)\bigg ]\Vert \varphi \Vert ^2_\sigma \\&\quad \times \frac{\prod _{t_0<t_k\le t}\max \big \{\xi _k,1\big \}}{\lambda _\mathrm{min}(P_1)(1+\varepsilon ^*(t-t_0)^2)} \end{aligned}$$
From condition (iii), there exists a constant \(\overline{M}\) such that
$$\begin{aligned} \frac{\displaystyle \prod\nolimits _{t_0<t_k\le t}\max \big \{\xi _k,1\big \}}{(1+\varepsilon ^*(t-t_0)^2)}<\overline{M} \end{aligned}$$
then
$$\begin{aligned} \Vert y(t)\Vert ^2\le & \bigg [\lambda _\mathrm{max}(P_1)+\bigg (\tau +\frac{\tau ^3}{3}\bigg )\frac{\lambda _\mathrm{max}(Q_2+R_2+2T^{*T}T^*)}{1-\rho }M \\&\quad +\lambda _\mathrm{max}(Q_3+R_3)\max _{1\le j\le n}N_j^2\displaystyle \int _{0}^{+\infty }K_j(\theta )\left(\theta +\frac{\theta ^3}{3}\right)\mathrm{d}\theta \\&\quad +\sigma \bigg (\frac{\sigma ^2}{2}+\frac{\sigma ^4}{12}\bigg )\lambda _\mathrm{max}(C(Q_4+R_4)C)\bigg ]\delta ^2\frac{\overline{M}}{\lambda _\mathrm{min}(P_1)} \\\le & \varepsilon ^2 \end{aligned}$$
which implies
$$\begin{aligned} \Vert y(t)\Vert \le \varepsilon \end{aligned}$$
It is concluded that the zero solution of system (1) is uniformly stable.
By condition (iv), it is easy that: \(\limsup _{t\rightarrow +\infty }\Vert y(t)\Vert ^2=0\), then the equilibrium point of system (1) is also uniformly asymptotically stable and globally asymptotically stable. This completes the proof of Theorem 3\(\blacksquare \).
1.4 D Proof of Theorem 4
By Theorem 1, system (1) has a unique equilibrium point, say, \(x^*=(x^*_1,x^*_2,\ldots ,x^*_n)^T\in \mathbb {R}^n\).
Let \(y(t)=(y_1(t),y_2(t),\ldots ,y_n(t))^T\) be a solution of system (5), \(\forall t_0\ge 0\). For any \(\varepsilon >0\), we choose \(\delta \) as: \(\delta =\frac{\varepsilon }{\sqrt{{\varTheta }}}\), \((\star )\) where
$$\begin{aligned}&{\varTheta }=\frac{M^*}{\lambda _\mathrm{min}(P_1)}\bigg [\lambda _\mathrm{max}(P_1)+\frac{\tau }{1-\rho }\lambda _\mathrm{max}(Q_2+R_2+2T^{*T}T^*)M \\&\quad +\displaystyle \sum _{j=1}^{n}[q^{(3)}_j+r^{(3)}_j]\max _{1\le j\le n}N_j^2\displaystyle \int _{0}^{+\infty }\theta K_j(\theta )\mathrm{d}\theta \\&\quad +\frac{\sigma ^3}{2}\lambda _\mathrm{max}\big (C(Q_4+R_4)C\big )\bigg ] \end{aligned}$$
Now we consider the Lyapunov functional:
$$\begin{aligned} V(t)=V_1(t)+V_2(t)+V_3(t)+V_4(t), \end{aligned}$$
where
$$\begin{aligned} V_1(t)= & y^T(t)P_1y(t) \\ V_2(t)= & \frac{1}{1-\rho }\displaystyle \int _{t-\tau (t)}^{t}G^T(y(s))[Q_2+R_2+2T^{*T}T^*]G(y(s))\mathrm {d}s \\ V_3(t)= & \displaystyle \sum _{j=1}^{n}[q^{(3)}_j+r^{(3)}_j]\displaystyle \int _{0}^{+\infty }K_j(\theta )\displaystyle \int _{t-\theta }^{t}H_j^2(y_j(s))\mathrm {d}s\mathrm{d}\theta \\ V_4(t)= & \sigma \displaystyle \int _{-\sigma }^{0}\int _{t+\theta }^{t}\dot{y}^T(s)C[Q_4+R_4]C\dot{y}(s)\mathrm {d}s\mathrm{d}\theta \end{aligned}$$
The remainder of the proof is similar to the proof of Theorem 2 and Theorem 3. \(\blacksquare \)
