Appendix A Proof of Theorem 1
Proof
Consider the following Lyapunov functional:
$$\begin{aligned} V_{1}(t)=\displaystyle \sum _{i=1}^{4}V_{1i}(t) \end{aligned}$$
(9)
where
$$\begin{aligned} V_{11}(t)= & {} \displaystyle \sum _{i=1}^{n}x_{i}(t)p_{i}x_{i}(t)+\displaystyle \sum _{i=1}^{n}y_{i}(t)q_{i}y_{i}(t),\\ V_{12}(t)= & {} \displaystyle \sum _{j=1}^{n}r^{1}_{ij}\int _{t-\tau (t)}^{t}g^{2}_{j}\big (y_{j}(s)\big )\mathrm{d}s+ \displaystyle \sum _{j=1}^{n}r^{2}_{ij}\int _{t-\rho (t)}^{t}g^{2}_{j}\big ({\dot{y}}_{j}(s)\big )\mathrm{d}s,\\ V_{13}(t)= & {} \displaystyle \sum _{j=1}^{n}\int _{t-\sigma (t)}^{t}\int _{s}^{t}g_{j}\big (y_{j}(s_{1})\big )w_{ij}g_{j}\big (y_{j}(s_{1})\big )\mathrm{d}s_{1}\mathrm{d}s\\&+\displaystyle \sum _{j=1}^{n}m^{1}_{ij}\int _{t-\eta (t)}^{t}\int _{s}^{t}x^{2}_{i}(s_{1})\mathrm{d}s_{1}\mathrm{d}s,\\ V_{14}(t)= & {} \displaystyle \sum _{j=1}^{n}m^{2}_{ij}\int _{t-\mu (t)}^{t}x_{i}^{2}(s)\mathrm{d}s+ \displaystyle \sum _{j=1}^{n}m^{3}_{ij}\int _{t-h(t)}^{t}{\dot{x}}^{2}_{j}(s)\mathrm{d}s. \end{aligned}$$
Calculating the time-derivative of \(V_{1}(\cdot )\) along the solution of system (3) and by using Lemma 1 we have:
$$\begin{aligned} {\dot{V}}_{11}(t)= & {} 2\displaystyle \sum _{i=1}^{n}x_i(t)p_{i}\big [-b_ix_i(t)+\displaystyle \sum _{j=1}^{n}c_{ij}g_j(y_j(t-\tau (t)))+\displaystyle \bigwedge _{j=1}^{n}\gamma _{ij}g_{j}(y_j(t-\tau (t)))\\&+\displaystyle \bigvee _{j=1}^{n}\delta _{ij}g_{j}(y_j(t-\tau (t))) +\displaystyle \bigwedge _{j=1}^{n}\alpha _{ij}\int _{t-\sigma (t)}^{t}g_j(y_j(s))\mathrm{d}s\\&+\displaystyle \bigvee _{j=1}^{n}\beta _{ij}\int _{t-\sigma (t)}^{t}g_j(y_j(s))\mathrm{d}s+\displaystyle \sum _{j=1}^{n}d_{ij}g_j({\dot{y}}_j(t-\rho (t)))\big ]\\&+2\displaystyle \sum _{i=1}^{n}y_i(t)q_{i}\big [-a_iy_i(t)+e_{i}x_i(t-\mu (t)) +\displaystyle \bigwedge _{j=1}^{n}{\bar{\gamma }}_{ij}x_{j}(t-\mu (t))\\&+\displaystyle \bigvee _{j=1}^{n}{\bar{\delta }}_{ij}x_{i}(t-\mu (t)) +\displaystyle \bigwedge _{j=1}^{n}{\bar{\alpha }}_{ij}\int _{t-\eta (t)}^{t}x_j(s)\mathrm{d}s+\displaystyle \bigvee _{j=1}^{n}{\bar{\beta }}_{ij}\int _{t-\eta (t)}^{t}x_{j}(s)\mathrm{d}s\\&+{\bar{d}}_{i}{\dot{x}}_i(t-h(t))]\\\le & {} -2\displaystyle \sum _{i=1}^{n}x_i(t)p_{i}b_ix_i(t)+2\displaystyle \sum _{i=1}^{n}\displaystyle \sum _{j=1}^{n}y_i(t)p_{i}[c_{ij}+|\gamma _{ij}|+|\delta _{ij}|]|g_j(y_j(t-\tau (t)))|\\&+2\displaystyle \sum _{i=1}^{n}\displaystyle \sum _{j=1}^{n}x_i(t)p_{i}[|\alpha _{ij}|+|\beta _{ij}|]|\int _{t-\sigma (t)}^{t}g_j(y_j(s))\mathrm{d}s|\\&+2\displaystyle \sum _{i=1}^{n}\displaystyle \sum _{j=1}^{n}x_i(t)p_{i}d_{ij}g_j({\dot{y}}_j(t-\rho (t)))\\&-2\displaystyle \sum _{i=1}^{n}y_i(t)q_{i}a_iy_i(t)+2\displaystyle \sum _{i=1}^{n}y_i(t)q_{i}[e_{i}+|{\bar{\gamma }}_{ij}|+|{\bar{\delta }}_{ij}|]|x_i(t-\mu (t))|\\&+2\displaystyle \sum _{i=1}^{n}\displaystyle \sum _{j=1}^{n}y_i(t)q_{i}[|{\bar{\alpha }}_{ij}|+|{\bar{\beta }}_{ij}|]|\int _{t-\eta (t)}^{t}x_j(s)\mathrm{d}s|+ 2\displaystyle \sum _{i=1}^{n}\displaystyle \sum _{j=1}^{n}y_i(t)q_{i}{\bar{d}}_{i}{\dot{x}}_i(t-h(t))\\\le & {} -2x^{T}(t)PBx(t)+2x^{T}(t)P[C+|\gamma |+|\delta |]|g(y(t-\tau (t)))|\\&+2x^{T}(t)P[|\alpha |+|\beta |]\int _{t-\sigma (t)}^{t}|g(y(s))|\mathrm{d}s+x^{T}(t)PDg({\dot{y}}(t-\rho (t)))\\&-2y^{T}(t)QAy(t)+2y^{T}(t)Q[E+|{\bar{\gamma }}|+|{\bar{\delta }}|]|x(t-\mu (t))|\\&+2y^{T}(t)Q[|{\bar{\alpha }}|+|{\bar{\beta }}|]\int _{t-\eta (t)}^{t}|x(s)|\mathrm{d}s+2y^{T}(t)Q{\bar{D}}{\dot{x}}(t-h(t)),\\ {\dot{V}}_{12}(t)= & {} \displaystyle \sum _{j=1}^{n}r^{1}_{ij}g^{2}_{j}\big (y_{j}(t)\big )-\displaystyle \sum _{j=1}^{n}(1-{\dot{\tau }}(t))r^{1}_{ij}g^{2}_{j}\big (y_{j}(t-\tau (t))\big )\\&+\displaystyle \sum _{j=1}^{n}r^{2}_{ij}g^{2}_{j}\big ({\dot{y}}_{j}(t)\big )-\displaystyle \sum _{j=1}^{n}(1-{\dot{\rho }}(t))r^{2}_{ij}g^{2}_{j}\big ({\dot{y}}_{j}(t-\rho (t))\big )\\\le & {} \displaystyle \sum _{j=1}^{n}r^{1}_{ij}g^{2}_{j}\big (y_{j}(t)\big )-\displaystyle \sum _{j=1}^{n}(1-{\tilde{\tau }})r^{1}_{ij}g^{2}_{j}\big (y_{j}(t-\tau (t))\big )\\&+\displaystyle \sum _{j=1}^{n}r^{2}_{ij}g^{2}_{j}\big ({\dot{y}}_{j}(t)\big )-\displaystyle \sum _{j=1}^{n}(1-{\tilde{\rho }})r^{2}_{ij}g^{2}_{j}\big ({\dot{y}}_{j}(t-\rho (t))\big )\\= & {} g^{T}\big (y(t)\big )R_{1}g\big (y(t)\big )-(1-{\tilde{\tau }})g^{T}\big (y(t-\tau (t))\big )R_{1}g\big (y(t-\tau (t))\big )\\&+g^{T}\big ({\dot{y}}(t)\big )R_{2}g\big ({\dot{y}}(t)\big )-(1-{\tilde{\rho }})g^{T}\big ({\dot{y}}(t-\rho (t))\big )R_{2}g\big ({\dot{y}}(t-\rho (t))\big )\\\le & {} g^{T}\big (y(t)\big )R_{1}g\big (y(t)\big )-(1-{\tilde{\tau }})g^{T}\big (y(t-\tau (t))\big )R_{1}g\big (y(t-\tau (t))\big )\\&+{\dot{y}}^{T}(t)G^{+}R_{2}G^{+}{\dot{y}}(t)-(1-{\tilde{\rho }})g^{T}\big ({\dot{y}}(t-\rho (t))\big )R_{2}g\big ({\dot{y}}(t-\rho (t))\big ),\\ \end{aligned}$$
$$\begin{aligned} {\dot{V}}_{13}(t)= & {} \displaystyle \sum _{j=1}^{n}\sigma (t)g_{j}\big (y_{j}(t)\big )w_{ij}g_{j}\big (y_{j}(t)\big ) -\displaystyle \sum _{j=1}^{n}\int _{t-\sigma (t)}^{t}g_{j}\big (y_{j}(s)\big )w_{ij}g_{j}\big (y_{j}(s)\big )\mathrm{d}s\\&+\displaystyle \sum _{j=1}^{n}m^{1}_{ij}\eta (t)x^{2}_{i}(t) -\displaystyle \sum _{j=1}^{n}m^{1}_{ij}\int _{t-\eta (t)}^{t}x^{2}_{i}(s)\mathrm{d}s\\\le & {} \sigma ^{*}g^{T}\big (y(t)\big )Wg\big (y(t)\big ) -\bigg (\displaystyle \int _{t-\sigma (t)}^{t}g\big (y(s)\big )\mathrm{d}s\bigg )^{T}W\bigg (\displaystyle \int _{t-\sigma (t)}^{t}g\big (y(s)\big )\mathrm{d}s\bigg )\\&+\eta ^{*}x^{T}(t)M_{1}x(t) -\bigg (\displaystyle \int _{t-\eta (t)}^{t}x(s)\mathrm{d}s\bigg )^{T}M_{1}\bigg (\displaystyle \int _{t-\eta (t)}^{t}x(s)\mathrm{d}s\bigg ),\\ {\dot{V}}_{14}(t)= & {} \displaystyle \sum _{j=1}^{n}m^{2}_{ij}x_{i}^{2}(t)-\displaystyle \sum _{j=1}^{n}(1-{\dot{\mu }}(t))m^{2}_{ij}x_{i}^{2}(t-\mu (t)) +\displaystyle \sum _{i=1}^{n}m^{3}_{ij}{\dot{x}}^{2}_{i}(t)\\&-\displaystyle \sum _{i=1}^{n}m^{3}_{ij}(1-{\dot{h}}(t)){\dot{x}}^{2}_{i}(t-h(t))\\\le & {} \displaystyle \sum _{j=1}^{n}m^{2}_{ij}x_{i}^{2}(t)-\displaystyle \sum _{j=1}^{n}(1-{\tilde{\mu }})m^{2}_{ij}x_{i}^{2}(t-\mu (t)) +\displaystyle \sum _{i=1}^{n}m^{3}_{ij}{\dot{x}}^{2}_{i}(t)\\&-\displaystyle \sum _{i=1}^{n}m^{3}_{ij}(1-{\tilde{h}}){\dot{x}}^{2}_{i}(t-h(t))\\= & {} x^{T}(t)M_{2}x(t)-(1-{\tilde{\mu }})x^{T}(t-\mu (t))M_{2}x(t-\mu (t)) +{\dot{x}}^{T}(t)M_{3}{\dot{x}}(t)\\&-(1-{\tilde{h}}){\dot{x}}^{T}(t-h(t))M_{3}{\dot{x}}(t-h(t)). \end{aligned}$$
Under the Assumption 3. the following inequality holds for any positive matrix \(\vartheta =(\vartheta _{ij})_{n\times n}\):
$$\begin{aligned} -g^{T}(y(t))\vartheta g(y(t))+y^{T}(t)G^{+}\vartheta G^{+}y(t)\ge 0. \end{aligned}$$
By using the Lemma 1 the following two inequalities are true
$$\begin{aligned} {\dot{x}}_i(t)= & {} -b_ix_i(t)+\displaystyle \sum _{j=1}^{n}c_{ij}g_j(y_j(t-\tau (t)))+\displaystyle \bigwedge _{j=1}^{n}\gamma _{ij}g_{j}(y_j(t-\tau (t)))\\&+\displaystyle \bigvee _{j=1}^{n}\delta _{ij}g_{j}(y_j(t-\tau (t))) +\displaystyle \bigwedge _{j=1}^{n}\alpha _{ij}\int _{t-\sigma (t)}^{t}g_j(y_j(s))\mathrm{d}s\\&+\displaystyle \bigvee _{j=1}^{n}\beta _{ij}\int _{t-\sigma (t)}^{t}g_j(y_j(s))\mathrm{d}s+\displaystyle \sum _{j=1}^{n}d_{ij}g_j({\dot{y}}_j(t-\rho (t)))\\\le & {} -Bx(t)+[C+|\gamma |+|\delta |]|g(y(t-\tau (t)))|+[|\alpha |+|\beta |]\int _{t-\sigma (t)}^{t}|g(y(s))|\mathrm{d}s\\&+Dg({\dot{y}}(t-\rho (t))),\\ \end{aligned}$$
and
$$\begin{aligned} {\dot{y}}_i(t)= & {} -a_iy_i(t)+e_{i}x_i(t-\mu _{i}(t)) +\displaystyle \bigwedge _{j=1}^{n}{\bar{\gamma }}_{ij}x_{j}(t-\mu (t))+\displaystyle \bigvee _{j=1}^{n}{\bar{\delta }}_{ij}x_{j}(t-\mu (t))\\&+\displaystyle \bigwedge _{j=1}^{n}{\bar{\alpha }}_{ij}\int _{t-\eta (t)}^{t}x_j(s)\mathrm{d}s +\displaystyle \bigvee _{j=1}^{n}{\bar{\beta }}_{ij}\int _{t-\eta (t)}^{t}x_{j}(s)\mathrm{d}s+{\bar{d}}_{i}{\dot{x}}_i(t-h(t)\\\le & {} -Ay(t)+[E+|{\bar{\gamma }}|+|{\bar{\delta }}|]|x(t-\mu (t))| +[|{\bar{\alpha }}|+|{\bar{\beta }}|]\int _{t-\eta (t)}^{t}|x(s)|\mathrm{d}s\\&+{\bar{D}}{\dot{x}}(t-h(t)), \end{aligned}$$
Therefore, for any positive matrix \(N_{1}=(n^{1}_{ij})_{n\times n}\) and \(N_{2}=(n^{2}_{ij})_{n\times n}:\)
$$\begin{aligned}&\big [2|{\dot{x}}(t)|^{T}N_{1}\big ]\big [-{\dot{x}}(t)-Bx(t)+[C+|\gamma |+|\delta |]|g(y(t-\tau (t)))|\\&\quad +[|\alpha |+|\beta |]\int _{t-\sigma (t)}^{t}|g(y(s))|\mathrm{d}s+Dg({\dot{y}}(t-\rho (t)))\big ]\ge 0,\\&\big [2|{\dot{y}}(t)|^{T}N_{2}\big ]\big [-{\dot{y}}(t)-Ay(t)+[E+|{\bar{\gamma }}|+|{\bar{\delta }}|]|x(t-\mu (t))|\\&\quad +[|{\bar{\alpha }}|+|{\bar{\beta }}|]\int _{t-\eta (t)}^{t}|x(s)|\mathrm{d}s\\&\quad +{\bar{D}}{\dot{x}}(t-h(t))\big ]\ge 0, \end{aligned}$$
which imply,
$$\begin{aligned} {\dot{V}}_{1}(t)\le & {} x^{T}(t)[-2PB+\eta ^{*}M_{1}+M_{2}]x(t)+2x^{T}(t)P(C+|\gamma |+|\delta |)|g(y(t-\tau (t)))|\\&+2x^{T}(t)P(|\alpha |+|\beta |)\int _{t-\sigma (t)}^{t}|g(y(s))|\mathrm{d}s+2x^{T}(t)PDg({\dot{y}}(t-\rho (t)))\\&+y^{T}(t)[-2QA+G^{+}\vartheta G^{+}]y(t) +2y^{T}(t)Q[E+|{\bar{\gamma }}|+|{\bar{\delta }}|]|x(t-\mu (t))|\\&+2y^{T}(t)Q[|{\bar{\alpha }}|+|{\bar{\beta }}|]\int _{t-\eta (t)}^{t}|x(s)|\mathrm{d}s+2y^{T}(t)Q{\bar{D}}{\dot{x}}(t-h(t))\\&+g^{T}\big (y(t)\big )[R_{1}+\sigma ^{*}W-\vartheta ]g\big (y(t)\big )-(1-{\tilde{\tau }})g^{T}\big (y(t-\tau (t))\big )R_{1}g\big (y(t-\tau (t))\big )\\&+|{\dot{y}}(t)|^{T}(G^{+}R_{2}G^{+}-2N_{2})|{\dot{y}}(t)|-(1-{\tilde{\rho }})g^{T}\big ({\dot{y}}(t-\rho (t))\big )R_{2}g\big ({\dot{y}}(t-\rho (t))\big )\\&-\bigg (\displaystyle \int _{t-\sigma (t)}^{t}|g\big (y(s)\big )|\mathrm{d}s\bigg )^{T}W\bigg (\displaystyle \int _{t-\sigma (t)}^{t}|g\big (y(s)\big )|\mathrm{d}s\bigg )\\&-\bigg (\displaystyle \int _{t-\eta (t)}^{t}|x(s)|\mathrm{d}s\bigg )^{T}M_{1}\bigg (\displaystyle \int _{t-\eta (t)}^{t}|x(s)|\mathrm{d}s\bigg )\\&-(1-{\tilde{\mu }})|x(t-\mu (t))|^{T}M_{2}|x(t-\mu (t))|+|{\dot{x}}(t)|^{T}(M_{3}-2N_{1})|{\dot{x}}(t)|\\&-(1-{\tilde{h}}){\dot{x}}^{T}(t-h(t))M_{3}{\dot{x}}(t-h(t)) -2|{\dot{x}}(t)|^{T}N_{1}Bx(t)\\&+2|{\dot{x}}(t)|^{T}N_{1}[C+|\gamma |+|\delta |]|g(y(t-\tau (t)))|\\&+2|{\dot{x}}(t)|^{T}N_{1}[|\alpha |+|\beta |]\int _{t-\sigma (t)}^{t}|g(y(s))|\mathrm{d}s\\&+2|{\dot{x}}(t)|^{T}N_{1}Dg({\dot{y}}(t-\rho (t)))-2 |{\dot{y}}(t)|^{T}N_{2}Ay(t)\\&+2|{\dot{y}}(t)|^{T}N_{2}[E+|{\bar{\gamma }}|+|{\bar{\delta }}|]|x(t-\mu (t))|\\&+2|{\dot{y}}(t)|^{T}N_{2}[|{\bar{\alpha }}|+|{\bar{\beta }}|]\int _{t-\eta (t)}^{t}|x(s)|\mathrm{d}s +2|{\dot{y}}(t)|^{T}N_{2}{\bar{D}}{\dot{x}}(t-h(t))\\\le & {} \Gamma ^{T}(t)\Sigma _{1}\Gamma (t), \end{aligned}$$
where,
$$\begin{aligned}&\Gamma ^{T}(t)=\bigg [x^{T}(t),\;g^{T}\big (y(t)\big ),\;|g\big (y(t-\tau (t))\big )|^{T},\;\bigg (\displaystyle \int _{t-\sigma (t)}^{t}|g\big (y(s)\big )|\mathrm{d}s\bigg )^{T},\\&\quad g^{T}\big ({\dot{y}}(t-\rho (t))\big ),\;|{\dot{x}}(t)|^{T},\;y^{T}(t),\;|x(t-\mu (t))|^{T},\;\bigg (\displaystyle \int _{t-\eta (t)}^{t}|x(s)|\mathrm{d}s\bigg )^{T},\\&\quad {\dot{x}}^{T}(t-h(t)),\;|{\dot{y}}(t)|^{T}\bigg ]^{T}. \end{aligned}$$
If the condition \(\Sigma _{1}<0\) holds, then \({\dot{V}}_{1}(t)<0.\) Therefore, the neutral-type FGRNs with mixed delays (3) is asymptotically stable. \(\square \)
Appendix B (Proof of Theorem 2)
Proof
Consider the following Lyapunov functional:
$$\begin{aligned} V_{2}(t)=\displaystyle \sum _{i=1}^{4}V_{2i}(t) \end{aligned}$$
(10)
where
$$\begin{aligned} V_{21}(t)= & {} \displaystyle \sum _{i=1}^{n}\zeta _{i}(t)p_{i}\zeta _{i}(t)+\displaystyle \sum _{i=1}^{n}\xi _{i}(t)q_{i}\xi _{i}(t),\\ V_{22}(t)= & {} \displaystyle \sum _{j=1}^{n}r^{1}_{ij}\int _{t-\tau (t)}^{t}f^{2}_{j}\big (\xi _{j}(s)\big )\mathrm{d}s+ \displaystyle \sum _{j=1}^{n}r^{2}_{ij}\int _{t-\rho (t)}^{t}f^{2}_{j}\big ({\dot{\xi }}_{j}(s)\big )\mathrm{d}s,\\ V_{23}(t)= & {} \displaystyle \sum _{j=1}^{n}\int _{t-\sigma (t)}^{t}\int _{s}^{t}f_{j}\big (\xi _{j}(s_{1})\big )w_{ij}f_{j}\big (\xi _{j}(s_{1})\big )\mathrm{d}s_{1}\mathrm{d}s\\&+\displaystyle \sum _{j=1}^{n}m^{1}_{ij}\int _{t-\eta (t)}^{t}\int _{s}^{t}\zeta ^{2}_{i}(s_{1})\mathrm{d}s_{1}\mathrm{d}s,\\ V_{24}(t)= & {} \displaystyle \sum _{j=1}^{n}m^{2}_{ij}\int _{t-\mu (t)}^{t}\zeta _{i}^{2}(s)\mathrm{d}s+ \displaystyle \sum _{j=1}^{n}m^{3}_{ij}\int _{t-h(t)}^{t}{\dot{\zeta }}^{2}_{j}(s)\mathrm{d}s. \end{aligned}$$
Calculating the time-derivative of \(V_{2}(\cdot )\) along the solution of system (1) and by using Lemma 1 we have:
$$\begin{aligned} {\dot{V}}_{21}(t)= & {} 2\displaystyle \sum _{i=1}^{n}\zeta _i(t)p_{i}\big [-b_i\zeta _i(t)+\displaystyle \sum _{j=1}^{n}c_{ij}f_j(\xi _j(t-\tau (t)))+\displaystyle \bigwedge _{j=1}^{n}\gamma _{ij}f_{j}(\xi _j(t-\tau (t)))\\&+\displaystyle \bigvee _{j=1}^{n}\delta _{ij}f_{j}(\xi _j(t-\tau (t))) +\displaystyle \bigwedge _{j=1}^{n}\alpha _{ij}\int _{t-\sigma (t)}^{t}f_j(\xi _j(s))\mathrm{d}s\\&+\displaystyle \bigvee _{j=1}^{n}\beta _{ij}\int _{t-\sigma (t)}^{t}f_j(\xi _j(s))\mathrm{d}s+\displaystyle \sum _{j=1}^{n}d_{ij}f_j\dot{(\xi }_j(t-\rho (t)))+\pi _{i}(t)\big ]\\&+2\displaystyle \sum _{i=1}^{n}\xi _i(t)q_{i}\big [-a_i\xi _i(t)+e_{i}\zeta _i(t-\mu (t)) +\displaystyle \bigwedge _{j=1}^{n}{\bar{\gamma }}_{ij}\zeta _{j}(t-\mu (t))\\&+\displaystyle \bigvee _{j=1}^{n}{\bar{\delta }}_{ij}\zeta _{i}(t-\mu (t)) +\displaystyle \bigwedge _{j=1}^{n}{\bar{\alpha }}_{ij}\int _{t-\eta (t)}^{t}\zeta _j(s) \mathrm{d}s+\displaystyle \bigvee _{j=1}^{n}{\bar{\beta }}_{ij}\int _{t-\eta (t)}^{t}\zeta _{j}(s)\mathrm{d}s\\&+{\bar{d}}_{i}{\dot{\zeta }}_i(t-h(t))]\\\le & {} -2\displaystyle \sum _{i=1}^{n}\zeta _i(t)p_{i}b_i\zeta _i(t)+2\displaystyle \sum _{i=1}^{n}\displaystyle \sum _{j=1}^{n}\zeta _i(t)p_{i}[c_{ij}+|\gamma _{ij}|+|\delta _{ij}|]|f_j(\xi _j(t-\tau (t)))|\\&+2\displaystyle \sum _{i=1}^{n}\displaystyle \sum _{j=1}^{n}\zeta _i(t)p_{i}[|\alpha _{ij}|+|\beta _{ij}|]|\int _{t-\sigma (t)}^{t}f_j(\xi _j(s))\mathrm{d}s|\\&+2\displaystyle \sum _{i=1}^{n}\displaystyle \sum _{j=1}^{n}\zeta _i(t)p_{i}d_{ij}f_j({\dot{\xi }}_j(t-\rho (t)))+2\displaystyle \sum _{i=1}^{n}\zeta _i(t)p_{i}\pi _{i}(t)\\&-2\displaystyle \sum _{i=1}^{n}\xi _i(t)q_{i}a_i\xi _i(t)+2\displaystyle \sum _{i=1}^{n}\xi _i(t)q_{i}[e_{i}+|{\bar{\gamma }}_{ij}|+|{\bar{\delta }}_{ij}|]|\zeta _i(t-\mu (t))|\\&+2\displaystyle \sum _{i=1}^{n}\displaystyle \sum _{j=1}^{n}\xi _i(t)q_{i}[|{\bar{\alpha }}_{ij}|+|{\bar{\beta }}_{ij}|]|\int _{t-\eta (t)}^{t}\zeta _j(s)\mathrm{d}s|+ 2\displaystyle \sum _{i=1}^{n}\displaystyle \sum _{j=1}^{n}\xi _i(t)q_{i}{\bar{d}}_{i}{\dot{\zeta }}_i(t-h_(t))\\\le & {} -2\zeta ^{T}(t)PB\zeta (t)+2\zeta ^{T}(t)P[C+|\gamma |+|\delta |]|f(\xi (t-\tau (t)))|\\&+2\zeta ^{T}(t)P[|\alpha |+|\beta |]\int _{t-\sigma (t)}^{t}|f(\xi (s))|\mathrm{d}s+\zeta ^{T}(t)PDf({\dot{\xi }}(t-\rho (t)))\\&+2\zeta ^{T}(t)P\pi (t)-2\xi ^{T}(t)QA\xi (t)+2\xi ^{T}(t)Q[E+|{\bar{\gamma }}|+|{\bar{\delta }}|]|\zeta (t-\mu (t))|\\&+2\xi ^{T}(t)Q[|{\bar{\alpha }}|+|{\bar{\beta }}|]\int _{t-\eta (t)}^{t}|\zeta (s)|\mathrm{d}s+2\xi ^{T}(t)Q{\bar{D}}{\dot{\zeta }}(t-h(t)),\\ {\dot{V}}_{22}(t)= & {} \displaystyle \sum _{j=1}^{n}r^{1}_{ij}f^{2}_{j}\big (\xi _{j}(t)\big )-\displaystyle \sum _{j=1}^{n}(1-{\dot{\tau }}_{j}(t))r^{1}_{ij}f^{2}_{j}\big (\xi _{j}(t-\tau _{j}(t))\big )\\&+\displaystyle \sum _{j=1}^{n}r^{2}_{ij}f^{2}_{j}\big ({\dot{\xi }}_{j}(t)\big )-\displaystyle \sum _{j=1}^{n}(1-{\dot{\rho }}_{j}(t))r^{2}_{ij}f^{2}_{j}\big ({\dot{\xi }}_{j}(t-\rho _{j}(t))\big )\\\le & {} \displaystyle \sum _{j=1}^{n}r^{1}_{ij}f^{2}_{j}\big (\xi _{j}(t)\big )-\displaystyle \sum _{j=1}^{n}(1-{\tilde{\tau }})r^{1}_{ij}f^{2}_{j}\big (\xi _{j}(t-\tau _{j}(t))\big )\\&+\displaystyle \sum _{j=1}^{n}r^{2}_{ij}f^{2}_{j}\big ({\dot{\xi }}_{j}(t)\big )-\displaystyle \sum _{j=1}^{n}(1-{\tilde{\rho }})r^{2}_{ij}f^{2}_{j}\big ({\dot{\xi }}_{j}(t-\rho _{j}(t))\big )\\= & {} f^{T}\big (\xi (t)\big )R_{1}f\big (\xi (t)\big )-(1-{\tilde{\tau }})f^{T}\big (\xi (t-\tau (t))\big )R_{1}f\big (\xi (t-\tau (t))\big )\\&+f^{T}\big ({\dot{\xi }}(t)\big )R_{2}f\big ({\dot{\xi }}(t)\big )-(1-{\tilde{\rho }})f^{T}\big ({\dot{\xi }}(t-\rho (t))\big )R_{2}f\big ({\dot{\xi }}(t-\rho (t))\big )\\\le & {} f^{T}\big (\xi (t)\big )R_{1}f\big (\xi (t)\big )-(1-{\tilde{\tau }})f^{T}\big (\xi (t-\tau (t))\big )R_{1}f\big (\xi (t-\tau (t))\big )\\&+{\dot{\xi }}^{T}(t)F^{+}R_{2}F^{+}{\dot{\xi }}(t)-(1-{\tilde{\rho }})f^{T}\big ({\dot{\xi }}(t-\rho (t))\big )R_{2}f\big ({\dot{\xi }}(t-\rho (t))\big ),\\ \end{aligned}$$
$$\begin{aligned} {\dot{V}}_{23}(t)= & {} \displaystyle \sum _{j=1}^{n}\sigma (t)f_{j}\big (\xi _{j}(t)\big )w_{ij}f_{j}\big (\xi _{j}(t)\big ) -\displaystyle \sum _{j=1}^{n}\int _{t-\sigma (t)}^{t}f_{j}\big (\xi _{j}(s)\big )w_{ij}f_{j}\big (\xi _{j}(s)\big )\mathrm{d}s\\&+\displaystyle \sum _{j=1}^{n}m^{1}_{ij}\eta (t)\zeta ^{2}_{i}(t) -\displaystyle \sum _{j=1}^{n}m^{1}_{ij}\int _{t-\eta (t)}^{t}\zeta ^{2}_{i}(s)\mathrm{d}s\\\le & {} \sigma ^{*}f^{T}\big (\xi (t)\big )Wf\big (\xi (t)\big ) -\bigg (\displaystyle \int _{t-\sigma (t)}^{t}f\big (\xi (s)\big )\mathrm{d}s\bigg )^{T}W\bigg (\displaystyle \int _{t-\sigma (t)}^{t}f\big (\xi (s)\big )\mathrm{d}s\bigg )\\&+\eta ^{*}\zeta ^{T}(t)M_{1}\zeta (t) -\bigg (\displaystyle \int _{t-\eta (t)}^{t}\zeta (s)\mathrm{d}s\bigg )^{T}M_{1}\bigg (\displaystyle \int _{t-\eta (t)}^{t}\zeta (s)\mathrm{d}s\bigg ),\\ {\dot{V}}_{24}(t)= & {} \displaystyle \sum _{j=1}^{n}m^{2}_{ij}\zeta _{i}^{2}(t)-\displaystyle \sum _{j=1}^{n}(1-{\dot{\mu }}(t))m^{2}_{ij}\zeta _{i}^{2}(t-\mu (t)) +\displaystyle \sum _{i=1}^{n}m^{3}_{ij}{\dot{\zeta }}^{2}_{i}(t)\\&-\displaystyle \sum _{i=1}^{n}m^{3}_{ij}(1-{\dot{h}}(t)){\dot{\zeta }}^{2}_{i}(t-h(t))\\\le & {} \displaystyle \sum _{j=1}^{n}m^{2}_{ij}\zeta _{i}^{2}(t)-\displaystyle \sum _{j=1}^{n}(1-{\tilde{\mu }})m^{2}_{ij}\zeta _{i}^{2}(t-\mu (t)) +\displaystyle \sum _{i=1}^{n}m^{3}_{ij}{\dot{\zeta }}^{2}_{i}(t)\\&-\displaystyle \sum _{i=1}^{n}m^{3}_{ij}(1-{\tilde{h}}){\dot{\zeta }}^{2}_{i}(t-h(t))\\= & {} \zeta ^{T}(t)M_{2}\zeta (t)-(1-{\tilde{\mu }})\zeta ^{T}(t-\mu (t))M_{2}\zeta (t-\mu (t)) +{\dot{\zeta }}^{T}(t)M_{3}{\dot{\zeta }}(t)\\&-(1-{\tilde{h}}){\dot{\zeta }}^{T}(t-h(t))M_{3}{\dot{\zeta }}(t-h(t)). \end{aligned}$$
Under the Assumption 1, the following inequality holds for any positive matrix \(\chi =(\chi _{ij})_{n\times n}:\)
$$\begin{aligned} -f^{T}(\xi (t))\chi f(\xi (t))+\xi ^{T}(t)F^{+}\chi F^{+}\xi (t)\ge 0. \end{aligned}$$
By utilizing the lemma 1, the following two inequalities are true
$$\begin{aligned} {\dot{\zeta }}_i(t)= & {} -b_i\zeta _i(t)+\displaystyle \sum _{j=1}^{n}c_{ij}g_j(\xi _j(t-\tau _{j}(t))) +\displaystyle \bigwedge _{j=1}^{n}\gamma _{ij}f_{j}(\xi _j(t-\tau (t)))\\&+\displaystyle \bigvee _{j=1}^{n}\delta _{ij}f_{j}(\xi _j(t-\tau (t))) +\displaystyle \bigwedge _{j=1}^{n}\alpha _{ij}\int _{t-\sigma (t)}^{t}f_j(\xi _j(s))\mathrm{d}s\\&+\displaystyle \bigvee _{j=1}^{n}\beta _{ij}\int _{t-\sigma (t)}^{t}f_j(\xi _j(s))\mathrm{d}s +\displaystyle \sum _{j=1}^{n}d_{ij}f_j({\dot{\xi }}_j(t-\rho (t)))+\pi _{i}(t)\\\le & {} -B\zeta (t)+[C+|\gamma |+|\delta |]|f(\xi (t-\tau (t)))|+[|\alpha |+|\beta |]\int _{t-\sigma (t)}^{t}|f(\xi (s))|\mathrm{d}s\\&+Df({\dot{\xi }}(t-\rho (t)))+\pi (t), \end{aligned}$$
and
$$\begin{aligned} {\dot{\xi }}_i(t)= & {} -a_i\xi _i(t)+e_{i}\zeta _i(t-\mu (t)) +\displaystyle \bigwedge _{j=1}^{n}{\bar{\gamma }}_{ij}\zeta _{j}(t-\mu (t))+\displaystyle \bigvee _{j=1}^{n}{\bar{\delta }}_{ij}\zeta _{j}(t-\mu (t))\\&+\displaystyle \bigwedge _{j=1}^{n}{\bar{\alpha }}_{ij}\int _{t-\eta (t)}^{t}\zeta _j(s)\mathrm{d}s +\displaystyle \bigvee _{j=1}^{n}{\bar{\beta }}_{ij}\int _{t-\eta (t)}^{t}\zeta _{j}(s)\mathrm{d}s+{\bar{d}}_{i}{\dot{\zeta }}_i(t-h(t)\\\le & {} -A\xi (t)+[E+|{\bar{\gamma }}|+|{\bar{\delta }}|]|\zeta (t-\mu (t))| +[|{\bar{\alpha }}|+|{\bar{\beta }}|]\int _{t-\eta (t)}^{t}|\zeta (s)|\mathrm{d}s\\&+{\bar{d}}{\dot{\zeta }}(t-h(t)), \end{aligned}$$
Therefore, for any positive diagonal matrix \(S_{1}=\text{ diag }\{s^{1},\cdots ,s^{n}\}\) and any positive matrix \(S_{2}=(s^{2}_{ij})_{n\times n},\) we have
$$\begin{aligned}&\big [2|{\dot{\zeta }}(t)|^{T}S_{1}\big ]\big [-{\dot{\zeta }}(t)-B\zeta (t)+[C+|\gamma |+|\delta |]|f(\xi (t-\tau (t)))|\\&\quad +[|\alpha |+|\beta |]\int _{t-\sigma (t)}^{t}|f(\xi (s))|\mathrm{d}s+Df({\dot{\xi }}(t-\rho (t)))+\pi (t)\big ]\ge 0,\\&\quad \big [2|{\dot{\xi }}(t)|^{T}S_{2}\big ]\big [-{\dot{\xi }}(t)-A\xi (t)+[E+|{\bar{\gamma }}|+|{\bar{\delta }}|]|\zeta (t-\mu (t))|\\&\quad +[|{\bar{\alpha }}|+|{\bar{\beta }}|]\int _{t-\eta (t)}^{t}|\zeta (s)|\mathrm{d}s\\&\quad +{\bar{D}}{\dot{\zeta }}(t-h(t))\big ]\ge 0. \end{aligned}$$
By using lemma 2
$$\begin{aligned}&2\zeta ^{T}(t)P\pi (t)\le \zeta ^{T}(t)P\zeta (t)+\pi ^{T}(t)P\pi (t),\\&\quad 2|{\dot{\zeta }}(t)|^{T}S_{1}\pi (t)\le |{\dot{\zeta }}(t)|^{T}S_{1}|{\dot{\zeta }}(t)|+\pi ^{T}(t)S_{1}\pi (t), \end{aligned}$$
which imply,
$$\begin{aligned} {\dot{V}}_{2}(t)\le & {} \zeta ^{T}(t)[-2PB+\eta ^{*}M_{1}+M_{2}+P+J_{1}]\zeta (t)\\&+2\zeta ^{T}(t)P(C+|\gamma |+|\delta |)|f(\xi (t-\tau (t)))|\\&+2\zeta ^{T}(t)P(|\alpha |+|\beta |)\int _{t-\sigma (t)}^{t}|f\xi (s))|\mathrm{d}s+2\zeta ^{T}(t)PDf({\dot{\xi }}(t-\rho (t)))\\&+\xi ^{T}(t)[-2QA+G^{+}\chi G^{+}+J_{2}]\xi (t)\\&+2\xi ^{T}(t)Q[E+|{\bar{\gamma }}|+|{\bar{\delta }}|]|\zeta (t-\mu (t))| +2\xi ^{T}(t)Q[|{\bar{\alpha }}|+|{\bar{\beta }}|]\int _{t-\eta (t)}^{t}|\zeta (s)|\mathrm{d}s\\&+2\xi ^{T}(t)Q{\bar{D}}{\dot{\zeta }}(t-h(t))+f^{T}\big (\xi (t)\big )[R_{1}+\sigma ^{*}W-\chi ]f\big (\xi (t)\big )\\&-(1-{\tilde{\tau }})f^{T}\big (\xi (t-\tau (t))\big )R_{1}f\big (\xi (t-\tau (t))\big ) +|{\dot{\xi }}(t)|^{T}(F^{+}R_{2}F^{+}-2S_{2})|{\dot{\xi }}(t)|\\&-(1-{\tilde{\rho }})f^{T}\big ({\dot{\xi }}(t-\rho (t))\big )R_{2}f\big ({\dot{\xi }}(t-\rho (t))\big )\\&-\bigg (\displaystyle \int _{t-\sigma (t)}^{t}|f\big (\xi (s)\big )|\mathrm{d}s\bigg )^{T}W\bigg (\displaystyle \int _{t-\sigma (t)}^{t}|f\big (\xi (s)\big )|\mathrm{d}s\bigg )\\&-\bigg (\displaystyle \int _{t-\eta (t)}^{t}|\zeta (s)|\mathrm{d}s\bigg )^{T}M_{1}\bigg (\displaystyle \int _{t-\eta (t)}^{t}|\zeta (s)|\mathrm{d}s\bigg )\\&-(1-{\tilde{\mu }})|\zeta (t-\mu (t))|^{T}M_{2}|\zeta (t-\mu (t))|+|{\dot{\zeta }}(t)|^{T}(M_{3}-S_{1})|{\dot{\zeta }}(t)|\\&-(1-{\tilde{h}}){\dot{\zeta }}^{T}(t-h(t))M_{3}{\dot{\zeta }}(t-h(t)) -2|{\dot{\zeta }}(t)|^{T}S_{1}B\zeta (t)\\&+2|{\dot{\zeta }}(t)|^{T}S_{1}[C+|\gamma |+|\delta |]|f(\xi (t-\tau (t)))|\\&+2|{\dot{\zeta }}(t)|^{T}S_{1}[|\alpha |+|\beta |]\int _{t-\sigma (t)}^{t}|f(\xi (s))|\mathrm{d}s\\&+2|{\dot{\zeta }}(t)|^{T}S_{1}Df({\dot{\xi }}(t-\rho (t)))\\&-2 |{\dot{\xi }}(t)|^{T}S_{2}A\xi (t)+2|{\dot{\xi }}(t)|^{T}S_{2}[E+|{\bar{\gamma }}|+|{\bar{\delta }}|]|\zeta (t-\mu (t))|\\&+2|{\dot{\xi }}(t)|^{T}S_{2}[|{\bar{\alpha }}|+|{\bar{\beta }}|]\int _{t-\eta (t)}^{t}|\zeta (s)|\mathrm{d}s +2|{\dot{\xi }}(t)|^{T}S_{2}{\bar{D}}{\dot{\xi }}(t-h(t))\\&+{\tilde{\pi }}^{T}P{\tilde{\pi }}+{\tilde{\pi }}^{T}S_{1}{\tilde{\pi }}-\zeta ^{T}(t)J_{1}\zeta (t)-\xi ^{T}(t)J_{2}\xi (t)\\\le & {} \triangle ^{T}(t)\Sigma _{2}\triangle (t)-\zeta ^{T}(t)J_{1}\zeta (t)+{\tilde{\pi }}^{T}S_{1}{\tilde{\pi }}-\xi ^{T}(t)J_{2}\xi (t)+{\tilde{\pi }}^{T}P{\tilde{\pi }}, \end{aligned}$$
where,
$$\begin{aligned}&\triangle ^{T}(t)=\bigg [\zeta ^{T}(t),\;f^{T}\big (\xi (t)\big ),\;|f\big (\xi (t-\tau (t))\big )|^{T},\;\bigg (\displaystyle \int _{t-\sigma (t)}^{t}|f\big (\xi (s)\big )|\mathrm{d}s\bigg )^{T},\\&\quad f^{T}\big ({\dot{\xi }}(t-\rho (t))\big ),\;|{\dot{\zeta }}(t)|^{T},\;\xi ^{T}(t),\;|\zeta (t-\mu (t))|^{T},\;\bigg (\displaystyle \int _{t-\eta (t)}^{t}|\zeta (s)|\mathrm{d}s\bigg )^{T},\\&\quad {\dot{\zeta }}^{T}(t-h(t)),\;|{\dot{\xi }}(t)|^{T}\bigg ]^{T}. \end{aligned}$$
We choose positive constants \({\mathcal {N}}_{1},\;{\mathcal {N}}_{2}\) such that \({\mathcal {N}}_{1}+{\mathcal {N}}_{2}=1,\;{\mathcal {N}}_{1}\ne 0,\;{\mathcal {N}}_{2}\ne 0.\) Then, by the condition \(\Sigma _{2}<0,\) one obtains:
$$\begin{aligned} {\dot{V}}_{2}(t)\le & {} -\zeta ^{T}(t)J_{1}\zeta (t)-\xi ^{T}(t)J_{2}\xi (t)+({\mathcal {N}}_{1}+{\mathcal {N}}_{2}){\tilde{\pi }}^{T}(P+S_{1}){\tilde{\pi }}, \end{aligned}$$
(11)
when, \((\zeta ^{T}(t),\;\xi ^{T}(t))^{T}\in {\mathbb {R}}^{2n}\diagdown \Upsilon ,\) i.e. \((\zeta ^{T}(t),\;\xi ^{T}(t))^{T}\notin \Upsilon .\) Accordingly, if \((\varphi ^{T}(s),\;\psi ^{T}(s))^{T}\in \Upsilon _{1},\) then \((\zeta ^{T}(t),\xi ^{T}(t))^{T}\subseteq \Upsilon ,\;t\ge 0,\) which results the set \(\Upsilon \) is a positive invariant set of (1). If \((\varphi ^{T}(s),\;\psi ^{T}(s))^{T}\notin \Upsilon ,\) there exist some \({\mathcal {T}}>0\) such that \((\zeta ^{T}(t),\;\xi ^{T}(t))^{T}\subseteq \Upsilon ,\;t\ge t_{0}+{\mathcal {T}},\) which results the system (1) is a globally dissipative system and \(\Upsilon \) is a globally attractive set and a positive invariant set of (1). \(\square \)
