1 Introduction

The neutral differential equation is a retarded system that often appears in many scientific and engineering fields such as aircraft, chemical and process control systems, and biological systems [1–3]. The problem of various stability analyses for dynamical systems with state delays has been intensively studied in the past years by several researchers in mathematics [1–37]. However, delay-dependent stability criteria for neutral differential equations have been attracting the attention of several researchers. Delay-dependent stability criteria make use of information on the length of delays. A certain neutral differential equation (CNDE) with constant delays is of the form

$$\begin{aligned} \frac{d}{dt} \bigl[x(t)+px(t-\tau) \bigr]= -ax(t)+b\tanh x(t- \sigma),\quad t\geq0, \end{aligned}$$
(1.1)

where a, b, τ, σ are positive real constants and \(|p|<1\). For each solution \(x(t)\) of (1.1), we assume the initial condition

$$\begin{aligned} x_{0}(t)=\phi(t), \quad t\in[-r,0], \end{aligned}$$

where \(\phi\in C([-r,0];R)\) denotes the space of all continuous vector functions mapping \([-r,0]\) into R with \(r=\max\{\tau,\sigma\}\). Over the past decades, the problem of asymptotic stability analysis for (1.1) has been discussed in [14, 21, 29, 30, 34, 35] by using several model transformation methods and the Lyapunov-Krasovskii functional approach, while the problem of exponential stability analysis has been studied with the use of the model transformation technique and the Lyapunov-Krasovskii functional approach in [34]. In [7, 8, 23], the authors studied the problem of exponential stability analysis for CNDE with time-varying delays of the form

$$\begin{aligned} \frac{d}{dt} \bigl[x(t)+px \bigl(t-\tau(t) \bigr) \bigr]= -ax(t)+b\tanh x \bigl(t-\sigma(t) \bigr),\quad t\geq0, \end{aligned}$$
(1.2)

where a, b are positive real constants and \(|p|<1\). \(\tau(t)\) and \(\sigma(t)\) are neutral and discrete time-varying delays, respectively,

$$\begin{aligned} &0 \leq\tau(t) \leq\tau, \qquad\dot{\tau}(t) < \tau_{d}, \\ &0 \leq\sigma(t) \leq\sigma,\qquad \dot{\sigma}(t) < \sigma_{d}, \end{aligned}$$

where τ, σ, \(\tau_{d}\), and \(\sigma_{d}\) are given positive real constants. For each solution \(x(t)\) of (1.2), we assume the initial condition

$$\begin{aligned} x_{0}(t)=\phi(t), \quad t\in[-r,0], \end{aligned}$$

where \(\phi\in C([-r,0];R)\). In [7], the results are derived without the use of the model transformation method and the bounding technique, while the authors have used the model transformation method, radially unboundedness, and the Lyapunov-Krasovskii functional approach in [23]. Stability analysis of uncertain neutral stochastic systems with time-varying delays has received the attention of a lot of theoreticians and engineers in this field over the last few decades [9–13]. Moreover, the authors have studied the problem of stability for systems with discrete and distributed delays such as [36], which presented some stability conditions for uncertain neutral systems with discrete and distributed delays. The robust stability of uncertain dynamical systems with discrete and distributed delays has been studied in [12, 17, 26, 27].

This research presents new criteria based on new methods with mixed model transformation techniques. We investigate the problem of exponential stability criteria for CNDE with discrete and distributed time-varying delays. The time-varying delays are assumed to belong to the given lower and upper bound delays and restrictions on the derivative of the time-varying delays are needed. Based on the combination of a mixed model transformation, decomposition technique of constant coefficients, utilization of a zero equation, and a new Lyapunov-Krasovskii functional, sufficient conditions for exponential stability are obtained and formulated in terms of LMIs for the systems. Finally, numerical examples suggest that the proposed criteria are effective and an improvement over previous ones.

2 Problem formulation and preliminaries

Consider the CNDE with mixed interval time-varying delays of the form

$$\begin{aligned} \frac{d}{dt} \bigl[x(t)+px \bigl(t-\tau(t) \bigr) \bigr]= -ax(t)+b\tanh x \bigl(t-\sigma(t) \bigr)+c \int ^{t}_{t-\rho(t)}x(s)\,ds,\quad t\geq0, \end{aligned}$$
(2.1)

where a, b, c are positive real constants and \(|p|<1\). \(\tau(t)\), \(\sigma(t)\), and \(\rho(t)\) are neutral, discrete, and distributed interval time-varying delays, respectively,

$$\begin{aligned} &0 \leq\tau_{1}\leq\tau(t) \leq\tau_{2} , \qquad\dot{\tau}(t) \leq \tau_{d}< \infty, \end{aligned}$$
(2.2)
$$\begin{aligned} &0 \leq\sigma_{1} \leq\sigma(t) \leq\sigma_{2},\qquad \dot{\sigma}(t) \leq\sigma_{d}< \infty, \end{aligned}$$
(2.3)
$$\begin{aligned} &0 \leq\rho_{1} \leq\rho(t) \leq\rho_{2}, \end{aligned}$$
(2.4)

where \(\tau_{1}\), \(\tau_{2}\), \(\sigma_{1}\), \(\sigma_{2}\), \(\rho_{1}\), \(\rho_{2}\), \(\tau_{d}\), and \(\sigma_{d}\) are given positive real constants. For each solution \(x(t)\) of (2.1), we assume the initial condition

$$x_{0}(t)=\phi(t), \quad t\in[-\omega,0], $$

where \(\phi\in C([-\omega,0];R)\) and \(\omega=\max\{\tau_{2},\sigma_{2},\rho_{2}\}\).

Definition 2.1

[20]

Equation (2.1) is exponentially stable, if there exist positive real constants α, β such that, for each \(\phi(t)\in C([-\omega,0],R)\), the solution \(x(t,\phi)\) of the system satisfies

$$\bigl\| x(t,\phi)\bigr\| \leq\beta\|\phi\|e^{-\alpha t},\quad t\geq0. $$

Lemma 2.2

[16] (Jensen’s inequality)

For any symmetric positive definite matrix Q, positive real number h, and vector function \(\dot{x}(t):[-h,0]\rightarrow R^{n}\) the following integral is well defined:

$$-h \int_{-h}^{0}\dot{x}^{T}(s+t)Q \dot{x}(s+t)\,ds\leq - \biggl( \int_{-h}^{0}\dot{x}(s+t)\,ds \biggr)^{T}Q \biggl( \int_{-h}^{0}\dot {x}(s+t)\,ds \biggr). $$

Lemma 2.3

For any constant symmetric positive definite matrix \(Q\in R^{n\times n}\), \(h(t)\) a discrete time-varying delay with (2.4), the vector function \(\omega:[-h_{2},0]\rightarrow {R}^{n}\) such that the integrations concerned are well defined, we have

$$\begin{aligned} &{-}[h_{2}-h_{1}] \int_{-h_{2}}^{-h_{1}}\omega^{T}(s)Q\omega(s)\,ds \\ &\quad\leq- \int_{-h(t)}^{-h_{1}}\omega^{T}(s)\,dsQ \int_{-h(t)}^{-h_{1}}\omega (s)\,ds- \int_{-h_{2}}^{-h(t)}\omega^{T}(s)\,dsQ \int_{-h_{2}}^{-h(t)}\omega (s)\,ds. \end{aligned}$$

Proof

It is easy to see that

$$\begin{aligned} &[h_{2}-h_{1}] \int_{-h_{2}}^{-h_{1}}\omega^{T}(s)Q\omega(s)\,ds \\ &\quad= [h_{2}-h_{1}] \int_{-h(t)}^{-h_{1}}\omega^{T}(s)Q\omega (s)\,ds+[h_{2}-h_{1}] \int_{-h_{2}}^{-h(t)}\omega^{T}(s)Q\omega (s)\,ds \\ &\quad\geq \bigl[h(t)-h_{1} \bigr] \int_{-h(t)}^{-h_{1}}\omega^{T}(s)Q\omega(s)\,ds+ \bigl[h_{2}-h(t) \bigr] \int_{-h_{2}}^{-h(t)}\omega^{T}(s)Q\omega(s)\,ds \\ &\quad= \frac{1}{2} \int_{-h(t)}^{-h_{1}} \int_{-h(t)}^{-h_{1}}\omega ^{T}(s)Q\omega(s)+ \omega^{T}(\xi)Q\omega(\xi)\,ds\,d\xi \\ &\qquad+\frac{1}{2} \int_{-h_{2}}^{-h(t)} \int_{-h_{2}}^{-h(t)}\omega ^{T}(s)Q\omega(s)+ \omega^{T}(\xi)Q\omega(\xi)\,ds\,d\xi \\ &\quad\geq \frac{1}{2} \int_{-h(t)}^{-h_{1}} \int_{-h(t)}^{-h_{1}}2\omega ^{T}(s){Q^{1/2}}^{T}{Q^{1/2}} \omega(\xi)\,ds\,d\xi \\ &\qquad+\frac{1}{2} \int_{-h_{2}}^{-h(t)} \int_{-h_{2}}^{-h(t)}2\omega ^{T}(s){Q^{1/2}}^{T}{Q^{1/2}} \omega(\xi)\,ds\,d\xi \\ &\quad= \int_{-h(t)}^{-h_{1}}\omega^{T}(s)\,dsQ \int_{-h(t)}^{-h_{1}}\omega (s)\,ds+ \int_{-h_{2}}^{-h(t)}\omega^{T}(s)\,dsQ \int_{-h_{2}}^{-h(t)}\omega (s)\,ds. \end{aligned}$$

This completes the proof. □

Remark 2.4

In Lemma 2.3, we have modified the method of [19].

3 Main results

In this section, we investigate the exponential stability problem for equation (2.1) with interval time-varying delays satisfying (2.2)-(2.4). From the model transformation method, we have the Leibniz-Newton formula of the form

$$\begin{aligned} &0=x(t)-x \bigl(t-\tau(t) \bigr)- \int_{t-\tau(t)}^{t}\dot{x}(s)\,ds, \end{aligned}$$
(3.1)
$$\begin{aligned} &0= x(t)- x \bigl(t-\gamma\tau(t) \bigr)- \int_{t-\gamma\tau(t)}^{t}\dot{x}(s)\,ds, \end{aligned}$$
(3.2)

where γ is a given positive real constant. We utilize the zero equations and obtain

$$\begin{aligned} &0=r_{1} x(t)-r_{1} x \bigl(t-\tau(t) \bigr)-r_{1} \int_{t-\tau(t)}^{t}\dot{x}(s)\,ds, \end{aligned}$$
(3.3)
$$\begin{aligned} &0=r_{2} x(t)-r_{2} x \bigl(t-\gamma\tau(t) \bigr)-r_{2} \int_{t-\gamma\tau(t)}^{t}\dot{x}(s)\,ds, \end{aligned}$$
(3.4)

where \(r_{1},r_{2}\in R\) will be chosen to guarantee the exponential stability of equation (2.1). By (3.1)-(3.4), equation (2.1) can be represented by the form

$$\begin{aligned} &\frac{d}{dt} \biggl[p_{1}x(t)+p_{2}x \bigl(t-\tau(t) \bigr)+x \bigl(t-\gamma\tau(t) \bigr)+ \int _{t-\gamma\tau(t)}^{t}\dot{x}(s)\,ds-p_{1} \int_{t-\tau(t)}^{t}\dot{x}(s)\,ds \biggr] \\ &\quad=-(a_{1}-r_{1}-r_{2})x(t)-(a_{2}+r_{1})x \bigl(t-\tau(t) \bigr)-(a_{2}+r_{1}) \int _{t-\tau(t)}^{t}\dot{x}(s)\,ds \\ &\quad\quad-r_{2} x \bigl(t- \gamma\tau(t) \bigr)-r_{2} \int_{t-\gamma\tau(t)}^{t}\dot{x}(s)\,ds \\ &\qquad+b\tanh x \bigl(t-\sigma(t) \bigr)+c \int^{t}_{t-\rho(t)}x(s)\,ds. \end{aligned}$$
(3.5)

For convenience, we define a new variable,

$$\begin{aligned} D(t)=p_{1}x(t)+p_{2}x \bigl(t-\tau(t) \bigr)+x \bigl(t-\gamma\tau(t) \bigr)+ \int_{t-\gamma\tau (t)}^{t}\dot{x}(s)\,ds-p_{1} \int_{t-\tau(t)}^{t}\dot{x}(s)\,ds. \end{aligned}$$
(3.6)

Rewrite equation (3.5) in the following equation:

$$\begin{aligned} \dot{D}(t) =&-(a_{1}-r_{1}-r_{2})x(t)-(a_{2}+r_{1})x \bigl(t-\tau (t) \bigr)-(a_{2}+r_{1}) \int_{t-\tau(t)}^{t}\dot{x}(s)\,ds \\ &{}-r_{2} x \bigl(t- \gamma\tau(t) \bigr)-r_{2} \int_{t-\gamma\tau(t)}^{t}\dot{x}(s)\,ds \\ &{}+b\tanh x \bigl(t-\sigma(t) \bigr)+c \int^{t}_{t-\rho(t)}x(s)\,ds. \end{aligned}$$
(3.7)

We introduce the following notations for later use:

$$\begin{aligned} \Sigma= [\Omega_{(i,j)}] _{25\times25}, \end{aligned}$$
(3.8)

where \(\Omega_{(i,j)}=\Omega_{(j,i)}\),

$$\begin{aligned} &\Omega_{(1,1)}=2k_{1}\alpha-2q_{1}, \\ &\Omega_{(1,2)}=q_{1}p_{1}-q_{2}+m_{1}+n_{1}+k_{1}r_{2}+k_{1}r_{1}-k_{1}a_{1}, \\ &\Omega_{(1,3)}=q_{1}p_{2}-q_{3}-m_{1}-k_{1}a_{2}-k_{1}r_{1}, \\ &\Omega_{(1,4)}=-q_{1}p_{1}-q_{4}-m_{1}-k_{1}a_{2}-k_{1}r_{1}, \\ &\Omega_{(1,5)}=q_{1}-q_{5}-n_{1}-k_{1}r_{2}, \\ &\Omega_{(1,6)}=q_{1}-q_{6}-n_{1}-k_{1}r_{2}, \\ &\Omega_{(1,7)}=\Omega_{(1,8)}=\Omega_{(1,9)}= \Omega_{(1,10)}=0, \\ &\Omega_{(1,11)}=k_{1}b, \\ &\Omega_{(1,12)}=\Omega_{(1,13)}=\Omega_{(1,14)}= \Omega_{(1,15)}=\Omega_{(1,16)}=\Omega_{(1,17)}= \Omega_{(1,18)}=\Omega_{(1,19)}=\Omega_{(1,20)}= \Omega_{(1,21)}=0, \\ &\Omega_{(1,22)}=-m_{9}, \\ &\Omega_{(1,23)}=\Omega_{(1,24)}=0, \\ &\Omega_{(1,25)}=k_{1}c, \\ &\Omega_{(2,2)}=2q_{2}p_{1}+k_{2}+k_{3}+k_{4}+k_{5}+k_{6} \tau_{2}^{2}+k_{7}\gamma^{2} \tau_{2}^{2}+k_{8}+k_{9} \sigma_{2}^{2}+2m_{2}+2n_{2} \\ &\hphantom{\Omega_{(2,2)}=}+w_{1}+w_{2}+w_{5}\tau_{1}^{2}+w_{6} \gamma^{2}\tau_{1}^{2}+w_{7}( \tau_{2}-\tau_{1})^{2}+w_{8} \gamma^{2}(\tau_{2}-\tau_{1})^{2}+w_{10}( \sigma_{2}-\sigma_{1})^{2}\\ &\hphantom{\Omega_{(2,2)}=}+k_{10}+ \omega(\rho_{2}-\rho_{1})^{2}, \\ &\Omega_{(2,3)}=q_{2}p_{2}+q_{3}p_{1}+m_{3}+n_{3}-m_{2}, \\ &\Omega_{(2,4)}=-q_{2}p_{1}+q_{4}p_{1}+m_{5}+n_{5}-m_{2}, \\ &\Omega_{(2,5)}=q_{2}+q_{5}p_{1}+m_{4}+n_{4}-n_{2}, \\ &\Omega_{(2,6)}=q_{2}+q_{6}p_{1}+m_{6}+n_{6}-n_{2}, \\ &\Omega_{(2,7)}=\Omega_{(2,8)}=\Omega_{(2,9)}= \Omega_{(2,10)}=0, \\ &\Omega_{(2,11)}=m_{7}+n_{7}, \\ &\Omega_{(2,12)}=m_{8}+n_{8}, \\ &\Omega_{(2,13)}=-q_{7}a_{1}, \\ &\Omega_{(2,14)}=q_{7}r_{1}+q_{7}r_{2}-q_{7}a_{1}, \\ &\Omega_{(2,15)}=\Omega_{(2,16)}=\Omega_{(2,17)}= \Omega_{(2,18)}=\Omega_{(2,19)}=\Omega_{(2,20)}= \Omega_{(2,21)}=0, \\ &\Omega_{(2,22)}=m_{9}+n_{9}, \\ &\Omega_{(2,23)}=\Omega_{(2,24)}=\Omega_{(2,25)}=0, \\ &\Omega_{(3,3)}=2q_{3}p_{2}-k_{3}e^{-2\alpha\tau_{2}}+k_{3}t_{d}-2m_{3}, \\ &\Omega_{(3,4)}=-q_{3}p_{1}+q_{4}p_{2}-m_{5}-m_{3}, \\ &\Omega_{(3,5)}=q_{3}+q_{5}p_{2}-m_{4}-n_{3}, \\ &\Omega_{(3,6)}=q_{3}+q_{6}p_{2}-m_{6}-n_{3}, \\ &\Omega_{(3,7)}=\Omega_{(3,8)}=\Omega_{(3,9)}= \Omega_{(3,10)}=0, \\ &\Omega_{(3,11)}=-m_{7}, \\ &\Omega_{(3,12)}=-m_{8}, \\ &\Omega_{(3,13)}=-q_{7}a_{2}, \\ &\Omega_{(3,14)}=\Omega_{(3,15)}=\Omega_{(3,16)}= \Omega_{(3,17)}=\Omega_{(3,18)}=\Omega_{(3,19)}= \Omega_{(3,20)}=\Omega_{(3,21)}=0, \\ &\Omega_{(3,22)}=-m_{9}, \\ &\Omega_{(3,23)}=\Omega_{(3,24)}=\Omega_{(3,25)}=0, \\ &\Omega_{(4,4)}=-2q_{4}p_{1}-2m_{5}, \\ &\Omega_{(4,5)}=q_{4}-q_{5}p_{1}-m_{4}-n_{5}, \\ &\Omega_{(4,6)}=q_{4}-q_{6}p_{1}-m_{6}-n_{5}, \\ &\Omega_{(4,7)}=\Omega_{(4,8)}=\Omega_{(4,9)}= \Omega_{(4,10)}=0, \\ &\Omega_{(4,11)}=-m_{7}, \\ &\Omega_{(4,12)}=-m_{8}, \\ &\Omega_{(4,13)}=-q_{7}a_{2}, \\ &\Omega_{(4,14)}=\Omega_{(4,15)}=\Omega_{(4,16)}= \Omega_{(4,17)}=\Omega_{(4,18)}=\Omega_{(4,19)}= \Omega_{(4,20)}=\Omega_{(4,21)}=0, \\ &\Omega_{(4,22)}=-m_{9}, \\ &\Omega_{(4,23)}=\Omega_{(4,24)}=\Omega_{(4,25)}=0, \\ &\Omega_{(5,5)}=2q_{5}-k_{5}e^{-2\alpha\gamma\tau_{2}}+k_{5}t_{d}-n_{4}, \\ &\Omega_{(5,6)}=q_{5}+q_{6}-n_{6}-n_{4}, \\ &\Omega_{(5,7)}=\Omega_{(5,8)}=\Omega_{(5,9)}= \Omega_{(5,10)}=0, \\ &\Omega_{(5,11)}=-n_{7}, \\ &\Omega_{(5,12)}=-n_{8}, \\ &\Omega_{(5,13)}=0, \\ &\Omega_{(5,14)}=\Omega_{(5,15)}=\Omega_{(5,16)}= \Omega_{(5,17)}=\Omega_{(5,18)}=\Omega_{(5,19)}= \Omega_{(5,20)}=\Omega_{(5,21)}=0, \\ &\Omega_{(5,22)}=-n_{9}, \\ &\Omega_{(5,23)}=\Omega_{(5,24)}=\Omega_{(5,25)}=0, \\ &\Omega_{(6,6)}=2q_{6}-2n_{6}, \\ &\Omega_{(6,7)}=\Omega_{(6,8)}=\Omega_{(6,9)}= \Omega_{(6,10)}=0, \\ &\Omega_{(6,11)}=-n_{7}, \\ &\Omega_{(6,12)}=-n_{8}, \\ &\Omega_{(6,13)}=0, \\ &\Omega_{(6,14)}=\Omega_{(6,15)}=\Omega_{(6,16)}= \Omega_{(6,17)}=\Omega_{(6,18)}=\Omega_{(6,19)}= \Omega_{(6,20)}=\Omega_{(6,21)}=0, \\ &\Omega_{(6,22)}=-n_{9}, \\ &\Omega_{(6,23)}=\Omega_{(6,24)}=\Omega_{(6,25)}=0, \\ &\Omega_{(7,7)}=-(k_{2}+w_{3})e^{-2\alpha\tau_{2}}, \\ &\Omega_{(7,8)}=\Omega_{(7,9)}=\Omega_{(7,10)}= \Omega_{(7,11)}=\Omega_{(7,12)}=\Omega_{(7,13)}= \Omega_{(7,14)}=\Omega_{(7,15)}=\Omega_{(7,16)}= \Omega_{(7,17)}\\ &\hphantom{\Omega_{(7,8)}}=\Omega_{(7,18)}=\Omega_{(7,19)}= \Omega_{(7,20)}=\Omega_{(7,21)}=\Omega_{(7,22)}= \Omega_{(7,23)}=\Omega_{(7,24)}=\Omega_{(7,25)}=0, \\ &\Omega_{(8,8)}=-k_{6}e^{-2\alpha\tau_{2}}, \\ &\Omega_{(8,9)}=\Omega_{(8,10)}=\Omega_{(8,11)}= \Omega_{(8,12)}=\Omega_{(8,13)}=\Omega_{(8,14)}= \Omega_{(8,15)}=\Omega_{(8,16)}=\Omega_{(8,17)}= \Omega_{(8,18)}\\ &\hphantom{\Omega_{(8,9)}}=\Omega_{(8,19)}=\Omega_{(8,20)}= \Omega_{(8,21)}=\Omega_{(8,22)}=\Omega_{(8,23)}= \Omega_{(8,24)}=\Omega_{(8,25)}=0, \\ &\Omega_{(9,9)}=-(k_{4}+w_{4})e^{-2\alpha\gamma\tau_{2}}, \\ &\Omega_{(9,10)}=\Omega_{(9,11)}=\Omega_{(9,12)}= \Omega_{(9,13)}=\Omega_{(9,14)}=\Omega_{(9,15)}= \Omega_{(9,16)}=\Omega_{(9,17)}=\Omega_{(9,18)}= \Omega_{(9,19)}\\ &\hphantom{\Omega_{(9,10)}}=\Omega_{(9,20)}=\Omega_{(9,21)}= \Omega_{(9,22)}=\Omega_{(9,23)}=\Omega_{(9,24)}= \Omega_{(9,25)}=0, \\ &\Omega_{(10,10)}=-k_{7}e^{-2\alpha\gamma\tau_{2}}, \\ &\Omega_{(10,11)}=\Omega_{(10,12)}=\Omega_{(10,13)}= \Omega_{(10,14)}=\Omega_{(10,15)}=\Omega_{(10,16)}= \Omega_{(10,17)}=\Omega_{(10,18)}=\Omega_{(10,19)}\\ &\hphantom{\Omega_{(10,11)}}= \Omega_{(10,20)}=\Omega_{(10,21)}=\Omega_{(10,22)}= \Omega_{(10,23)}=\Omega_{(10,24)}=\Omega_{(10,25)}=0, \\ &\Omega_{(11,11)}=-k_{8}e^{-2\alpha\sigma_{2}}+k_{8}s_{d}-k_{10}, \\ &\Omega_{(11,12)}=0, \\ &\Omega_{(11,13)}=q_{7}b, \\ &\Omega_{(11,14)}=\Omega_{(11,15)}=\Omega_{(11,16)}= \Omega_{(11,17)}=\Omega_{(11,18)}=\Omega_{(11,19)}= \Omega_{(11,20)}=\Omega_{(11,21)}=\Omega_{(11,22)}\\ &\hphantom{\Omega_{(11,14)}}= \Omega_{(11,23)}=\Omega_{(11,24)}=\Omega_{(11,25)}=0, \\ &\Omega_{(12,12)}=-k_{9}e^{-2\alpha\sigma_{2}}, \\ &\Omega_{(12,13)}=\Omega_{(12,14)}=\Omega_{(12,15)}= \Omega_{(12,16)}= \Omega_{(12,17)}=\Omega_{(12,18)}= \Omega_{(12,19)}=\Omega_{(12,20)}=\Omega_{(12,21)}\\ &\hphantom{\Omega_{(12,13)}}= \Omega_{(12,22)}=\Omega_{(12,23)}=\Omega_{(12,24)}= \Omega_{(12,25)}=0, \\ &\Omega_{(13,13)}=-2q_{7}, \\ &\Omega_{(13,14)}=\Omega_{(13,15)}=\Omega_{(13,16)}= \Omega_{(13,17)}=\Omega_{(13,18)}=\Omega_{(13,19)}= \Omega_{(13,20)}=\Omega_{(13,21)}=\Omega_{(13,22)}\\ &\hphantom{\Omega_{(13,14)}}= \Omega_{(13,23)}=\Omega_{(13,24)}=\Omega_{(13,25)}=0, \\ &\Omega_{(14,14)}=(w_{3}-w_{1})e^{-2\alpha\tau_{1}}, \\ &\Omega_{(14,15)}=\Omega_{(14,16)}=\Omega_{(14,17)}= \Omega_{(14,18)}=\Omega_{(14,19)}=\Omega_{(14,20)}= \Omega_{(14,21)}=\Omega_{(14,22)}=\Omega_{(14,23)}\\ &\hphantom{\Omega_{(14,15)}}= \Omega_{(14,24)}=\Omega_{(14,25)}=0, \\ &\Omega_{(15,15)}=(w_{4}-w_{2})e^{-2\alpha\gamma\tau_{1}}, \\ &\Omega_{(15,16)}=\Omega_{(15,17)}=\Omega_{(15,18)}= \Omega_{(15,19)}=\Omega_{(15,20)}=\Omega_{(15,21)}= \Omega_{(15,22)}=\Omega_{(15,23)}=\Omega_{(15,24)}\\ &\hphantom{\Omega_{(15,16)}}= \Omega_{(15,25)}=0, \\ &\Omega_{(16,16)}=-w_{5}e^{-2\alpha\tau_{1}}, \\ &\Omega_{(16,17)}=\Omega_{(16,18)}=\Omega_{(16,19)}= \Omega_{(16,20)}=\Omega_{(16,21)}=\Omega_{(16,22)}= \Omega_{(16,23)}=\Omega_{(16,24)}\\ &\hphantom{\Omega_{(16,17)}}=\Omega_{(16,25)}=0, \\ &\Omega_{(17,17)}=-w_{6}e^{-2\alpha\gamma\tau_{1}}, \\ &\Omega_{(17,18)}=\Omega_{(17,19)}=\Omega_{(17,20)}= \Omega_{(17,21)}=\Omega_{(17,22)}=\Omega_{(17,23)}= \Omega_{(17,24)}=\Omega_{(17,25)}=0, \\ &\Omega_{(18,18)}=-w_{7}e^{-2\alpha\tau_{2}}, \\ &\Omega_{(18,19)}=\Omega_{(18,20)}=\Omega_{(18,21)}= \Omega_{(18,22)}=\Omega_{(18,23)}=\Omega_{(18,24)}= \Omega_{(18,25)}=0, \\ &\Omega_{(19,19)}=-w_{7}e^{-2\alpha\tau_{2}}, \\ &\Omega_{(19,20)}=\Omega_{(19,21)}=\Omega_{(19,22)}= \Omega_{(19,23)}=\Omega_{(19,24)}=\Omega_{(19,25)}=0, \\ &\Omega_{(20,20)}=-w_{8}e^{-2\alpha\gamma\tau_{2}}, \\ &\Omega_{(20,21)}=\Omega_{(20,22)}=\Omega_{(20,23)}= \Omega_{(20,24)}=\Omega_{(20,25)}=0, \\ &\Omega_{(21,21)}=-w_{8}e^{-2\alpha\gamma\tau_{2}}, \\ &\Omega_{(21,22)}=\Omega_{(21,23)}=\Omega_{(21,24)}= \Omega_{(21,25)}=0, \\ &\Omega_{(22,22)}=-w_{9}e^{-2\alpha\sigma_{1}}, \\ &\Omega_{(22,23)}=\Omega_{(22,24)}=\Omega_{(22,25)}=0, \\ &\Omega_{(23,23)}=-w_{10}e^{-2\alpha\sigma_{2}}, \\ &\Omega_{(23,24)}=\Omega_{(23,25)}=0, \\ &\Omega_{(24,24)}=-w_{10}e^{-2\alpha\sigma_{2}}, \\ &\Omega_{(24,25)}=0, \\ &\Omega_{(25,25)}=-\omega e^{-2\alpha\rho_{2}}. \end{aligned}$$

The exponential stability for the CNDE with time-varying delays in equation (2.1) will be represented as follows.

Theorem 3.1

For given positive real constants \(\sigma_{1}\), \(\sigma_{2}\), \(\sigma_{d}\), \(\tau_{1}\), \(\tau_{2}\), \(\tau_{d}\), \(\rho_{1}\), \(\rho_{2}\) and γ, equation (2.1) is exponentially stable with a decay rate α if there exist positive real constants ω, \(k_{i}\), \(w_{i}\) where \(i=1,2,\ldots,10\), and real constants \(r_{1}\), \(r_{2}\), \(m_{k}\), \(n_{k}\) where \(k=1,2,\ldots,9\) such that the following symmetric linear matrix inequality holds:

$$\begin{aligned} \Sigma< 0. \end{aligned}$$
(3.9)

Proof

For ω, \(k_{i}\) and \(w_{i}\) are positive real constants where \(i=1,2,\ldots,10\), we consider the Lyapunov-Krasovskii functional candidate for equation (3.7) of the form

$$\begin{aligned} V(t,x_{t})= \sum^{5}_{i=1}V_{i}(t,x_{t}), \end{aligned}$$
(3.10)

where

$$\begin{aligned} &V_{1}(t,x_{t})=k_{1}D^{2}(t), \\ &V_{2}(t,x_{t})=k_{2} \int_{t-\tau_{2}}^{t}e^{2\alpha (s-t)}x^{2}(s)\,ds+k_{3} \int_{t-\tau(t)}^{t}e^{2\alpha (s-t)}x^{2}(s)\,ds \\ &\hphantom{V_{2}(t,x_{t})=}+k_{4} \int_{t-\gamma\tau_{2}}^{t}e^{2\alpha (s-t)}x^{2}(s)\,ds+k_{5} \int_{t-\gamma\tau(t)}^{t}e^{2\alpha (s-t)}x^{2}(s)\,ds \\ & \hphantom{V_{2}(t,x_{t})=}+w_{1} \int_{t-\tau_{1}}^{t}e^{2\alpha(s-t)}x^{2}(s)\,ds+w_{2} \int _{t-\gamma\tau_{1}}^{t}e^{2\alpha(s-t)}x^{2}(s)\,ds \\ & \hphantom{V_{2}(t,x_{t})=}+w_{3} \int_{t-\tau_{2}}^{t-\tau_{1}}e^{2\alpha (s-t)}x^{2}(s)\,ds+w_{4} \int_{t-\gamma\tau_{2}}^{t-\gamma\tau _{1}}e^{2\alpha(s-t)}x^{2}(s)\,ds, \\ &V_{3}(t,x_{t})=k_{6}\tau_{2} \int_{-\tau_{2}}^{0}{ \int _{t+s}^{t}e^{2\alpha(\theta-t)} x^{2}( \theta)\,d\theta}\,ds+k_{7}\gamma\tau _{2} \int_{-\gamma\tau_{2}}^{0}{ \int_{t+s}^{t}e^{2\alpha(\theta-t)} x^{2}( \theta)\,d\theta}\,ds \\ & \hphantom{V_{3}(t,x_{t})=}+w_{5}\tau_{1} \int_{-\tau_{1}}^{0}{ \int_{t+s}^{t}e^{2\alpha(\theta -t)} x^{2}( \theta)\,d\theta}\,ds\\ &\hphantom{V_{3}(t,x_{t})=}+w_{6}\gamma\tau_{1} \int_{-\gamma\tau _{1}}^{0}{ \int_{t+s}^{t}e^{2\alpha(\theta-t)} x^{2}( \theta)\,d\theta }\,ds \\ & \hphantom{V_{3}(t,x_{t})=}+w_{7}(\tau_{2}-\tau_{1}) \int_{-\tau_{2}}^{-\tau_{1}}{ \int _{t+s}^{t}e^{2\alpha(\theta-t)} x^{2}( \theta)\,d\theta}\,ds \\ & \hphantom{V_{3}(t,x_{t})=}+w_{8}\gamma(\tau_{2}-\tau_{1}) \int_{-\gamma\tau_{2}}^{-\gamma\tau _{1}}{ \int_{t+s}^{t}e^{2\alpha(\theta-t)} x^{2}( \theta)\,d\theta }\,ds, \\ &V_{4}(t,x_{t})=k_{8} \int_{t-\sigma(t)}^{t}e^{2\alpha(s-t)}\tanh ^{2}x(s)\,ds \\ & \hphantom{V_{4}(t,x_{t})=}+k_{9}\sigma_{2} \int_{-\sigma_{2}}^{0}{ \int_{t+s}^{t}e^{2\alpha (\theta-t)}\tanh^{2}x( \theta)\,d\theta}\,ds \\ & \hphantom{V_{4}(t,x_{t})=}+w_{9}\sigma_{1} \int_{-\sigma_{1}}^{0}{ \int_{t+s}^{t}e^{2\alpha (\theta-t)}\tanh^{2}x( \theta)\,d\theta}\,ds \\ & \hphantom{V_{4}(t,x_{t})=}+w_{10}(\sigma_{2}-\sigma_{1}) \int_{-\sigma_{2}}^{-\sigma_{1}}{ \int _{t+s}^{t}e^{2\alpha(\theta-t)}\tanh^{2}x( \theta)\,d\theta}\,ds, \\ &V_{5}(t,x_{t})=\omega(\rho_{2}- \rho_{1}) \int_{-\rho_{2}}^{-\rho _{1}}{ \int_{t+s}^{t}e^{2\alpha(\theta-t)} x^{2}( \theta)\,d\theta}\,ds. \end{aligned}$$

Calculating the time derivatives of \(V(t,x_{t})\) along the solution of equation (3.7) yields

$$\begin{aligned} \dot{V}(t,x_{t})= \sum^{5}_{i=1} \dot{V_{i}}(t,x_{t}). \end{aligned}$$
(3.11)

The time derivatives of \(V_{1}(t,x_{t})\) and \(V_{2}(t,x_{t})\) are calculated as

$$\begin{aligned} &\dot{V}_{1}(t,x_{t})=2k_{1}D(t)\dot{D}(t) \\ &\hphantom{\dot{V}_{1}(t,x_{t})}=2k_{1}D(t) \biggl[-(a_{1}-r_{1}-r_{2})x(t)-(a_{2}+r_{1})x \bigl(t-\tau (t) \bigr)-(a_{2}+r_{1}) \int_{t-\tau(t)}^{t}\dot{x}(s)\,ds \\ &\hphantom{\dot{V}_{1}(t,x_{t})=}-r_{2} x \bigl(t-\gamma\tau(t) \bigr)-r_{2} \int_{t-\gamma\tau(t)}^{t}\dot{x}(s)\,ds+b\tanh x \bigl(t-\sigma(t) \bigr)+c \int^{t}_{t-\rho(t)}x(s)\,ds \biggr] \\ &\hphantom{\dot{V}_{1}(t,x_{t})=}+2q_{1}D(t) \biggl[-D(t)+p_{1}x(t)+p_{2}x \bigl(t-\tau(t) \bigr)+x \bigl(t-\gamma\tau (t) \bigr) \\ & \hphantom{\dot{V}_{1}(t,x_{t})=}+ \int_{t-\gamma\tau(t)}^{t}\dot{x}(s)\,ds-p_{1} \int_{t-\tau(t)}^{t}\dot{x}(s)\,ds \biggr] \\ & \hphantom{\dot{V}_{1}(t,x_{t})=}+2q_{2}x(t) \biggl[-D(t)+p_{1}x(t)+p_{2}x \bigl(t-\tau(t) \bigr)+x \bigl(t-\gamma\tau (t) \bigr) \\ & \hphantom{\dot{V}_{1}(t,x_{t})=}+ \int_{t-\gamma\tau(t)}^{t}\dot{x}(s)\,ds-p_{1} \int_{t-\tau(t)}^{t}\dot{x}(s)\,ds \biggr] \\ & \hphantom{\dot{V}_{1}(t,x_{t})=}+2q_{3}x \bigl(t-\tau(t) \bigr) \biggl[-D(t)+p_{1}x(t)+p_{2}x \bigl(t-\tau(t) \bigr)+x \bigl(t-\gamma \tau(t) \bigr) \\ & \hphantom{\dot{V}_{1}(t,x_{t})=}+ \int_{t-\gamma\tau(t)}^{t}\dot{x}(s)\,ds-p_{1} \int_{t-\tau(t)}^{t}\dot{x}(s)\,ds \biggr] \\ & \hphantom{\dot{V}_{1}(t,x_{t})=}+2q_{4} \int_{t-\tau(t)}^{t}\dot{x}(s)\,ds \biggl[-D(t)+p_{1}x(t)+p_{2}x \bigl(t-\tau(t) \bigr)+x \bigl(t-\gamma\tau(t) \bigr) \\ &\hphantom{\dot{V}_{1}(t,x_{t})=}+ \int_{t-\gamma\tau(t)}^{t}\dot{x}(s)\,ds-p_{1} \int_{t-\tau(t)}^{t}\dot{x}(s)\,ds \biggr] \\ & \hphantom{\dot{V}_{1}(t,x_{t})=}+2q_{5}x \bigl(t-\gamma\tau(t) \bigr) \biggl[-D(t)+p_{1}x(t)+p_{2}x \bigl(t-\tau (t) \bigr)+x \bigl(t-\gamma\tau(t) \bigr) \\ & \hphantom{\dot{V}_{1}(t,x_{t})=}+ \int_{t-\gamma\tau(t)}^{t}\dot{x}(s)\,ds-p_{1} \int_{t-\tau(t)}^{t}\dot{x}(s)\,ds \biggr] \\ & \hphantom{\dot{V}_{1}(t,x_{t})=}+2q_{6} \int_{t-\gamma\tau(t)}^{t}\dot{x}(s)\,ds \biggl[-D(t)+p_{1}x(t)+p_{2}x \bigl(t-\tau(t) \bigr)+x \bigl(t-\gamma\tau(t) \bigr) \\ &\hphantom{\dot{V}_{1}(t,x_{t})=}+ \int_{t-\gamma\tau(t)}^{t}\dot{x}(s)\,ds-p_{1} \int_{t-\tau(t)}^{t}\dot{x}(s)\,ds \biggr] \\ & \hphantom{\dot{V}_{1}(t,x_{t})=}+2q_{7}\dot{D}(t) \biggl[-\dot{D}(t)-a_{1}x(t)-a_{2}x \bigl(t-\tau (t) \bigr) \\ & \hphantom{\dot{V}_{1}(t,x_{t})=}-a_{2} \int_{t-\tau(t)}^{t}\dot{x}(s)\,ds +b\tanh x \bigl(t-\sigma(t) \bigr)+c \int^{t}_{t-\rho(t)}x(s)\,ds \biggr] \\ & \hphantom{\dot{V}_{1}(t,x_{t})=}+2\alpha k_{1}D^{2}(t)-2\alpha V_{1}(t), \end{aligned}$$
(3.12)
$$\begin{aligned} &\dot{V}_{2}(t,x_{t})=(k_{2}+k_{3}+k_{4}+k_{5}+w_{1}+w_{2})x^{2}(t)-(k_{2}+w_{3})e^{-2\alpha \tau_{2}}x^{2}(t- \tau_{2}) \\ &\hphantom{\dot{V}_{2}(t,x_{t})=}-k_{3} \bigl(1-\dot{\tau}(t) \bigr)e^{-2\alpha\tau(t)}x^{2} \bigl(t-\tau (t) \bigr)-(k_{4}+w_{4})e^{-2\alpha\gamma\tau_{2}}x^{2}(t- \gamma\tau _{2}) \\ & \hphantom{\dot{V}_{2}(t,x_{t})=}-k_{5} \bigl(1-\gamma\dot{\tau}(t) \bigr)e^{-2\alpha\gamma\tau(t)}x^{2} \bigl(t-\gamma \tau(t) \bigr) \\ &\hphantom{\dot{V}_{2}(t,x_{t})=}+(w_{3}-w_{1})e^{-2\alpha\tau_{1}}x^{2}(t- \tau_{1})(w_{4}-w_{2})e^{-2\alpha\gamma\tau_{1}}x^{2}(t-\gamma \tau _{1})-2\alpha V_{2}(t) \\ &\hphantom{\dot{V}_{2}(t,x_{t})}\leq (k_{2}+k_{3}+k_{4}+k_{5}+w_{1}+w_{2})x^{2}(t)-(k_{2}+w_{3})e^{-2\alpha \tau_{2}}x^{2}(t- \tau_{2}) \\ & \hphantom{\dot{V}_{2}(t,x_{t})=}-k_{3}e^{-2\alpha\tau_{2}}x^{2} \bigl(t-\tau(t) \bigr)+k_{3}\tau_{d}x^{2} \bigl(t-\tau (t) \bigr) \\ & \hphantom{\dot{V}_{2}(t,x_{t})=}-(k_{4}+w_{4})e^{-2\alpha\gamma\tau_{2}}x^{2}(t-\gamma \tau _{2})-k_{5}e^{-2\alpha\gamma\tau_{2}}x^{2} \bigl(t- \gamma\tau(t) \bigr) \\ & \hphantom{\dot{V}_{2}(t,x_{t})=}+k_{5}\gamma\tau_{d}x^{2} \bigl(t-\gamma \tau(t) \bigr) \\ & \hphantom{\dot{V}_{2}(t,x_{t})=}+(w_{3}-w_{1})e^{-2\alpha \tau_{1}}x^{2}(t- \tau_{1})(w_{4}-w_{2})e^{-2\alpha\gamma\tau_{1}}x^{2}(t-\gamma \tau _{1})-2\alpha V_{2}(t). \end{aligned}$$
(3.13)

Obviously, for any a scalar \(s\in[t-\tau_{2},t]\), we get \(e^{-2\alpha\tau_{2}}\leq e^{2\alpha(s-t)}\leq1\) and \(e^{-2\alpha\gamma\tau_{2}}\leq e^{2\alpha(s-t)}\leq1\), for any a scalar \(s\in[t-\gamma\tau_{2},t]\). Together with Lemma 2.2 and 2.3, we obtain

$$\begin{aligned} \dot{V}_{3}(t,x_{t}) =&k_{6}\tau_{2} \int_{-\tau_{2}}^{0} x^{2}(t)\,ds-k_{6} \tau_{2} \int_{-\tau_{2}}^{0}e^{2s\alpha} x^{2}(t+s)\,ds \\ &{}+k_{7}\gamma\tau_{2} \int_{-\gamma\tau_{2}}^{0} x^{2}(t)\,ds-k_{7} \gamma \tau_{2} \int_{-\gamma\tau_{2}}^{0}e^{2s\alpha} x^{2}(t+s)\,ds \\ &{}+w_{5}\tau_{1} \int_{-\tau_{1}}^{0} x^{2}(t)\,ds-w_{5} \tau_{1} \int _{-\tau_{1}}^{0}e^{2s\alpha} x^{2}(t+s)\,ds \\ &{}+w_{6}\gamma\tau_{1} \int_{-\gamma\tau_{1}}^{0} x^{2}(t)\,ds-w_{6} \gamma \tau_{1} \int_{-\gamma\tau_{1}}^{0}e^{2s\alpha} x^{2}(t+s)\,ds \\ &{}+w_{7}(\tau_{2}-\tau_{1}) \int_{-\tau_{2}}^{-\tau_{1}} x^{2}(t)\,ds-w_{7}( \tau_{2}-\tau_{1}) \int_{-\tau_{2}}^{-\tau _{1}}e^{2s\alpha} x^{2}(t+s)\,ds \\ &{}+w_{8}(\gamma\tau_{2}-\gamma\tau_{1}) \int_{-\gamma\tau_{2}}^{-\gamma \tau_{1}} x^{2}(t)\,ds-w_{8}( \gamma\tau_{2}-\gamma\tau_{1}) \int_{-\gamma \tau_{2}}^{-\gamma\tau_{1}}e^{2s\alpha} x^{2}(t+s)\,ds \\ &{}-2\alpha V_{3}(t) \\ \leq&k_{6}(\tau_{2})^{2} x^{2}(t)-k_{6}e^{-2\alpha\tau_{2}} \biggl( \int _{t-\tau_{2}}^{t} x(s)\,ds \biggr)^{2} \\ &{}+k_{7}(\gamma\tau_{2})^{2} x^{2}(t)-k_{7}e^{-2\alpha\gamma\tau _{2}} \biggl( \int_{t-\gamma\tau_{2}}^{t} x(s)\,ds \biggr)^{2} \\ &{}+w_{5}(\tau_{1})^{2} x^{2}(t)-w_{5}e^{-2\alpha\tau_{1}} \biggl( \int _{t-\tau_{1}}^{t} x(s)\,ds \biggr)^{2} \\ &{}+w_{6}(\gamma\tau_{1})^{2} x^{2}(t)-w_{6}e^{-2\alpha\gamma\tau _{1}} \biggl( \int_{t-\gamma\tau_{1}}^{t} x(s)\,ds \biggr)^{2} \\ &{}+w_{7}(\tau_{2}-\tau_{1})^{2}x^{2}(t)+w_{8}( \gamma\tau_{2}-\gamma\tau _{1})^{2}x^{2}(t) \\ &{}-w_{7}e^{-2\alpha\tau_{2}} \biggl( \int_{t-\tau(t)}^{t-\tau_{1}} x(s)\,ds \biggr)^{2}-w_{7}e^{-2\alpha\tau_{2}} \biggl( \int_{t-\tau_{2}}^{t-\tau (t)} x(s)\,ds \biggr)^{2} \\ &{}-w_{8}e^{-2\gamma\alpha\tau_{2}} \biggl( \int_{t-\gamma\tau(t)}^{t-\gamma \tau_{1}} x(s)\,ds \biggr)^{2}-w_{8}e^{-2\gamma\alpha\tau_{2}} \biggl( \int _{t-\gamma\tau_{2}}^{t-\gamma\tau(t)} x(s)\,ds \biggr)^{2} \\ &{}-2\alpha V_{3}(t). \end{aligned}$$
(3.14)

From Lemma 2.2, 2.3, and \(\tanh^{2}x(t)\leq x^{2}(t)\), we have

$$\begin{aligned} &\dot{V}_{4}(t,x_{t})=k_{8} \tanh^{2}x(t)-k_{8} \bigl(1-\dot{\sigma}(t) \bigr)e^{-2\alpha\sigma(t)}\tanh^{2}x \bigl(t-\sigma(t) \bigr) \\ & \hphantom{\dot{V}_{4}(t,x_{t})=}+k_{9}\sigma_{2} \int_{-\sigma_{2}}^{0}\tanh^{2}x(t)\,ds-k_{9} \sigma _{2} \int_{-\sigma_{2}}^{0}e^{2\alpha s}\tanh^{2}x(t+s)\,ds \\ & \hphantom{\dot{V}_{4}(t,x_{t})=}+w_{9}\sigma_{1} \int_{-\sigma_{1}}^{0}\tanh^{2}x(t)\,ds-w_{9} \sigma _{1} \int_{-\sigma_{1}}^{0}e^{2\alpha s}\tanh^{2}x(t+s)\,ds \\ & \hphantom{\dot{V}_{4}(t,x_{t})=}+w_{10}(\sigma_{2}-\sigma_{1}) \int_{-\sigma_{2}}^{-\sigma_{1}}\tanh ^{2}x(t)\,ds \\ & \hphantom{\dot{V}_{4}(t,x_{t})=}-w_{10}(\sigma_{2}-\sigma_{1}) \int_{-\sigma_{2}}^{-\sigma _{1}}e^{2\alpha s}\tanh^{2}x(t+s)\,ds \\ & \hphantom{\dot{V}_{4}(t,x_{t})=}-2\alpha V_{4}(t) \\ &\hphantom{\dot{V}_{4}(t,x_{t})}\leq k_{8}\tanh^{2}x(t)-k_{8}e^{-2\alpha\sigma_{2}} \tanh^{2}x \bigl(t-\sigma(t) \bigr) +k_{8}s_{d} \tanh^{2}x \bigl(t-\sigma(t) \bigr) \\ &\hphantom{\dot{V}_{4}(t,x_{t})=}+k_{9}\sigma_{2}^{2}x^{2}(t)-k_{9}e^{-2\alpha\sigma_{2}} \biggl( \int _{t-\sigma_{2}}^{t}\tanh x(s)\,ds \biggr)^{2} \\ & \hphantom{\dot{V}_{4}(t,x_{t})=}+w_{9}\sigma_{1}^{2}x^{2}(t)-w_{9}e^{-2\alpha\sigma_{1}} \biggl( \int _{t-\sigma_{1}}^{t}\tanh x(s)\,ds \biggr)^{2} \\ &\hphantom{\dot{V}_{4}(t,x_{t})=}+w_{10}(\sigma_{2}-\sigma_{1})^{2}x^{2}(t)-w_{10}e^{-2\alpha\tau _{2}} \biggl( \int_{t-\tau(t)}^{t-\tau_{1}} \tanh x(s)\,ds \biggr)^{2} \\ & \hphantom{\dot{V}_{4}(t,x_{t})=}+k_{10}x^{2}(t)-k_{10}\tanh^{2}x(t) \\ & \hphantom{\dot{V}_{4}(t,x_{t})=}-w_{10}e^{-2\alpha\tau_{2}} \biggl( \int_{t-\tau_{2}}^{t-\tau(t)} \tanh x(s)\,ds \biggr)^{2} \\ & \hphantom{\dot{V}_{4}(t,x_{t})=}-2 \alpha V_{4}(t), \end{aligned}$$
(3.15)
$$\begin{aligned} &\dot{V}_{5}(t,x_{t})\leq\omega(\rho_{2}- \rho_{1})^{2}x^{2}(t)-\omega e^{-2\alpha\rho_{2}} \biggl( \int_{t-\rho(t)}^{t} x(s)\,ds \biggr)^{2}. \end{aligned}$$
(3.16)

From the Leibniz-Newton formula, the following equations are true for any real constants \(m_{i}\), \(n_{i}\), \(i=1,2,\ldots,9\) with appropriate dimensions:

$$\begin{aligned} &2 \biggl[m_{1}D(t)+m_{2}x(t)+m_{3}x \bigl(t- \tau(t) \bigr)+m_{4}x \bigl(t-\gamma\tau(t) \bigr)+m_{5} \int _{t-\tau(t)}^{t}\dot{x}(s)\,ds \\ & \quad+m_{6} \int_{t-\gamma\tau(t)}^{t}\dot{x}(s)\,ds+m_{7}\tanh x \bigl(t-\sigma (t) \bigr)+m_{8} \int_{t-\sigma_{2}}^{t}\tanh x(s)\,ds \\ & \quad+m_{9} \int_{t-\sigma_{1}}^{t}\tanh x(s)\,ds \biggr] \times \biggl[x(t)-x \bigl(t-\tau(t) \bigr)- \int_{t-\tau(t)}^{t}\dot{x}(s)\,ds \biggr]=0, \end{aligned}$$
(3.17)
$$\begin{aligned} &2 \biggl[n_{1}D(t)+n_{2}x(t)+n_{3}x \bigl(t-\tau(t) \bigr)+n_{4}x \bigl(t-\gamma\tau(t) \bigr)+n_{5} \int _{t-\tau(t)}^{t}\dot{x}(s)\,ds \\ & \quad+n_{6} \int_{t-\gamma\tau(t)}^{t}\dot{x}(s)\,ds+n_{7}\tanh x \bigl(t-\sigma (t) \bigr)+n_{8} \int_{t-\sigma_{2}}^{t}\tanh x(s)\,ds \\ & \quad+n_{9} \int_{t-\sigma_{1}}^{t}\tanh x(s)\,ds \biggr] \times \biggl[x(t)-x \bigl(t-\gamma\tau(t) \bigr)- \int_{t-\gamma \tau(t)}^{t}\dot{x}(s)\,ds \biggr]=0. \end{aligned}$$
(3.18)

According to (3.11)-(3.18), it is straightforward to see that

$$\begin{aligned} \dot{V}(t,x_{t})+2\alpha V(t,x_{t}) \leq \xi^{T}(t) \Sigma\xi(t), \end{aligned}$$
(3.19)

where

$$\begin{aligned} \zeta^{T}(t)={}& \biggl[D(t),x(t),x\bigl(t-\tau(t)\bigr), \int _{t-\tau(t)}^{t}\dot{x}(s)\,ds, x\bigl(t-\gamma\tau(t) \bigr), \int_{t-\gamma\tau (t)}^{t}\dot{x}(s)\,ds,x(t-\tau_{2}), \\ & \int_{t-\tau_{2}}^{t} x(s)\,ds,x(t-\gamma\tau_{2}), \int_{t-\gamma\tau _{2}}^{t} x(s)\,ds,\tanh x\bigl(t-\sigma(t)\bigr), \int_{t-\sigma_{2}}^{t}\tanh x(s)\,ds,\dot{D}(t), \\ & x(t-\tau_{1}),x(t-\gamma\tau_{1}), \int_{t-\tau_{1}}^{t} x(s)\,ds, \int _{t-\gamma\tau_{1}}^{t} x(s)\,ds, \int_{t-\tau(t)}^{t-\tau_{1}} x(s)\,ds, \int_{t-\tau_{2}}^{t-\tau(t)} x(s)\,ds, \\ & \int_{t-\gamma\tau(t)}^{t-\gamma\tau_{1}} x(s)\,ds, \int_{t-\gamma\tau _{2}}^{t-\gamma\tau(t)} x(s)\,ds, \int_{t-\sigma_{1}}^{t}\tanh x(s)\,ds, \int_{t-\sigma(t)}^{t-\sigma_{1}}\tanh x(s)\,ds, \\ & \int_{t-\sigma_{2}}^{t-\sigma(t)}\tanh x(s)\,ds, \int_{t-\rho _{2}}^{t} \dot{x}(s)\,ds \biggr], \end{aligned}$$

and Σ is defined in (3.8). It is true that if condition (3.9) holds, then

$$\begin{aligned} \dot{V}(t,x_{t})+2\alpha V(t,x_{t})\leq0, \quad \forall t\in R^{+}. \end{aligned}$$
(3.20)

From (3.20), it is easy to see that

$$\bigl\Arrowvert x(t,\phi)\bigr\Arrowvert \leq\beta\Arrowvert\phi\Arrowvert e^{-\alpha t}, \quad t\in R^{+}. $$

This means that equation (2.1) is exponentially stable. The proof of the theorem is complete. □

We now present the delay-dependent criteria for exponential stability of equation (2.1) where \(\tau_{1}=\sigma_{1}=c=0\). We introduce the following notations for later use:

$$\begin{aligned} \tilde{\Sigma}= [\tilde{\Omega}_{(i,j)}] _{13\times13}, \end{aligned}$$
(3.21)

where \(\tilde{\Omega}_{(i,j)}=\Omega_{(i,j)}\), except

$$\begin{aligned}& \tilde{\Omega}_{(2,2)}=2q_{2}p_{1}+k_{2}+k_{3}+k_{4}+k_{5}+k_{6} \tau _{2}^{2}+k_{7}\gamma^{2} \tau_{2}^{2}+k_{8}+k_{9}\sigma _{2}^{2}+2m_{2}+2n_{2}+k_{10}, \\& \tilde{\Omega}_{(7,7)}=-k_{2}e^{-2\alpha\tau_{2}}, \\& \tilde{\Omega}_{(9,9)}=-k_{4}e^{-2\alpha\gamma\tau_{2}}. \end{aligned}$$

Corollary 3.2

For given positive real constants \(\sigma_{2}\), \(\tau_{2}\), \(\sigma_{d}\), \(\tau_{d}\) and γ, equation (2.1) where \(\tau_{1}=\sigma_{1}=c=0\) is exponentially stable with a decay rate α if there exist positive real constants \(k_{i}\) where \(i=1,2,\ldots,10\) and we have the real constants \(r_{1}\), \(r_{2}\), \(m_{k}\), \(n_{k}\) where \(k=1,2,\ldots,8\) such that the following symmetric linear matrix inequality holds:

$$\begin{aligned} \tilde{\Sigma}< 0. \end{aligned}$$
(3.22)

Proof

For \(k_{i}\), \(i=1,2,\ldots,9\) are positive real numbers, we consider the Lyapunov-Krasovskii functional candidate for equation (2.1) where \(\tau_{1}=\sigma_{1}=c=0\) of the form

$$\begin{aligned} V(t,x_{t})= \sum^{4}_{i=1}V_{i}(t,x_{t}), \end{aligned}$$
(3.23)

where

$$\begin{aligned} &V_{1}(t,x_{t})=k_{1}D^{2}(t), \\ &V_{2}(t,x_{t})=k_{2} \int_{t-\tau_{2}}^{t}e^{2\alpha (s-t)}x^{2}(s)\,ds+k_{3} \int_{t-\tau(t)}^{t}e^{2\alpha (s-t)}x^{2}(s)\,ds \\ & \hphantom{V_{2}(t,x_{t})=}+k_{4} \int_{t-\gamma\tau_{2}}^{t}e^{2\alpha (s-t)}x^{2}(s)\,ds+k_{5} \int_{t-\gamma\tau(t)}^{t}e^{2\alpha (s-t)}x^{2}(s)\,ds, \\ &V_{3}(t,x_{t})=k_{6}\tau_{2} \int_{-\tau_{2}}^{0}{ \int _{t+s}^{t}e^{2\alpha(\theta-t)} x^{2}( \theta)\,d\theta}\,ds \\ & \hphantom{V_{3}(t,x_{t})=}+k_{7}\gamma\tau_{2} \int_{-\gamma\tau_{2}}^{0}{ \int _{t+s}^{t}e^{2\alpha(\theta-t)} x^{2}( \theta)\,d\theta}\,ds, \\ &V_{4}(t,x_{t})=k_{8} \int_{t-\sigma(t)}^{t}e^{2\alpha(s-t)}\tanh ^{2}x(s)\,ds \\ & \hphantom{V_{4}(t,x_{t})=}+k_{9}\sigma_{2} \int_{-\sigma_{2}}^{0}{ \int_{t+s}^{t}e^{2\alpha (\theta-t)}\tanh^{2}x( \theta)\,d\theta}\,ds. \end{aligned}$$

According to Theorem 3.1, we have the delay-dependent exponential stability criteria (3.22) of equation (2.1) where \(\tau _{1}=\sigma_{1}=c=0\). □

4 Numerical examples

In this section, four numerical examples are given to present the effectiveness of our main results by comparing the upper bounds of the delays σ and the parameter b as well as investigating the rate of convergence.

Example 4.1

Consider the following equation with mixed interval time-varying delays:

$$\begin{aligned} \frac{d}{dt} \bigl[x(t)+0.1x \bigl(t-\tau(t) \bigr) \bigr]= -1.5x(t)+b\tanh x \bigl(t-\sigma(t) \bigr)+0.5 \int^{t}_{t-\rho(t)}x(s)\,ds. \end{aligned}$$
(4.1)

Decompose the constants a and p as \(a=a_{1}+a_{2}\) and \(p=p_{1}+p_{2}\), respectively, where

$$a_{1}=1,\qquad a_{2}=0.5,\qquad p_{1}=0.05,\qquad p_{2}=0.05. $$

Solving the LMI (3.9) when \(b=0.2\), \(\alpha=0.4\), \(\tau (t)=0.1+\frac{\sin^{2}(t)}{2}\), \(\sigma(t)=0.2+\frac{\cos^{2}(t)}{2}\), and \(\rho(t)=0.2+\frac{|\cos(t)|}{5}\), we can obtain a set of parameters guaranteeing exponential stability as follows:

$$\begin{aligned}& q_{1}=26.7604,\qquad q_{2}=-47.2411,\qquad q_{3}=27.1931,\qquad q_{4}=24.6455, \\& q_{5}=-7.5706,\qquad q_{6}=-9.9581,\qquad q_{7}=1.8709,\qquad k_{1}=36.9295,\qquad k_{2}=3.8691, \\& k_{3}=8.2250,\qquad k_{4}=3.4622,\qquad k_{5}=3.2629,\qquad k_{6}=7.4445,\qquad k_{7}=8.0889, \\& k_{8}=3.4072,\qquad k_{9}=6.9838,\qquad \omega=30.6074,\qquad w_{1}=7.4709,\qquad w_{2}=7.0912, \\& w_{3}=3.9990,\qquad w_{4}=3.6515,\qquad w_{5}=8.2545,\qquad w_{6}=8.0142,\qquad w_{7}=9.2084, \\& w_{8}=8.1243,\qquad w_{9}=8.0686,\qquad w_{10}=9.2084. \end{aligned}$$

Moreover, the upper bounds of the parameter b which guarantees the exponential and asymptotic stabilities are 0.9830 and 1.4834, respectively. The maximum upper bounds b of this example can be found in Table 1 for different values of α, \(\tau_{d}\), \(\sigma_{d}\). The maximum upper bounds as regards the rate of convergent α for this example can be found in Table 2 with different values of b, \(\tau_{d}\), \(\sigma_{d}\). The maximum upper bounds b for exponential and asymptotic stabilities of Example 4.1 are listed in Table 3 for different values of α, c. The maximum upper bounds α for exponential and asymptotic stabilities of Example 4.1 are listed in Table 4 for different values of c, b.

Table 1 Upper bounds of b for Example 4.1 when \(\pmb{\gamma=0.05}\)
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Table 2 Upper bounds of α for Example 4.1 when \(\pmb{\gamma=0.05}\)
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Table 3 Upper bounds of b for Example 4.1 when \(\pmb{\gamma=0.05}\)
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Table 4 Upper bounds of α for Example 4.1 when \(\pmb{\gamma=0.05}\)
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Example 4.2

Consider the following equation studied in [8, 23]:

$$\begin{aligned} \frac{d}{dt} \bigl[x(t)+0.2x \bigl(t-\tau(t) \bigr) \bigr]= -0.6x(t)+0.5\tanh x \bigl(t-\sigma(t) \bigr), \end{aligned}$$
(4.2)

when \(\sigma(t)=\frac{\sin^{2}(t)}{10}\) and \(\tau_{d}=0.2\).

Decompose the constants a and p thus: \(a=a_{1}+a_{2}\) and \(p=p_{1}+p_{2}\), respectively, where

$$a_{1}=0.3,\qquad a_{2}=0.3,\qquad p_{1}=0.1,\qquad p_{2}=0.1. $$

By solving the linear matrix inequality (3.22), the maximum upper bounds \(\sigma_{2}\) for exponential stability of this example is listed fore comparison in Table 5, for different values of α. We can see that our results in Corollary 3.2 are much less conservative than those obtained in [8, 23].

Table 5 The upper bound of time delay \(\pmb{\sigma(t)}\) for Example 4.2 when \(\pmb{\gamma=0.02}\)
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Example 4.3

Consider the following equation, which is considered in [4, 7, 8, 14, 15, 21, 23, 24, 29, 32, 34]:

$$\begin{aligned} \frac{d}{dt} \bigl[x(t)+0.35x(t-0.5) \bigr]= -1.5x(t)+b \tanh x(t-0.5). \end{aligned}$$
(4.3)

Decompose constants a and p as \(a=a_{1}+a_{2}\) and \(p=p_{1}+p_{2}\), respectively, where

$$a_{1}=1.3,\qquad a_{2}=0.2,\qquad p_{1}=0.15,\qquad p_{2}=0.2. $$

Table 6 lists for comparison the upper bounds b for asymptotic stability (\(\alpha=0\)) and exponential stability (\(\alpha=0.177\)) of equation (4.3) by different methods. We can see from Table 6 that our result (Corollary 3.2) is better than other existing work.

Table 6 Upper bounds of b for Example 4.3 when \(\pmb{\gamma=0.5}\)
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Example 4.4

Consider the following equation in [7, 8, 21, 23, 24, 29, 30, 32, 34]:

$$\begin{aligned} \frac{d}{dt} \bigl[x(t)+0.2x(t-0.1) \bigr]= -0.6x(t)+0.3 \tanh x(t-\sigma_{2}). \end{aligned}$$
(4.4)

Decompose the matrix a and p in the same way as the decomposition in Example 4.3. Table 7 lists for comparison the upper bounds delay for asymptotic stability (\(\alpha=0\)) and exponential stability (\(\alpha=0.0038\)) of (4.4) by different methods. It is clear that our results (Corollary 3.2) are significantly better than some existing criteria.

Table 7 Upper bounds of \(\pmb{\sigma_{2}}\) for Example 4.4 when \(\pmb{\gamma=0.05}\)
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5 Conclusions

In this paper, we proposed the delay-range-dependent exponential stability criteria for certain NDE with discrete and distributed interval time-varying delays. Then we presented the delay-dependent exponential stability criteria for certain NDE with time-varying delays. The method combining an augmented Lyapunov-Krasovskii functional, a mixed model transformation, the decomposition technique of constant coefficients, and utilization of zero equations has been adopted to study the paper. New stability criteria have been formulated in terms of LMI. Finally, four numerical examples are given to show that the proposed criteria are less conservative than some existing stability criteria.