In this section, we give our main results.

**Theorem 1** *For a given positive constant* \kappa >0, *the zero solution of equation* (1) *is globally exponentially stable if there exist some positive scalars*: {\alpha}_{0}, {\alpha}_{1}, {\alpha}_{2}, {\alpha}_{3} *and* {\alpha}_{4}, *such that the following linear matrix inequalities* (*LMIs*) *hold*:

\begin{array}{r}{\mathrm{\Omega}}_{1}=\left[\begin{array}{cccc}{\mathrm{\Omega}}_{11}^{1}& {\mathrm{\Omega}}_{12}^{1}& {\lambda}_{1}{\alpha}_{0}b& 0\\ \ast & {\mathrm{\Omega}}_{22}^{1}& {\lambda}_{1}{\alpha}_{0}pb& 0\\ \ast & \ast & -{\lambda}_{1}{\alpha}_{3}(1-{\mu}_{2})& 0\\ \ast & \ast & \ast & -{\alpha}_{2}\tau \end{array}\right]<0,\\ {\mathrm{\Omega}}_{2}=\left[\begin{array}{cccc}{\mathrm{\Omega}}_{11}^{2}& {\mathrm{\Omega}}_{12}^{2}& {\lambda}_{2}{\alpha}_{0}b& 0\\ \ast & {\mathrm{\Omega}}_{22}^{2}& {\lambda}_{2}{\alpha}_{0}pb& 0\\ \ast & \ast & -{\alpha}_{3}{\lambda}_{2}(1-{\mu}_{2})& 0\\ \ast & \ast & \ast & -{\alpha}_{4}\sigma \end{array}\right]<0,\end{array}

(3)

*where*

*Proof* Consider the Lyapunov functional defined by

\begin{array}{rcl}V(t,x(t))& =& {e}^{2\kappa t}{\alpha}_{0}{[x(t)+px(t-\tau (t))]}^{2}\\ +{\alpha}_{1}{\int}_{t-\tau (t)}^{t}{e}^{2\kappa (s+\tau )}{x}^{2}(s)\phantom{\rule{0.2em}{0ex}}ds+{\alpha}_{2}{\int}_{-\tau}^{0}{\int}_{t+\theta}^{t}{e}^{2\kappa (s-\theta )}{x}^{2}(s)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}d\theta \\ +{\alpha}_{3}{\int}_{t-\sigma (t)}^{t}{e}^{2\kappa (s+\sigma )}{tanh}^{2}x(s)\phantom{\rule{0.2em}{0ex}}ds+{\alpha}_{4}{\int}_{-\sigma}^{0}{\int}_{t+\theta}^{t}{e}^{2\kappa (s-\theta )}{tanh}^{2}x(s)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}d\theta ,\end{array}

(4)

where {\alpha}_{i} (i=0,1,2,3,4) are positive scalars to be chosen later.

The derivative of V(t,x(t)) along the solution of equation (1) is determined by

The fact that {tanh}^{2}x(t)\le {x}^{2}(t) implies

[{\alpha}_{3}{e}^{2\kappa \sigma}+\frac{{\alpha}_{4}}{2\kappa}({e}^{2\kappa \sigma}-1)]{tanh}^{2}x(t)\le [{\alpha}_{3}{e}^{2\kappa \sigma}+\frac{{\alpha}_{4}}{2\kappa}({e}^{2\kappa \sigma}-1)]{x}^{2}(t).

(6)

Substituting (6) into (5), we have

\frac{dV(t,x(t))}{dt}\le \frac{1}{\tau}{\int}_{t-\tau}^{t}{\xi}_{1}^{T}(t,s){\mathrm{\Omega}}_{1}{\xi}_{1}(t,s)\phantom{\rule{0.2em}{0ex}}ds+\frac{1}{\sigma}{\int}_{t-\sigma}^{t}{\xi}_{2}^{T}(t,s){\mathrm{\Omega}}_{2}{\xi}_{2}(t,s)\phantom{\rule{0.2em}{0ex}}ds,

(7)

where {\xi}_{1}(t,s)={[{x}^{T}(t),{x}^{T}(t-\tau (t)),{tanh}^{T}x(t-\sigma (t)),{x}^{T}(s)]}^{T} and {\xi}_{2}(t,s)={[{x}^{T}(t),{x}^{T}(t-\tau (t)),{tanh}^{T}x(t-\sigma (t)),{tanh}^{T}x(s)]}^{T}.

From (3), we have \frac{dV(t,x(t))}{dt}<0, which implies V(t,x(t))\le V(0,x(0)). And from the definition of the Lypunov function V(t,x(t)), we have

\begin{array}{rcl}V(0,x(0))& =& {\alpha}_{0}{[x(0)+px(-\tau (0))]}^{2}\\ +{\alpha}_{1}{\int}_{-\tau (0)}^{0}{e}^{2\kappa (s+\tau )}{x}^{2}(s)\phantom{\rule{0.2em}{0ex}}ds+{\alpha}_{2}{\int}_{-\tau}^{0}{\int}_{\theta}^{0}{e}^{2\kappa (s-\theta )}{x}^{2}(s)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}d\theta \\ +{\alpha}_{3}{\int}_{-\sigma (0)}^{0}{e}^{2\kappa (s+\sigma )}{tanh}^{2}x(s)\phantom{\rule{0.2em}{0ex}}ds+{\alpha}_{4}{\int}_{-\sigma}^{0}{\int}_{\theta}^{0}{e}^{2\kappa (s-\theta )}{tanh}^{2}x(\theta )\phantom{\rule{0.2em}{0ex}}d\theta \phantom{\rule{0.2em}{0ex}}ds\\ \le & [4{\alpha}_{0}+({\alpha}_{1}+{\alpha}_{3})r{e}^{2\kappa r}+({\alpha}_{2}+{\alpha}_{4})\frac{{e}^{2\kappa r}}{4{\kappa}^{2}}]\underset{\theta \in [-r,0]}{sup}{|\varphi (\theta )|}^{2}\\ \equiv & M.\end{array}

So, we obtain

{|x(t)+px(t-\tau (t))|}^{2}\le {M}_{1}{e}^{-2\kappa t},

(8)

where {M}_{1}=\frac{M}{{\alpha}_{0}}>0. For \mathrm{\forall}\epsilon \in (0,min\{2\kappa ,-\frac{2}{r}log|p|\}) and \nu >0, the fundamental inequality xy\le \nu {x}^{2}+\frac{1}{\nu}{y}^{2} for any x,y\in R implies

\begin{array}{rcl}{e}^{\epsilon t}{|x(t)|}^{2}& \le & (1+\nu ){e}^{\epsilon t}{|x(t)+px(t-\tau (t))|}^{2}+\frac{1+\nu}{\nu}{e}^{\epsilon t}{\left|px(t-\tau (t))\right|}^{2}\\ \le & (1+\nu ){M}_{1}+\frac{1+\nu}{\nu}{|p|}^{2}{e}^{\epsilon r}{e}^{\epsilon (t-\tau (t))}{\left|x(t-\tau (t))\right|}^{2}.\end{array}

And from \epsilon \in (0,min\{2\kappa ,-\frac{2}{r}log|p|\}), we have {|p|}^{2}{e}^{\epsilon r}<1. Thus, as \nu >0 is chosen sufficiently large,

\u03f5=\frac{{|p|}^{2}(1+\nu ){e}^{\epsilon r}}{\nu}<1.

Therefore, we have

{e}^{\epsilon t}{|x(t)|}^{2}\le (1+\nu ){M}_{1}+\u03f5{e}^{\epsilon (t-\tau (t))}{\left|x(t-\tau (t))\right|}^{2}.

(9)

About (9), for \mathrm{\forall}T\ge 0,

\underset{0\le t\le T}{sup}\left\{{e}^{\epsilon t}{|x(t)|}^{2}\right\}\le (1+\nu ){M}_{1}+\u03f5\underset{\theta \in [-r,0]}{sup}{|\varphi (\theta )|}^{2}+\u03f5\underset{0\le t\le T}{sup}\left\{{e}^{\epsilon t}{|x(t)|}^{2}\right\}.

Consequently,

\underset{0\le t\le T}{sup}\left\{{e}^{\epsilon t}{|x(t)|}^{2}\right\}\le \frac{(1+\nu ){M}_{1}+\u03f5{sup}_{\theta \in [-r,0]}{|\varphi (\theta )|}^{2}}{1-\u03f5}.

(10)

As T\to +\mathrm{\infty}, it follows from (10) that

\underset{0\le t<+\mathrm{\infty}}{sup}\left[{e}^{\epsilon t}{|x(t)|}^{2}\right]\le \frac{(1+\nu ){M}_{1}+\u03f5{sup}_{\theta \in [-r,0]}{|\varphi (\theta )|}^{2}}{1-\u03f5},

that is,

|x(t)|\le {M}_{2}{e}^{-\alpha t},

where {M}_{2}=\sqrt{\frac{(1+\nu ){M}_{1}+\u03f5{sup}_{\theta \in [-r,0]}{|\varphi (\theta )|}^{2}}{1-\u03f5}}>0 and \alpha =\frac{\epsilon}{2}>0. The proof of this theorem is completed. □

When \tau (t)\equiv \tau ,\sigma (t)\equiv \sigma, we can easily derive the following corollary.

**Corollary 2** *For a given positive constant* \kappa >0, *the zero solution of equation* (2) *is globally exponentially stable if there exist some positive scalars*: {\alpha}_{0}, {\alpha}_{1}, {\alpha}_{2}, {\alpha}_{3}, {\alpha}_{4}, *such that the following linear matrix inequalities* (*LMIs*) *hold*:

*where*

*and* {\mathrm{\Omega}}_{11}^{1}, {\mathrm{\Omega}}_{11}^{2}, {\lambda}_{1}, {\lambda}_{2} *are given in Theorem* 1.

**Remark 1** The proofs of Theorem 1 and Corollary 2 are completed by utilizing the technique involved in [19], so that the mode-transformation technique and the bounding technique are not employed. Although one LMI-based sufficient condition ensuring the exponential stability for equation (2) has been obtained in [16], this condition is more conservative since the mode-transformation technique and the bounding technique are both used, and the obtained result has narrow applications since the condition

|p|+(\sigma -\tau )|b|<1,

(11)

must be imposed. Besides, note that Theorem 4 in [16] involves seven decision variables, while Corollary 2 involves five decision variables. Thus, Corollary 2 needs fewer decision variables than Theorem 4 in [16]. What is more, this restrictive condition (11) is removed in this paper. Thus, our LMIs-based sufficient conditions are less conservative than those provided in [16], which is shown by Example 1 and Example 2 in Section 3. And the technique employed in this paper is different from the previous ones introduced in [1, 3, 7–11, 16, 17].

**Remark 2** Although the delay-independent sufficient condition for the global exponential stability of equation (2) has been obtained, the technique used in [8] is only suitable for constant delay, not for time-varying delays. So, our result can complement the result in [8]. Besides, in [8], the delay-independent sufficient condition for the global exponential stability of equation (2) has been given in the form

**Remark 3** If \kappa =0, the criteria about the global asymptotical stability for equation (1) are presented as follows:

**Corollary 3** *The zero solution of equation* (1) *is globally asymptotically stable if there exist some positive scalars*: {\alpha}_{0}, {\alpha}_{1}, {\alpha}_{2}, {\alpha}_{3} *and* {\alpha}_{4}, *such that the following linear matrix inequalities* (*LMIs*) *hold*:

\begin{array}{r}{\mathrm{\Omega}}_{1}=\left[\begin{array}{cccc}{\mathrm{\Omega}}_{11}^{1}& {\mathrm{\Omega}}_{12}^{1}& {\lambda}_{1}{\alpha}_{0}b& 0\\ \ast & {\mathrm{\Omega}}_{22}^{1}& {\lambda}_{1}{\alpha}_{0}pb& 0\\ \ast & \ast & -{\lambda}_{1}{\alpha}_{3}(1-{\mu}_{2})& 0\\ \ast & \ast & \ast & -{\alpha}_{2}\tau \end{array}\right]<0,\\ {\mathrm{\Omega}}_{2}=\left[\begin{array}{cccc}{\mathrm{\Omega}}_{11}^{2}& {\mathrm{\Omega}}_{12}^{2}& {\lambda}_{2}{\alpha}_{0}b& 0\\ \ast & {\mathrm{\Omega}}_{22}^{2}& {\lambda}_{2}{\alpha}_{0}pb& 0\\ \ast & \ast & -{\lambda}_{2}{\alpha}_{3}(1-{\mu}_{2})& 0\\ \ast & \ast & \ast & -{\alpha}_{4}\sigma \end{array}\right]<0,\end{array}

(13)

where

{\lambda}_{i} (i=1,2) *are given in Theorem * 1.