1 Introduction

The Cohen–Grossberg neural network model, proposed by Cohen and Grossberg in 1983 [1], has been attracting much attention because of its wide application in various engineering fields and because of it being highly inclusive of other neural networks such as Hopfield neural network, cellular neural network, recurrent neural network, and so on. Hence many scholars devoted themselves to the research of this aspect (see [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]). In some practical applications and hardware implementations of artificial neural networks, time delays are inevitable due to the finite switching speed of the amplifiers and the interneurons conduction distances, and they are even time-varying and unbounded in some cases such as the memory activation function of the human brain neural network model. Therefore, it is more suitable to introduce unbounded time-varying delays to the neural network, especially to Cohen–Grossberg models, and some results have been reported recently, for example, [19,20,21,22,23,24,25,26,27,28].

In the applications of pattern recognition, the addressable memories of patterns are stored as stable equilibrium points. Thus it is necessary that there exist multiple stable equilibrium points for neural networks. The coexistence of multiple equilibrium points and their local stability, which is usually referred to as the multistability of neural network models, has been reported in depth in the last years (see [29,30,31,32,33,34,35,36,37,38,39,40,41,42,43] and the references therein). Wang et al. in [35] studied a class of neural networks with r-level piecewise linear nondecreasing activation functions and showed that the n-neuron dynamical system had exact \((2r+1)^{n}\) equilibrium points, of which \((r +1)^{n}\) were locally exponentially stable and the others were unstable. By using the partition space method, [41] proved that neural networks with unbounded time-varying delays could exhibit at least \(3^{n}\) equilibrium points, \(2^{n}\) of them are locally μ-stable and others are unstable. In [43], based on the geometrical configuration of activation functions and mathematic tools, some novel algebraic criteria were proposed to guarantee the coexistence of \(25^{n}\) equilibrium points, in which \(9^{n}\) equilibrium points are locally μ-stable, for the memristor-based complex-valued neural networks with non-monotonic piecewise nonlinear activation functions and unbounded time-varying delays. From the references mentioned above, we find that the multistability of Cohen–Grossberg neural networks with unbounded time-varying delays is a challenging problem.

Motivated by the challenging problem, we investigate the multistability of a Cohen–Grossberg neural network with unbounded time-varying delay and nondecreasing activation functions in this paper and prove that the considered model has \(3^{n}\) equilibrium points, and \(2^{n}\) of them are locally μ-stable, the remaining ones are unstable. Compared with the literature [41], the results are more general. The rest of this paper is organized as follows. In Sect. 2, the Cohen–Grossberg model and some preliminaries are given. The main results are presented and proved in Sect. 3. The corollaries and comparison with the results of existing literature are presented in Sect. 4. A numerical example with its simulation is showed in Sect. 5 to illustrate the effectiveness the proposed results. Finally, conclusions are drawn in Sect. 6.

2 Preliminaries

In this paper, the following Cohen–Grossberg neural network is considered.

$$ \frac{\mathrm{d}x_{i}(t)}{\mathrm{d}t} =-a_{i} \bigl(x_{i}(t) \bigr) \Biggl[b_{i} \bigl(x _{i}(t) \bigr)-\sum _{j=1}^{n}c_{ij}g_{j} \bigl(x_{j}(t) \bigr)-\sum_{j=1}^{n}d_{ij}f _{j} \bigl(x_{j} \bigl(t-\tau (t) \bigr) \bigr)+I_{i} \Biggr], \quad t \ge 0, $$
(1)

where \(i=1,2,\ldots ,n\), \(x_{i}(t)\) denotes the state variable associated with the ith neuron at time t; \(a_{i}(x_{i}(t))\) represents an amplification function at time t; \(b_{i}(x_{i}(t))\) is an appropriate inhibition behavior function at time t such that the solutions of model (1) remain bounded; \(g_{j}(x_{j}(t))\) and \(f_{j}(x_{j}(t-\tau (t)))\) denote the activation functions of the jth neuron unit at time t without and with time delays, respectively, and \(C=(c_{ij})_{n\times n}\) and \(D=(d_{ij})_{n\times n}\) are the corresponding connection weights matrices; \(\tau (t)\) corresponds to the transmission delay and satisfies \(\tau (t)\geq 0\); \(I_{i}\) is the constant external input of the network on the ith neuron.

The initial conditions of model (1) are assumed to be \(x_{i}(s)= \varphi _{i}(s)\), \(s\leq 0\), \(i=1,2,\ldots ,n\), where \(\varphi _{i}(s)\) is a real-valued continuous function bounded on \((-\infty , 0]\), except that finite points existing at the left and right limits are continuous to the right. Throughout this paper, we make the following assumptions.

  1. (H1)

    For each \(i \in 1,2,\ldots ,n \), the amplification function \(a_{i}(u)\) is nonnegative continuous and satisfies

    $$ 0< \underline{a}_{i}\leq a_{i}(u)\leq \bar{a}_{i}< \infty ,\quad u\in R, i=1,2, \ldots ,n. $$

    And let two n-dimensional positive diagonal matrices \(\hat{A}= \operatorname{diag}\{\bar{a}_{1},\bar{a}_{2},\ldots ,\bar{a}_{n}\}\) and \(\check{A}=\operatorname{diag}\{ \underline{a}_{1}, \underline{a}_{2}, \ldots , \underline{a}_{n}\}\).

  2. (H2)

    \(b_{i}(u)\) is an odd function and monotone increasing, and there exists an n-dimensional positive diagonal matrix \(B=\operatorname{diag}\{ {b}_{1},{b}_{2},\ldots ,{b}_{n}\}\) such that

    $$ \frac{b_{i}(u)-b_{i}(v)}{u-v}\geq b_{i},\quad u, v\in R, u\neq v, i=1,2, \ldots ,n. $$
  3. (H3)

    \(g_{j}(\cdot )\) and \(f_{j}(\cdot )\) are nondecreasing sigmoid continuous nonlinear function or nondecreasing piecewise continuous linear function, and there exist constants \(p_{j}\leq q_{j}\), \(m_{j} \leq M_{j}\), \(m_{j}^{\prime }\leq M_{j}^{\prime }\), \(m_{j}^{\prime \prime } \leq M_{j}^{\prime \prime }\), so that

    $$\begin{aligned}& m_{j}^{\prime }= {\lim_{x \rightarrow -\infty }} {g_{j}(x)},\qquad M_{j}^{\prime }= {\lim _{x \rightarrow +\infty }} {g_{j}(x)}, \\& m_{j}^{\prime \prime }= {\lim_{x \rightarrow -\infty }} {f_{j}(x)},\qquad M_{j}^{\prime \prime }= {\lim _{x \rightarrow +\infty }} {f_{j}(x)}, \\& 0\leq \underline{\sigma }_{j}^{l} \leq \frac{g_{j}(u)-g_{j}(v)}{u-v}\leq \bar{\sigma }_{j}^{l},\qquad 0\leq \underline{ \delta }_{j}^{l} \leq \frac{f_{j}(u)-f_{j}(v)}{u-v}\leq \bar{\delta } _{j}^{l},\quad \forall u,v \in (-\infty , p_{j}), \\& 0\leq \underline{\sigma }_{j}^{m} \leq \frac{g_{j}(u)-g_{j}(v)}{u-v}\leq \bar{\sigma }_{j}^{m},\qquad 0\leq \underline{ \delta }_{j}^{m} \leq \frac{f_{j}(u)-f_{j}(v)}{u-v}\leq \bar{\delta } _{j}^{m},\quad \forall u,v \in [p_{j},q_{j}], \\& 0\leq \underline{\sigma }_{i}^{r} \leq \frac{g_{j}(u)-g_{j}(v)}{u-v}\leq \bar{\sigma }_{j}^{r},\qquad 0\leq \underline{ \delta }_{j}^{r} \leq \frac{f_{j}(u)-f_{j}(v)}{u-v}\leq \bar{\delta } _{j}^{r},\quad \forall u,v \in ( p_{j},+\infty ), \end{aligned}$$

    where \(m_{j}=\min \{ m_{j}^{\prime }, m_{j}^{\prime \prime }\}\), \(M_{j}=\min \{M_{j}^{\prime }, M_{j}^{\prime \prime }\}\), \(\bar{\sigma }_{j}=\max \{\bar{\sigma }_{j}^{l}, \bar{\sigma }_{j}^{m}, \bar{\sigma }_{j}^{r} \}\), \(\bar{\delta }_{j}=\max \{{\bar{\delta }}_{j}^{l},{\bar{ \delta }}_{j}^{m},{\bar{\delta }}_{j}^{r} \}\), \(j=1,2,\ldots ,n\), and define two n-dimensional positive diagonal matrices \(\varSigma ^{g}= \operatorname{diag}\{\bar{\sigma }_{1},\bar{\sigma }_{2},\ldots ,\bar{ \sigma }_{n}\}\) and \(\Delta ^{f}=\operatorname{diag}\{\bar{\delta }_{1},\bar{ \delta }_{2},\ldots , \bar{\delta }_{n}\}\). The superscripts “l”, “m”, “r” denote “left”, “middle”, and “right”, respectively.

It is not hard to find such activation functions as the sigmoid continuous nonlinear function \(f(x)=\tanh (x)=\frac{e^{x}-e^{-x}}{e ^{x}+e^{-x}}\), piecewise continuous linear function \(g(x)= \frac{|x+1|-|x+1|}{2}\), which are different functions, but the properties of the functions can be discussed by common interval separation points. Based on the geometric structure of the activation function, we can define the interval of a one-dimensional real number space as follows:

$$ (-\infty ,+\infty )=(-\infty ,p_{i})\cup [p_{i},q_{i}] \cup (q_{i},+ \infty ),\quad i=1,2,\ldots ,n, $$

then the n-dimensional real number space \(R^{n}\) can be divided into \(3^{n}\) non-intersection subregions. For convenience, let Φ denote the set of these subregions, and so

$$ \varPhi = \Biggl\{ \prod_{i=1}^{n} w_{i}\mid w_{i}=(-\infty ,p_{i}), [p _{i},q_{i}]\mbox{ or }(q_{i},+\infty ) \Biggr\} . $$

For each \(\prod_{i=1}^{n} w_{i}\in \varPhi \), we define the following index subsets with respect to different interval as \(N_{1}=\{i\mid w_{i}=(-\infty , p_{i}), i=1,2,\ldots ,n\}\), \(N_{2}=\{i\mid w_{i}=[p_{i}, q_{i}],i=1,2,\ldots ,n\}\), \(N_{3}=\{i \mid w_{i}=(q_{i}, +\infty ), i=1,2,\ldots ,n\}\).

Furthermore, Φ can be separated into two parts: \(\varPhi _{1}=\{ \prod_{i=1}^{n} w_{i}\mid w_{i}=(-\infty ,p_{i})\mbox{ or }(q_{i}, + \infty )\}\), \(\varPhi _{2}=\varPhi -\varPhi _{1}\). Obviously, \(\varPhi _{1}\) is composed of \(2^{n}\) subregions and \(\varPhi _{2}\) contains \(3^{n}-2^{n}\) subregions.

To facilitate showing the existence of equilibrium points of model (1), we define new sets

$$\begin{aligned}& \varOmega = \Biggl\{ \prod_{i=1}^{n} v_{i}\mid v_{i}=[-E_{i},p_{i}], [p _{i},q_{i}]\mbox{ or }[q_{i},E_{i}] \Biggr\} , \\& \varOmega _{1}= \Biggl\{ \prod_{i=1}^{n} v _{i}\mid v_{i}=[-E_{i},p_{i}]\mbox{ or } [q_{i},E_{i}] \Biggr\} , \end{aligned}$$

where \(E_{i}=2 b_{i}^{-1}[\sum_{j=1}^{n}(|c_{ij}|+|d_{ij}|) \max \{m_{j}, M_{j}\}+|I_{i}|+\max \{|b_{i}(p_{i})|, |b_{i}(q_{i}|\}]\), \(i=1,2,\ldots ,n\).

For each \(\prod_{i=1}^{n} v_{i}\in \varOmega \), we can similarly define its index subsets: \(N_{1}^{\prime }=\{i\mid v_{i}=[-E_{i}, p_{i}], i=1,2, \ldots ,n\}\), \(N_{2}^{\prime }=\{i\mid v_{i}=[p_{i}, q_{i}], i=1,2,\ldots ,n \}\), \(N_{3}^{\prime }=\{i\mid v_{i}=[q_{i}, E_{i}], i=1,2,\ldots ,n\}\).

3 Main results

Because an equilibrium point of system (1) is a constant satisfying the equation \(b_{i}(x_{i}(t))-\sum_{j=1}^{n}c_{ij}g_{j}(x_{j}(t))- \sum_{j=1}^{n}d_{ij}f_{j}(x_{j}(t))+I_{i}=0\), it is obvious that of model (1) has the same equilibrium point with the following system:

$$ \frac{\mathrm{d}x_{i}(t)}{\mathrm{d}t} =-a_{i} \bigl(x_{i}(t) \bigr) \Biggl[b_{i} \bigl(x _{i}(t) \bigr)-\sum _{j=1}^{n}c_{ij}g_{j} \bigl(x_{j}(t) \bigr)-\sum_{j=1}^{n}d_{ij}f _{j}(x_{j}(t)+I_{i} \Biggr], \quad t \ge 0, $$
(2)

for \(i=1,2,\ldots ,n\). Therefore, we can investigate the existence of multiple equilibrium points of model (2) instead of (1).

Theorem 1

For any \(\prod_{i=1}^{n} w_{i}\in \varPhi \), if

$$ \textstyle\begin{cases} -b_{i}(p_{i})+c_{ii}g_{i}(p_{i})+d_{ii}f_{i}(p_{i})+\sum^{n} _{\stackrel{j=1}{j\neq i}}\max \{(c_{ij}+d_{ij})m_{j},(c_{ij}+d_{ij})M _{j}\}-I_{i} < 0,\\ \quad i \in N_{1}\cup N_{2}, \\ -b_{i}(q_{i})+c_{ii}g_{i}(q_{i})+d_{ii}f_{i}(q_{i})+\sum^{n} _{\stackrel{j=1}{j\neq i}}\min \{(c_{ij}+d_{ij})m_{j},(c_{ij}+d_{ij})M _{j}\}-I_{i} > 0,\\ \quad i \in N_{2}\cup N_{3}, \end{cases} $$
(3)

where \(N_{i}\cap N_{j}=\phi \), \(i\neq j\), \(i,j=1,2,3\), \(N_{1}\cup N _{2}\cup N_{3}=\{1,2,\ldots ,n\}\), then there is at least one equilibrium point of model (1) in \(\prod_{i=1}^{n} w_{i}\).

Proof

Let \((x_{1},x_{2},\ldots ,x_{n})\) be a point of \(\prod_{i=1} ^{n} v_{i}\in \varOmega \). Then, for the ith component \(x_{i}\), fixing other components \(x_{1},\ldots ,x_{i-1},x_{i+1},\ldots ,x_{n}\), we can define a function as follows:

$$ F_{i}(u)=-b_{i}(u)+c_{ii}g_{i}(u)+d_{ii}f_{i}(u)+ \sum^{n}_{\stackrel{j=1}{j \neq i}}c_{ij}g_{j}(x_{j}) +\sum^{n}_{ \stackrel{j=1}{j\neq i}}d_{ij}f_{j}(x_{j})-I_{i}, $$

where \(i=1,2,\ldots ,n\). Then, by (3), we can deduce the following:

  1. (1)

    The function \(F_{i}(u)\) is continuous in the interval \([-E_{i},p_{i}]\), and \(F_{i}(-E_{i})>0\), \(F_{i}(p_{i})<0\). Then by the zero point theorem, there exists at least a point \(\bar{x}_{i}\in (-E_{i},p _{i})\) such that \(F_{i}(\bar{x}_{i})=0\). On account of \(a_{i}({x}_{i})>0\), by (H1), \(\bar{x}_{i}\) is the equilibrium point of system (2) for the state component \(x_{i}(t)\) of the ith neuron in the interval \((-E_{i},p_{i})\).

  2. (2)

    If \(u\in [p_{i}, q_{i}]\), then we have \(F_{i}(p_{i})>0\), \(F_{i}(q _{i})<0\). In view of \(a_{i}({x}_{i})>0\), there exists at least a point \(\bar{x}_{i}\in (p_{i}, q_{i})\), which is the equilibrium point of system (2) for the state component \(x_{i}(t)\) of the ith neuron in the interval \((p_{i}, q_{i})\).

  3. (3)

    There exists at least a point \(\bar{x}_{i}\in (q_{i}, E_{i})\), which is the equilibrium point of system (2) for the state component \(x_{i}(t)\) of the ith neuron in the interval \((q_{i}, E_{i})\) because of \(F_{i}(q_{i})>0\), \(F_{i}(E_{i})<0\), and \(a_{i}({x}_{i})>0\).

Above all, the function \(F_{i}(u)\) has at least one zero point in any subinterval. It follows that there is at least one equilibrium point of system (2) in \(\prod_{i=1}^{n} v_{i}\in \varOmega \).

In addition, we can define a mapping \(\wp :\prod_{i=1}^{n} v _{i} \rightarrow \prod_{i=1}^{n} v_{i}\) for any given \(x=(x_{1},x_{2},\ldots , x_{n}) \in \prod_{i=1}^{n} v_{i}\) such that \(\wp (x_{1},x_{2},\ldots ,x_{n})=(\bar{x}_{1},\bar{x}_{2},\ldots ,\bar{x}_{n})\), where \(\bar{x}_{i}\) is a solution of equation \(F_{i}(u)=0\), \(j=1,2,\ldots ,n\). Since \(f_{j}(\cdot ), g_{j}(\cdot ), j=1,2, \ldots ,n\), are continuous by assumption (H3), the mapping ℘ is also continuous, then by Brouwer’s fixed point theorem, there exists at least one fixed equilibrium point, denoted as \(x^{\star }=({x}_{1} ^{\star },{x}_{2}^{\star },\ldots ,{x}_{n}^{\star })\), such that \(\wp (x^{\star })={x}^{\star }\). Furthermore, components of the equilibrium point \(x^{\star }\) satisfy \(-E_{i}<{x}_{i}^{\star }<p_{i}\) for \(i\in N_{1}^{\prime }\), \(p_{i}<{x}_{i}^{\star }<q_{i}\) for \(i\in N_{2} ^{\prime }\), \(q_{i}<{x}_{i}^{\star }<E_{i}\) for \(i\in N_{3}^{\prime }\). It means that all the equilibrium points in \(\prod_{i=1}^{n} v_{i}\) are located in its interior. In view of the arbitrariness of the region \(\prod_{i=1}^{n} v_{i}\) and the sameness of the equilibrium points between model (1) and model (2), model (1) has at least \(3^{n}\) equilibrium points in Ω. From the definition of set Φ and Ω, we know that the corresponding region of \(\prod_{i=1}^{n} v_{i}\) is \(\prod_{i=1}^{n} w_{i}\) and satisfies \(\prod_{i=1}^{n} v_{i}\subseteq \prod_{i=1} ^{n} w_{i}\). Hence it is easy to see that \(x^{\star }\) is also an equilibrium of model (1) in \(\prod_{i=1}^{n} w_{i}\), and so there is at least one equilibrium point of model (1) in \(\prod_{i=1}^{n} w_{i}\). □

For each equilibrium point \(x^{\star }=(x^{\star }_{1},\ldots ,x^{ \star }_{n})\) of \(\prod_{i=1}^{n} w_{i}\in \varPhi _{1}\), we define its μ-stability in \(\prod_{i=1}^{n} w_{i}\) (local μ-stability in \(\varPhi _{1}\)), and prove the μ-stability of all equilibrium points in \(\varPhi _{1}\) in the following Definition 1 and Theorem 2, respectively.

Definition 1

Let \((x_{1}(t),x_{2}(t),\ldots ,x_{n}(t))\) be an arbitrary solution of model (1) located in \(\prod_{i=1}^{n} w _{i}\in \varPhi _{1}\) with the initial state \(x_{i}(s)=\varphi _{i}(s)\), \(s \in (-\infty , 0]\), \(i=1,2,\ldots ,n\), and \(\mu (t)\) be a nondecreasing function with \(\mu (t)\rightarrow +\infty\) (\(t \rightarrow +\infty \)). Then \(x^{\star }\) is said to be μ-stable in \(\prod_{i=1} ^{n} w_{i}\) (locally μ-stable in \(\varPhi _{1}\)) if there is a positive constant M such that

$$ \bigl\vert x_{i}(t)-x^{\star }_{i} \bigr\vert \leq \frac{M}{\mu (t)}. $$

Theorem 2

For any \(\prod_{i=1}^{n} w_{i}\in \varPhi _{1}\), given that

$$ \textstyle\begin{cases} -b_{i}(p_{i})+c_{ii}g_{i}(p_{i})+\sum^{n}_{\stackrel{j=1}{j \neq i}}\max \{c_{ij}m_{j},c_{ij}M_{j}\}+\sum^{n}_{j=1}\max \{d_{ij}m_{j},d_{ij}M_{j}\}-I_{i} < 0,\\ \quad i \in N_{1}, \\ -b_{i}(q_{i})+c_{ii}g_{i}(q_{i})+\sum^{n}_{\stackrel{j=1}{j \neq i}}\min \{c_{ij}m_{j},c_{ij}M_{j}\}+\sum^{n}_{j=1}\min \{d_{ij}m_{j},d_{ij}M_{j}\}-I_{i} > 0,\\ \quad i \in N_{3}, \end{cases} $$
(4)

and the nondecreasing function \(\mu (t)>0\) with

$$ \lim_{t\rightarrow +\infty }\mu (t)=+\infty ,\qquad 0\leq \sup _{t\geq T^{\ast }} \frac{\dot{\mu }(t)}{\mu (t)}\leq \alpha ,\qquad \sup _{t\geq T^{\ast }}\frac{\mu (t)}{\mu (t-\tau (t))}\leq 1+ \beta , $$
(5)

where α, β, \(T^{\ast }\) are nonnegative constants, \(N_{1}\cap N_{3}=\phi \), \(N_{1}\cup N_{3}=\{1,2,\ldots ,n\}\). Then the equilibrium point \(x^{\star }=(x_{1}^{\star },\ldots ,x_{n}^{ \star })\) is μ-stable in \(\prod_{i=1}^{n} w_{i}\) (locally μ-stable in \(\varPhi _{1}\)) if there are positive constants \(\zeta _{1},\zeta _{2},\ldots ,\zeta _{n}\) such that

$$\begin{aligned}& (-\underline{a}_{i}\beta _{i}+\alpha )\zeta _{i}+\sum_{j\in N _{1}}\zeta _{j} \bar{a}_{i}\bar{\sigma }_{j}^{l} \vert c_{{i}{j}} \vert +\sum_{j\in N_{3}}\zeta _{j}\bar{a}_{i}\bar{\sigma }_{j}^{r} \vert c_{{i}{j}} \vert \\& \quad {} +(1+\beta ) \biggl(\sum_{j\in N_{1}}\zeta _{j}\bar{a}_{i}\bar{\delta }_{j}^{l} \vert d_{{i}{j}} \vert + \sum_{j\in N_{3}}\zeta _{j}\bar{a}_{i}\bar{\delta }_{j}^{r} \vert d_{{i}{j}} \vert \biggr)< 0, \end{aligned}$$
(6)

where \(i=1,2,\ldots ,n\).

Proof

From the comparison of (3) and (4), we know that there is at least one equilibrium point in \(\prod_{i=1}^{n} w_{i}\) for model (1). The following proof will show that the equilibrium point is unique and μ-stable in \(\prod_{i=1}^{n} w_{i}\in \varPhi _{1}\).

Let \(x(t)=(x_{1}(t),x_{2}(t),\ldots ,x_{n}(t))\) be an arbitrary solution of model (1) in \(\prod_{i=1}^{n} v_{i}\in \varOmega _{1}\) with the initial condition \(x_{i}(s)=\varphi _{i}(s)\), \(s\in (-\infty , 0]\). We claim that \(x(t)\) would remain in \(\prod_{i=1}^{n} v_{i}\) for all \(t\geq 0\).

For \(x_{i}(t)\), suppose that \(i \in N_{1}^{\prime }\). Then \(i \in N_{1}\) on account of \([-E_{i},p_{i}]\subset (-\infty ,p_{i}]\). Further, by (4), one has \(\varepsilon > 0\) small enough such that

$$\begin{aligned} -b_{i}(p_{i}-\varepsilon )+c_{ii}g_{i}(p_{i}- \varepsilon )+\sum ^{n}_{\stackrel{j=1}{j\neq i}}\max \{c_{ij}m_{j},c_{ij}M_{j} \}+ \sum^{n}_{j=1}\max \{d_{ij}m_{j},d_{ij}M_{j} \}-I_{i} < 0. \end{aligned}$$
(7)

And for ε above, we can find some \(t^{\ast } \geq 0\) such that \(p_{i}-\varepsilon \leq x_{i}(t^{\ast }) \leq p_{i}\). It follows that

$$\begin{aligned} \frac{\mathrm{d}x_{i}(t)}{\mathrm{d}t}\biggm| _{t=t^{\ast }} =& -a_{i} \bigl(x _{i} \bigl(t^{\ast } \bigr) \bigr) \Biggl[b_{i} \bigl(x_{i} \bigl(t^{\ast } \bigr) \bigr)-\sum _{j=1}^{n}c_{ij}g _{j} \bigl(x_{j} \bigl(t^{\ast } \bigr) \bigr)-\sum _{j=1}^{n}d_{ij}f_{j} \bigl(x_{j} \bigl(t^{\ast }- \tau \bigl(t^{\ast } \bigr) \bigr) \bigr)+I_{i} \Biggr] \\ \leq &a_{i} \bigl(x_{i} \bigl(t^{\ast } \bigr) \bigr) \Biggl[-b_{i}(p_{i}-\varepsilon ) \\ &{}+c_{ii}g_{i}(p_{i}-\varepsilon )+\sum ^{n}_{\stackrel{j=1}{j \neq i}}\max \{c_{ij}m_{j},c_{ij}M_{j} \}+\sum^{n}_{j=1}\max \{d_{ij}m_{j},d_{ij}M_{j} \}-I_{i} \Biggr] \\ < &0. \end{aligned}$$
(8)

On the other hand, we also can find some \(t^{\ast } \geq 0\) such that \(x_{i}(t^{\ast })= -E_{i}\). By (H1), we get that

$$\begin{aligned} \frac{\mathrm{d}x_{i}(t)}{\mathrm{d}t}\biggm| _{t=t^{\ast }} \geq &a_{i}(-E _{i}) \Biggl[b_{i}(E_{i}) +c_{ii}g_{i}(-E_{i})+ \sum ^{n}_{\stackrel{j=1}{j \neq i}}\min \{c_{ij}m_{j},c_{ij}M_{j} \} \\ &{}+\sum^{n}_{j=1}\min \{d_{ij}m_{j},d_{ij}M_{j} \}-I_{i} \Biggr] \\ >&0. \end{aligned}$$
(9)

Similarly, for \(x_{i}(t),i \in N_{3}^{\prime }\), there exists \(\varepsilon > 0\) small enough such that

$$ -b_{i}(q_{i}+\varepsilon )+c_{ii}g_{i}(q_{i}+ \varepsilon )+\sum^{n}_{\stackrel{j=1}{j\neq i}}\min \{c_{ij}m_{j},c_{ij}M_{j}\}+ \sum ^{n}_{j=1}\min \{d_{ij}m_{j},d_{ij}M_{j} \}-I_{i} > 0. $$

And for ε above, we can discover some \(t^{\ast } \geq 0\) such that \(q_{i} \leq x_{i}(t^{\ast }) \leq q_{i}+\varepsilon \). It follows that

$$\begin{aligned} \frac{\mathrm{d}x_{i}(t)}{\mathrm{d}t}\biggm| _{t=t^{\ast }} \geq & a _{i} \bigl(x_{i} \bigl(t^{\ast } \bigr) \bigr) \Biggl[-b_{i}(q_{i}+ \varepsilon ) +c_{ii}g_{i}(q_{i}+ \varepsilon )+ \sum^{n}_{\stackrel{j=1}{j\neq i}}\min \{c_{ij}m _{j},c_{ij}M_{j}\} \\ &{}+\sum^{n}_{j=1}\min \{d_{ij}m_{j},d_{ij}M _{j} \}-I_{i} \Biggr] > 0. \end{aligned}$$
(10)

We also can find some \(t^{\ast } \geq 0\) so that \(x_{i}(t^{\ast })= E _{i}\) and obtain the following inequality:

$$\begin{aligned} \frac{\mathrm{d}x_{i}(t)}{\mathrm{d}t}\biggm| _{t=t^{\ast }} \leq &a_{i}(E _{i}) \Biggl[-b_{i}(E_{i}) +c_{ii}g_{i}(-E_{i})+ \sum^{n}_{\stackrel{j=1}{j \neq i}}\max \{c_{ij}m_{j},c_{ij}M_{j} \} \\ &{}+\sum^{n}_{j=1}\max \{d_{ij}m_{j},d_{ij}M_{j} \}-I_{i} \Biggr] \\ < &0. \end{aligned}$$
(11)

From (8)–(11), we see that \(x_{i}(t)\) would not escape from \([-E_{i},p_{i}]\) when \(i\in N_{1}^{\prime }\), and \(x_{i}(t)\) would remain in \([q_{i}, E_{i}]\) when \(i\in N_{3}^{\prime }\). Let i go through \(1,2, \ldots ,n\), we get \(x(t)\) is located in \(\varOmega _{1}\). In view of \(\varOmega _{1}\subseteq \varPhi _{1}\), it implies that \(\prod_{i=1} ^{n} w_{i}\in \varPhi _{1}\) is an invariant set of model (1) with the initial condition \(x_{i}(s)=\varphi _{i}(s)\), \(s\in (-\infty , 0]\).

Denote \(u_{i}(t)=x_{i}(t)-x_{i}^{\star }\), \(i=1,2,\ldots ,n\), then we have

$$\begin{aligned} \frac{\mathrm{d}u_{i}(t)}{\mathrm{d}t} =&-a_{i} \bigl(x_{i}(t) \bigr) \Biggl[ \bigl(b_{i} \bigl(x _{i}(t) \bigr)-b_{i} \bigl(x_{i}^{\star } \bigr) \bigr) \\ &{}-\sum^{n}_{j=1}c_{{i}{j}} \bigl(g_{j} \bigl(x_{j}(t) \bigr)-g_{j} \bigl(x_{j}^{ \star } \bigr) \bigr)-\sum ^{n}_{j=1}d_{{i}{j}} \bigl(f_{j} \bigl(x_{j} \bigl(t-\tau (t) \bigr) \bigr)-f _{j} \bigl(x_{j}^{\star } \bigr) \bigr) \Biggr]. \end{aligned}$$
(12)

Let \(U_{i}(t)=\mu (t)u_{i}(t)\) and \(U(t)=\sup_{s\leq t}( \max_{i=1,2,\ldots ,n}(\zeta _{i}^{-1}|U_{i}(s)|))\), \(t\geq T \geq T^{\ast }\). Then, for any \(t^{\ast }\), \(t^{\ast }\geq T \geq T ^{\ast }\), we have \(\max_{i=1,2,\ldots ,n}(\zeta _{i}^{-1}|U _{i}(t^{\ast })|)\leq U(t^{\ast })\), which implies that \(U(t)\) is bounded.

Let \(i_{t^{\ast }}=i_{t^{\ast }}(t^{\ast })\) when \(\max_{i=1,2,\ldots ,n}(\zeta _{i}^{-1}|U_{i}(t^{\ast })|)= U(t^{ \ast })\) holds. Differentiating \(|U_{i_{t^{\ast }}}(t)|\) at time \(t^{\ast }\), we can deduce that

$$\begin{aligned} \frac{\mathrm{d} \vert U_{{i}_{t^{\ast }}}(t) \vert }{\mathrm{d}t} \biggm| _{t=t^{\ast }} =&\operatorname{sign} \bigl(U_{{i}_{t^{\ast }}} \bigl(t^{\ast } \bigr) \bigr)\dot{\mu } \bigl(t ^{\ast } \bigr)u_{{i}_{t^{\ast }}} \bigl(t^{\ast } \bigr) + \operatorname{sign} \bigl(U_{{i}_{t^{\ast }}} \bigl(t ^{\ast } \bigr) \bigr) \mu \bigl(t^{\ast } \bigr) \\ &{}\cdot \Biggl\{ -a_{{i}_{t^{\ast }}} \bigl(x_{{i}_{t^{\ast }}} \bigl(t^{\ast } \bigr) \bigr) \Biggl[ \bigl(b_{{i}_{t ^{\ast }}} \bigl(x_{{i}_{t^{\ast }}} \bigl(t^{\ast } \bigr) \bigr)-b_{{i}_{t^{\ast }}} \bigl(x_{ {i}_{t^{\ast }}}^{\star } \bigr) \bigr) \\ &{} - \sum^{n}_{j=1}c_{{{i}_{t^{\ast }}} {j}} \bigl(g_{j} \bigl(x_{j} \bigl(t^{\ast } \bigr) \bigr)-g_{j} \bigl(x_{j}^{\star } \bigr) \bigr) \\ &{}-\sum^{n}_{j=1}d_{{{i}_{t^{\ast }}}{j}} \bigl(f_{j} \bigl(x_{j} \bigl(t^{ \ast }-\tau \bigl(t^{\ast } \bigr) \bigr) \bigr)-f_{j} \bigl(x_{j}^{\star } \bigr) \bigr) \Biggr] \Biggr\} . \end{aligned}$$
(13)

By hypothesis (H2), there exists a positive constant \(\beta _{i}\), so that

$$ b_{i}(x_{i})-b_{i}(y_{i})= \beta _{i}(x_{i}-y_{i}), $$
(14)

where \(\beta _{i}\geq b_{i}\), \(i=1,2,\ldots ,n\). Substituting (5), (6), (14) into (13) and applying (H1) and (H3), we can obtain the following inequality.

$$\begin{aligned} \frac{\mathrm{d} \vert U_{{i}_{t^{\ast }}}(t) \vert }{\mathrm{d}t} \biggm| _{t=t^{\ast }} =&\operatorname{sign} \bigl(U_{{i}_{t^{\ast }}} \bigl(t^{\ast } \bigr) \bigr)\dot{\mu } \bigl(t ^{\ast } \bigr)u_{{i}_{t^{\ast }}} \bigl(t^{\ast } \bigr) + \operatorname{sign} \bigl(U_{{i}_{t^{\ast }}} \bigl(t ^{\ast } \bigr) \bigr) \mu \bigl(t^{\ast } \bigr) \\ &{}\cdot \Biggl\{ -a_{{i}_{t^{\ast }}} \bigl(x_{{i}_{t^{\ast }}} \bigl(t^{\ast } \bigr) \bigr) \Biggl[ \bigl(b_{{i}_{t ^{\ast }}} \bigl(x_{{i}_{t^{\ast }}} \bigl(t^{\ast } \bigr) \bigr)-b_{{i}_{t^{\ast }}} \bigl(x_{ {i}_{t^{\ast }}}^{\star } \bigr) \bigr) \\ &{} - \sum^{n}_{j=1}c_{{{i}_{t^{\ast }}} {j}} \bigl(g_{j} \bigl(x_{j} \bigl(t^{\ast } \bigr) \bigr)-g_{j} \bigl(x_{j}^{\star } \bigr) \bigr) \\ &{}-\sum^{n}_{j=1}d_{{{i}_{t^{\ast }}}{j}} \bigl(f_{j} \bigl(x_{j} \bigl(t^{ \ast }-\tau \bigl(t^{\ast } \bigr) \bigr) \bigr)-f_{j} \bigl(x_{j}^{\star } \bigr) \bigr) \Biggr] \Biggr\} \\ \leq & \biggl(-\underline{a}_{{i}_{t^{\ast }}}\beta _{{i}_{t^{\ast }}}+ \frac{ \dot{\mu }(t^{\ast })}{\mu (t^{\ast })} \biggr) \bigl\vert U_{{i}_{t^{\ast }}} \bigl(t ^{\ast } \bigr) \bigr\vert +\sum_{j\in N_{1}}\bar{a}_{{i}_{t^{\ast }}} \vert c_{{{i}_{t^{\ast }}}{j}} \vert \bar{\sigma }_{j}^{l} \bigl\vert U _{j} \bigl(t^{\ast } \bigr) \bigr\vert \\ &{}+\sum_{j\in N_{3}}\bar{a}_{{i}_{t^{ \ast }}} \vert c_{{{i}_{t^{\ast }}}{j}} \vert \bar{\sigma }_{j}^{r} \bigl\vert U_{j} \bigl(t^{\ast } \bigr) \bigr\vert \\ &{}+\sum_{j\in N_{1}}\bar{a}_{{i}_{t^{\ast }}} \frac{\mu (t^{ \ast })}{\mu (t^{\ast }-\tau (t^{\ast }))} \vert d_{{{i}_{t^{\ast }}} {j}} \vert \bar{\delta }_{j}^{l} \bigl\vert U_{j} \bigl(t^{\ast }- \tau \bigl(t^{ \ast } \bigr) \bigr) \bigr\vert \\ &{}+\sum_{j\in N_{3}}\bar{a}_{{i}_{t^{\ast }}} \frac{\mu (t^{ \ast })}{\mu (t^{\ast }-\tau (t^{\ast }))} \vert d_{{{i}_{t^{\ast }}} {j}} \vert \bar{\delta }_{j}^{r} \bigl\vert U_{j} \bigl(t^{\ast }- \tau \bigl(t^{ \ast } \bigr) \bigr) \bigr\vert \\ \leq & \biggl\{ \biggl(-\underline{a}_{{i}_{t^{\ast }}}\beta _{{i}_{t^{\ast }}}+ \frac{ \dot{\mu }(t^{\ast })}{\mu (t^{\ast })} \biggr)\zeta _{{i}_{t^{\ast }}} + \sum _{j\in N_{1}}\bar{a}_{{i}_{t^{\ast }}} \vert c_{{{i}_{t ^{\ast }}}{j}} \vert \bar{\sigma }_{j}^{l}\zeta _{j} +\sum _{j\in N_{3}}\bar{a}_{{i}_{t^{\ast }}} \vert c_{{{i}_{t^{ \ast }}}{j}} \vert \bar{\sigma }_{j}^{r}\zeta _{j} \\ &{}+\frac{\mu (t^{\ast })}{\mu (t^{\ast }-\tau (t^{\ast }))} \biggl(\sum_{j\in N_{1}} \bar{a}_{{i}_{t^{\ast }}} \vert d_{{{i}_{t^{ \ast }}}{j}} \vert \bar{\delta }_{j}^{l}\zeta _{j} +\sum _{j\in N_{3}}\bar{a}_{{i}_{t^{\ast }}} \vert d_{{{i}_{t^{ \ast }}}{j}} \vert \bar{\delta }_{j}^{r}\zeta _{j} \biggr) \biggr\} \cdot U \bigl(t^{ \ast } \bigr) \\ < &0. \end{aligned}$$
(15)

From (15), we can see that there exists a positive constant \(\delta _{1}\) such that \(U(t)=U(t^{\ast })\) for \(t\in (t^{\ast },t^{ \ast }+\delta _{1})\). Because of the arbitrariness of \(t^{\ast }\), we can get that \(U(t)=U(T)\) for all \(t\geq T \geq T^{\ast }\), which implies \(\zeta _{i}^{-1}|\mu (t)(x_{i}(t)-x_{i}^{\star })|<\max_{i=1,2,\ldots ,n}(\zeta _{i}^{-1}|U_{i}(t)|)<U(t)\). Therefore, by Definition 1, the equilibrium \(x^{\star }\) is μ-stable in \(\prod_{i=1}^{n} w_{i}\in \varPhi _{1}\). □

Next, we show that there exists an unstable equilibrium point in \(\varPhi _{2}\).

Theorem 3

For any \(\prod_{i=1}^{n} w_{i}\in \varPhi _{2}\), given that (3) holds. If there are positive constants \(\xi _{1},\ldots ,\xi _{n}\) such that

$$\begin{aligned}& \min_{i\in N_{2}} \biggl\{ \bigl(-\beta _{i}+c_{{i}{i}} \sigma _{i}^{m^{ \ast }} \bigr)\xi _{i} -\sum_{j\in N_{1}}\xi _{j} \vert c_{{i}{j}} \vert \bar{\sigma }_{j}^{l} -\sum _{j\in N_{2}} \xi _{j} \vert c_{{i}{j}} \vert \bar{ \sigma }_{j}^{m} -\sum _{j\in N_{3}}\xi _{j} \vert c_{{i}{j}} \vert \bar{\sigma }_{j}^{r} \\& \quad {}-\sum_{j\in N_{1}}\xi _{j} \vert d_{{i}{j}} \vert \bar{ \delta }_{j}^{l} -\sum _{j\in N_{2}}\xi _{j} \vert d_{{i}{j}} \vert \bar{\delta }_{j}^{m} -\sum _{j\in N_{3}}\xi _{j} \vert d_{{i}{j}} \vert \bar{ \delta }_{j}^{r} \biggr\} >\max \{\lambda ,0\}, \end{aligned}$$
(16)

where

$$\begin{aligned} \lambda \triangleq &\max_{i\in N_{1}\cup N_{3}} \biggl\{ (-\beta _{i} \xi _{i} +\sum_{j\in N_{1}}\xi _{j} \vert c_{{i}{j}} \vert \bar{\sigma }_{j}^{l} +\sum_{j\in N_{2}}\xi _{j} \vert c_{{i} {j}} \vert \bar{\sigma }_{j}^{m} +\sum _{j\in N_{3}}\xi _{j} \vert c_{{i}{j}} \vert \bar{\sigma }_{j}^{r} \\ &{}+\sum_{j\in N_{1}}\xi _{j} \vert d_{{i}{j}} \vert \bar{ \delta }_{j}^{l} +\sum _{j\in N_{2}}\xi _{j} \vert d_{{i}{j}} \vert \bar{\delta }_{j}^{m} +\sum _{j\in N_{3}}\xi _{j} \vert d_{{i}{j}} \vert \bar{ \delta }_{j}^{r} \biggr\} , \end{aligned}$$
(17)

then the equilibrium point \(x^{\star }\) of model (1) in \(\prod_{i=1}^{n} w_{i}\in \varPhi _{2}\) is unstable.

Proof

Let \(x(t)=(x_{1}(t),x_{2}(t),\ldots ,x_{n}(t))\) be an arbitrary solution of model (1) with the initial condition \(x(s)=\varphi (s) \in \varPhi _{2}\), \(s\in (-\infty ,0]\), and let \(u_{i}(t)=x_{i}(t)-x_{i} ^{\star }\), \(i=1\), \(2,\ldots ,n,\ t\geq 0\), where \(x^{\star }=(x_{1}^{ \star },\ldots ,x_{n}^{\star })\) is one equilibrium point of model (1) in \(\prod_{i=1}^{n} w_{i}\in \varPhi _{2}\). Without loss of generality, suppose that \(x(t)\) remains in \(\prod_{i=1}^{n} w _{i}\in \varPhi _{2}\). And we define \(H(t)=\sup_{{t-\tau (t)} \leq s\leq t}\{\max_{i=1,2,\ldots ,n}\xi _{i}^{-1} \vert u_{i}(s) \vert \}\), \(t\geq 0\). If \(\max_{i\in {N_{2}}}\xi _{i}^{-1} \vert u_{i}(t) \vert =H(t)\) holds, denote a sequence item \({{i}_{t}}\) of \({N_{2}}\) such that \(\xi _{{i} _{t}}^{-1} \vert u_{{i}_{t}}(t) \vert =\max_{i\in {N_{2}}}\xi _{i}^{-1} \vert u_{i}(t) \vert \), and differentiate \(\vert u_{{i} _{t}}(t) \vert \) at time t, then we can deduce

$$\begin{aligned} \frac{\mathrm{d} \vert u_{{i}_{t}}(t) \vert }{\mathrm{d}t}\biggm| _{(1)} =&\operatorname{sign} \bigl(x _{{i}_{t}}(t)-x_{{i}_{t}}^{\star } \bigr)\cdot \Biggl\{ -a_{{i}_{t}} \bigl(x_{{i}_{t}}(t) \bigr) \Biggl[ \bigl(b _{{i}_{t}} \bigl(x_{{i}_{t}}(t) \bigr)-b_{{i}_{t}} \bigl(x_{{i}_{t}}^{\star } \bigr) \bigr) \\ &-\sum^{n}_{j=1}c_{{{i}_{t}}{j}} \bigl(g_{j} \bigl(x_{j}(t) \bigr)-g_{j} \bigl(x_{j} ^{\star } \bigr) \bigr) -\sum ^{n}_{j=1}d_{{{i}_{t}}{j}} \bigl(f_{j} \bigl(x_{j} \bigl(t-\tau _{ij}(t) \bigr) \bigr)-f_{j} \bigl(x_{j}^{\star } \bigr) \bigr) \Biggr] \Biggr\} . \end{aligned}$$

On account of hypothesis (H3), that is, \(0\leq \underline{\sigma} _{i}^{m} \leq \frac{g_{i}(u)-g_{i}(v)}{u-v}\leq \bar{\sigma }_{i}^{m}\), \(\forall u,v \in [p_{i},q_{i}]\), we can find a positive constant \(\sigma _{i}^{m^{\ast }}\) such that

$$ \frac{g_{i}(u)-g_{i}(v)}{u-v}=\sigma _{i}^{m^{\ast }}. $$
(18)

Therefore by (14), (18), hypothesis (H1) and (H3), Eq. (17) can be further converted to

$$\begin{aligned} \frac{\mathrm{d} \vert u_{{i}_{t}}(t) \vert }{\mathrm{d}t}\biggm| _{(1)} =& \Biggl\{ a_{{i}_{t}} \bigl(x_{{i}_{t}}(t) \bigr) \Biggl[-\operatorname{sign} \bigl(u_{{i}_{t}}(t) \bigr)\cdot \beta _{{i}_{t}}\cdot u_{{i}_{t}}(t)+\operatorname{sign} \bigl(u_{{i}_{t}}(t) \bigr)\cdot c_{ {{i}_{t}}{{i}_{t}}} \bigl(\sigma _{i}^{m^{\ast }}\cdot u_{{i}_{t}}(t) \bigr) \\ &{}+\operatorname{sign} \bigl(u_{{i}_{t}}(t) \bigr)\cdot \sum ^{n}_{\stackrel{j=1}{j\neq i _{t}}}c_{{{i}_{t}}{j}} \bigl(g_{j} \bigl(u_{j}(t)+x_{j}^{\star } \bigr)-g_{j} \bigl(x_{j}^{ \star } \bigr) \bigr) \\ &{}+\operatorname{sign} \bigl(u_{{i}_{t}}(t) \bigr)\cdot \sum ^{n}_{j=1}d_{{{i}_{t}}{j}} \bigl(f _{j} \bigl(u_{j} \bigl(t-\tau _{ij}(t) \bigr)+x_{j}^{\star } \bigr)-f_{j} \bigl(x_{j}^{\star } \bigr) \bigr) \Biggr] \Biggr\} \\ \geq & \biggl\{ a_{{i}_{t}} \bigl(x_{{i}_{t}}(t) \bigr) \biggl[ \bigl(- \beta _{{i}_{t}}+c_{{{i}_{t}} {{i}_{t}}}\sigma _{i}^{m^{\ast }} \bigr) \bigl\vert u_{{i}_{t}}(t) \bigr\vert - \sum _{j\in N_{1}} \vert c_{{{i}_{t}}{j}} \vert \bar{\sigma } _{j}^{l} \bigl\vert u_{j}(t) \bigr\vert \\ &{}-\sum_{\stackrel{j\in N_{2}}{j\neq{i}_{t}}} \vert c_{{{i}_{t}} {j}} \vert \bar{\sigma }_{j}^{m} \bigl\vert u_{j}(t) \bigr\vert -\sum_{j\in N_{3}} \vert c_{{{i}_{t}}{j}} \vert \bar{ \sigma }_{j}^{r} \bigl\vert u_{j}(t) \bigr\vert -\sum_{j\in N_{1}} \vert d_{{{i}_{t}}{j}} \vert \bar{\delta }_{j}^{l} \bigl\vert u_{j} \bigl(t- \tau _{ij}(t) \bigr) \bigr\vert \\ &{}-\sum_{j\in N_{2}} \vert d_{{{i}_{t}}{j}} \vert \bar{ \delta }_{j}^{m} \bigl\vert u_{j} \bigl(t- \tau _{ij}(t) \bigr) \bigr\vert -\sum_{j\in N_{3}} \vert d_{{{i}_{t}}{j}} \vert \bar{\delta }_{j} ^{r} \bigl\vert u_{j} \bigl(t-\tau _{ij}(t) \bigr) \bigr\vert \biggr] \biggr\} \\ \geq &a_{{i}_{t}} \bigl(x_{{i}_{t}}(t) \bigr) \biggl\{ \bigl(-\beta _{{i}_{t}}+c_{{{i}_{t}} {{i}_{t}}}\sigma _{i}^{m^{\ast }} \bigr)\xi _{{i}_{t}} - \biggl[\sum_{\stackrel{j\in N_{1}}{j\neq{i}_{t}}}\xi _{j} \vert c_{{{i} _{t}}{j}} \vert \bar{\sigma }_{j}^{m} +\sum_{j\in N_{2}}\xi _{j} \vert d_{{{i}_{t}}{j}} \vert \bar{\delta }_{j}^{m} \biggr] \\ &{}- \biggl[\sum_{j\in N_{1}}\xi _{j} \vert c_{{{i}_{t}}{j}} \vert \bar{ \sigma }_{j}^{l} +\sum _{j\in N_{3}}\xi _{j} \vert c_{{{i}_{t}} {j}} \vert \bar{\sigma }_{j}^{r} +\sum _{j\in N_{1}}\xi _{j} \vert d_{{{i}_{t}}{j}} \vert \bar{\delta }_{j}^{l} +\sum_{j\in N_{3}} \xi _{j} \vert d_{{{i}_{t}}{j}} \vert \bar{ \delta }_{j}^{r} \biggr] \biggr\} H(t) \\ >&a_{{i}_{t}} \bigl(x_{{i}_{t}}(t) \bigr)\max \{\lambda ,0\}H(t). \end{aligned}$$
(19)

Inequality (19) implies that there exists a number \(r>0\) such that \(\vert u_{{i}_{t}}(s) \vert > \vert u_{{i}_{t}}(t) \vert \), \(s \in (t, t+r)\).

Besides, suppose that there exists some time point \(t'\) such that

$$\begin{aligned}& \sup_{{t^{\prime }-\tau (t^{\prime })} \leq s\leq t^{\prime }}\Bigl\{ \max_{i\in N_{1}\cup N_{3}}\xi _{i}^{-1} \bigl\vert u_{i}(s) \bigr\vert \Bigr\} =\sup_{{t^{\prime }-\tau (t^{\prime })} \leq s\leq t^{\prime }}\Bigl\{ \max_{i\in N_{2}}\xi _{i}^{-1} \bigl\vert u_{i}(s) \bigr\vert \Bigr\} , \\ & \sup_{{t^{\prime }-\tau (t^{\prime })} \leq s\leq t^{\prime }}\Bigl\{ \max_{i\in N_{1}\cup N_{3}}\xi _{i}^{-1} \bigl\vert u_{i}(s) \bigr\vert \Bigr\} =\max_{i\in {N_{1}\cup N_{3}}}\xi _{i}^{-1} \bigl\vert u_{i}\bigl(t'\bigr) \bigr\vert , \end{aligned}$$

and denote a sequence item \(i'\in N_{1}\cup N_{3}\) such that \(\xi _{i'}^{-1} \vert u_{i'}(t') \vert =\max_{i\in {N_{1}\cup N_{3}}}\xi _{i}^{-1} \vert u_{i}(t') \vert \). Hence we can get

$$\begin{aligned} \frac{\mathrm{d} \vert u_{i'}(t) \vert }{\mathrm{d}t}\biggm| _{t=t'} =& \Biggl\{ a_{i'} \bigl(x_{i'} \bigl(t' \bigr) \bigr) \Biggl[- \operatorname{sign} \bigl(u_{i'}(t) \bigr)\cdot \beta _{i'} \cdot u_{i'} \bigl(t' \bigr) \\ &{}+\operatorname{sign} \bigl(u_{i'} \bigl(t' \bigr) \bigr) \cdot \sum^{n}_{j=1}c_{{i'}{j}} \bigl(g_{j} \bigl(u_{j} \bigl(t' \bigr)+x _{j}^{r} \bigr)-g_{j} \bigl(x_{j}^{r} \bigr) \bigr) \\ &{}+\operatorname{sign} \bigl(u_{i'} \bigl(t' \bigr) \bigr) \cdot \sum^{n}_{j=1}d_{{i'}{j}} \bigl(f_{j} \bigl(u_{j} \bigl(t- \tau _{i'j}(t) \bigr)+x_{j}^{r} \bigr)-f_{j} \bigl(x_{j}^{r} \bigr) \bigr) \Biggr] \Biggr\} \\ \leq & \biggl\{ a_{i'} \bigl(x_{i'} \bigl(t' \bigr) \bigr) \biggl[-\beta _{i'} \bigl\vert u_{i'} \bigl(t' \bigr) \bigr\vert + \sum_{j\in N_{1}} \vert c_{{i'}{j}} \vert \bar{\sigma }_{j} ^{l} \bigl\vert u_{j} \bigl(t' \bigr) \bigr\vert +\sum _{j\in N_{2}} \vert c_{{i'} {j}} \vert \bar{\sigma }_{j}^{m} \bigl\vert u_{j} \bigl(t' \bigr) \bigr\vert \\ &{}+\sum_{j\in N_{3}} \vert c_{{i'}{j}} \vert \bar{ \sigma } _{j}^{r} \bigl\vert u_{j} \bigl(t' \bigr) \bigr\vert +\sum_{j\in N_{1}} \vert d _{{i'}{j}} \vert \bar{\delta }_{j}^{l} \bigl\vert u_{j} \bigl(t'-\tau _{i'j} \bigl(t' \bigr) \bigr) \bigr\vert \\ &{}+\sum_{j\in N_{2}} \vert d_{{i'}{j}} \vert \bar{ \delta } _{j}^{m} \bigl\vert u_{j} \bigl(t'-\tau _{i'j} \bigl(t' \bigr) \bigr) \bigr\vert +\sum_{j\in N _{3}} \vert d_{{i'}{j}} \vert \bar{\delta }_{j}^{r} \bigl\vert u_{j} \bigl(t'- \tau _{ij} \bigl(t' \bigr) \bigr) \bigr\vert \biggr] \biggr\} \\ \leq & \biggl\{ a_{i'} \bigl(x_{i'} \bigl(t' \bigr) \bigr) \biggl[(-\beta _{i'}\xi _{i'} +\sum _{j \in N_{2}}\xi _{j} \vert c_{{i'}{j}} \vert \bar{\sigma }_{j}^{m} + \sum_{j\in N_{2}} \xi _{j} \vert d_{{i'}{j}} \vert \bar{\delta }_{j}^{m} +\sum_{j\in N_{1}}\xi _{j} \vert c_{{i'}{j}} \vert \bar{ \sigma }_{j}^{l} \\ &{}+\sum_{j\in N_{3}}\xi _{j} \vert c_{{i'}{j}} \vert \bar{ \sigma }_{j}^{r} +\sum _{j\in N_{1}}\xi _{j} \vert d_{{i'}{j}} \vert \bar{ \delta }_{j}^{l} +\sum _{j\in N_{3}}\xi _{j} \vert d_{{i'}{j}} \vert \bar{ \delta }_{j}^{r} \biggr] \biggr\} H \bigl(t' \bigr) \\ \leq & a_{i'} \bigl(x_{i'} \bigl(t' \bigr) \bigr) \lambda H \bigl(t' \bigr). \end{aligned}$$
(20)

In the meantime, we can obtain the derivative of \(\vert u_{{i'}_{t'}}(t) \vert \) at time \(t'\) by (19) when \(i'\in N_{2}\)

$$\begin{aligned} \frac{\mathrm{d} \vert u_{{i'}_{t'}}(t) \vert }{\mathrm{d}t}\biggm| _{t=t'} > a_{{i'}_{t'}} \bigl(x_{{i'}_{t'}} \bigl(t' \bigr) \bigr)\max \{\lambda ,0\}H \bigl(t' \bigr), \end{aligned}$$
(21)

where \(\xi _{{i'}_{t'}}^{-1} \vert u_{{i'}_{t'}}(t') \vert =\max_{i'\in N_{2}}\xi _{i'}^{-1} \vert u_{i'}(t') \vert \).

Given the above, we can conclude that

$$H(t)=\sup_{{t-\tau (t)} \leq s\leq t}\Bigl\{ \max_{i\in N_{2}}\xi _{i} ^{-1} \bigl\vert u_{i}(s) \bigr\vert \Bigr\} \geq \sup_{{-\tau (t)} \leq s \leq 0}\Bigl\{ \max_{i\in N_{2}}\xi _{i}^{-1} \bigl\vert u_{i}(s) \bigr\vert \Bigr\} $$

holds for all \(t\geq 0\), and there exists an increasing time sequence \(\{t_{l}\}_{l=1}^{\infty }\) with \(\lim_{l\rightarrow \infty }t_{l}=+\infty \) such that \(\sup_{{t_{l}-\tau (t_{l})} \leq s\leq t_{l}}\{\max_{i\in N _{2}}\xi _{i}^{-1} \vert u_{i}(s) \vert \}=\max_{i\in N_{2}}\xi _{i}^{-1} \vert u_{i}(t_{l}) \vert \). Correspondingly, there exists an increasing time subsequence \(\{t_{{l}_{k}}\}_{k=1}^{\infty }\) with \(\lim_{k\rightarrow \infty }t_{{l}_{k}}=+\infty \) such that \(\xi _{i''}^{-1} \vert u_{i''}(t_{{l}_{k}}) \vert =\sup_{{t_{{l}_{k}}-\tau (t_{{l}_{k}})} \leq s\leq t_{{l}_{k}}}\{ \max_{i\in N_{2}}\xi _{i}^{-1} \vert u_{i}(s) \vert \}\geq \sup_{{-\tau (t)} \leq s\leq 0}\{\max_{i\in N_{2}}\xi _{i}^{-1} \vert u_{i}(s) \vert \}>0\), \(k=1,2,\ldots \) . Thus \(u_{i''}(t)\) would not converge to 0 when \(t\rightarrow +\infty \) means that the equilibrium point \(x^{\star }\) is unstable in \(\prod_{i=1}^{n} w_{i}\in \varPhi _{2}\). □

4 Corollaries and comparison

According to the above theorems, we have the following two corollaries.

Corollary 1

If conditions (3)–(6) and (16) hold, then model (1) has at least \(3^{n}\) equilibrium points in \(R^{n}\), \(2^{n}\) of which are locally μ-stable in \(\varPhi _{1}\), the remaining \(3^{n}-2^{n}\) points are unstable in \(\varPhi _{2}\).

Corollary 2

When \(a_{i}(x_{i}(t))=1\), model (1) becomes the Hopfield neural network model

$$ \frac{\mathrm{d}x_{i}(t)}{\mathrm{d}t}=-b_{i} \bigl(x_{i}(t) \bigr)+\sum _{j=1} ^{n}c_{ij}g_{j} \bigl(x_{j}(t) \bigr)+\sum_{j=1}^{n}d_{ij}f_{j} \bigl(x_{j} \bigl(t-\tau (t) \bigr) \bigr)-I _{i}, \quad t \ge 0. $$
(22)

On the basis of conditions (3)–(6) and (16), model (22) has at least \(3^{n}\) equilibrium points in \(R^{n}\), \(2^{n}\) of which are locally μ-stable in \(\varPhi _{1}\), the remaining \(3^{n}-2^{n}\) points are unstable equilibrium points in \(\varPhi _{2}\).

Remark 1

The present paper investigates the multistability of a Cohen–Grossberg neural network with unbounded time-varying delay and nondecreasing activation functions. Compared with [41], the obtained results are more general.

Remark 2

The net self-inhibition function \(b_{i}(x_{i}(t))\) in this paper is an odd function and monotonically increasing, which includes the case of [41]. Thus model (22) is more general than [41].

Remark 3

The activation functions of [41] are identical whether with or without time delay, but the activation functions in this paper are different. Therefore the conclusion of this paper is closer to the practical application.

5 Simulation example

Example

We consider the following two-dimensional Cohen–Grossberg neural network model:

$$ \frac{\mathrm{d}x_{i}(t)}{\mathrm{d}t}=-a_{i} \bigl(x_{i}(t) \bigr) \Biggl[b_{i} \bigl(x _{i}(t) \bigr)-\sum _{j=1}^{n}c_{ij}g_{j} \bigl(x_{j}(t) \bigr)-\sum_{j=1}^{n}d_{ij}f _{j} \bigl(x_{j} \bigl(t-\tau _{ij}(t) \bigr) \bigr)+I_{i}(t) \Biggr], \quad t \ge 0, $$

where \(i=1,2\), \(a(x)=1+0.2\sin (x)\), \(b_{1}(x_{1}(t))=x_{1}(t)\), \(b _{2}(x_{2}(t))=-1.2x_{2}(t)\),

$$\begin{aligned}& g(x)=\textstyle\begin{cases} \tanh (0.2x)-\tanh (1)+\tanh (0.2),&x< -1, \\ \tanh (x),&-1\leq x\leq 1, \\ \tanh (0.2x)+\tanh (1)-\tanh (0.2),&x>1, \end{cases}\displaystyle \qquad f(x)=\frac{| x+1| -|x-1| }{2},\\& C=(c_{ij})= \begin{pmatrix} 3.5& 0.2 \\ 0.4& 4.8 \end{pmatrix},\qquad D=(d_{ij})= \begin{pmatrix} 0.4& 0.2 \\ 0.2& 0.5 \end{pmatrix},\qquad I= \begin{pmatrix} -0.3 \\ -0.6 \end{pmatrix},\\& \tau _{ij}(t)=\tau (t)=0.2t. \end{aligned}$$

After a simple calculation, we know that the above hypothesis satisfies the conditions of Theorems 1–3. Therefore there are at least nine equilibrium points in model (1) from Corollary 1, 4 of which are μ-stable equilibrium points and others are unstable points. The solution of model (1) is traced with 150 initial conditions, the simulation results are showed in the above three graphs of Fig. 1.

Figure 1
figure 1

The state trajectories and phase plots of the neural network model (1). The above three graphs are the state trajectories and phrase plots, where the third graph is a 3D phase plot of the state variable \((t,x_{2},x_{1})\)

Full size image

6 Conclusion

Stability of multiple unstable Cohen–Grossberg neural networks with unbounded time-varying delays is discussed analytically in this paper. Based on the geometric structure of two different activation functions and some rigorous mathematical analysis, the present paper proved that there exist multiple equilibrium points in the model, some of which are unstable, others are μ-stable. One numerical example and its simulation show the effectiveness of the conclusion. Here, we also need point out the following. On the one hand, the impulsive control is rarely used to deal with cases of unbounded time-varying delays, especially for multiple unstable Cohen–Grossberg neural networks with unbounded time-varying delays. Therefore, the stability under impulsive control of multiple unstable Cohen–Grossberg neural networks with unbounded time-varying delays is still a challenging problem. On the other hand, we use something like positivity-based method to study the stability of Cohen–Grossberg neural networks in this article. The positivity-based method is a valid approach for difference and delay differential systems (see [24,25,26,27,28, 44,45,46,47,48]). Therefore, the research on stability with positivity-based approach is an interesting and meaningful topic, and we will also consider the stability of other neural networks by employing the positivity-based approach in the near future.