1 Introduction

Over the past decades, the neural networks have been widely investigated and found many applications in different areas such as image processing, signal processing, pattern recognition and optimization. The dynamical behaviors of neural networks such as stability, oscillation and convergence issues have been extensively studied. In general, many applications of neural networks are built upon the existence and stability of the equilibrium point. For example, if a neural network is used to solve an optimization problem, it is desirable for the neural network to have a unique globally stable equilibrium. Therefore, the stability analysis and synchronization problem of neural networks has caught many researchers’ attention [1–18].

In many biological and artificial neural networks, time delays always exist due to varieties of reasons such as the finite speed of information transmission and processing. As is well known, the time delay is one of main sources for causing instability and bad performances of neural networks [11]. Consequently, the stability analysis problems for delayed neural networks have received considerable research attention. Recently, a great deal of results have been reported in the literature, see e.g. [1, 8–10, 12, 14] and references therein, where the time delays considered can be categorized as constant delays, time-varying delays and distributed delays, the methods used include the M-matrix theory, linear matrix inequality (LMI) approach, Lyapunov functional method and techniques of inequality analysis, and the stability criteria derived contain delay-independent conditions and delay-dependent conditions.

On the other hand, it is common in engineering systems that the time delay occurs not only in system states but also in the derivatives of system states. The systems containing the information of past state derivatives are called neutral-type systems, and such systems can be found in many engineering systems, e.g. chemical reactors, transmission lines, partial element equivalent circuits in very large scale integration (VLSI) systems and Lotka-Volterra systems. Due to the fact that neutral delays may exist in VLSI implementations of neural networks, the stability analysis of neural networks with neutral terms has received increasing attention and a rich body of results has been reported [19–23]. In [20], the delay-dependent exponential stability have been studied for a class of neural networks described by nonlinear delay differential equations of neutral type by means of linear matrix inequalities (LMIs). By utilizing the Lyapunov-Krasovkii functional and the LMI approach, the global exponential stability have been analyzed in [24] for a kind of neutral-type impulsive neural networks. By constructing the new Lyapunov-Krasovskii functional, a unified framework has been established in [25] to derive sufficient conditions for the global exponential mean square stability of a class of Markovian jumping neutral-type neural networks with mode-dependent mixed time-delays.

As has been pointed by Hale [26], the properties of difference operator are crucial for the existence and stability of solutions to neutral functional differential equations (NFDEs). In order to obtain solutions of NFDEs, the definition of stability for difference operator has been introduced in [26]. The properties of difference operator has been studied in [27] when it is not stable. By using the results derived in [27], some results on the existence of periodic solutions to NFDEs have been obtained in [28–30]. However, to the best of the authors’ knowledge, the problems of existence and stability of periodic conditions for delayed neural networks with difference operator have not been fully addressed, which constitutes the main motivation of the current research. In this paper, we aim to investigate the existence and stability of periodic solutions for a class of neutral neural networks by using the properties of difference operator. Three fundamental issues emerge as follows: (1) how to prove the existence of the periodic solution of the delayed neural networks with difference operator; (2) how to construct a feasible Lyapunov functional to reflect the influence of the neutral operator in neural networks; (3) how to analyze the stability of the periodic solution for the neutral-type neural networks with difference operator. By using the Mawhin’s continuation theorem and Lyapunov functional method, some new sufficient conditions are derived to guarantee the existence, uniqueness, and global exponential stability of the periodic solution for neutral neural networks.

The main contributions of this paper are highlighted as follows. (1) The neural network under consideration shows the neutral features characterized by the operator \(A_i\), which is different from other papers. Hence, when the neutral term is studied as a neutral operator \(A_i\), novel analysis technique is developed since the conventional analysis tool no longer applies; (2) By employing the Mawhin’s continuation theorem and Lemma 1 in [27], the sufficient condition is obtained to guarantee the existence of the periodic solution for a class of neutral-type neural networks with delays; and (3) By constructing a novel Lyapunov functional, the sufficient conditions are derived for the concerned systems to be globally exponentially stable.

The following sections are organized as follows: In Sect. 2, the problem under consideration is formulated and some useful lemmas are introduced. In Sect. 3, sufficient conditions are established for the existence of a unique periodic solution of neutral neural networks. The global exponential stability of the periodic solution are investigated in Sect. 4. In Sect. 5, a numerical example is provided to show the feasibility of our results. Finally, we conclude the paper in Sect. 6.

2 Preliminaries

Consider the following neutral-type neural networks with delays:

$$\begin{aligned} \left\{ \begin{aligned}&(A_ix_i)'(t) =-a_i(t)x_i(t)+\sum _{j=1}^n[b_{ij}(t)f_j(t,x_j(t))+d_{ij}(t)g_j(t,x_j(t-\tau _{ij}(t)))]+I_i(t),\\&x_i(t) =\phi _i(t),~t\in [-\tau ,0],~i=1,2,\ldots ,n, \end{aligned}\right. \end{aligned}$$
(1)

where \(A_i\) is a difference operator defined by

$$\begin{aligned} (A_ix_i)(t)=x_i(t)-c_i(t)x_i(t-\gamma ), \end{aligned}$$
(2)

\(x_i(t)\) denotes the state of the ith unit at time t, and \(I_i(t)\) is the external bias on the ith at time t, \(a_i(t)\) represents the rate with which the ith unit will reset its potential to the resting state when disconnected from the network and external inputs at time t, \(\tau _{ij}(t)\) corresponds to the finite speed of the axonal transmission of signal, \(b_{ij}(t)\) denotes the strength of the jth unit on the ith unit at time t, \(d_{ij}(t)\) denotes the strength of the jth unit on the ith unit at time \(t-\tau _{ij}(t)\) and \(f_j\) is the signal transmission function. Throughout this paper, it is assumed that \(c_i(t),~a_{i}(t),~b_{ij}(t),~d_{ij}(t),~\tau _{ij}(t),~I_i(t)\) are continuously periodic functions defined on \(t\in [0,\infty )\) with a common period \(\omega >0\). Moreover, \(\gamma ,~a_i(t),~b_{ij}(t),~d_{ij}(t)\) are positive everywhere, \(f_j(t,x),~g_j(t,x)\) are continuous and \(\omega -\)periodic with respect to t.

Let \(\tau =\max \{\gamma ,~\tau _{ij}(t),~1\le i,j\le n,~t\in [0,\infty )\}.\) The initial-value functions are as follow:

$$\begin{aligned} \phi (t)=(\phi _1(t),\phi _2(t),\cdots ,\phi _n(t))^T\in C([-\tau ,0], {\mathbb {R}}^n), \end{aligned}$$

where \(C([-\tau ,0], {\mathbb {R}}^n)\) is the Banach space of continuous functions on \([-\tau ,0]\) with norm

$$\begin{aligned} ||\phi ||=\sup _{t\in [-\tau ,0]}\max _{1\le i\le n}|\phi _i(t)|. \end{aligned}$$

Denote that

$$\begin{aligned} c_{0}= & {} \max _{t\in [0,T]}|c(t)|,~~\sigma =\min _{t\in [0,T]}|c(t)|,\\ C_{T}= & {} \{x|x\in C({\mathbb {R}}, {\mathbb {R}}),~x(t+T)\equiv x(t),~ \forall t\in {\mathbb {R}}\} \end{aligned}$$

with the norm

$$\begin{aligned} |\varphi |_{0}=\max \limits _{t\in [0,T]}|\varphi (t)|,~~\forall \varphi \in C_{T}. \end{aligned}$$

Clearly, \(C_{T}\) is a Banach space. Define a linear operator as follow:

$$\begin{aligned} A:C_{T}\rightarrow C_{T},~~[Ax](t)=x(t)-c(t)x(t-\tau ),~~\forall t\in \mathbb {R}. \end{aligned}$$

Lemma 2.1

[27] If  \(|c(t)|\ne 1\), then the operator A  has the continuous inverse \(A^{-1}\) on \(C_T\) which satisfies

  1. (i)
    $$\begin{aligned}{}[A^{-1}f](t)=\left\{ \begin{aligned}&f(t)+\sum \limits _{j= 1}^{\infty }\prod \limits _{i=1}^{j}c(t-(i-1)\tau )f(t-j\tau ),~c_{0}< 1,~\forall f\in C_{T} ,&\hbox {} \\&-\frac{f(t+\tau )}{c(t+\tau )}-\sum \limits _{j= 1}^{\infty }\prod \limits _{i=1}^{j+1}\frac{1}{c(t+i\tau )}f(t+j\tau +\tau ),~\sigma > 1,~\forall f\in C_{T} .&\hbox {} \end{aligned} \right. \end{aligned}$$
  2. (ii)
    $$\begin{aligned} \int _{0}^{T}|[A^{-1}f](t)|dt\le \left\{ \begin{aligned}&\frac{1}{1-c_{0} }\int _{0}^{T}|f(t)|dt,~c_{0}< 1 ,~\forall f\in C_{T},&\hbox {} \\&\frac{1}{\sigma -1}\int _{0}^{T}|f(t)|dt,~\sigma > 1 ,~\forall f\in C_{T} .&\hbox {} \end{aligned} \right. \end{aligned}$$
  3. (iii)
    $$\begin{aligned}|A^{-1}f|_0\le \left\{ \begin{aligned}&\frac{1}{1-c_{0} }|f|_0,~c_{0}< 1 ,~\forall f\in C_{T},&\hbox {} \\&\frac{1}{\sigma -1}|f|_0,~\sigma > 1 ,~\forall f\in C_{T} .&\hbox {} \end{aligned} \right. \end{aligned}$$

The famous Mawhin’s continuation theorem is recalled as follows.

Lemma 2.2

[31] Suppose that X and Y are two Banach spaces, and \(L:D(L)\subset X\rightarrow Y\) is a Fredholm operator with index zero. Furthermore, \(\Omega \subset X\) is an open bounded set and \(N:\bar{\Omega }\rightarrow Y\) is L-compact on \(\bar{\Omega }\). If all the following conditions hold

  1. (i)

    \(Lx\ne \lambda Nx,\forall x\in \partial \Omega \cap D(L),\forall \lambda \in (0,1),\)

  2. (ii)

    \(Nx\notin ImL,\forall x\in \partial \Omega \cap KerL,\)

  3. (iii)

    \( deg\{JQN,\Omega \cap KerL,0\}\ne 0\),

where \(J:ImQ\rightarrow KerL\) is an isomorphism, then the equation \(Lx=Nx\) has a solution on \(\bar{\Omega }\cap D(L).\)

3 Existence of Periodic Solution

For convenience, the following notations will be used in this paper:

$$\begin{aligned} \bar{f}=\frac{1}{\omega }\int _0^\omega f(t)dt, ~~f^+=\max _{t\in [0,\omega ]}|f(t)|,~~f^-=\min _{t\in [0,\omega ]}|f(t)|, \end{aligned}$$

where f is a continuous \(\omega -\)periodic function.

Denote by \(C_\omega \) (respectively, \(C_\omega ^1\)) the set of all continuous (respectively, differentiable) \(\omega \)-periodic functions with respect to \(x(t)=(x_1(t),x_2(t),\cdots ,x_n(t))^\mathrm {T}\) defined on \(\mathbb {R}.\) Moreover, denote that

$$\begin{aligned} |x|_0=\max _{1\le i\le n}\{x_i^+\},~~|x|_1=\max \{|x|_0,|x'|_0\}. \end{aligned}$$

Then \(C_\omega \) and \(C_\omega ^1\) are Banach spaces with the norms \(|\cdot |\) and \(|\cdot |_1\), respectively. Define

$$\begin{aligned} A:&C_{\omega }\rightarrow C_{\omega }, (Ax)(t)=x(t)-C(t)x(t-\gamma ), \forall t\in \mathbb {R},\\ L:&D(L)\subset C_{\omega }\rightarrow C^{1}_{\omega }, (Lx)(t)=(Ax)'(t),\\ N:&C^{1}_{\omega }\rightarrow C_{\omega }, (Nx)_i(t) =-a_i(t)x_i(t)+\sum _{j=1}^n[b_{ij}(t)f_j(t,x_j(t))\\&+\,d_{ij}(t)g_j(t,x_j(t-\tau _{ij}(t)))]+I_i(t),~i=1,2,\cdots ,n, \end{aligned}$$

where \(C(t)=\mathrm {diag}(c_1(t),c_2(t),\cdots ,c_n(t))\) and \(D(L)=\{x: x\in C^{1}_{\omega }\}.\)

Then system (1) is the operator equation \(Lx=Nx\). It is easy to see

$$\begin{aligned} \text {Im}L=\left\{ y: y\in C_{\omega }, \int _{0}^{\omega }y(s)ds=0\right\} . \end{aligned}$$

We have \((x(t)-C(t)x(t-\tau ))'=0\) \(\forall x\in KerL\). Therefore,

$$\begin{aligned} x(t)-C(t)x(t-\tau )=\widetilde{c}, \end{aligned}$$
(3)

where \(\widetilde{c}\in {\mathbb {R}}^n\) is a constant vector. Let \(\varphi (t)\) be the unique \(\omega -\)periodic solution of (3), then \(\varphi (t)\ne \mathbf {0}\) and

$$\begin{aligned} \text {Ker}L=\{a_{0}\varphi (t): a_{0}\in \mathbb {R}\}. \end{aligned}$$

It is now a position to state the result on the existence of the periodic solution.

Theorem 3.1

Assume that \(c_i^+<1\) or \(c_i^->1\) for \(t>0\) and \(i=1,2,\cdots ,n.\) Moreover, suppose that \(\int _{0}^{T}\varphi ^{2}(t)dt\ne 0\) where \(\varphi (t)\) is defined in (3), and there exist non-negative constants \(p_j\) and \(q_j\) such that

$$\begin{aligned} |f_j(t,x)|\le p_j,~~|g_j(t,x)|\le q_j,~j=1,2,\cdots ,n. \end{aligned}$$

Then system (1) has at least one \(\omega -\)periodic solution.

Proof

Obviously, ImL is a closed in \(C_{\omega }\) and \(dimKerL=condimImL=n\). So L is a Fredholm operator with index zero. From \(\int _{0}^{T}\varphi ^{2}(t)dt\ne 0\), define continuous projectors P, Q

$$\begin{aligned} P: C_{\omega }\rightarrow KerL, (Px)(t)=\frac{\int _{0}^{\omega }x(t)\varphi (t)dt}{\int _{0}^{\omega }\varphi ^{2}dt}\varphi (t) \end{aligned}$$

and

$$\begin{aligned} Q: C_{\omega }\rightarrow C_{\omega }/ImL, Qy=\frac{1}{\omega }\int _{0}^{\omega }y(s)ds. \end{aligned}$$

Let

$$\begin{aligned} L_{P}=L|_{D(L)\cap KerP}: D(L)\cap KerP\rightarrow ImL, \end{aligned}$$

then

$$\begin{aligned} L^{-1}_{P}=K_{P}: ImL\rightarrow D(L)\cap KerP. \end{aligned}$$

Since \(ImL\subset C_{T}\) and \(D(L)\cap KerP\subset C^{1}_{T}\), we know that \(K_{P}\) is an embedding operator. Hence, \(K_{P}\) is a completely operator in ImL. According to the definitions of Q and N, it can be found that \(QN(\overline{\Omega })\) is bounded on \(\overline{\Omega }\). Then, nonlinear operator N is \(L-\)compact on \(\overline{\Omega }\). Next, we will complete the proof in three steps.

Step 1. Let \(\Omega _{1}=\{x\in D(L)\subset C^{1}_{\omega }: Lx=\lambda Nx, \lambda \in (0, 1)\}.\) We show that \(\Omega _{1}\) is a bounded set. If \(\forall x\in \Omega _{1}\), then \(Lx=\lambda Nx\), i.e., for \(i=1,2,\cdots ,n,\)

$$\begin{aligned}&(A_ix_i(t))'+\lambda a_i(t)x_i(t)-\lambda \sum _{j=1}^n[b_{ij}(t)f_j(t,x_j(t))+d_{ij}(t)g_j(t,x_j(t-\tau _{ij}(t)))]\nonumber \\&\quad -\lambda I_i(t)=0. \end{aligned}$$
(4)

Notice that there exists \(t_i\in [0,\omega ]\) such that \(Ax_i(t_i)=[Ax_i(t)]^+.\) Hence \((A_ix_i)'(t_i)=0\) which implies that

$$\begin{aligned} a_i(t_i)x_i(t_i)&=\sum _{j=1}^n[b_{ij}(t_i)f_j(t_i,x_j(t_i))\nonumber \\&\quad +d_{ij}(t_i)g_j(t_i,x_j(t_i-\tau _{ij}(t_i)))]+ I_i(t_i). \end{aligned}$$
(5)

From (5), we have

$$\begin{aligned} |x_i(t_i)|&=\left| \sum _{j=1}^n\left[ \frac{b_{ij}(t_i)}{a_i(t_i)}f_j(t_i,x_j(t_i))\right. \right. \nonumber \\&\left. \left. \quad +\frac{d_{ij}(t_i)}{a_i(t_i)}g_j(t_i,x_j(t_i-\tau _{ij}(t_i)))\right] + \frac{I_i(t_i)}{a_i(t_i)}\right| \nonumber \\&\le \sum _{j=1}^np_j\bigg [\frac{b_{ij}(t_i)}{a_i(t_i)}\bigg ]^++\sum _{j=1}^nq_j\bigg [\frac{d_{ij}(t_i)}{a_i(t_i)}\bigg ]^++ \bigg [\frac{I_i(t_i)}{a_i(t_i)}\bigg ]^+,\quad i=1,2,\cdots ,n. \end{aligned}$$
(6)

By (6), we obtain

$$\begin{aligned}{}[x_i(t)]^+\le h_i,~~i=1,2,\cdots ,n, \end{aligned}$$

where \(h_i\) is the ith component of vector h, and it is independent of \(\lambda \). Moreover, it follows from (4) that

$$\begin{aligned}{}[(A_ix_i(t))']^{+}&\le \max _{t\in [0,\omega ]}[a_i(t)|x_i(t)|+\sum _{j=1}^n[|b_{ij}(t)f_j(t,x_j(t))|\nonumber \\&\quad +|d_{ij}(t)g_j(t,x_j(t-\tau _{ij}(t)))|]+|I_i(t)|]\nonumber \\&\le [a_i(t)]^+h_i+\sum _{j=1}^n[p_j|b_{ij}(t)]^++q_i[d_{ij}(t)]^+]+[I_i(t)]^+:=\hbar _i. \end{aligned}$$
(7)

From (7) and Lemma 2.1, if \(c_i^+<1,\) we get

$$\begin{aligned} |x_i'(t)|&=|A_i^{-1}A_ix_i'(t)|\nonumber \\&\le \frac{1}{1-c_i^+}\max _{t\in [0,\omega ]}\{|A_ix_i'(t)|\}\nonumber \\&=\frac{1}{1-c_0}\max _{t\in [0,\omega ]}\{|(A_ix_i)'(t)+c_i'(t)x_i(t-\gamma )|\}\nonumber \\&\le \frac{\hbar _i+[c_i'(t)]^+h_i}{1-c_i^+}. \end{aligned}$$
(8)

Similarly, if \(c_i^->1,\) we have

$$\begin{aligned} |x_i'(t)|\le \frac{\hbar _i+[c_i'(t)]^+h_i}{c_i^--1}. \end{aligned}$$
(9)

From (8) and (9), we obtain

$$\begin{aligned}{}[x_i'(t)]^+\le \ell _i. \end{aligned}$$

Step 2. From the above proof, it can be found that there exists some \(d>1\) such that

$$\begin{aligned} dh_i>A~~\mathrm {for}~~i=1,2,\cdots ,n, \end{aligned}$$

where \(A=\max _{1\le i\le n}\{h_i+1,\hbar _i+1\}.\) Let \(\Omega _2=\{x\in C_\omega ^1: -dh<x(t)<dh\}.\) We shall prove that if \(x\in \partial \Omega _{2}\subset KerL,\) then

$$\begin{aligned} |(QNx)_i|\ne 0~~\mathrm {for}~~i=1,2,\cdots ,n \end{aligned}$$
(10)

where

$$\begin{aligned} (QNx)_i= & {} \frac{1}{\omega }\int _0^\omega \Big [-a_i(t)x_i(t)+\sum _{j=1}^n \left[ b_{ij}(t)f_j(t,x_j(t))\right. \\&\left. +\,d_{ij}(t)g_j(t,x_j(t-\tau _{ij}(t)))\right] +I_i(t)\Big ]dt. \end{aligned}$$

Considering \(x\in \partial \Omega _{2}\subset KerL,\) it is obvious that x is a constant vector in \({\mathbb {R}}^n\) with \(|x_i|=dh_i\) for \(i=1,2,\cdots ,n.\)

Next, the proof by contradiction will be used. Suppose that there exists some \(i\in \{1,2,\cdots ,n\}\) such that \(|(QNx)_i|=0,\) i.e.,

$$\begin{aligned} \int _0^\omega \Big [-a_i(t)x_i(t)+\sum _{j=1}^n \left[ b_{ij}(t)f_j(t,x_j(t))+d_{ij}(t)g_j(t,x_j(t-\tau _{ij}(t)))\right] +I_i(t)\Big ]dt=0. \end{aligned}$$

Then there exists some \(\xi \in [0,\omega ]\) such that

$$\begin{aligned} -a_i(\xi )x_i+\sum _{j=1}^n \left[ b_{ij}(\xi )f_j(\xi ,x_j)+d_{ij}(\xi )g_j(\xi ,x_j)\right] +I_i(\xi )=0. \end{aligned}$$

Therefore, we have

$$\begin{aligned} dh_i=|x_i|\le \sum _{j=1}^n \left[ \frac{|b_{ij}(\xi )|}{a_i(\xi )}|f_j(\xi ,x_j)|+\frac{|d_{ij}(\xi )|}{a_i(\xi )}|g_j(\xi ,x_j)|\right] +\frac{|I_i(\xi )|}{a_i(\xi )}\le h_i. \end{aligned}$$

In view of \(d>1,\) it follows from the above inequality that \(dh_i < dh_i\), which is a contradiction. Thus, (10) holds and hence

$$\begin{aligned} QNx\ne 0,~~\forall x\in \partial \Omega _{2}\subset KerL. \end{aligned}$$

Step 3. We shall prove that the third condition in Lemma 2.2 holds. Take the homotopy

$$\begin{aligned} H(x, \mu )=\mu \mathrm { diag}(-\bar{a}_1,-\bar{a}_2,\cdots ,-\bar{a}_n) x+(1-\mu )QNx,~ x\in \overline{\Omega }\cap KerL,~ \mu \in [0, 1]. \end{aligned}$$

When \(x\in \partial \Omega _{2}\subset KerL,\) one has \(|x_i|=dh_i\) for \(i=1,2,\cdots ,n.\) Thus,

$$\begin{aligned} |H(x, \mu )|_0=\max _{1\le i\le n}\bigg \{-\bar{a}_ix_i +\frac{(1-\mu )}{\omega }\Sigma _{j=1}^n\int _0^\omega [b_{ij}(t)f_j(t,x_j)+d_{ij}(t)f_j(t,x_j)]dt+\bar{I}_i\bigg \}. \end{aligned}$$

We claim that

$$\begin{aligned} |H(x, \mu )|_0>0. \end{aligned}$$
(11)

Suppose that \(|H(x, \mu )|_0=0\), then for \(i=1,2,\cdots ,n\) we have

$$\begin{aligned} -\bar{a}_ix_i +\frac{(1-\mu )}{\omega }\Sigma _{j=1}^n\int _0^\omega [b_{ij}(t)f_j(t,x_j)+d_{ij}(t)f_j(t,x_j)]dt+(1-\mu )\bar{I}_i=0. \end{aligned}$$
(12)

According to the integral mean value theorem and (12), there is some \(\eta \in [0,\omega ]\) such that

$$\begin{aligned} -\bar{a}_ix_i +(1-\mu )\Sigma _{j=1}^n[b_{ij}(\eta )f_j(\eta ,x_j)+d_{ij}(\eta )f_j(\eta ,x_j)]+(1-\mu )\bar{I}_i=0. \end{aligned}$$

Then, one has

$$\begin{aligned} dh_i=|x_i|\le (1-\mu )\sum _{j=1}^n \left[ \frac{|b_{ij}(\eta )|}{\bar{a}_i}|f_j(\eta ,x_j)|+\frac{|d_{ij}(\eta )|}{\bar{a}_i}|g_j(\eta ,x_j)|\right] +(1-\mu )\frac{|\bar{I}_i|}{\bar{a}_i}\le h_i \end{aligned}$$

which contradicts that \(d>1.\) Therefore, Eq. (11) holds. Using the property of topological degree and taking J to be the identity mapping \(I: ImQ\rightarrow KerL\), we have

$$\begin{aligned} \begin{aligned} \deg \{JQN,\Omega \cap KerL,0\}&=\deg \{H(\cdot ,0),\Omega \cap KerL,0\}\\&=\deg \{H(\cdot ,1),\Omega \cap KerL,0\}\\&=\deg \{\mathrm { diag}(-\bar{a}_1,-\bar{a}_2,\cdots ,-\bar{a}_n),\Omega \cap KerL,0\}\\&=1\ne 0. \end{aligned} \end{aligned}$$

Then, by using Lemma 2.2, we obtain that the equation \(Lx=Nx\) has at least one \(\omega -\)periodic solution x in \(\bar{\Omega }.\) Namely, system (1) has at least one \(\omega -\)periodic solution. \(\square \)

Remark 3.2

The neural network system (1) shows the neutral features characterized by the \(A_i\) operator, which is different from the corresponding results of other papers, see, e.g., [1, 22, 24, 25]. And also the main results are derived by fully taking advantage of the some properties of operators, such as the characterization of the kernel space of operator L.

If we further assume that \(f_j(t,x)\) and \(g_j(t,x)~(j=1,2,\cdots ,n)\) are globally Lipschitz with respect to the second variables, then we shall obtain a unique \(\omega -\)periodic solution for system (1). Then we have the following corollary.

Corollary 3.3

Assume that all the conditions of Theorem 3.1 hold and there exist nonnegative constants \(L_j^f\) and \(L_j^g\) such that \(|f_j(t,x)-f_j(t,y)|\le L_j^f|x-y|\) and \(|g_j(t,x)-g_j(t,y)|\le L_j^g|x-y|,\) where \(t>0,~x,y\in {\mathbb {R}},~j=1,2,\cdots ,n.\) Then system (1) has a unique \(\omega -\)periodic solution x(t) on \([-\tau ,\infty ]\).

Since the proof of Corollary 3.3 is similar to Ref. [32], we omit it here.

4 Exponential Stability

In this section, we shall deal with the exponential stability of the periodic solutions.

Definition 4.1

Let \(z^*(t)=(z_1^*(t), z_2^*(t), \cdots , z_n^*(t))^\top \) be a periodic solution of system (1). Then the periodic solution \(z^*(t)\) is globally exponentially stable if there exist constants \(\alpha >0\) and \(\beta \ge 1\) such that, for the any solution of system (1), the following holds

$$\begin{aligned} |z(t)-z^*(t)|\le \beta e^{-\alpha t}|z(0)-z^*(0)|,~~t>0. \end{aligned}$$

Denote that

$$\begin{aligned} \mathbb {D}= & {} \mathrm {diag}\left( a_1^--\left( \frac{a_1^+c_1^+}{1-c_1^+}+1\right) , a_2^--\left( \frac{a_2^+c_2^+}{1-c_2^+}+1\right) ,\cdots ,a_n^--\left( \frac{a_n^+c_n^+}{1-c_n^+}+1\right) \right) ,\\ {\mathbb {E}}= & {} \bigg (\frac{b_{ij}^+L_j^f+d_{ij}^+L_j^g}{1-c_j^+}\bigg )_{n\times n}, \end{aligned}$$

then we have the following theorem for the stability of system (1).

Theorem 4.2

Assume that all the conditions of Theorem 3.1 hold, \(\rho ({\mathbb {D}}^{-1}{\mathbb {E}})<1,\) \(f_j(t,0)=g_j(t,0)=0\) and there exist nonnegative constants \(L_j^f\) and \(L_j^g\) such that \(|f_j(t,x)-f_j(t,y)|\le L_j^f|x-y|\) and \(|g_j(t,x)-g_j(t,y)|\le L_j^g|x-y|,\) where \(t>0,~x,y\in {\mathbb {R}},~j=1,2,\cdots ,n.\) Then system (1) has a unique \(\omega -\)periodic solution x(t) on \([-\tau ,\infty ]\), which is globally exponentially stable.

Proof

Since \(\rho ({\mathbb {D}}^{-1}{\mathbb {E}})<1,\) i.e., the spectral radius of \({\mathbb {D}}^{-1}{\mathbb {E}}\) is less than 1, there exists a constant vector \(\vartheta =(\vartheta _1,\vartheta _2,\cdots ,\vartheta _n)^\mathrm {T}>0\) such that

$$\begin{aligned} ({\mathbb {E}}_n-{\mathbb {D}}^{-1}{\mathbb {E}})\vartheta >0,~~[I_i(t)]^+\le \vartheta _i,~i=1,2,\cdots ,n, \end{aligned}$$

where \({\mathbb {E}}_n\) is an \(n\times n\) unit matrix. Thus

$$\begin{aligned} \left( -a_i^-+\frac{a_i^+c_i^+}{1-c_i^+}+1\right) \vartheta _i +\sum _{j=1}^n\frac{(b_{ij}^+L_j^f+d_{ij}^+L_j^g)}{1-c_j^+}\vartheta _j<0,~~i=1,2,\cdots ,n. \end{aligned}$$
(13)

For \(t>0,\) define

$$\begin{aligned} \Xi _i(t)=\bigg [t-a_i^-+\left( \frac{a_i^+c_i^+}{1-c_i^+}+1\right) e^{t\gamma }\bigg ]\vartheta _i +\sum _{j=1}^n\frac{(b_{ij}^+L_j^f+d_{ij}^+L_j^ge^{t\tau _{ij}^+})}{1-c_i^+}\vartheta _j,~~i=1,2,\cdots ,n. \end{aligned}$$

It is easy to verify that \(\Xi _i(t),~i=1,2,\cdots ,n\) are all continuous on interval \([0,\lambda _0].\) Then we have

$$\begin{aligned} \Xi _i(0)=\left( -a_i^-+\frac{a_i^+c_i^+}{1-c_i^+}+1\right) \vartheta _i +\sum _{j=1}^n\frac{(b_{ij}^+L_j^f+d_{ij}^+L_j^g)}{1-c_j^+}\vartheta _j<0,~~i=1,2,\cdots ,n \end{aligned}$$

and there is a constant \(\lambda \in [0,\lambda _0]\) such that

$$\begin{aligned}&\Xi _i(\lambda )=\bigg [\lambda -a_i^-+\left( \frac{a_i^+c_i^+}{1-c_i^+}+1\right) e^{\lambda \gamma }\bigg ]\vartheta _i +\sum _{j=1}^n\frac{(b_{ij}^+L_j^f+d_{ij}^+L_j^ge^{\lambda \tau _{ij}^+})}{1-c_j^+}\vartheta _j\\&\quad <0,~~i=1,2,\cdots ,n. \end{aligned}$$

For the above \(\lambda \), we choose the following Lyapunov functional:

$$\begin{aligned} V_i(t)=|(A_ix_i)(t)|e^{\lambda t},~~t>0,~i=1,2,\cdots ,n. \end{aligned}$$

We claim that

$$\begin{aligned} V_i(t)=|(A_ix_i)(t)|e^{\lambda t}<\xi _i,~~t>0,~i=1,2,\cdots ,n. \end{aligned}$$
(14)

Otherwise, there must exist an \(i\in \{1,2,\cdots ,n\}\) and \(t_i>0\) with \(t_i\le \gamma \) such that

$$\begin{aligned} V_i(t_i)=\xi _i~\mathrm {and}~V_j(t)<\xi _j,~i=1,2,\cdots ,n,~t<t_i. \end{aligned}$$

Calculating the time derivative of \(V_i(t)\) along the trajectories of system (1), we have

$$\begin{aligned} 0\le D^+V_i(t_i)&=\mathrm {sgn}\{(A_ix_i)(t_i)\}(A_ix_i)'(t_i)e^{\lambda t_i}+\lambda |(A_ix_i)(t_i)|e^{\lambda t_i}\nonumber \\&=\mathrm {sgn}\{(A_ix_i)(t_i)\}\big [-a_i(t_i)x_i(t_i)\nonumber \\&\quad +\sum _{j=1}^n[b_{ij}(t_i)f_j(t,x_j(t_i))+d_{ij}(t_i)g_j(t_i,x_j(t_i-\tau _{ij}(t_i)))]+I_i(t_i)\big ]e^{\lambda t_i}\nonumber \\&\quad +\lambda |(A_ix_i)(t_i)|e^{\lambda t_i}\nonumber \\&\le (\lambda -a_i^-)|(A_ix_i)(t_i)|e^{\lambda t_i} +a_i^+c_i^+|x_i(t_i-\gamma )|e^{\lambda (t_i-\gamma )}e^{\lambda \gamma }+[I_i(t)]^+e^{\lambda \gamma }\nonumber \\&\quad +\sum _{j=1}^nb_{ij}^+L_j^f|x_j(t_i)|e^{\lambda t_i}+\sum _{j=1}^nd_{ij}^+L_j^g|x_j(t_i-\tau _{ij}(t_i))|e^{\lambda (t_i-\tau _{ij}(t_i))}e^{\lambda \tau _{ij}(t_i)}\nonumber \\&\le \bigg [\lambda -a_i^-+(a_i^+c_i^+\frac{1}{1-c_i^+}+1)e^{\lambda \gamma }\bigg ]\xi _i\nonumber \\&\quad +\sum _{j=1}^n(b_{ij}^+L_j^f+d_{ij}^+L_j^ge^{\lambda \tau _{ij}^+})\frac{1}{1-c_j^+}\xi _j, \end{aligned}$$
(15)

where the following inequalities are used

$$\begin{aligned} |x_i(t_i-\gamma )|&=|A_i^{-1}A_ix_i(t_i-\gamma )|\le \bigg |\frac{A_ix_i(t_i-\gamma )}{1-c_i^+}\bigg |,\\ |x_j(t_i)|&=|A_j^{-1}A_jx_j(t_i)|\le \bigg |\frac{A_jx_j(t_i)}{1-c_j^+}\bigg |,~~|x_j(t_i)|=|A_j^{-1}A_jx_j(t_i-\tau _{ij}(t_i))|\\&\le \bigg |\frac{A_jx_j(t_i-\tau _{ij}(t_i))}{1-c_j^+}\bigg |. \end{aligned}$$

It is obvious that (15) contradicts (13), and then (14) holds. It follows from (14) that, for \(i=1,2,\cdots ,n\) and \(t>0,\)

$$\begin{aligned} |x_i(t)|=|A_i^{-1}A_ix_i(t)|\le \bigg | \frac{A_ix_i(t)}{1-c_i^+}\bigg |\le \frac{\vartheta _i}{1-c_i^+}e^{-\lambda t}\le M_\phi ||\phi ||e^{-\lambda t}, \end{aligned}$$

where \(M_\phi \) is a constant such that \(M_\phi ||\phi ||\ge \frac{\vartheta _i}{1-c_i^+}.\) The proof is now completed. \(\square \)

If we denote that

$$\begin{aligned} {\mathbb {\tilde{D}}}= & {} \mathrm {diag}\bigg (a_1^--\left( \frac{a_1^+c_1^+}{c_1^--1}+1\right) , a_2^--\left( \frac{a_2^+c_2^+}{c_2^--1}+1\right) ,\cdots ,a_n^--\left( \frac{a_n^+c_n^+}{c_n^--1}+1\right) \bigg ),\quad \\ {\mathbb {\tilde{E}}}= & {} \bigg (\frac{b_{ij}^+L_j^f+d_{ij}^+L_j^g}{c_j^--1}\bigg )_{n\times n}, \end{aligned}$$

then we get the following corollary.

Corollary 4.3

Assume that all the conditions of Theorem 3.1 hold, \(\rho ({\mathbb {\tilde{D}}}^{-1}{\mathbb {\tilde{E}}})<1,\) \(f_j(t,0)=g_j(t,0)=0\) and there exist nonnegative constants \(L_j^f\) and \(L_j^g\) such that \(|f_j(t,x)-f_j(t,y)|\le L_j^f|x-y|\) and \(|g_j(t,x)-g_j(t,y)|\le L_j^g|x-y|,\) where \(t>0,~x,y\in {\mathbb {R}},~j=1,2,\cdots ,n.\) Then system (1) has a unique \(\omega -\)periodic solution x(t) on \([-\tau ,\infty ]\), which is globally exponentially stable.

5 Numerical Example

In order to verify the feasibility of our results, consider the following neutral-type neural network:

$$\begin{aligned} \left\{ \begin{aligned} (A_1x_1)'(t)=-a_1(t)x_1(t)+\sum _{j=1}^2[b_{1j}(t)f_j(t,x_j(t))+d_{1j}(t)g_j(t,x_j(t-\tau _{1j}(t)))]+I_1(t),\\ (A_2x_2)'(t)=-a_2(t)x_2(t)+\sum _{j=1}^2[b_{2j}(t)f_j(t,x_j(t))+d_{2j}(t)g_j(t,x_j(t-\tau _{2j}(t)))]+I_2(t), \end{aligned}\right. \end{aligned}$$

where

$$\begin{aligned}&(A_1x_1)(t)=x_1(t)-c_1(t)x_1(t-\gamma ),~ (A_2x_2)(t)=x_2(t)-c_2(t)x_2(t-\gamma ),~\omega =2\pi ,~\gamma =100,\\&I_1(t)=I_2(t)=\sin t,~a_1(t)=a_2(t)=2,~c_1(t)=c_2(t)=0.01\cos t,~b_{ij}(t)=d_{ij}(t)=0.1,\\&\tau _{ij}(t)=\frac{1}{2\pi }\sin t,~f_j(t,u)=g_j(t,u)=0.2\sin u. \end{aligned}$$

For \(i,j=1,2,\) we have

$$\begin{aligned}&c_i^+ =0.01<1,~L_j^f=L_j^g=0.2,~{\mathbb {D}}=\frac{97}{99}\left( \begin{array}{cc} 1 &{} 0 \\ 0 &{}1\\ \end{array} \right) ,~{\mathbb {E}}=\frac{4}{99}\left( \begin{array}{cc} 1 &{} 1 \\ 1 &{} 1 \\ \end{array} \right) ,\\&{\mathbb {D}}^{-1}{\mathbb {E}}=\frac{4}{97}\left( \begin{array}{cc} 1 &{} 1 \\ 1 &{} 1 \\ \end{array} \right) , \end{aligned}$$

then \(\rho ({\mathbb {D}}^{-1}{\mathbb {E}})=\frac{8}{97}<1.\) It follows from Theorem 4.2 that the periodic solution of the above system is globally exponentially stable.

6 Conclusions

In this paper, the problem of stability analysis has been discussed for a class of neutral-type neural networks with time delays and difference operator. By using Mawhin’s continuation theorem and Lyapunov functional method, some results have been derived for the existence, uniqueness and global exponential stability of periodic solution for the concerned systems. A numerical example has been provided to illustrate the effectiveness of the obtained results. The results about existence and global exponential stability of periodic solution for neutral-type neural networks proposed in this paper could be further utilized for other related problems, such as control and filtering [35–37] the non-fragile state estimation in [38, 39], the distributed state estimation for sensor networks as considered in [40, 41]. The results derived in this paper can be also extended into Sensor networks or social networks, which is now a hot research topic [42–44].