Anti-periodic solutions problem for inertial competitive neutral-type neural networks via Wirtinger inequality

By using the Wirtinger inequality and topology degree theory, we investigate the anti-periodic solutions problem for inertial competitive neutral-type neural networks and obtain the existence of anti-periodic solutions to the above system. Our results are completely new.

The Wirtinger inequality was first used in Fourier analysis, then was used in 1904 to prove the isoperimetric inequality; see [1, 2]. Since the Wirtinger inequality has been recognized as a powerful tool to estimate the prior bounds of solutions, it has been used in many research areas, such as Hamiltonian system, delay equations, biomathematics, neural networks, partial differential equation; see [3,4,5,6,7,8] and the relevant references therein.

In this paper, we will obtain the existence of anti-periodic solutions for inertial competitive neutral-type neural networks by using the Wirtinger inequality and topology degree theory. In 1996, Meyer, Ohl and Scheich [9] first proposed the competitive neural network, the behavior of this network is characterized by an equation of neural activity as a fast phenomenon and an equation of synaptic modification as a slow part of the neural system. Since by a competitive neural network one can study the dynamics of complex neural networks including the aspects of long- and short-term memory, it has received great attention. Meyer, Pilyugin and Chen [10] studied global exponential stability of competitive neural networks with different time scales. They presented a new method of analyzing the dynamics of a biological relevant system with different time scales based on the theory of flow invariance. After that, Meyer, Reberts and Thmmle [11] further studied the local uniform stability of competitive neural networks with different time scales under vanishing perturbations. Gu, Jiang and Teng [12] obtained the existence and global exponential stability of a unique equilibrium point of competitive neural networks with different time scales and multiple delays by using a nonlinear Lipschitz measure (NLM) method and constructing a suitable Lyapunov functional. Liu et al. [13] obtained the existence of a periodic solution for competitive neural networks with time-varying and distributed delays on time scales.

On the other hand, the anti-periodic solution problem for dynamic systems is an interesting topic which has been investigated by many researchers. Okochi [14] first considered the anti-periodic solution of nonlinear abstract parabolic equations. Then anti-periodic problems of neural networks have attracted much attention by many authors. Li, Yang and Wu [15] discussed an anti-periodic solution for impulsive BAM neural networks with time-varying leakage delays on time scales. Then Li et al. [16,17,18] further studied anti-periodic solutions for different types of neural networks. Xu, Chen and Guo [19] studied anti-periodic oscillations of bidirectional associative memory (BAM) neural networks with leakage delays. The existence and exponentially stability of anti-periodic solutions for neutral BAM neural networks with time-varying delays in the leakage terms have been obtained by Xu and Guo [20]. However, to the best of our knowledge there are only few results on the anti-periodic solutions for neutral-type competitive neural networks with inertial terms.

The above discussions constitute the motivation for the present paper. In this paper, we will study a kind of neutral-type competitive neural networks with inertial terms. Based on the Wirtinger inequality and topology degree theory, we prove the existence of anti-periodic solutions of the above neural network. We list the main contributions of this paper as follows:

1. (1)

   We propose a class of inertial competitive neutral-type neural networks which is different from the existing competitive neural networks; see [9,10,11, 13, 15, 19].

2. (2)

   Since the model of the present paper contains neutral terms, it is very difficult to estimate an a priori bound. In order to overcome this difficulty, we use the Wirtinger inequality and develop some new mathematical analysis techniques.

3. (3)

   A unified framework is established to handle competitive neural networks with neutral-type terms, time-varying delays and inertial terms.

The subsequent sections are organized as follows: In Sect. 2, the description of the model, some useful lemmas and notations are given. In Sect. 3, sufficient conditions are established for existence of anti-periodic solutions of considered neural networks. In Sect. 4, an example is given to show the feasibility of our results. Finally, Sect. 5 concludes the paper.

Consider the following generalized inertial competitive neutral-type neural networks with time-varying delays:

where \(i,j=1,2,\ldots ,n\); \((A_{i}x_{i})(t)\) is a neutral operator which is defined by

\(c_{i}\) and σ are constants with \(|c_{i}|\neq 1\) and \(\sigma >0\); the second order derivative is an inertial term; \(x_{i}(t)\) is the neuron current activity level; \(\varepsilon >0\) is a fast time scale; \(\alpha _{i}(t)>0\) is a variable coefficient; \(\beta _{i}(t)>0\) is a damping coefficient; \(b_{ij}(t)\) and \(c_{ij}(t)\) represent the connection weight and the synaptic weight of delayed synaptic efficiency; \(m_{ij}(t)\) the is synaptic efficiency; \(d_{j}\) is the constant external stimulus; \(B_{i}(t)\) is the strength of the external stimulus; \(f_{j}(x_{j}(t))\) is the output of neurons; \(\gamma _{ij}(t)>0\) is a transmission delay.

The initial values of system (2.1) are given by

where \(i,j=1,2,\ldots ,n\) and \(s\in [-\tau ,0]\) with \(\tau = \max_{1\leq i,j\leq n} \{\sigma ,\gamma _{ij}(t),t\in \mathbb{R} \}\).

Let \(\varepsilon =1\) and \(z_{i}(t)=\sum_{j=1}^{n}d_{j}m_{ij}(t)\), then (2.1) is transformed into

where \(d=\sum_{j=1}^{n}d_{j}^{2}>0\). Let

then (2.4) can be written as follows:

In view of the initial values of system (2.1), we can obtain the initial values of system (2.6)

where \(i,j=1,2,\ldots ,n\) and \(s\in [-\tau ,0]\).

Now, we give the famous Wirtinger inequality.

Lemma 2.1

([21, 22] Wirtinger inequality)

If u is a \(C^{1}\) function such that \(u(0) = u(T)\), then

where \(\|u'\|_{L_{2}}= (\int _{0}^{T}|u(t)|^{2}\,dt )^{ \frac{1}{2}}\) and \(\bar{u}=\frac{1}{T}\int _{0}^{T}|u(t)|\,dt\).

Remark 2.1

When u is an anti-periodic function, i.e., \(u(t+T)=-u(t)\), \(\forall t \in \mathbb{R}\), then

and the Wirtinger inequality is given by

In this paper, we also need the following lemmas.

Lemma 2.2

([22])

Let X and Y be Banach spaces, and let \(L: \operatorname{Dom}L \subset X \rightarrow Y \) be linear, \(N: X \rightarrow Y \) be continuous. Assume that L is one-to-one and \(K:=L^{-1}N\) is compact. Furthermore, assume there exists a bounded and open subset \(\varOmega \subset X\) with \(0\in \varOmega \) such that the equation \(Lu=\lambda Nu\) has no solutions in \(\partial \varOmega \cup \operatorname{Dom}L\) for any \(\lambda \in (0, 1)\). Then the problem \(Lu = Nu\) has at least one solution in Ω̄.

Lemma 2.3

([23, 24])

Define A on \(C_{T}\)

where \(C_{T}=\{x:x\in C(\mathbb{R},\mathbb{R}),x(t+T)\equiv x(t)\}\), c is constant. When \(|c|\neq 1\), then A has a unique continuous bounded inverse \(A^{-1}\) satisfying

Obviously, we have

1. (1)

   \(\|A^{-1}\|\le \frac{1}{ \vert 1- \vert c \vert \vert }\);

2. (2)

   \(\int _{0}^{T}|[A^{-1}f](t)|\,dt\le \frac{1}{ \vert 1- \vert c \vert \vert }\int _{0}^{T}|f(t)|\,dt\), \(\forall f\in C_{T}\);

3. (3)

   \(\int _{0}^{T}|[A^{-1}f](t)|^{2}\,dt\le \frac{1}{ \vert 1- \vert c \vert \vert }\int _{0} ^{T}|f(t)|^{2}\,dt\), \(\forall f\in C_{T}\).

In what follows, for \(u=(u_{1},u_{2},\ldots ,u_{3n})^{\top }\), denote

For \(i,j=1,2,\ldots ,n\), we list the following notations which will be used in this paper:

Throughout this paper, we need the following assumptions:

(H₁):

For \(i,j=1,2,\ldots ,n\) and \(t,x\in \mathbb{R}\), \(\alpha _{i},\beta _{i},b_{ij},c_{ij},B_{i},\gamma _{ij}\in C(\mathbb{R}, \mathbb{R})\) with

$$\begin{aligned}& \alpha _{i}(t+\omega )=\alpha _{i}(t),\qquad \beta _{i}(t+\omega )=\beta _{i}(t), \\& B _{i}(t+\omega )=B_{i}(t),\qquad \gamma _{ij}(t+\omega )=\gamma _{ij}(t), \\& b_{ij}(t+\omega )f_{j}(x)=-b_{ij}(t)f_{j}(-x),\qquad c_{ij}(t+\omega )f _{j}(x)=-c_{ij}(t)f_{j}(-x). \end{aligned}$$

(H₂):

For \(j=1,2,\ldots ,n\) and \(x,y\in \mathbb{R}\), there exist positive constants \(L_{j}>0\)

$$ \bigl\vert f_{j}(x)-f_{j}(y) \bigr\vert \leq L_{j} \vert x-y \vert . $$

(H₃):

For \(i,j=1,2,\ldots ,n\)

$$ 1-\gamma _{ij}'(t)>0,\quad t\in \mathbb{R}. $$

Remark 2.2

From \(1-\gamma _{ij}'(t)>0\), \(t\in \mathbb{R}\), it is easy to see that \(t-\gamma _{ij}(t)\) has the inverse function \(\varGamma _{ij}\) for \(i,j=1,2,\ldots ,n\) and \(t\in \mathbb{R}\). Hence, let \(t-\gamma _{ij}(t)=u _{ij}\), then \(t=\varGamma _{ij}(u_{ij})\).

Let

with the norm

Clearly, X is a Banach space. Let

where \(D(L)=\{u:u\in X,(A_{i}x_{i})',y_{i}',z_{i}'\in X\}\). Let there be a nonlinear operator \(N:X\rightarrow X\):

Clearly,

For \(u=(x_{1},\ldots ,x_{n},y_{1},\ldots ,y_{n},z_{1},\ldots ,z_{n})^{ \top }\in \operatorname{Im}L\), let the inverse operator of L be \(L^{-1}\) as follows:

where

Theorem 3.1

Assume that the assumptions (H₁)–(H₃) hold. Furthermore, the following assumption holds:

(H₄):

$$\begin{aligned} \varTheta =& \min_{1\leq i\leq n} \Biggl\{ 1-\sum _{i=1}^{n} \frac{\omega \vert 1- \alpha _{i}^{-} \vert }{ \vert 1- \vert c_{i} \vert \vert \pi } \\ &{}-\sum_{i=1}^{n} \Biggl[\beta _{i} ^{+}+\sum_{j=1}^{n} \biggl(b_{ij}^{+}L_{j} +c_{ij}^{+}L_{j} \max_{s\in \mathbb{R}}\frac{1}{ \vert 1-\gamma _{ij}'(s) \vert } \biggr) \Biggr] \frac{ \omega ^{2}}{ \vert 1- \vert c_{i} \vert \vert \pi ^{2}} \\ &{}-\sum_{i=1}^{n} \frac{B_{i}^{+}dL _{i}\omega ^{3}}{ \vert 1- \vert c_{i} \vert \vert \pi ^{3}} \Biggr\} >0. \end{aligned}$$

Proof

Consider the operator equation

where L and N are defined by (3.1) and (3.2). Let \(u\in D(L)\) be an arbitrary solution of (3.3), then

Multiplying by \(y_{i}'(t)\) on both sides of the second equation of (3.4) and integrating it over \([0,2\omega ]\), we have

From (3.5) and the Hölder inequality, we have

which results in

Consider the fourth term \(\sum_{j=1}^{n}c_{ij}^{+}L_{j} (\int _{0} ^{2\omega } |x_{j}(t-\gamma _{ij}(t))|^{2}\,dt )^{\frac{1}{2}}\) in (3.6). In view of Remark 2.2, we have

From (3.6) and (3.7), we have

Multiplying by \(A_{i}x_{i}'(t)\) on both sides of the first equation of (3.4) and integrating it over \([0,2\omega ]\), we have

which results in

Using Lemma 2.3 and (3.9), we have

The Wirtinger inequality, (3.9) and (3.10) result in

Using again the Wirtinger inequality and (3.11), we have

Multiplying by \(z_{i}'(t)\) on both sides of the third equation of (3.4) and integrating it over \([0,2\omega ]\), we have

which results in

From (3.12) and (3.13), we have

which together with the Wirtinger inequality results in

From (3.8), (3.11), (3.12) and (3.15), we have

By using assumption (H₄) and (3.16), we obtain

Equations (3.11) and (3.17) result in

Equations (3.14) and (3.17) result in

Since \(u\in X\) is an ω-anti-periodic function, there exist \(\xi _{i},\eta _{i},\zeta _{i}\in [0,2\omega ]\) such that

which results in

and

Similarly, using (3.17) and (3.19), we also find that there exist positive constants \(\widetilde{M}_{2}\), \(\widetilde{M}_{3}\) such that

From (3.19)–(3.21), we get

Let

From Lemma 2.2, the operator equation \(Lu=Nu\) has at least one ω-anti-periodic solution in X. Thus, system (2.1) has at least one ω-anti-periodic solution. □

Remark 3.1

We very much want to obtain the globally exponential stability of system (2.1) with initial values conditions (2.3). But transforming system (2.6) of system (2.1) contains a neutral term \(A_{i}x_{i}\) which makes constructing the appropriate Lyapinov function very difficult. Hence, we wish that some authors will develop new methods to derive globally exponential stability of system (2.1) in the future.

In this section, a numerical example is given to illustrate the effectiveness of the results obtained in this paper.

Example 4.1

Consider the following inertial competitive neutral-type neural networks:

where \(i,j=1,2\),

After a simple calculation, we have

Thus,

and

Then

By Theorem 3.1, system (4.1) has at least a unique π-anti-periodic solution. For the trajectories of \(x_{i}(t)\), \(y_{i}(t)\), \(z_{i}(t)\) and \(m_{ij}(t)\) in system (4.1), see Figs. 1–3.

For \(i,j=1,2\), trajectories of \(x_{i}(t)\) and \(m_{ij}(t)\) in system (4.1)

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For \(i,j=1,2\), trajectories of \(y_{i}(t)\) and \(m_{ij}(t)\) in system (4.1)

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For \(i,j=1,2\), trajectories of \(x_{i}(t)\), \(y_{i}(t)\) and \(z_{i}(t)\) in system (4.1)

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In this paper, we study a class of he anti-periodic solutions problem for inertial competitive neutral-type neural networks. By employing the Wirtinger inequality, topology degree theory and some analytic techniques, we have presented some new sufficient criteria for the existence of anti-periodic for the above neural networks. These criteria possess adjustable parameters which are important in some applied fields. Finally, an example is given to demonstrate the effectiveness of the obtained theoretical results. However, there exist many problems for further study such as the problems of the stability and other dynamic properties of anti-periodic solutions to neutral-type neural networks.

Acknowledgements

The author would like to thank the editor and the referees for their valuable comments and suggestions, which improved the quality of our paper.

Availability of data and materials

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

Funding

The work is supported by Natural Science Foundation of Jiangsu High Education Institutions of China (Grant No. 17KJB110001).

Affiliations

1. Department of Mathematics, Huaiyin Normal University, Huaian, China

   • Bo Du

Contributions

Only the author contributed to the writing of this paper. The author read and approved the final manuscript.

Corresponding author

Correspondence to Bo Du.

Competing interests

The author declares to have no competing interests.

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