Periodicity and multi-periodicity of generalized Cohen–Grossberg neural networks via functional differential inclusions

In this paper, we investigate the periodic dynamical behaviors for a class of general Cohen–Grossberg neural networks with discontinuous right-hand sides and mixed time delays involving both time-varying delays and distributed delays. In view of functional differential inclusions theory, we obtain the existence of global solutions. By means of functional differential inclusions theory and fixed-point theorem of multi-valued maps, the existence of one and multiple positive periodic solutions for the neural networks is given. It is worthy to point out that, without assuming the boundedness or under linear growth condition of the discontinuous neuron activation functions, our results on the existence of one and multiple positive periodic solutions will also be valid. We derive some sufficient conditions for the global exponential stability and convergence of the discontinuous neural networks, in terms of non-smooth analysis theory with generalized Lyapunov approach. Finally, we give some numerical examples to show the applicability and effectiveness of our main results.

Authors and Affiliations

1. School of Mathematical Sciences, Huaqiao University, Quanzhou, Fujian, 362021, People’s Republic of China

   Dongshu Wang

2. College of Mathematics and Econometrics, Hunan University, Changsha, Hunan, 410082, People’s Republic of China

   Dongshu Wang & Lihong Huang

3. Changsha University of Science and Technology, Changsha, Hunan, 410014, People’s Republic of China

   Lihong Huang

Corresponding author

Correspondence to Lihong Huang.

This work was jointly supported by the National Natural Science Foundation of China (11371127, 11401228, 11501221, 61573004), the Natural Science Foundation of Fujian Province (2015J01584), the Promotion Program for Young and Middle-aged Teacher in Science and Technology Research of Huaqiao University (ZQN-PY201, ZQN-YX301).

Appendix

1.1 Appendix A. Proof of Proposition 2.1

Proof

It is easy to see that the multi-valued map

where

is upper semi-continuous with non-empty compact convex values, the local existence of a solution \(x(t)=(x_1(t),x_2(t),\ldots ,x_n(t))^T\) of (2.2) can be guaranteed [46, 47]. That is, the IVP of system (2.2) has at least a solution \(x(t)=(x_1(t),x_2(t),\ldots ,x_n(t))^T\) on \([0,\mathcal {T})\) for some \(\mathcal {T}\in (0,+\infty ]\) and the derivative of x(t) is a measurable selection from \(Q(x(t))\mathcal {F}(t,f)\). For a real matrix \(\Lambda =(\lambda _{ij})_{m\times n}\), denote \(||\Lambda ||_{\mathcal {M}}=\max _{1\leqslant i\leqslant n}\sum _{j=1}^m|\lambda _{ij}|\). From the Continuation Theorem (Theorem 2, P78, [46]), we have that either \(\mathcal {T}=+\infty \), or \(\mathcal {T}<+\infty \) and \(\lim _{t\rightarrow T^{-}}||x(t)||_{\mathcal {M}}=+\infty \). In the following, we will prove that \(\lim _{t\rightarrow T^{-}}||x(t)||_{\mathcal {M}}<+\infty \) if \(\mathcal {T}<+\infty \), which means that the maximal existing interval of x(t) can be extended to \(+\infty \).

First, we’ll show that x(t) is uniformly bounded about \(t\in [0,\mathcal {T})\).

Let \(r_0=||x(0)||_{\mathcal {M}}\), \(r_s=\max _{0\leqslant t\leqslant s}||x(t)||_{\mathcal {M}}\). And the interval \([r_0, r_s]\) can be divided as follows

Denote \(t_l^*=\min \{t\in [0,s]\mid ||x(t)||_{\mathcal {M}}=r_l\}, l=0,1,2,\ldots ,m;\) and \(t_l^{**}=\max \{t\in [0,t_{l+1}^*]\mid ||x(t)||_{\mathcal {M}}=r_l\}, l=0,1,2,\ldots ,m-1.\) Note that \(||x(t)||_{\mathcal {M}}\) is a continuous function, we have

and

Thus, we have

Since x(t) is a solution of the differential inclusions (2.2) with the initial condition \([\phi ,\psi ]\), we can obtain

where

and \(q^M=\max _{1\leqslant i\leqslant n}\{q_i^M\}\), \(||\psi ||_{\mathcal {M},\infty }=\max _{1{\leqslant } i{\leqslant } n}\{||\psi _i||_{\infty }\}{=}\max _{1{\leqslant } i{\leqslant } n}\{\hbox {ess}\sup _{s\in ({-}\infty ,0]}|\psi _i(s)|\}\).

It follows from (7.1) and (7.2) that

where \(||x(\eta _l)||_{\mathcal {M}}+W(||x(\eta _l)||_{\mathcal {M}})=\max _{t\in [t_l^{**},t_{l+1}^*]}\{||x(t)||_{\mathcal {M}}+W(||x(t)||_{\mathcal {M}})\}\). From (7.3), we have

Summing on both sides of (7.4) from 0 to \(m-1\) with respect to l, we can derive

Since the division of segment \([r_0,r_s]\) is arbitrary, and from the definition of integration, we have

which, together with the condition (H3) imply that \(r_s\) is uniformly bounded about \(s\in [0,\mathcal {T})\). Hence, x(t) is uniformly bounded about \(t\in [0,\mathcal {T})\).

Next, we will show that \(\lim _{t\rightarrow T^-}||x(t)||_{\mathcal {M}}<+\infty \). There exists \(M_2\,{>}\,0\), such that \(||x(t)||_{\mathcal {M}}\leqslant M_2\), \(\forall t\in [0,\mathcal {T})\), since x(t) is uniformly bounded about \(t\in [0,\mathcal {T})\). For \(0\leqslant t_1<t_2<\mathcal {T}\), we have

which implies that \(\lim _{t\rightarrow T^-}x(t)\) exist, i.e., \(\lim _{t\rightarrow T^-}||x(t)||_{\mathcal {M}}<+\infty \). The proof is completed. \(\square \)

1.2 Appendix B. Proof of Lemma 3.1

Suppose that \(x(t)=(x_1(t),x_2(t),\ldots ,x_n(t))^T\) is a \(\omega \)-periodic solution of system (1.1) in the sense of Filippov, in view of Definition 2.1, we can obtain from (2.2) that

Thus

Note that the periodicity of x(t), by integrating both sides of differential inclusion (7.5) over the interval \([t, t+\omega ](0\leqslant t\leqslant \omega )\), we get the following integral inclusions

That is, x(t) is a \(\omega \)-periodic solution of integral inclusions (3.1).

On the other hand, suppose that x(t) is a \(\omega \)-periodic solution of integral inclusions (3.1). By the integral representation theorem [48], there exist a measurable function \(\gamma =(\gamma _1,\gamma _2,\ldots ,\gamma _n)^T:[0,\mathcal {T})\rightarrow \mathbb {R}^n\) such that \(\gamma _i(t)\in \overline{co}[f_i(x_i(t))]\) for a.e. \(t\in [0, \mathcal {T})\) and

Since the right-hand sides of (7.6) is absolutely continuous, deviating the two sides of (7.6) about t, for a.e. \(t\in [0, \mathcal {T})\), we obtain

Note that the periodicity of x(t) and \(\gamma (t)\), we have

In view of Definition 2.1, we know that \(x(t)=(x_1(t),x_2(t),\ldots ,x_n(t))^T\) is a \(\omega \)-periodic solution of system (1.1) in the sense of Filippov.

The proof of Lemma 3.1 is completed.

1.3 Appendix C. Proof of Lemma 3.2

First, for any \(x(t)=(x_1(t),x_2(t),\ldots ,x_n(t))^T\in \overline{\Omega }_R\cap \mathbb {P}\) and \(y(t)=(y_1(t),y_2(t),\ldots ,y_n(t))^T\in \varphi (x)\). There exists a measurable function \(\gamma =(\gamma _1,\gamma _2,\ldots ,\gamma _n)^T:[0,\mathcal {T})\rightarrow \mathbb {R}^n\) such that \(\gamma _i(t)\in \overline{co}[f_i(x_i(t))]\) with \( |\gamma _i(t)|\leqslant \max _{0\leqslant s\leqslant R}\{W_i(s)\} (i=1,2,\ldots ,n)\) for a.e. \(t\in [0, \mathcal {T})\) and

Thus, for \(t\leqslant v\leqslant t+\omega \) and \(x\in \overline{\Omega }_R\cap \mathbb {P}\), we have

which implies

From (7.7), for \(t\leqslant v\leqslant t+\omega \) and \(x\in \overline{\Omega }_R\cap \mathbb {P}\), we also have

Therefore, for any \(x\in \overline{\Omega }_R\cap \mathbb {P}\) and \(y\in \varphi (x)\), we have \(y\in \mathbb {P}\). That is, \(\varphi (x)\in \mathbb {P}\) for every fixed \(x\in \mathbb {P}\), i.e., \(\varphi :\overline{\Omega }_R\cap \mathbb {P}\rightarrow \mathbb {P}\).

Next, we prove that \(\varphi (x)\) is convex for each \(x\in \overline{\Omega }_R\cap \mathbb {P}\). In fact, for any \(x=(x_1,x_2,\ldots ,x_n)^T\), if \(y=(y_1,y_2,\ldots ,y_n)^T, z=(z_1,z_2,\ldots ,z_n)^T\in \varphi (x)\), then there exist \(\gamma =(\gamma _1,\gamma _2,\ldots ,\gamma _n)^T:[0,\mathcal {T})\rightarrow \mathbb {R}^n\) and \(\eta =(\eta _1,\eta _2,\ldots ,\eta _n)^T:[0,\mathcal {T})\rightarrow \mathbb {R}^n\) with \(\gamma _i(t)\in \overline{co}[f_i(x_i(t))]\) and \(\eta _i(t)\in \overline{co}[f_i(x_i(t))]\) for a.e. \(t\in [0,\mathcal { T})\), such that for each \(t\in [0,\mathcal { T})\) we have

and

Let \(0\leqslant \alpha \leqslant 1\). Then for each \(t\in [0,\omega ]\) we have

that is

Hence,

Finally, it is easy to see that \(\varphi (x)\) is closed. The proof of Lemma 3.2 is completed.

1.4 Appendix D. Proof of Lemma 3.3

It is enough to show that \(\varphi :\overline{\Omega }_R\cap \mathbb {P}\rightarrow P_{kc}(\mathbb {P})\) is a compact map. According to the Ascoli-Arzela Theorem, it suffices to show that \(\varphi (\overline{\Omega }_R\cap \mathbb {P})\) is an uniformly bounded and equi-continuous set. For any \(x=(x_1,x_2,\ldots ,x_n)^T\in \overline{\Omega }_R\cap \mathbb {P}\) and \(y=(y_1,y_2,\ldots ,y_n)^T\in \varphi (x)\). There exists a measurable function \(\gamma =(\gamma _1,\gamma _2,\ldots ,\gamma _n)^T:[0,\mathcal {T})\rightarrow \mathbb {R}^n\) such that \(\gamma _i(t)\in \overline{co}[f_i(x_i(t))]\) with \( |\gamma _i(t)|\leqslant \max _{0\leqslant s\leqslant R}\{W_i(s)\}\ (i=1,2,\ldots ,n)\) for a.e. \(t\in [0, \mathcal {T})\) and

Hence

where \(||\psi ||_{\infty }=\hbox {ess}\sup _{s\in (-\infty ,0]}|\psi (s)|\). Which yields

Thus, \(\varphi (\overline{\Omega }_R\cap \mathbb {P})\) is an uniformly bounded set for all \(x\in \overline{\Omega }_R\cap \mathbb {P}\).

Let \(t_1, t_2\in [0,\omega ]\), then for any \(x\in \overline{\Omega }_R\cap \mathbb {P}\) and each \(i=1,2, \ldots ,n\), we have

According to the mean value theorem of derivations, we obtain

where \(0<\lambda <1\). And

Hence

where \(M_i=G_iq_i^M(q_i^Md_i^M+2)[\sum _{j=1}^n(a_{ij}^M+b_{ij}^M+c_{ij}^M)(\max _{0\leqslant s\leqslant R}\{W_i(s)\}+||\psi ||_{\infty })+I_i^M]\).

As a result we have that

Hence, \(\varphi (\overline{\Omega }_R\cap \mathbb {P})\) is an equi-continuous set in X.

Therefore, the multi-valued map \(\varphi :\overline{\Omega }_R\cap \mathbb {P}\rightarrow P_{cp,cv}(\mathbb {P})\) is a k-set-contractive map with \(k=0\).

1.5 Appendix E. Proof of Lemma 3.4

We will show that \(\varphi \) has closed graph. Denote

Let \(|||\digamma (t,x)|||=\sup \{|u|: u\in \digamma (t,x)\}\) and \(L^1([0,\omega ],\mathbb {R}^n)\) be the Banach space of all functions \(u=(u_1,u_2,\ldots ,u_n)^T:[0,\omega ]\rightarrow \mathbb {R}^n\) which are Lebesgue integrable. Define the multi-valued operator

by letting

It is easy to show that \(\digamma (t,x)\) is a \(L^1\)-Carathéodory map and the set F(x) is non-empty for each fixed \(x\in \overline{\Omega }_R\cap \mathbb {P}\).

Consider the linear continuous operator \(\mathfrak {L}: L^1([0,\omega ], \mathbb {R}^n)\rightarrow C([0,\omega ], \mathbb {R}^n)\),

Hence, it follows from Lemma 2.3 that \(\varphi =\mathfrak {L}\circ F\) is a closed graph operator. We should be point out that USC is equivalent to the condition of being a closed graph multi-valued map when the map has non-empty compact values, that is to say, we have shown that \(\varphi \) is USC.

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