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Existence and stability of periodic solutions for impulsive fuzzy BAM Cohen-Grossberg neural networks on time scales
Advances in Difference Equations volume 2016, Article number: 64 (2016)
Abstract
This paper is concerned with the existence and global exponential stability of periodic solutions for a kind of impulsive fuzzy Cohen-Grossberg BAM neural networks on time scales. Applying the method of coincidence degree and constructing some suitable Lyapunov functional, we obtain some sufficient conditions for the existence and global exponential stability of periodic solutions for a kind of impulsive fuzzy Cohen-Grossberg BAM neural networks on time scales. Moreover, we give an example to illustrate the results obtained.
1 Introduction
In recent years, Cohen and Grossberg BAM neural networks have been extensively studied and applied in many different fields such as associative memory, signal processing, and some optimization problems. They have been widely studied both in theory and applications [1, 2]. Many results for the existence of their periodic solutions and the exponential convergence properties for Cohen-Grossberg neural networks have been reported in the literature (see, e.g., [3–14] and the references therein).
In this paper, we would like to integrate fuzzy operations into Cohen-Grossberg BAM neural networks. Speaking of fuzzy operations, Yang and Yang [15] first introduced fuzzy cellular neural networks (FCNNs) combining those operations with cellular neural networks. So far researchers have found that FCNNs and fuzzy Cohen-Grossberg neural networks are useful in image processing, and some results have been reported on stability, periodicity, and antiperiodicity (see, e.g., [15–23] and the references therein).
In fact, both continuous and discrete systems are very important in implementing and applications. But it is troublesome to study the existence and stability of periodic solutions for continuous and discrete systems, respectively. Therefore, it is meaningful to study that on time scales, which can unify the continuous and discrete situations. In this paper, we consider the following fuzzy Cohen-Grossberg BAM neural networks with impulses on time scales:
where \(x_{i}(t)\), \(y_{j}(t)\) are the activations of the ith neuron in X-layer and the jth neuron in Y-layer, the functions \(a_{i}\), \(b_{j}\) represent the abstract amplification functions, whereas the functions \(c_{i}\), \(d_{j}\) represent the self-excitation rate functions; time delays \(\tau_{ji}\) and \(\sigma_{ij}\) are positive constants, which correspond to the finite speed of the axonal signal transmission; \(\alpha_{ji}(t)\), \(\beta _{ji}(t)\), \(p_{ij}(t)\), \(q_{ij}(t)\) are elements of fuzzy feedback MIN template and fuzzy feedback MAX template in X-layer and Y-layer, respectively; ⋀ and ⋁ denote the fuzzy AND and fuzzy OR operations, respectively; \(E_{i}(t)\) and \(F_{j}(t)\) denote the ith and jth components of an external input source introduced from outside the network to the cell i in X-layer and the cell j in Y-layer, respectively; \(\mathbb{T}\) is an ω-periodic time scale, which has the subspace topology inherited from the standard topology on \(\mathbb{R}\). We denote \(I_{\mathbb{T}}=I\cap\mathbb{T}\), \(\Delta x_{i}(t_{k})=x_{i}(t_{k}^{+})-x_{i}(t_{k}^{-})\), \(\Delta y_{j}(t_{k})=y_{j}(t_{k}^{+})-y_{j}(t_{k}^{-})\), where \(x_{i}(t_{k}^{+})\), \(x_{i}(t_{k}^{-})\), \(y_{j}(t_{k}^{+})\), \(y_{j}(t_{k}^{-})\) (\(i=1,2,\ldots,n\), \(j=1,2,\ldots,m\)) represent the right and left limits of \(x_{i}(t_{k})\) and \(y_{j}(t_{k})\) in the sense of time scales; \(\{t_{k}\}\) is a sequence of real numbers such that \(t_{1}< t_{2}<\cdots\) and \(\lim_{k\rightarrow+\infty}t_{k}=+\infty\). There exists a positive integer q such that \(t_{k+q}=t_{k}+\omega\), \(I_{ik+q}=I_{ik}\), \(J_{jk+q}=J_{j}k\), \(k\in\mathbb{N}\). Without loss of generality, we also assume that \(\left .[0,\omega) \right ._{\mathrm{T}}\cap\{t_{k},k\in\mathbb{N}\}=\{t_{1},t_{2},\ldots ,t_{q}\}\). Let \(\mathbb{R}^{+}=(0,+\infty)\) and \(\mathbb{T}^{+}=\mathbb{R}^{+}\cap \mathbb{T}\).
The initial conditions associated with system (1) are of the form
where \(\varphi_{i}(t)\in C([-\tau,0]_{\mathbb{T}},\mathbb{R})\), \(\psi _{j}(t)\in C([-\sigma,0]_{\mathbb{T}},\mathbb{R})\). For convenience, we introduce the notation
where f is an ω-periodic function.
Throughout this paper, we make the following assumptions:
-
(A1)
\(E_{i}, F_{j},\alpha_{ji},\beta_{ji},p_{ij},q_{ij}\in C(\mathbb {T},\mathbb{R})\) are ω-periodic functions, \(\tau_{ji},\sigma_{ij}\in\mathbb {R}^{+}\), \(i=1,2,\ldots,n\), \(j=1,2,\ldots,m\).
-
(A2)
\(a_{i},b_{j}\in C(\mathbb{R},\mathbb{R}^{+})\) are bounded functions, namely, there exist positive constants \(\underline{a}_{i}\), \(\overline{a}_{i}\), \(\underline{b}_{j}\), \(\overline {b}_{j}\) such that \(\underline{a}_{i}\le a_{i}(\cdot)\le\overline{a}_{i}\), \(\underline{b}_{j}\le b_{j}(\cdot)\le\overline{b}_{j}\), \(i=1,2,\ldots,n\), \(j=1,2,\ldots,m\).
-
(A3)
\(c_{i},d_{j}\in C(\mathbb{R},\mathbb{R}^{+})\) are delta differentiable, and \(0<\varrho_{i}\le c_{i}^{\Delta}\le\delta_{i}\), \(0<\varrho_{j}'\le d_{j}^{\Delta}\le\delta_{j}'\), \(c_{i}(0)=0\), \(b_{j}(0)=0\), \(i=1,2,\ldots,n\), \(j=1,2,\ldots,m\).
-
(A4)
\(f_{j},g_{i}\in C(\mathbb{R},\mathbb{R})\), and there exist \(M_{j}\), \(N_{i}\), \(\kappa_{j}\), \(\nu_{i}\) (\(i=1,2,\ldots,n\), \(j=1,2,\ldots ,m\)) such that
$$\begin{aligned}& \vert f_{j}\vert \le M_{j},\qquad \vert g_{i}\vert \le N_{i}, \\& \bigl\vert f_{j}(u)-f_{j}(v)\bigr\vert \le\kappa_{j} \vert u-v\vert ,\qquad \bigl\vert g_{i}(u)-g_{i}(v)\bigr\vert \le\nu_{i}\vert u-v\vert . \end{aligned}$$ -
(A5)
\(I_{ik}, J_{ik}\in C(\mathbb{R},\mathbb{R})\), and there exist positive constants \(\rho_{ik}\), \(\rho_{ik}'\) such that \(|I_{ik}|\le \rho_{ik}\), \(|J_{jk}|\le \rho_{jk}'\), \(k\in\mathbb{N}\), \(i=1,2,\ldots,n\), \(j=1,2,\ldots,m\).
The organization of the paper is as follows. In Section 2, we introduce some definitions and lemmas. In Section 3, by using coincidence degree we establish sufficient conditions for the existence of the periodic solutions of system (1). In Section 4, by constructing Lyapunov functional we derive sufficient conditions for the global exponential stability of periodic solutions of system (1). An example is given to demonstrate the effectiveness of our results in Section 5. Conclusions are drawn in Section 6.
2 Preliminaries
In this section, we shall first recall some basic definitions and lemmas, which are used in what follows.
Let \(\mathbb{T}\) be a nonempty closed subset (time scale) of \(\mathbb{R}\). The forward and backward jump operators \(\sigma,\rho:\mathbb{T}\rightarrow\mathbb{T}\) and the graininess \(\mu: \mathbb{T}\rightarrow\mathbb{R}^{+}\) are defined, respectively, by
A point \(t\in\mathbb{T}\) is called left-dense if \(t>\inf\mathbb{T}\) and \(\rho(t)=t\), left-scattered if \(\rho(t)< t\), right-dense if \(t<\sup\mathbb{T}\) and \(\sigma(t)=t\), and right-scattered if \(\sigma(t)>t\). If \(\mathbb{T} \) has a left-scattered maximum m, then \(\mathbb{T}^{k}=\mathbb{T}\setminus\{m\}\); otherwise, \(\mathbb{T}^{k}=\mathbb{T}\). If \(\mathbb{T}\) has a right-scattered minimum m, then \(\mathbb{T}_{k}=\mathbb{T}\setminus \{m\}\); otherwise, \(\mathbb{T}_{k}=\mathbb{T}\).
Let \(\omega\in\mathbb{R}^{+}\); then \(\mathbb{T}\) is an ω-periodic time scale if \(\mathbb{T}\) is a nonempty closed subset of \(\mathbb{R}\) such that \(t+\omega\in\mathbb{T}\) and \(\mu(t+\omega)=\mu(t)\) for all \(t\in\mathbb{T}\).
A function \(f:\mathbb{T}\rightarrow\mathbb{R}\) is right-dense continuous if it is continuous at right-dense points in \(\mathbb{T}\) and its left-side limits exist at left-dense points in \(\mathbb{T}\). If f is continuous at each right-dense point and each left-dense point, then f is said to be a continuous function on \(\mathbb{T}\).
For \(y:\mathbb{T}\rightarrow\mathbb{R}\) and \(t\in\mathbb{T}^{k}\), we define the delta derivative \(y^{\Delta}(t)\) of \(y(t)\) as the number (if exists) with the property that for given \(\varepsilon>0\), there exists a neighborhood U of t such that \(|[y(\sigma(t))-y(s)]-y^{\Delta}(t)[\sigma(t)-y(s)]|<\varepsilon|\sigma(t)-s|\) for all \(s\in U\). If y is continuous, then y is right-dense continuous, and y is delta differentiable at t, then y is continuous at t. Let y be right-dense continuous. If \(Y^{\Delta}(t)=y(t)\), then we define the delta integral by \(\int_{a}^{t}y(s)\Delta s=Y(t)-Y(a)\).
Definition 2.1
[24]
If \(a\in\mathbb{T}\), \(\sup\mathbb{T}=\mathbb{R}\), and f is rd-continuous on \([0,\infty)\), then we define the improper integral by
provided that this limit exists, and in this case, we say that the improper integral converges. If this limit does not exist, then we say that the improper integral diverges.
Definition 2.2
[25]
For each \(t\in\mathbb{T}\), let N be a neighborhood of t. Then, for \(V\in C_{\mathrm{rd}}[\mathbb {T}\times\mathbb{R}^{n},\mathbb{R}^{+})\), we define \(D^{+}V^{\Delta}(t,x(t))\) so that, for every \(\varepsilon>0\), there exists a right neighborhood \(N_{\varepsilon}\subset N\) of t such that
for each \(s\in N_{\varepsilon}\), \(s>t\), where \(\mu(t,s)=\sigma(t)-s\). If t is rd and \(V(t,x(t))\) is continuous at t, this reduces to
Definition 2.3
[25]
Let \(\mathbb{T}\neq\mathbb{R}\) be a periodic time scale with periodic p. We say that a function \(f:\mathbb{T}\rightarrow\mathbb{R}\) is ω periodic if there exists a natural number n such that \(\omega=np\), \(f(t+\omega)=f(t)\) for all \(t\in\mathbb{T}\), and ω is the least number such that \(f(t+\omega)=f(t)\). If \(\mathbb{T}=\mathbb{R}\), then we say that f is \(\omega>0\) periodic if ω is the least positive number such that \(f(t+\omega)=f(t)\) for all \(t\in\mathbb{T}\).
A function \(r: \mathbb{T}\rightarrow R\) is called regressive if \(1+\mu(t)r(t)\neq0\) for all \(t\in\mathbb{T}^{k}\).
If r is a regressive function, then the generalized exponential function \(e_{r}\) is defined by
with the cylinder transformation
For two regressive functions \(p,q:\mathbb{T}\rightarrow\mathbb{R}\), we define
Lemma 2.1
[26]
Let p, q be regressive functions on \(\mathbb{T}\). Then
-
(i)
\(e_{0}(t,s)=1\) and \(e_{p}(t,t)=1\);
-
(ii)
\(e_{p}(\sigma(t),s)=(1+\mu(t)p(t))e_{p}(t,s)\);
-
(iii)
\(e_{p}(t,s)e_{p}(s,r)=e_{p}(t,r)\);
-
(iv)
\(e_{p}^{\Delta}(\cdot,s)=pe_{p}(\cdot,s)\).
Lemma 2.2
[27]
Assume that \(f, g:\mathbb{T}\rightarrow\mathbb{R}\) are delta differentiable at \(t\in\mathbb{T}^{k}\). Then
Lemma 2.3
[28]
Let \(t_{1},t_{2}\in[0,\omega]_{\mathrm{T}}\). If \(x:\mathbb{T}\rightarrow\mathbb{R}\) is ω periodic, then
Lemma 2.4
[29]
Let \(a,b\in\mathbb{T}\). For rd-continuous functions \(f,g:[a,b]_{\mathbb{T}}\rightarrow\mathbb{R}\), we have
Lemma 2.5
[29]
Let \(\mathbb{T}\) be an ω-periodic time scale. Then \(\sigma(t+\omega)=\sigma(t)+\omega\) for all \(t\in\mathbb{T}\).
Lemma 2.6
[29]
Let f be a continuous function on \([a,b]_{\mathbb{T}}\) that is Δ differentiable on \(\left .[a,b) \right ._{\mathbb{T}}\). Then there exist \(\xi,\tau\in \left .[a,b) \right ._{\mathbb{T}}\) such that
Lemma 2.7
[15]
Let x and y be two states of system (1). Then we have
and
Definition 2.4
The periodic solution \(u^{*}(t)=(x_{1}^{*}(t),\ldots,x_{n}^{*}(t),y_{1}(t),\ldots,y_{m}(t))^{T}\) of system (1) with initial value \((\varphi^{*}(t),\psi^{*}(t))^{T}=(\varphi _{1}^{*}(t),\ldots,\varphi_{n}^{*}(t),\psi_{1}^{*}(t),\ldots, \psi_{m}^{*}(t))^{T}\) is said to be globally exponentially stable if there exists a constant \(M\geq1\) and \(\varepsilon>0\) such that, for every \(\eta\in\mathbb{T}\),
where \(\eta\in[-\max\{\tau,\sigma\},0]_{\mathbb{T}}\).
3 Existence of periodic solution
In this section, based on Mawhin’s continuation theorem, we study the existence of at least one periodic solution of (1). To do so, we shall make some preparations.
Let \(\mathbb{X}\), \(\mathbb{Y}\) be two Banach space, \(L:\operatorname{Dom}L\subset \mathbb{X}\rightarrow\mathbb{Y}\) be a linear mapping, and \(N:\mathbb{X}\rightarrow\mathbb{Y}\) be a continuous mapping. Then L is called a Fredholm mapping of index zero if \(\operatorname{dim}\operatorname{Ker}L=\operatorname{codim}\operatorname{Im}L<+\infty \) and ImL is closed in \(\mathbb{Y}\). If L is a Fredholm mapping of index zero and there exist continuous projectors \(P:\mathbb{X}\rightarrow\mathbb{X}\) and \(Q:\mathbb {Y}\rightarrow\mathbb{Y}\) such that \(\operatorname{Im}P=\operatorname{Ker}L\), \(\operatorname{Ker}Q=\operatorname{Im}(I-Q)\), then the mapping \(L|_{\operatorname{Dom}L\cap\operatorname{Ker}P}: (I-P)\mathbb{X}\rightarrow \operatorname{Im}L\) is invertible. We denote its inverse by \(K_{p}\). If Ω is an open bounded subset of \(\mathbb{X}\), then the mapping N is called L-compact on Ω̅ if \(QN(\overline{\Omega})\) is bounded and \(K_{p}(I-Q)N: \overline{\Omega}\rightarrow\mathbb{X}\) is compact. Since ImQ is isomorphic to KerL, there exists an isomorphism \(J: \operatorname{Im}Q\rightarrow\operatorname{Ker}L\).
Lemma 3.1
[30]
Let \(\mathbb{X}\), \(\mathbb{Y}\) be two Banach spaces, and let \(\Omega\subset\mathbb{X}\) be open bounded. Suppose that \(L:\operatorname{Dom}L\subset\mathbb {X}\rightarrow\mathbb{Y}\) is a linear Fredholm operator of index zero with \(\operatorname{Dom}L\cap\overline{\Omega}\neq\phi\) and \(N:\overline{\Omega}\rightarrow\mathbb{Y}\) is L-compact. Furthermore, suppose that:
-
(a)
for each \(\lambda\in(0,1)\), \(x\in\partial\Omega\cap\operatorname{Dom}L\), \(Lx\neq\lambda Nx\);
-
(b)
for each \(x\in\partial\Omega\cap\operatorname{Ker}L\), \(QNx\neq 0\);
-
(c)
\(\operatorname{deg}\{JQNx,\Omega\cap\operatorname{Ker}L,0\}\neq0\).
Then the equation \(Lx=Nx\) has at least one solution in \(\overline{\Omega}\cap\operatorname{Dom}L\), where Ω̅ is the closure of Ω, and ∂Ω is the boundary of Ω.
Definition 3.1
A real matrix \(A=(a_{ij})_{n\times n}\) is said to be a nonsingular M-matrix if \(a_{ij}\le0\), \(i,j=1,2,\ldots,n\) and all successive principal minors of A are positive.
Theorem 3.1
Under conditions (A1)-(A5), let H be a nonsingular M-matrix of the form
where
Then system (1) has at least one ω-periodic solution.
Proof
Let \(C[0,\omega;t_{1},\ldots,t_{q}]_{\mathbb{T}}\) = {\(u:[0,\omega]_{\mathbb{T}}\rightarrow R^{n+m}\) is a piecewise continuous map with first-class discontinuity points in \([0,\omega]_{\mathbb{T}}\cap\{t_{k}\}\), and at each discontinuity point, it is continuous on the left}. Take
with the norm \(\|u\|_{\mathbb{X}}=\sum_{i=1}^{n}|x_{i}|_{0}+\sum_{j=1}^{m}|y_{j}|_{0}\), where \(|x_{i}|_{0}=\max_{t\in[0,\omega]_{T}}|x_{i}(t)|\) and \(|y_{j}|_{0}=\max_{t\in[0,\omega]_{T}}|y_{j}(t)|\). Then \(\mathbb{X}\) is a Banach space.
Set
where
It is easy to see that
Thus, \(\operatorname{dim}\operatorname{Ker}L=\operatorname{codim}\operatorname{Im}L=n+m\). So, ImL is closed in \(\mathbb{Y}\), and L is a Fredholm mapping of index zero. Define the project operators P and Q as
Obviously, P and Q are continuous projectors and satisfy
Denoting \(L_{p}^{-1}=L|_{\operatorname{Dom}L\cap\operatorname{Ker}P}\) and generalized inverse by \(K_{p}=L_{P}^{-1}\), we have
Similarly to [31], it is not difficult to show that \(QN(\overline{\Omega})\) and \(K_{P} (I-Q)N(\overline{\Omega})\) are relatively compact for any open bounded set \(\Omega\subset\mathbb{X}\). Therefore, N is L-compact on Ω̅ for any open bounded set \(\Omega\subset\mathbb{X}\).
Now, to apply Lemma 3.1, we only need to look for an appropriate open bounded subset Ω. Correspondingly to the operator equation \(Lx= \lambda Nx\), \(\lambda\in(0, 1)\), we have
Suppose that \(u = (x_{1}, \ldots, x_{n},y_{1},\ldots,y_{m})^{T}\) is a solution of system (3) for a certain \(\lambda\in(0,1)\). Multiplying both sides of the first and third equations in system (3) by \(x_{i}^{\Delta}\) and \(y_{j}^{\Delta}\), respectively, and integrating over \([0,\omega]_{\mathbb{T}}\), we get
In view of Lemma 2.4, Lemma 2.7, and (A2)-(A4), we have
namely,
where \(B_{i}=\overline{a}_{i} [\sum_{j=1}^{m}(\overline{\alpha }_{ji}+\overline{\beta}_{ji})\sqrt{\omega}M_{j} +\overline{E}_{i}\sqrt{\omega} ]\). Similarly, we obtain that
where \(B_{j}'=\overline{b}_{j} [\sum_{i=1}^{n}(\overline{p}_{ij}+\overline {q}_{ij})\sqrt{\omega}N_{i} +\overline{F}_{j}\sqrt{\omega} ]\). Setting \(t_{0}=t_{0}^{+}=0\) and \(t_{q+1}=\omega\), in view of (3), (A2)-(A5), Lemma 2.4, and Lemma 2.7, we have
and
Integrating both sides of (3) from 0 to ω, we have
and
Applying Lemma 2.6 and (A3), we obtain
and
From Lemma 2.3, for any \(t_{1}^{i},t_{2}^{i}, t_{3}^{j},t_{4}^{j}\in[0,\omega]_{\mathbb{T}}\), \(i=1,2,\ldots,n\), \(j=1,2,\ldots,m\), we have
and
Dividing by \(\int_{0}^{\omega}a_{i}(x_{i}(t))\Delta t\) and \(\int_{0}^{\omega}b_{j}(y_{j}(t))\Delta t\) both sides of (10) and (11), respectively, we obtain, for \(i=1,2,\ldots,n\),
and, for \(j=1,2,\ldots,m\),
Let \(\overline{t}_{i},\underline{t}_{i},\overline{t}_{j}',\underline{t}_{j}'\in [0,\omega]_{\mathbb{T}}\) be such that \(x_{i}(\overline{t}_{i})=\max_{t\in[0,\omega]_{\mathbb{T}}}x_{i}(t)\), \(x_{i}(\underline{t}_{i})=\min_{t\in[0,\omega]_{\mathbb{T}}}x_{i}(t)\), \(y_{j}(\overline{t}_{j}')=\max_{t\in[0,\omega]_{\mathbb{T}}}y_{j}(t)\), \(y_{j}(\underline{t}_{j}')=\min_{t\in[0,\omega]_{\mathbb{T}}}y_{j}(t)\). From (12) we have, for \(i=1,2,\ldots,n\),
and
Similarly, from (13) we obtain, for \(j=1,2,\ldots,m\),
and
Therefore, we obtain that, for \(i=1,2,\ldots,n\),
and, for \(j=1,2,\ldots,m\),
In addition, we have that
By (14) we obtain, for \(i=1,2,\ldots,n\),
that is,
where, for \(i=1,2,\ldots,n\),
Similarly, we have
where, for \(j=1,2,\ldots,m\),
Denote \(\|u\|_{2}=(\|x_{1}\|_{2},\ldots,\|x_{n}\|_{2},\|y_{1}\|_{2},\ldots,\|y_{m}\| _{2})^{T}\) and
Then (16) and (17) can be written in the matrix form
From the conditions of Theorem 3.1 we have that H is a nonsingular M-matrix, so that
that is, \(\|x_{i}\|_{2}\le D_{i}\), \(i=1,2,\ldots,n\), and \(|y_{j}\|_{2}\le D_{j}'\), \(j=1,2,\ldots,m\).
From (14) and (15) we have, for \(i=1,2,\ldots,n\),
and, for \(j=1,2,\ldots,m\),
Let \(G=\sum_{i=1}^{n}G_{i}+\sum_{j=1}^{m}G_{j}'+G_{0}\), where \(G_{0}\) is a positive constant. Clearly, F is independent of λ. Take \(\Omega=\{u\in\mathbb{X}|\|u\|_{\mathbb{X}}< G\}\). Obviously, Ω satisfies condition (a) of Lemma 3.1.
When \(u(t)\in\partial\Omega\cap\operatorname{Ker}L=\partial\Omega\cap\mathbb{R}^{n+m}\), u is a constant vector with \(\|u\|=G\). Furthermore, take \(J:\operatorname{Im}Q\rightarrow\operatorname{Ker}L\). Then
for \(i=1,2,\ldots,n\), and
for \(j=1,2,\ldots,m\).
We can take G large enough such that
Hence, for any \(x\in\partial\Omega\cap\operatorname{Ker}L\), \(QNu\neq0\), namely, condition (b) in Lemma 3.1 is satisfied.
Furthermore, let \(\Psi(\gamma;u)=-\gamma u+(1-\gamma)QNu\). Then, for any \(u\in\partial\Omega\cap\operatorname{Ker}L\), \(u^{T}\Psi(\gamma;u)<0\), and we get
This shows that condition (c) in Lemma 3.1 is satisfied. Thus, by Lemma 3.1 we conclude that \(Lu = Nu\) has at least one solution in \(\mathbb{X}\), that is, system (1) has at least one ω-periodic solution. This completes the proof. □
4 Global exponential stability of periodic solutions
Suppose that \(u^{*}(t)=(x_{1}^{*}(t),,\ldots,x_{n}^{*}(t),y_{1}(t),\ldots ,y_{m}(t))^{T}\) is an ω-periodic solution of system (1). We will construct some suitable Lyapunov functions to prove the global exponential stability of this periodic solution.
Theorem 4.1
Assume that all conditions of Theorem 3.1 are satisfied. Suppose further that:
-
(A6)
The impulsive operators \(I_{ik}(x_{i}(t_{k}))\), \(J_{jk}(y_{j}(t_{k}))\) satisfy
$$\left \{ \textstyle\begin{array}{l} I_{ik}(x_{i}(t_{k})) = -\gamma_{ik}(x_{i}(t_{k})),\quad 0< \gamma_{ik}< 2,i=1,2,\ldots,n,k\in\mathbb{N}, \\ J_{jk}(y_{j}(t_{k})) = -\overline{\gamma}_{jk}(y_{j}(t_{k})),\quad 0< \overline{\gamma }_{jk}< 2, j=1,2,\ldots,m, k\in\mathbb{N}. \end{array}\displaystyle \right . $$ -
(A7)
$$\left \{ \textstyle\begin{array}{l} \underline{a}_{i}\varrho_{i}-\sum_{j=1}^{m}(\overline{p}_{ij}+\overline {q}_{ij})\nu_{i} \overline{b}_{j} > 0,\quad i=1,2,\ldots,n, \\ \underline{b}_{j}\varrho_{j}'-\sum_{i=1}^{n}(\overline{\alpha}_{ji}+\overline {\beta}_{ji})\kappa_{j} \overline{a}_{i} > 0,\quad j=1,2,\ldots,m. \end{array}\displaystyle \right . $$
Then the ω-periodic solution of (1) is globally exponentially stable.
Proof
According to Theorem 3.1, we know that system (1) has an ω-periodic solution \(u^{*}(t)=(x_{1}^{*}(t),\ldots,x_{n}^{*}(t),y_{1}^{*}(t),\ldots,y_{m}^{*}(t))^{T}\).
Let \(u(t)=(x_{1}(t),\ldots,x_{n}(t),y_{1}(t),\ldots,y_{m}(t))^{T}\) be an arbitrary solution of system (1). In view of (A6), from (1) we have
In view of this system, for \(t>0\), \(t\neq t_{k}\), \(k\in\mathbb{N}\), we have
for \(i=1,2,\ldots,n\), \(j=1,2,\ldots,m\). From (A6) we have that
for \(i=1,2,\ldots,n\), \(j=1,2,\ldots,m\), \(k\in\mathbb{N}\).
Let \(F_{i}\) and \(G_{j}\) be defined by
where \(\theta_{i},\xi_{j}\in[0,\infty)\), \(i=1,2,\ldots,n\), \(j=1,2,\ldots,m\). It is clear that
for \(i=1,2,\ldots,n\), \(j=1,2,\ldots,m\).
Since \(F_{i}\), \(G_{j}\) are continuous on \([0,\infty)\) and \(F_{i}(\theta _{i})\rightarrow-\infty\), \(G_{j}(\xi_{j})\rightarrow-\infty\) as \(\theta_{i}\rightarrow+\infty\), \(\xi_{j}\rightarrow+\infty\), there exist \(\theta_{i}^{*}, \xi_{j}^{*}>0\) such that \(F_{i}(\theta_{i}^{*})=0\), \(G_{j}(\xi_{j}^{*})=0\) and \(F_{i}(\theta_{i})>0\), \(G_{j}(\xi_{j})>0\) for \(\theta_{i}\in(0,\theta_{i}^{*})\), \(\xi_{j}\in(0,\xi_{j}^{*})\). By choosing \(\varepsilon=\min_{1\le i\le n,1\le j\le m}\{\theta_{i}^{*},\xi _{j}^{*}\}\) we have
for \(i=1,2,\ldots,n\), \(j=1,2,\ldots,m\).
Now let us define
where \(\delta\in[-\max\{\tau,\sigma\},0]\), for \(t>0\), \(t\neq t_{k}\), \(k\in \mathbb{N}\), \(i=1,2,\ldots,n\), \(j=1,2,\ldots,m\).
It follows from (19) and (21) that
Also, we have
Consider the Lyapunov functional
Calculating the Δ-derivatives of V along (22) and (23), we get
Also,
On other hand, we note that \(V(t)>0\) for \(t>0\) and \(V(0)\) is positive and finite. Therefore, it follows that \(V(t)\le V(0)\) for \(t>0\). From this and from (24) we obtain
for \(t>0\). In view of (21) and the last inequality, we have, for \(t>0\),
where
By Definition 2.4 the periodic solution of system (1) is globally exponentially stable. This completes the proof. □
5 An example
Example 5.1
Consider the following fuzzy Cohen-Grossberg BAM neural networks with impulses:
where \(\mathbb{T}\) is a 2π-periodic time scale, \(a_{1}(u)=\frac{1}{9\pi}(2+\cos u)\), \(a_{2}(u)=\frac{1}{9\pi}(2-\cos u)\), \(b_{1}(u)=\frac{1}{12\pi}(2+\sin u)\), \(b_{2}(u)=\frac{1}{12\pi}(2-\sin u)\), \(c_{1}(u)=\frac{1}{3}u\), \(c_{2}(u)=\frac{2}{3}u\), \(d_{1}(u)=\frac{3}{2}u\), \(d_{2}(u)=\frac{1}{2}u\), \(f_{i}(u)= g_{i}(u)=\frac{1}{2}(|u+1|-|u-1|)\), \(\alpha _{11}(t)=\frac{1}{20}\cos t\), \(\alpha_{12}(t)=\alpha_{21}(t)=0\), \(\alpha _{22}(t)=\frac{1}{20}\sin t\), \(\beta_{11}(t)=\frac{1}{20} \sin t\), \(\beta _{12}(t)=\beta_{21}(t)=0\), \(\beta_{22}(t)=\frac{1}{20}\sin t\), \(p_{11}(t)=\frac{1}{18}\sin t\), \(p_{12}(t)=p_{21}(t)=0\), \(p_{22}(t)=\frac {1}{18}\sin t\), \(q_{11}(t)=\frac{1}{18}\cos t\), \(q_{12}(t)=q_{21}(t)=0\), \(q_{22}(t)=\frac{1}{18}\sin u\), \(\tau_{ji}(t)=\frac{1}{2}\sin t \sigma _{ij}(t)=\frac{2}{3}\cos t\), \(\kappa_{j}=\nu_{i}=1\) (\(i,j=1,2\)). By calculating we have \(\overline{a}_{1}=\overline{a}_{2}=\frac{1}{3\pi}\), \(\underline{a}_{1}=\underline{a}_{2}=\frac{1}{9\pi}\), \(\overline{b}_{1}=\overline {b}_{2}=\frac{1}{4\pi}\), \(\underline{b}_{1}=\underline{b}_{2}=\frac{1}{12\pi}\), \(\overline{\alpha}_{11}=\frac{1}{20}\), \(\overline{\alpha}_{22}=\frac {1}{20}\), \(\overline{\alpha}_{12}=\overline{\alpha}_{21}=0\), \(\overline {\beta}_{11}=\frac{1}{20}\), \(\overline{\beta}_{22}=\frac{1}{20}\), \(\overline {\beta}_{12}=\overline{\beta}_{21}=0\), \(\overline{p}_{11}= \overline {p}_{22}=\frac{1}{18}\), \(\overline{p}_{12}=\overline{p}_{21}=0\), \(\overline {q}_{11}=\overline{q}_{22}=\frac{1}{18}\), \(\overline{q}_{12}=\overline{q}_{21}=0\).
It is easy to compute
and
Hence, we have that \(E =(e_{ij})_{4\times4}\) is a nonsingular M-matrix. From Theorem 3.1 and Theorem 4.1 we know that system (25) has at least one 2π-periodic solution, which is globally exponentially stable.
6 Conclusions
In this paper, we have studied the existence and globally exponential stability of the periodic solution for fuzzy Cohen-Grossberg BAM neural networks with impulses on time scales. Some sufficient conditions set up here are easily verified, and these conditions are correlated with parameters of system (1). The obtained criteria can be applied to design globally exponential stability of periodic continuous and discrete fuzzy Cohen-Grossberg BAM neural networks.
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Acknowledgements
The authors would like to thank the editor and anonymous reviewers for their helpful comments and valuable suggestions, which have greatly improved the quality of this paper. This work is partially supported by the National Natural Science Foundation of China (Grants Nos. 11264005, 11361012), the Scientific Research Foundation of Guizhou Science and Technology Department ([2013]J2083), and ‘125’ Science and Technology Grand Project Sponsored by the Department of Education of Guizhou Province ([2012]011).
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Cai, S., Zhang, Q. Existence and stability of periodic solutions for impulsive fuzzy BAM Cohen-Grossberg neural networks on time scales. Adv Differ Equ 2016, 64 (2016). https://doi.org/10.1186/s13662-016-0762-1
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DOI: https://doi.org/10.1186/s13662-016-0762-1
Keywords
- periodic solutions
- fuzzy Cohen-Grossberg BAM neural networks
- coincidence degree
- impulses
- time scales