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On anti-periodic solutions for neutral shunting inhibitory cellular neural networks with time-varying delays and D operator
Communicated by Dr. N. Ozcan
Accepted Manuscript
On anti-periodic solutions for neutral shunting inhibitory cellular neural networks with time-varying delays and D operator Changjin Xu, Peiluan Li PII: DOI: Reference:
S0925-2312(17)31451-0 10.1016/j.neucom.2017.08.030 NEUCOM 18822
To appear in:
Neurocomputing
Received date: Revised date: Accepted date:
14 December 2016 27 June 2017 25 August 2017
Please cite this article as: Changjin Xu, Peiluan Li, On anti-periodic solutions for neutral shunting inhibitory cellular neural networks with time-varying delays and D operator, Neurocomputing (2017), doi: 10.1016/j.neucom.2017.08.030
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ACCEPTED MANUSCRIPT
On anti-periodic solutions for neutral shunting
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inhibitory cellular neural networks with time-varying delays and D operator Changjin Xu1 , Peiluan Li2 1
∗
Guizhou Key Laboratory of Economics System Simulation
2
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Guizhou University of Finance and Economics, Guiyang 550004, PR China
School of Mathematics and Statistics, Henan University of Science and Technology Luoyang 471023, PR China
Abstract: This paper deals with a class of neutral shunting inhibitory cellular neural networks with timevarying delays and D operator. Using the differential inequality theory and Lyaunov functional method,
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a set of sufficient conditions which ascertain that the existence and exponential stability of anti-periodic solutions of neutral shunting inhibitory cellular neural networks with time-varying delays and D operator are
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derived. Computer simulations are delineated to substantiate the correctness of our theoretical predictions. The obtained results of this paper are new and complement some earlier works. Keywords:
Neutral shunting inhibitory cellular neural networks; Anti-periodic solution; Exponential
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stability; D operator; Time-varying delays
Mathematics Subject Classification 2000: 34C25; 34K13; 34K25
Introduction
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1.
Shunting inhibitory cellular neural networks (SICNNs) were introduced by Bouzerdoum and Pinter [33].
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The important property of SICNNs lies in the differential response to stimuli moving in opposite directions. Some studies reveal that this directional response adapts to mean luminance levels and changes with size and speed of moving objects and coupling order among elements of the networks [33]. It is well known that SICNNs play a vital role in wide world of science such as adaptive pattern recognition, psychophysics, image processing perception, speech, robotics, vision, etc. [1-3]. Delays usually inevitably appear in the signal transmission among the neurons of neural networks due to the finite switching speed of information processing and the inherent communication time of neurons. Thus numerous scholars focus on the dynamics of SICNNs with delays and some excellent achievements on SICNNs with delays have been reported. For instance, Liu [4] considered the convergence behavior of solutions of SICNNs with unbounded delays and time-varying coefficients, Fen and Fen [5] studied the SICNNs with Li-Yorke chaotic outputs on a time scale, Peng and Wang [6] analyzed the anti-periodic solutions for SICNNs with time-varying delays in ∗
This work is supported by National Natural Science Foundation of China (No.61673008 and No.11261010) and
Project of High-level Innovative Talents of Guizhou Province ([2016]5651). E-mail:[email protected]
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leakage terms. For more related works on these aspects, we refer the readers to [7-18] and the references cited therein. On the one hand, considering the complexity of neural cells, we think that in many cases, neural networks contain some information about the derivative of the past state to characterize the complex neural reactions [19-21]. Thus it is important for us to handle the neutral SICNNs. In recent years, there are some
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papers that consider this aspect. For example, Li and Yang [22] studied the almost automorphic solution for neutral type high-order Hopfield neural networks with delays in leakage terms on time scales, Peng et al. [23] addressed the LMI-based global exponential stability of neutral delayed BAM neural networks with delays in leakage terms, Li et al. [24] focused on the dynamics of SICNNs with continuously distributed delays of neutral type, etc. On the other hand, anti-periodic solutions of neural networks can be applied to describe the dynamical behavior of neural networks and they play an important role in designing the
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neural networks. Here we would like to mention that the applications and design of SICNNs heavily depend on the existence and exponential stability of anti-periodic solutions for neural networks [25-31]. Therefore, it is worthwhile to investigate the existence and exponential stability of anti-periodic solutions of neutral SICNNs with time-varying delays and D operator. However, so far, there is few existing papers on the existence and exponential stability of anti-periodic solutions of neutral SICNNs with time-varying delays and D operator.
Motivated by the above ideas, it is necessary for us to study existence and exponential stability of
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anti-periodic solution for neutral SICNNs with time-varying delays and D operator. In this paper, we will
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consider the following neutral SICNNs with time-varying delays and D operator 0 [xij (t) − ζij (t)xij (t − γij (t))] = −aij (t)xij (t) + Lij (t) X kl − Cij (t)f (xkl (t − σkl (t)))xij (t)
(1.1)
Ckl ∈Nr (i,j)
PT
where ij ∈ Λ = {11, 12, · · · , 1n, 21, 22, · · · , 2n, · · · , m1, · · · , mn}, Cij accounts for the cell at the (i, j) position of the lattice, the r-neighborhood Nr (i, j) of Cij is
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Nr (i, j) = {Ckl : max(|k − i|, |l − j|) ≤ r, 1 ≤ k ≤ m, 1 ≤ l ≤ n}, 0
xij is the activity of the cell Cij , xij (t) denotes the derivative of xij (t), Lij (t) is the external input to Cij , kl aij (t) denotes the passive decay rate of the cell activity, Cij (t) is the connection or coupling strength of
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postsynaptic activity of the cell transmitted to the cell Cij , f (xkl ) is the activity function which represents the output or firing rate of the cell Ckl , γij (t) and σij (t) denote the transmission delays, ζij (t) is a continuous function with respect to t. The initial values of (1.1) are given by xij (s) = ϕij (s), s ∈ [−ρij , 0],
(1.2)
where ρij = mint∈R {γij (t), σij (t)} and ϕij (s) is a real-value bounded and continuous function defined on
[−ρij , 0]. For convenience, we present some notations. Denote
g + = sup |g(t)|, g − = inf |g(t)|, t∈R
t∈R
where g is a bounded and continuous function defined on R. Let x = {xij } = (x11 , x12 , · · · , xmn )T ∈ Rmn , |x|
is the absolute-value vector given by |x| = {|xij |} = (|x11 |, |x12 |, · · · , |xmn |)T and ||x|| = maxij∈Λ supt∈R |xij (t)|.
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Let BC(R, Rmn ) be the set of bounded continuous functions from R to Rmn . Then (BC(R, Rmn ), ||.||∞ ) is a Banach space, where ||g||∞ := supt∈R ||f (g)||. Denote by
AP T (R, Rmn ) = {f ∈ BC(R, Rmn )|f (t + ω) = −f (t), t ∈ R}
(1.3)
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the set of ω-anti-periodic functions from R to Rmn . kl For ij, kl ∈ Λ, we assume that aij , Lij , Cij , ζij : R → R and γij , σij : R → [0, +∞) are bounded
+ kl and continuous functions and ζij < 1, and aij (t + ω) = aij (t), ζij (t + ω) = ζij (t), Cij (t + ω)f (u) = kl Cij (t)f (−u), σkl (t + ω) = σkl (t), γij (t + ω) = γij (t), Lij (t + ω) = −Lij (t), t, u ∈ R.
Furthermore, we also make the following assumptions:
(H1) For ij ∈ Λ, there exist a bounded and continuous function: a∗ij : R → (0, +∞) and a positive constant κij such that e−
Rt s
aij (θ)dθ
≤ κij e−
Rt s
a∗ ij (θ)dθ
for all t, s ∈ R and t − s ≥ 0.
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(H2) There exist constants L ≥ 0, M ≥ 0 such that |f (u) − f (v)| ≤ L|u − v|, |f (u)| ≤ M for all u, v ∈ R.
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(H3) There exist positive constants A0 and % such that Z t Rt A = max sup e− s aij (θ)dθ Lij (s)ds > 0, ij∈Λ t∈R −∞ κ X % ij + kl µij = sup |aij (t)ζij (t)| + |Cij (t)|(L(% + A0 ) + f (0)) < − ζij , ∗ a (t) % + A0 t∈R ij Ckl ∈Nr (i,j) κ X ij + kl νij = sup |aij (t)ζij (t)| + |Cij (t)|(L(% + A0 ) + M ) < 1 − ζij , ∗ t∈R aij (t) Ckl ∈Nr (i,j) κ X 1 ij kl |aij (t)ζij (t)| + |Cij υij = sup (t)|M + ∗ 1 − ζij t∈R aij (t) Ckl ∈Nr (i,j) X 1 kl + |Cij (t)|L + (% + A0 ) < 1. 1 − ζkl C ∈N (i,j) r
kl
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The key object of this article is to establish a set of sufficient conditions to guarantee the existence
and exponential stability of anti-periodic solutions of (1.1). The main contributions of this paper conclude the following aspects: (a) the study on the existence and exponential stability of anti-periodic solutions
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of neutral SICNNs with time-varying delays and D operator is firstly proposed; (b) a set of new sufficient
conditions which guarantee the existence and exponential stability of anti-periodic solutions of neutral SICNNs with time-varying delays and D operator are derived; (c) the obtained results of this paper are valid for some other similar neural networks with time-varying delays and D operator. The remainder of the paper is organized as follows. In Section 2, using the differential inequality theory and Lyaunov functional method, we will establish a set of sufficient conditions which ensure the existence and exponential stability of anti-periodic solutions of (1.1). In Section 3, an example with its computer simulations is given to show the correctness of theoretical results. We end this paper with a brief conclusion in Section 4. Remark 1.1 D operator based on shunting inhibitory cellular neural networks can be defined as the form: 0
[u(t) − a(t)u(t − τ (t))] = f (u(t), t), where a(t), f (t) and τ (t) are functions with respect to t.
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2.
Main results
In this section, we will establish a sufficient condition to ensure the existence and exponential stability of anti-periodic solutions of (1.1). Theorem 2.1 If (H1)-(H3) are fulfilled, then system (1.1) with the initial value (1.2) has a unique ω-anti-
Proof.
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periodic solution, which is globally stable.
Let x(t) = (x11 (t), x12 (t), · · · , xmn (t))T be an arbitrary solution of system (1.1) associated with
initial condition ϕ = (ϕ11 , ϕ12 , · · · , ϕmn )T . Set
uij (t) = xij (t) − ζij (t)xij (t − γij (t)), ij ∈ Λ.
0
uij (t)
=
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Then 0
[xij (t) − ζij (t)xij (t − γij (t))]
= −aij (t)uij (t) − aij (t)ζij (t)xij (t − γij (t)) X kl − Cij (t)f (xkl (t − σkl (t)))xij (t) + Lij (t), ij ∈ Λ. Ckl ∈Nr (i,j)
Let
Z
t
e−
−∞
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H(t) = {Hij (t)} = and = =
{Uijϕ } (Z t
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Uϕ (t)
e−
−∞
X
Rt
PT −
(2.1)
s
aij (θ)dθ
Rt s
aij (θ)dθ
Lij (s)ds , ij ∈ Λ,
[−aij (s)ζij (s)ϕij (s − γij (s))
kl Cij (s)f (ϕkl (s
Ckl ∈Nr (i,j)
)
− σkl (s)))ϕij (s) + Lij (s)]ds , ϕ ∈ AP T .
(2.2)
(2.3)
(2.4)
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In view of the proof of Lemma 2.1 of Long [32], we know that Uϕ ∈ AP T and H = U0 ∈ AP T . Denote A0 = ||H||∞ , Ω = {ϕ|||ϕ − H||∞ ≤ %, ϕ ∈ AP T }.
(2.5)
||ϕ||∞ ≤ ||ϕ − H||∞ + ||H||∞ ≤ % + A0 .
(2.6)
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Obviously,
Now we define a mapping Γ : Ω → AP T as follows: (Γϕ)(t) = {ζij (t)ϕij (t − γij (t))} + Uϕ (t), ϕ ∈ Ω.
(2.7)
In the sequel, we will show that for each ϕ ∈ Ω, Γϕ ∈ Ω. In view of (2.4), (2.7) and (H1)-(H3), we have ( Z t Rt |(Γϕ)(t) − H(t)| = e− s aij (θ)dθ [−aij (s)ζij (s)ϕij (s − γij (s)) ζij (t)ϕij (t − γij (t)) + −∞ ) X kl − Cij (s)f (ϕkl (s − σkl (s)))ϕij (s)]ds Ckl ∈Nr (i,j) ( Z ≤
+ ζij ||ϕ||∞ +
t
−∞
e−
Rt s
a∗ ij (θ)dθ
4
κij [|aij (s)ζij (s)|||ϕ||∞
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≤
(
Ckl ∈Nr (i,j) + ζij ||ϕ||∞
≤
Ckl ∈Nr (i,j) + ζij
+
Z
t
−∞
Ckl ∈Nr (i,j)
≤
(
≤
{%}, t ∈ R,
+ ζij
+
Z
Rt
e−
s
a∗ ij (θ)dθ
kl |Cij (s)|(L||ϕ||∞
e−
X
+
t
−∞
X
+ (
+
Z
Rt s
a∗ ij (θ)dθ
e
−
−∞
Rt s
a∗ ij (θ)dθ
+ |f (0)|)]ds||ϕ||∞
)
κij [|aij (s)ζij (s)|
kl |Cij (s)|(L(%
t
κij [|aij (s)ζij (s)|||ϕ||∞
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+
) kl |Cij (s)|(|f (ϕkl (s − σkl (s))) − f (0)| + |f (0)|)||ϕ||∞ ]ds
)
+ A0 ) + |f (0)|)]ds (% + A0 )
) % + ∗ − ζij aij (s)ds (% + A0 ) % + A0
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X
(2.8)
which implies that Γϕ ∈ Ω. Next we will prove that Γ : Ω → Ω is a contract operator. According to (2.4), (2.7) and (H1)-(H3), one has
M
ED
=
|(Γϕ)(t) − (Γψ)(t)| = {|((Γϕ)(t) − (Γψ)(t))ij |} ( ζij (t)[ϕij (t − γij (t)) − ψij (t − γij (t))] Z t Rt + e− s aij (θ)dθ [−aij (s)ζij (s)(ϕij (s − γij (s)) − ψij (s − γij (s))) −∞
−
kl Cij (s)(f (ϕkl (s
Ckl ∈Nr (i,j)
PT
≤
(
X
+ ζij ||ϕ − ψ||∞ +
X
CE
+
AC
Ckl ∈Nr (i,j)
≤
≤ ≤
Z
t
e−
−∞
Rt s
) − σkl (s)))ϕij (s) − f (ψkl (s − σkl (s)))ψij (s))]ds a∗ ij (θ)dθ
κij [|aij (s)ζij (s)|||ϕ − ψ||∞
kl |Cij (s)|(|f (ϕkl (s − σkl (s)))||ϕij − ψij (s)|
)
+|f (ϕkl (s − σkl (s))) − f (ψkl (s − σkl (s)))||ψij (s)|)]ds (
+
+
X
+ ζij
(
t
+ ζij
e−
−∞
Ckl ∈Nr (i,j)
+ ζij
(
Z
+
Z
e )
s
a∗ ij (θ)dθ
κij [|aij (s)ζij (s)||
kl |Cij (s)|(L(%
t
−∞
Rt
−
Rt s
a∗ ij (θ)dθ
)
+ A0 ) + M )]ds ||ϕ − ψ||∞ )
νij a∗ij (s)ds
||ϕ − ψ||∞
+ νij ||ϕ − ψ||∞ ,
(2.9)
where t ∈ R, ϕ, ψ ∈ Ω. Then it follows from (2.9) that + ||Γϕ − Γψ||∞ < max{ζij + νij }||ϕ − ψ||∞ . ij∈Λ
5
(2.10)
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+ Notice that maxij∈Λ {ζij + νij } < 1, we can conclude that Γ has a unique fixed point x ¯ = {¯ xij (t)} ∈ Ω such
that
{¯ xij (t)} = x ¯(t) = Γ(¯ x)(t) = {ζij (t)¯ xij (t − γij (t))} + Ux¯ (t) = {ζij (t)¯ xij (t − γij (t))} + {Uij x¯ (t)},
(2.11)
x ¯ij (t)
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and = ζij (t)¯ xij (t − γij (t)) + Uij x¯ (t) Z t Rt xij (s − γij (s)) = ζij (t)¯ xij (t − γij (t)) + e− s aij (θ)dθ [−aij (s)ζij (s)¯ −∞
X
Ckl ∈Nr (i,j)
kl Cij (s)f (¯ xkl (s − σkl (s)))¯ xij (s) + Lij (s)]ds, ij ∈ Λ.
Then 0
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−
(2.12)
0
[¯ xij (t) − ζij (t)¯ xij (t − γij (t))] = [Uij x¯ (t)]
= −aij (t)Uij x¯ (t) − aij (t)ζij (t)¯ xij (t − γij (t)) X kl − Cij (t)f (¯ xkl (t − σkl (t)))¯ xij (t) + Lij (t) = −aij (t)¯ xij (t) −
X
Ckl ∈Nr (i,j)
kl Cij (t)f (¯ xkl (t − σkl (t)))¯ xij (t) + Lij (t), ij ∈ Λ,
(2.13)
M
Ckl ∈Nr (i,j)
and x ¯ is an ω-anti-periodic solution of (1.1). At last, we will prove the global exponential stability of x ¯. Let
Then 0
ED
yij (t) = xij (t) − x ¯ij (t), Yij (t) = yij (t) − ζij (t)yij (t − γij (t)), ij ∈ Λ.
CE
0
=
[yij (t) − ζij (t)yij (t − γij (t))]
=
−aij (t)Yij (t) − aij (t)ζij (t)yij (t − γij (t)) X kl − Cij (t)[f (xkl (t − σkl (t)))xij (t)
PT
Yij (t)
(2.14)
Ckl ∈Nr (i,j)
−f (¯ xkl (t − σkl (t)))¯ xij (t)], ij ∈ Λ.
(2.15)
+
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+ λγij By (H3), there exists a constant λ ∈ (0, minij∈Λ (a∗ij )− ) such that 1 − ζij e > 0 and X + κij 1 1 λγij kl sup + |Cij (t)|M + |aij (t)ζij (t)|e + ∗ + λγij + λγij t∈R aij (t) − λ 1 − ζij e 1 − ζij e Ckl ∈Nr (i,j) + λσkl X e kl < 1, ij ∈ Λ. + |Cij (t)|L (% + A ) 0 + 1 − ζ + eλγkl
(2.16)
kl
Ckl ∈Nr (i,j)
Let
||φ||0 = sup max |[φij (t) − x ¯ij (t)] − ζij (t)[φij (t − γij (t)) − x ¯ij (t − γij (t))]|.
(2.17)
||Y (0)|| < ||φ||0 + , > 0
(2.18)
||Y (t)|| < (||φ||0 + )e−λt < β(||φ||0 + )e−λt ,
(2.19)
t≤0 ij∈Λ
Then
and
6
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where β > 1 + κij and t ∈ (−∞, 0]. In the following, we will show that ||Y (t)|| < β(||φ||0 + )e−λt
(2.20)
for all t > 0. If (2.20) does not hold, then there exist ij ∈ Λ and t∗ such that ∗
(2.21)
||Y (t)|| < β(||φ||0 + )e−λt , t ∈ (−∞, t∗ ).
(2.22)
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|Yij (t∗ )| = ||Y (t∗ )|| = β(||φ||0 + )e−λt and
In addition,
≤ eλς |yij (ς) − ζij (ς)yij (ς − γij (ς))| + eλς |ζij (ς)yij (ς − γij (ς))| +
+ λγij λ(ς−γij (ς)) ≤ eλς |Yij (ς)| + ζij e e |yij (ς − γij (ς))|
AN US
eλς |yij (ς)|
+
+ λγij ≤ β(||φ||0 + ) + ζij e
sup
s∈(−∞,t]
eλs |yij (s)|,
(2.23)
where ς ∈ (−∞, t], t ∈ (−∞, t∗ ), ij ∈ Λ. It follows from (2.23) that
where t ∈ (−∞, t∗ ], ij ∈ Λ. Notice that Rs 0
≤e
Rs 0
0
aij (θ)dθ
X
Ckl ∈Nr (i,j)
CE
one has
AC
Yij (t)
= Yij (0)e− −
β(||φ||0 + ) +
+ λγij 1 − ζij e
,
[Yij (s) + aij (s)Yij (s)] (
− aij (t)ζij (s)yij (s − γij (s))
Rt 0
X
(2.24)
)
kl Cij (s)[f (xkl (s − σkl (s)))xij (s) − f (¯ xkl (s − σkl (s)))¯ xij (s)] ,
PT
−
aij (θ)dθ
eλs |yij (s)| ≤
ED
e
sup
s∈(−∞,t]
M
eλt |yij (t)| ≤
aij (θ)dθ
+
Z
0
t R t
e
s
kl Cij (s)[f (xkl (s
Ckl ∈Nr (i,j)
aij (θ)dθ
(
− aij (s)ζij (s)yij (s − γij (s))
)
− σkl (s)))xij (s) − f (¯ xkl (s − σkl (s)))¯ xij (s)] ds,
where t ∈ [0, t∗ ]. Then
( Z t∗ R ∗ R ∗ t − 0t aij (θ)dθ a (θ)dθ |Yij (t )| = Yij (0)e + e s ij − aij (s)ζij (s)yij (s − γij (s)) 0 ∗
) kl − Cij (s)[f (xkl (s − σkl (s)))xij (s) − f (¯ xkl (s − σkl (s)))¯ xij (s)] ds Ckl ∈Nr (i,j) Z t∗ R ∗ R ∗ t ∗ − 0t a∗ (θ)dθ ij e s aij (θ)dθ κij − aij (s)ζij (s)yij (s − γij (s)) ≤ |Yij (0)|κij e + 0 X kl − Cij (s)[f (xkl (s − σkl (s)))xij (s) − f (¯ xkl (s − σkl (s)))¯ xij (s)] ds X
Ckl ∈Nr (i,j)
7
(2.25)
(2.26)
ACCEPTED MANUSCRIPT
≤
|Yij (0)|κij e +
X
−
Ckl ∈Nr (i,j)
R t∗ 0
a∗ ij (θ)dθ
+
Z
t∗
0
e
R t∗ s
a∗ ij (θ)dθ
(
κij |aij (s)ζij (s)||yij (s − γij (s))|
kl |Cij (s)|[|f (xkl (s − σkl (s)))||xij (s)|
)
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+|f (xkl (s − σkl (s))) − f (¯ xkl (s − σkl (s)))||¯ xij (s)|] ds
∗
R t∗
∗
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Z t∗ R ∗ R t∗ ∗ t ∗ ∗ ≤ (||φ||0 + )e−λt κij e− 0 (aij (θ)−λ)dθ + e s (aij (θ)−λ)dθ κij 0 ( X + 1 1 λγij kl + |Cij (s)|M × + |aij (s)ζij (s)|e + + λγij + λγij 1 − ζij e 1 − ζij e Ckl ∈Nr (i,j) ) + X ∗ eλσkl kl β(||φ||0 + )e−λt + |Cij (s)|L + (% + A0 ) + λγij 1 − ζkl e Ckl ∈Nr (i,j) ≤ (||φ||0 + )e−λt κij e− 0 (aij (θ)−λ)dθ Z t∗ R ∗ t ∗ ∗ + e s (aij (θ)−λ)dθ [a∗ij (s) − λ]dsβ(||φ||0 + )e−λt 0 R ∗ t ∗ ∗ κij = (||φ||0 + )e−λt − 1 e− 0 (aij (θ)−λ)dθ + 1 β ∗
(||φ||0 + )e−λt ,
(2.27)
M
≤
which contracts (2.21). Then (2.20) holds. Let → 0, it follows from (2.27) that
ED
|Y (t)| ≤ (||φ||0 + )e−λt , t > 0.
(2.28)
In a similar way, in view of the proof of (2.23) and (2.24), we can know that
PT
eλt |yij (t)| ≤
CE
and
|yij (t)| ≤
sup
s∈(−∞,t]
eλs |yij (s)| ≤
β||φ||0 +
+ λγij 1 − ζij e
β||φ||0 +
+ λγij 1 − ζij e
e−λt , t > 0, ij ∈ Λ.
(2.29)
(2.30)
This ends the proof of Theorem 2.1.
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Remark 2.1 All the authors in [6-8,10,13,15-16,25-32] investigated the anti-periodic solution of neural
networks without D operator. In [21], Yao studied the global exponential convergence of neutral type SICNNs
with D operator, but it does not involve anti-periodic solution. In this paper, we study the existence and
exponential stability of anti-periodic solutions of neutral SICNNs with time-varying delays and D operator. All the obtained results in [6-8,10,13,15-16,21,25-32] can not be applicable to model (1.1) to obtain the existence and exponential stability of anti-periodic solutions of system (1.1). Up to now, there are no results on the existence and exponential stability of anti-periodic solutions of neutral SICNNs with time-varying delays and D operator. From the viewpoint, our results on the existence and exponential stability of antiperiodic solutions for neutral SICNNs with time-varying delays and D operator are essentially new and complement previously known results to some extent. Remark 2.2 Generally, it is hard to construct a suitable Lyapunov functional to obtain the result we need. In this paper, we skillfully apply the inequality techniques and construct a suitable Lyapunov functional which differ from Yao [21] to achieve our goal. In this sense, the paper has novelty of techniques.
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3.
Examples
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Considering the following SICNNs with time-varying delays and D operator X 0 kl C11 [x11 (t) − ζ11 (t)x11 (t − γ11 (t))] = −a11 (t)x11 (t) − (t)f (xkl (t − σkl t))x11 (t) + L11 (t), Ckl ∈N1 (1,1) X 0 kl [x12 (t) − ζ12 (t)x12 (t − γ12 (t))] = −a12 (t)x12 (t) − C12 (t)f (xkl (t − σkl t))x12 (t) + L12 (t), Ckl ∈N1 (1,2) X 0 kl [x21 (t) − ζ21 (t)x21 (t − γ21 (t))] = −a21 (t)x21 (t) − C21 (t)f (xkl (t − σkl t))x21 (t) + L21 (t), C ∈N (2,1) 1 kl X 0 kl [x (t) − ζ (t)x (t − γ (t))] = −a (t)x (t) − C22 (t)f (xkl (t − σkl t))x22 (t) + L22 (t), 22 22 22 22 22 22 Ckl ∈N1 (2,2)
(3.1)
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where f (u) = 0.5(|u + 1) − |u − 1|), σkl = 0.02, r = 1, t ≥ 1, xij (s) = ϕij (s), s ∈ [0.0025, 1], i, j = 1, 2, # # " " 0.8(1 + 0.1 cos t) 0.4(1 + 0.2 sin t) a11 (t) a12 (t) , = 0.8(1 + 0.1 cos t) 0.4(1 + 0.2 sin t) a21 (t) a22 (t) # # " " C11 (t) C12 (t) 0.02 cos(10πt) 0.02 sin(10πt) , = 0.01 cos(10πt) 0.02 sin(10πt) C21 (t) C22 (t) # # " " 0.02 sin(10πt)et 0.02 sin(10πt)et L11 (t) L12 (t) , = 0.03 sin(10πt)et 0.04 sin(10πt)et L21 (t) L22 (t) # # " " 0.0012 sin(t) 0.0012 cos(t) ζ11 (t) ζ12 (t) = , 0.0012 sin(t) 0.0012 cos(t) ζ21 (t) ζ22 (t) # # " " 0.0012 sin t 0.0012 cos t γ11 (t) γ12 (t) . = 0.0012 sin t 0.0012 cos t γ21 (t) γ22 (t)
2
e 5 , then e− e e
−(t−s)
s
,t ≥
a11 (θ)dθ + s, ζ11
=
4
≤ e 5 e−(t−s) , e− + ζ12
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2 5
Rt
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P P kl Hence M = 1. Let a∗11 (t) = a∗21 (t) = 0.8, a∗12 (t) = a∗22 (t) = 0.4, Ckl ∈N1 (1,1) |C11 (t)| ≤ 0.07, Ckl ∈N1 (1,2) P P 4 kl kl kl |C12 (t)| ≤ 0.07, Ckl ∈N1 (2,1) |C21 (t)| ≤ 0.07, Ckl ∈N1 (2,2) |C22 (t)| ≤ 0.07, κ11 = κ21 = e 5 , κ21 = κ22 = =
+ ζ21
=
Rt
+ ζ22
s
a12 (θ)dθ
4
≤ e 5 e−(t−s) , e−
Rt s
a21 (θ)dθ
2
≤ e 5 e−(t−s) , e−
Rt s
a22 (θ)dθ
≤
= 0.0012. By direct computation, we can check that all the as-
sumptions in Theorem 2.1 are fulfilled, then system (3.1) has a unique anti-periodic solution, which is
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globally stable. This results are illustrated in Figure 1 and Figure 2.
4.
Conclusions
During the past decades, shunting inhibitory cellular neural networks with delays attract much attention from many researchers due to the potential applications in numerous disciplines. This paper is concerned with a class of neutral shunting inhibitory cellular neural networks with time-varying delays and D operator.
Applying the differential inequality theory and Lyaunov functional method, a set of sufficient criteria which guarantee the the existence and exponential stability of anti-periodic solutions of neutral shunting inhibitory cellular neural networks with time-varying delays and D operator are obtained. The sufficient criteria can be easily tested in practice by simple algebra computation. The obtained results play an important role in designing neural networks. The obtained results also complement many previous publications (for instance,[6-8,10,13,15-16,21,25-32]). The results solved in this paper can not be applied to [6-8,10,13,1516,21,25-32]. Furthermore, the analysis method in this paper can be applied to some other similar neural
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60
x11(t), x12(t)
40
20
0
−40
−60
0
10
20
30
40 t
50
60
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−20
70
80
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Figure 1: Numerical solutions x11 and x12 of system (3.1). The red line stands for x11 and the
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blue line stands for x12 .
100
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80 60
x21(t), x22(t)
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40 20
0
−20 −40 −60 −80
−100
0
10
20
30
40 t
50
60
70
80
Figure 2. Numerical solutions x21 and
x22 of system (3.1). The red line stands for x21 and the blue line stands for x22 .
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networks with D operator. For example, fuzzy cellular neural networks with the following form: n n X X 0 [x (t) − c (t)x (t − σ (t))] = −a (t)x (t) b (t)f (x (t − τ (t))) + dij (t)uj (t) i i i i i i ij j j ij j=1 j=1 n n n _ ^ ^ + αij (t)gj (xj (t)) + βij (t)gj (xj (t)) + Tij (t)uj (t) j=1 j=1 j=1 n _ + Sij (t)uj (t) + Ii (t), t ∈ R, i = 1, 2, · · · , n. j=1
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work:Computation in Neural Systems 25 (4) (2014) 168-192. [15] C.J. Xu, Q.M. Zhang, Existence and global exponential stability of anti-periodic solutions for BAM neural networks with inertial term and delay, Neurocomputing 153 (2015) 106-116.
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[21] L.G. Yao, Global exponential convergence of neutral type shunting inhibitory cellular neural networks
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with D operator, Neural processing Letters (2016) doi: 10.1007/s11063-016-9529-7. [22] Y.K. Li, L. Yang, Almost automorphic solution for neutral type high-order Hopfield neural networks
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with delays in leakage terms on time scales, Applied Mathematics and Computation 242 (2014) 679-693. [23] W.L. Peng, Q.X. Wu, Z.Q. Zhang, LMI-based global exponential stability of equilibrium point for neutral delayed BAM neural networks with delays in leakage terms via new inequality technique,
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[30] Y.K. Li, J.Y. Shu, Anti-periodic solutions to impulsive shunting inhibitory cellular neural networks with distributed delays on time scales, Communications in Nonlinear Science and Numerical Simulation 16 (8) (2011) 3326-3336. [31] M. Sayli, E. Yilmaz, Anti-periodic solutions for state-dependent impulsive recurrent neural networks
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with time-varying and continuously distributed delays, Annals of Operations Research (2016) in press. [32] Z. Long, New results on anti-periodic solutions for SICNNs with oscillating coefficients in leakage terms, Neuraocomputing 171 (1) (2016) 503-509.
[33] A. Bouzerdoum, R.B. Pinter, Analysis and analog implementation of directionally sensitive shunting
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inhibitory neural networks, Proceedings of Spie 1473 (1991) 29-38.
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Changjin Xu graduated from Huaihua University, China, in 1994. He received the M. S. degree from Kunming University of Science and Technology in 2004 and the Ph.D. degree from Central South University, China, in 2010. He is currently a professor at the Guizhou Key Laboratory of Economics System Simulation at Guizhou University of Finance and Economics. He has published about 100 refereed journal papers. He is a Reviewer of Mathematical Reviews and Zentralbatt-Math. His research
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interests include nonlinear systems, neural networks, anti-periodic solution, stability and bifurcation theorey.
Peiluan Li graduated from Wuhan University, China, in 2001. He received the M. S. degree
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from Wuhan University in 2004 and the Ph.D. degree from Central South University, China, in 2010. He was a postdoctoral from 2011 to 2013 in Hunan Normal University, China. He is currently an associate
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professor at the School of Mathematics and Statistics of Henan University of Science and Technology. He has published about 70 refereed journal papers. His research interests include nonlinear systems, functional
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CE
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differential equations, boundary value problems.
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Accepted Manuscript
On anti-periodic solutions for neutral shunting inhibitory cellular neural networks with time-varying delays and D operator Changjin Xu, Peiluan Li PII: DOI: Reference:
S0925-2312(17)31451-0 10.1016/j.neucom.2017.08.030 NEUCOM 18822
To appear in:
Neurocomputing
Received date: Revised date: Accepted date:
14 December 2016 27 June 2017 25 August 2017
Please cite this article as: Changjin Xu, Peiluan Li, On anti-periodic solutions for neutral shunting inhibitory cellular neural networks with time-varying delays and D operator, Neurocomputing (2017), doi: 10.1016/j.neucom.2017.08.030
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On anti-periodic solutions for neutral shunting
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inhibitory cellular neural networks with time-varying delays and D operator Changjin Xu1 , Peiluan Li2 1
∗
Guizhou Key Laboratory of Economics System Simulation
2
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Guizhou University of Finance and Economics, Guiyang 550004, PR China
School of Mathematics and Statistics, Henan University of Science and Technology Luoyang 471023, PR China
Abstract: This paper deals with a class of neutral shunting inhibitory cellular neural networks with timevarying delays and D operator. Using the differential inequality theory and Lyaunov functional method,
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a set of sufficient conditions which ascertain that the existence and exponential stability of anti-periodic solutions of neutral shunting inhibitory cellular neural networks with time-varying delays and D operator are
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derived. Computer simulations are delineated to substantiate the correctness of our theoretical predictions. The obtained results of this paper are new and complement some earlier works. Keywords:
Neutral shunting inhibitory cellular neural networks; Anti-periodic solution; Exponential
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stability; D operator; Time-varying delays
Mathematics Subject Classification 2000: 34C25; 34K13; 34K25
Introduction
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1.
Shunting inhibitory cellular neural networks (SICNNs) were introduced by Bouzerdoum and Pinter [33].
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The important property of SICNNs lies in the differential response to stimuli moving in opposite directions. Some studies reveal that this directional response adapts to mean luminance levels and changes with size and speed of moving objects and coupling order among elements of the networks [33]. It is well known that SICNNs play a vital role in wide world of science such as adaptive pattern recognition, psychophysics, image processing perception, speech, robotics, vision, etc. [1-3]. Delays usually inevitably appear in the signal transmission among the neurons of neural networks due to the finite switching speed of information processing and the inherent communication time of neurons. Thus numerous scholars focus on the dynamics of SICNNs with delays and some excellent achievements on SICNNs with delays have been reported. For instance, Liu [4] considered the convergence behavior of solutions of SICNNs with unbounded delays and time-varying coefficients, Fen and Fen [5] studied the SICNNs with Li-Yorke chaotic outputs on a time scale, Peng and Wang [6] analyzed the anti-periodic solutions for SICNNs with time-varying delays in ∗
This work is supported by National Natural Science Foundation of China (No.61673008 and No.11261010) and
Project of High-level Innovative Talents of Guizhou Province ([2016]5651). E-mail:[email protected]
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leakage terms. For more related works on these aspects, we refer the readers to [7-18] and the references cited therein. On the one hand, considering the complexity of neural cells, we think that in many cases, neural networks contain some information about the derivative of the past state to characterize the complex neural reactions [19-21]. Thus it is important for us to handle the neutral SICNNs. In recent years, there are some
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papers that consider this aspect. For example, Li and Yang [22] studied the almost automorphic solution for neutral type high-order Hopfield neural networks with delays in leakage terms on time scales, Peng et al. [23] addressed the LMI-based global exponential stability of neutral delayed BAM neural networks with delays in leakage terms, Li et al. [24] focused on the dynamics of SICNNs with continuously distributed delays of neutral type, etc. On the other hand, anti-periodic solutions of neural networks can be applied to describe the dynamical behavior of neural networks and they play an important role in designing the
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neural networks. Here we would like to mention that the applications and design of SICNNs heavily depend on the existence and exponential stability of anti-periodic solutions for neural networks [25-31]. Therefore, it is worthwhile to investigate the existence and exponential stability of anti-periodic solutions of neutral SICNNs with time-varying delays and D operator. However, so far, there is few existing papers on the existence and exponential stability of anti-periodic solutions of neutral SICNNs with time-varying delays and D operator.
Motivated by the above ideas, it is necessary for us to study existence and exponential stability of
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anti-periodic solution for neutral SICNNs with time-varying delays and D operator. In this paper, we will
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consider the following neutral SICNNs with time-varying delays and D operator 0 [xij (t) − ζij (t)xij (t − γij (t))] = −aij (t)xij (t) + Lij (t) X kl − Cij (t)f (xkl (t − σkl (t)))xij (t)
(1.1)
Ckl ∈Nr (i,j)
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where ij ∈ Λ = {11, 12, · · · , 1n, 21, 22, · · · , 2n, · · · , m1, · · · , mn}, Cij accounts for the cell at the (i, j) position of the lattice, the r-neighborhood Nr (i, j) of Cij is
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Nr (i, j) = {Ckl : max(|k − i|, |l − j|) ≤ r, 1 ≤ k ≤ m, 1 ≤ l ≤ n}, 0
xij is the activity of the cell Cij , xij (t) denotes the derivative of xij (t), Lij (t) is the external input to Cij , kl aij (t) denotes the passive decay rate of the cell activity, Cij (t) is the connection or coupling strength of
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postsynaptic activity of the cell transmitted to the cell Cij , f (xkl ) is the activity function which represents the output or firing rate of the cell Ckl , γij (t) and σij (t) denote the transmission delays, ζij (t) is a continuous function with respect to t. The initial values of (1.1) are given by xij (s) = ϕij (s), s ∈ [−ρij , 0],
(1.2)
where ρij = mint∈R {γij (t), σij (t)} and ϕij (s) is a real-value bounded and continuous function defined on
[−ρij , 0]. For convenience, we present some notations. Denote
g + = sup |g(t)|, g − = inf |g(t)|, t∈R
t∈R
where g is a bounded and continuous function defined on R. Let x = {xij } = (x11 , x12 , · · · , xmn )T ∈ Rmn , |x|
is the absolute-value vector given by |x| = {|xij |} = (|x11 |, |x12 |, · · · , |xmn |)T and ||x|| = maxij∈Λ supt∈R |xij (t)|.
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Let BC(R, Rmn ) be the set of bounded continuous functions from R to Rmn . Then (BC(R, Rmn ), ||.||∞ ) is a Banach space, where ||g||∞ := supt∈R ||f (g)||. Denote by
AP T (R, Rmn ) = {f ∈ BC(R, Rmn )|f (t + ω) = −f (t), t ∈ R}
(1.3)
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the set of ω-anti-periodic functions from R to Rmn . kl For ij, kl ∈ Λ, we assume that aij , Lij , Cij , ζij : R → R and γij , σij : R → [0, +∞) are bounded
+ kl and continuous functions and ζij < 1, and aij (t + ω) = aij (t), ζij (t + ω) = ζij (t), Cij (t + ω)f (u) = kl Cij (t)f (−u), σkl (t + ω) = σkl (t), γij (t + ω) = γij (t), Lij (t + ω) = −Lij (t), t, u ∈ R.
Furthermore, we also make the following assumptions:
(H1) For ij ∈ Λ, there exist a bounded and continuous function: a∗ij : R → (0, +∞) and a positive constant κij such that e−
Rt s
aij (θ)dθ
≤ κij e−
Rt s
a∗ ij (θ)dθ
for all t, s ∈ R and t − s ≥ 0.
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(H2) There exist constants L ≥ 0, M ≥ 0 such that |f (u) − f (v)| ≤ L|u − v|, |f (u)| ≤ M for all u, v ∈ R.
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(H3) There exist positive constants A0 and % such that Z t Rt A = max sup e− s aij (θ)dθ Lij (s)ds > 0, ij∈Λ t∈R −∞ κ X % ij + kl µij = sup |aij (t)ζij (t)| + |Cij (t)|(L(% + A0 ) + f (0)) < − ζij , ∗ a (t) % + A0 t∈R ij Ckl ∈Nr (i,j) κ X ij + kl νij = sup |aij (t)ζij (t)| + |Cij (t)|(L(% + A0 ) + M ) < 1 − ζij , ∗ t∈R aij (t) Ckl ∈Nr (i,j) κ X 1 ij kl |aij (t)ζij (t)| + |Cij υij = sup (t)|M + ∗ 1 − ζij t∈R aij (t) Ckl ∈Nr (i,j) X 1 kl + |Cij (t)|L + (% + A0 ) < 1. 1 − ζkl C ∈N (i,j) r
kl
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The key object of this article is to establish a set of sufficient conditions to guarantee the existence
and exponential stability of anti-periodic solutions of (1.1). The main contributions of this paper conclude the following aspects: (a) the study on the existence and exponential stability of anti-periodic solutions
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of neutral SICNNs with time-varying delays and D operator is firstly proposed; (b) a set of new sufficient
conditions which guarantee the existence and exponential stability of anti-periodic solutions of neutral SICNNs with time-varying delays and D operator are derived; (c) the obtained results of this paper are valid for some other similar neural networks with time-varying delays and D operator. The remainder of the paper is organized as follows. In Section 2, using the differential inequality theory and Lyaunov functional method, we will establish a set of sufficient conditions which ensure the existence and exponential stability of anti-periodic solutions of (1.1). In Section 3, an example with its computer simulations is given to show the correctness of theoretical results. We end this paper with a brief conclusion in Section 4. Remark 1.1 D operator based on shunting inhibitory cellular neural networks can be defined as the form: 0
[u(t) − a(t)u(t − τ (t))] = f (u(t), t), where a(t), f (t) and τ (t) are functions with respect to t.
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2.
Main results
In this section, we will establish a sufficient condition to ensure the existence and exponential stability of anti-periodic solutions of (1.1). Theorem 2.1 If (H1)-(H3) are fulfilled, then system (1.1) with the initial value (1.2) has a unique ω-anti-
Proof.
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periodic solution, which is globally stable.
Let x(t) = (x11 (t), x12 (t), · · · , xmn (t))T be an arbitrary solution of system (1.1) associated with
initial condition ϕ = (ϕ11 , ϕ12 , · · · , ϕmn )T . Set
uij (t) = xij (t) − ζij (t)xij (t − γij (t)), ij ∈ Λ.
0
uij (t)
=
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Then 0
[xij (t) − ζij (t)xij (t − γij (t))]
= −aij (t)uij (t) − aij (t)ζij (t)xij (t − γij (t)) X kl − Cij (t)f (xkl (t − σkl (t)))xij (t) + Lij (t), ij ∈ Λ. Ckl ∈Nr (i,j)
Let
Z
t
e−
−∞
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H(t) = {Hij (t)} = and = =
{Uijϕ } (Z t
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Uϕ (t)
e−
−∞
X
Rt
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(2.1)
s
aij (θ)dθ
Rt s
aij (θ)dθ
Lij (s)ds , ij ∈ Λ,
[−aij (s)ζij (s)ϕij (s − γij (s))
kl Cij (s)f (ϕkl (s
Ckl ∈Nr (i,j)
)
− σkl (s)))ϕij (s) + Lij (s)]ds , ϕ ∈ AP T .
(2.2)
(2.3)
(2.4)
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In view of the proof of Lemma 2.1 of Long [32], we know that Uϕ ∈ AP T and H = U0 ∈ AP T . Denote A0 = ||H||∞ , Ω = {ϕ|||ϕ − H||∞ ≤ %, ϕ ∈ AP T }.
(2.5)
||ϕ||∞ ≤ ||ϕ − H||∞ + ||H||∞ ≤ % + A0 .
(2.6)
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Obviously,
Now we define a mapping Γ : Ω → AP T as follows: (Γϕ)(t) = {ζij (t)ϕij (t − γij (t))} + Uϕ (t), ϕ ∈ Ω.
(2.7)
In the sequel, we will show that for each ϕ ∈ Ω, Γϕ ∈ Ω. In view of (2.4), (2.7) and (H1)-(H3), we have ( Z t Rt |(Γϕ)(t) − H(t)| = e− s aij (θ)dθ [−aij (s)ζij (s)ϕij (s − γij (s)) ζij (t)ϕij (t − γij (t)) + −∞ ) X kl − Cij (s)f (ϕkl (s − σkl (s)))ϕij (s)]ds Ckl ∈Nr (i,j) ( Z ≤
+ ζij ||ϕ||∞ +
t
−∞
e−
Rt s
a∗ ij (θ)dθ
4
κij [|aij (s)ζij (s)|||ϕ||∞
ACCEPTED MANUSCRIPT
≤
(
Ckl ∈Nr (i,j) + ζij ||ϕ||∞
≤
Ckl ∈Nr (i,j) + ζij
+
Z
t
−∞
Ckl ∈Nr (i,j)
≤
(
≤
{%}, t ∈ R,
+ ζij
+
Z
Rt
e−
s
a∗ ij (θ)dθ
kl |Cij (s)|(L||ϕ||∞
e−
X
+
t
−∞
X
+ (
+
Z
Rt s
a∗ ij (θ)dθ
e
−
−∞
Rt s
a∗ ij (θ)dθ
+ |f (0)|)]ds||ϕ||∞
)
κij [|aij (s)ζij (s)|
kl |Cij (s)|(L(%
t
κij [|aij (s)ζij (s)|||ϕ||∞
CR IP T
+
) kl |Cij (s)|(|f (ϕkl (s − σkl (s))) − f (0)| + |f (0)|)||ϕ||∞ ]ds
)
+ A0 ) + |f (0)|)]ds (% + A0 )
) % + ∗ − ζij aij (s)ds (% + A0 ) % + A0
AN US
X
(2.8)
which implies that Γϕ ∈ Ω. Next we will prove that Γ : Ω → Ω is a contract operator. According to (2.4), (2.7) and (H1)-(H3), one has
M
ED
=
|(Γϕ)(t) − (Γψ)(t)| = {|((Γϕ)(t) − (Γψ)(t))ij |} ( ζij (t)[ϕij (t − γij (t)) − ψij (t − γij (t))] Z t Rt + e− s aij (θ)dθ [−aij (s)ζij (s)(ϕij (s − γij (s)) − ψij (s − γij (s))) −∞
−
kl Cij (s)(f (ϕkl (s
Ckl ∈Nr (i,j)
PT
≤
(
X
+ ζij ||ϕ − ψ||∞ +
X
CE
+
AC
Ckl ∈Nr (i,j)
≤
≤ ≤
Z
t
e−
−∞
Rt s
) − σkl (s)))ϕij (s) − f (ψkl (s − σkl (s)))ψij (s))]ds a∗ ij (θ)dθ
κij [|aij (s)ζij (s)|||ϕ − ψ||∞
kl |Cij (s)|(|f (ϕkl (s − σkl (s)))||ϕij − ψij (s)|
)
+|f (ϕkl (s − σkl (s))) − f (ψkl (s − σkl (s)))||ψij (s)|)]ds (
+
+
X
+ ζij
(
t
+ ζij
e−
−∞
Ckl ∈Nr (i,j)
+ ζij
(
Z
+
Z
e )
s
a∗ ij (θ)dθ
κij [|aij (s)ζij (s)||
kl |Cij (s)|(L(%
t
−∞
Rt
−
Rt s
a∗ ij (θ)dθ
)
+ A0 ) + M )]ds ||ϕ − ψ||∞ )
νij a∗ij (s)ds
||ϕ − ψ||∞
+ νij ||ϕ − ψ||∞ ,
(2.9)
where t ∈ R, ϕ, ψ ∈ Ω. Then it follows from (2.9) that + ||Γϕ − Γψ||∞ < max{ζij + νij }||ϕ − ψ||∞ . ij∈Λ
5
(2.10)
ACCEPTED MANUSCRIPT
+ Notice that maxij∈Λ {ζij + νij } < 1, we can conclude that Γ has a unique fixed point x ¯ = {¯ xij (t)} ∈ Ω such
that
{¯ xij (t)} = x ¯(t) = Γ(¯ x)(t) = {ζij (t)¯ xij (t − γij (t))} + Ux¯ (t) = {ζij (t)¯ xij (t − γij (t))} + {Uij x¯ (t)},
(2.11)
x ¯ij (t)
CR IP T
and = ζij (t)¯ xij (t − γij (t)) + Uij x¯ (t) Z t Rt xij (s − γij (s)) = ζij (t)¯ xij (t − γij (t)) + e− s aij (θ)dθ [−aij (s)ζij (s)¯ −∞
X
Ckl ∈Nr (i,j)
kl Cij (s)f (¯ xkl (s − σkl (s)))¯ xij (s) + Lij (s)]ds, ij ∈ Λ.
Then 0
AN US
−
(2.12)
0
[¯ xij (t) − ζij (t)¯ xij (t − γij (t))] = [Uij x¯ (t)]
= −aij (t)Uij x¯ (t) − aij (t)ζij (t)¯ xij (t − γij (t)) X kl − Cij (t)f (¯ xkl (t − σkl (t)))¯ xij (t) + Lij (t) = −aij (t)¯ xij (t) −
X
Ckl ∈Nr (i,j)
kl Cij (t)f (¯ xkl (t − σkl (t)))¯ xij (t) + Lij (t), ij ∈ Λ,
(2.13)
M
Ckl ∈Nr (i,j)
and x ¯ is an ω-anti-periodic solution of (1.1). At last, we will prove the global exponential stability of x ¯. Let
Then 0
ED
yij (t) = xij (t) − x ¯ij (t), Yij (t) = yij (t) − ζij (t)yij (t − γij (t)), ij ∈ Λ.
CE
0
=
[yij (t) − ζij (t)yij (t − γij (t))]
=
−aij (t)Yij (t) − aij (t)ζij (t)yij (t − γij (t)) X kl − Cij (t)[f (xkl (t − σkl (t)))xij (t)
PT
Yij (t)
(2.14)
Ckl ∈Nr (i,j)
−f (¯ xkl (t − σkl (t)))¯ xij (t)], ij ∈ Λ.
(2.15)
+
AC
+ λγij By (H3), there exists a constant λ ∈ (0, minij∈Λ (a∗ij )− ) such that 1 − ζij e > 0 and X + κij 1 1 λγij kl sup + |Cij (t)|M + |aij (t)ζij (t)|e + ∗ + λγij + λγij t∈R aij (t) − λ 1 − ζij e 1 − ζij e Ckl ∈Nr (i,j) + λσkl X e kl < 1, ij ∈ Λ. + |Cij (t)|L (% + A ) 0 + 1 − ζ + eλγkl
(2.16)
kl
Ckl ∈Nr (i,j)
Let
||φ||0 = sup max |[φij (t) − x ¯ij (t)] − ζij (t)[φij (t − γij (t)) − x ¯ij (t − γij (t))]|.
(2.17)
||Y (0)|| < ||φ||0 + , > 0
(2.18)
||Y (t)|| < (||φ||0 + )e−λt < β(||φ||0 + )e−λt ,
(2.19)
t≤0 ij∈Λ
Then
and
6
ACCEPTED MANUSCRIPT
where β > 1 + κij and t ∈ (−∞, 0]. In the following, we will show that ||Y (t)|| < β(||φ||0 + )e−λt
(2.20)
for all t > 0. If (2.20) does not hold, then there exist ij ∈ Λ and t∗ such that ∗
(2.21)
||Y (t)|| < β(||φ||0 + )e−λt , t ∈ (−∞, t∗ ).
(2.22)
CR IP T
|Yij (t∗ )| = ||Y (t∗ )|| = β(||φ||0 + )e−λt and
In addition,
≤ eλς |yij (ς) − ζij (ς)yij (ς − γij (ς))| + eλς |ζij (ς)yij (ς − γij (ς))| +
+ λγij λ(ς−γij (ς)) ≤ eλς |Yij (ς)| + ζij e e |yij (ς − γij (ς))|
AN US
eλς |yij (ς)|
+
+ λγij ≤ β(||φ||0 + ) + ζij e
sup
s∈(−∞,t]
eλs |yij (s)|,
(2.23)
where ς ∈ (−∞, t], t ∈ (−∞, t∗ ), ij ∈ Λ. It follows from (2.23) that
where t ∈ (−∞, t∗ ], ij ∈ Λ. Notice that Rs 0
≤e
Rs 0
0
aij (θ)dθ
X
Ckl ∈Nr (i,j)
CE
one has
AC
Yij (t)
= Yij (0)e− −
β(||φ||0 + ) +
+ λγij 1 − ζij e
,
[Yij (s) + aij (s)Yij (s)] (
− aij (t)ζij (s)yij (s − γij (s))
Rt 0
X
(2.24)
)
kl Cij (s)[f (xkl (s − σkl (s)))xij (s) − f (¯ xkl (s − σkl (s)))¯ xij (s)] ,
PT
−
aij (θ)dθ
eλs |yij (s)| ≤
ED
e
sup
s∈(−∞,t]
M
eλt |yij (t)| ≤
aij (θ)dθ
+
Z
0
t R t
e
s
kl Cij (s)[f (xkl (s
Ckl ∈Nr (i,j)
aij (θ)dθ
(
− aij (s)ζij (s)yij (s − γij (s))
)
− σkl (s)))xij (s) − f (¯ xkl (s − σkl (s)))¯ xij (s)] ds,
where t ∈ [0, t∗ ]. Then
( Z t∗ R ∗ R ∗ t − 0t aij (θ)dθ a (θ)dθ |Yij (t )| = Yij (0)e + e s ij − aij (s)ζij (s)yij (s − γij (s)) 0 ∗
) kl − Cij (s)[f (xkl (s − σkl (s)))xij (s) − f (¯ xkl (s − σkl (s)))¯ xij (s)] ds Ckl ∈Nr (i,j) Z t∗ R ∗ R ∗ t ∗ − 0t a∗ (θ)dθ ij e s aij (θ)dθ κij − aij (s)ζij (s)yij (s − γij (s)) ≤ |Yij (0)|κij e + 0 X kl − Cij (s)[f (xkl (s − σkl (s)))xij (s) − f (¯ xkl (s − σkl (s)))¯ xij (s)] ds X
Ckl ∈Nr (i,j)
7
(2.25)
(2.26)
ACCEPTED MANUSCRIPT
≤
|Yij (0)|κij e +
X
−
Ckl ∈Nr (i,j)
R t∗ 0
a∗ ij (θ)dθ
+
Z
t∗
0
e
R t∗ s
a∗ ij (θ)dθ
(
κij |aij (s)ζij (s)||yij (s − γij (s))|
kl |Cij (s)|[|f (xkl (s − σkl (s)))||xij (s)|
)
CR IP T
+|f (xkl (s − σkl (s))) − f (¯ xkl (s − σkl (s)))||¯ xij (s)|] ds
∗
R t∗
∗
AN US
Z t∗ R ∗ R t∗ ∗ t ∗ ∗ ≤ (||φ||0 + )e−λt κij e− 0 (aij (θ)−λ)dθ + e s (aij (θ)−λ)dθ κij 0 ( X + 1 1 λγij kl + |Cij (s)|M × + |aij (s)ζij (s)|e + + λγij + λγij 1 − ζij e 1 − ζij e Ckl ∈Nr (i,j) ) + X ∗ eλσkl kl β(||φ||0 + )e−λt + |Cij (s)|L + (% + A0 ) + λγij 1 − ζkl e Ckl ∈Nr (i,j) ≤ (||φ||0 + )e−λt κij e− 0 (aij (θ)−λ)dθ Z t∗ R ∗ t ∗ ∗ + e s (aij (θ)−λ)dθ [a∗ij (s) − λ]dsβ(||φ||0 + )e−λt 0 R ∗ t ∗ ∗ κij = (||φ||0 + )e−λt − 1 e− 0 (aij (θ)−λ)dθ + 1 β ∗
(||φ||0 + )e−λt ,
(2.27)
M
≤
which contracts (2.21). Then (2.20) holds. Let → 0, it follows from (2.27) that
ED
|Y (t)| ≤ (||φ||0 + )e−λt , t > 0.
(2.28)
In a similar way, in view of the proof of (2.23) and (2.24), we can know that
PT
eλt |yij (t)| ≤
CE
and
|yij (t)| ≤
sup
s∈(−∞,t]
eλs |yij (s)| ≤
β||φ||0 +
+ λγij 1 − ζij e
β||φ||0 +
+ λγij 1 − ζij e
e−λt , t > 0, ij ∈ Λ.
(2.29)
(2.30)
This ends the proof of Theorem 2.1.
AC
Remark 2.1 All the authors in [6-8,10,13,15-16,25-32] investigated the anti-periodic solution of neural
networks without D operator. In [21], Yao studied the global exponential convergence of neutral type SICNNs
with D operator, but it does not involve anti-periodic solution. In this paper, we study the existence and
exponential stability of anti-periodic solutions of neutral SICNNs with time-varying delays and D operator. All the obtained results in [6-8,10,13,15-16,21,25-32] can not be applicable to model (1.1) to obtain the existence and exponential stability of anti-periodic solutions of system (1.1). Up to now, there are no results on the existence and exponential stability of anti-periodic solutions of neutral SICNNs with time-varying delays and D operator. From the viewpoint, our results on the existence and exponential stability of antiperiodic solutions for neutral SICNNs with time-varying delays and D operator are essentially new and complement previously known results to some extent. Remark 2.2 Generally, it is hard to construct a suitable Lyapunov functional to obtain the result we need. In this paper, we skillfully apply the inequality techniques and construct a suitable Lyapunov functional which differ from Yao [21] to achieve our goal. In this sense, the paper has novelty of techniques.
8
ACCEPTED MANUSCRIPT
3.
Examples
CR IP T
Considering the following SICNNs with time-varying delays and D operator X 0 kl C11 [x11 (t) − ζ11 (t)x11 (t − γ11 (t))] = −a11 (t)x11 (t) − (t)f (xkl (t − σkl t))x11 (t) + L11 (t), Ckl ∈N1 (1,1) X 0 kl [x12 (t) − ζ12 (t)x12 (t − γ12 (t))] = −a12 (t)x12 (t) − C12 (t)f (xkl (t − σkl t))x12 (t) + L12 (t), Ckl ∈N1 (1,2) X 0 kl [x21 (t) − ζ21 (t)x21 (t − γ21 (t))] = −a21 (t)x21 (t) − C21 (t)f (xkl (t − σkl t))x21 (t) + L21 (t), C ∈N (2,1) 1 kl X 0 kl [x (t) − ζ (t)x (t − γ (t))] = −a (t)x (t) − C22 (t)f (xkl (t − σkl t))x22 (t) + L22 (t), 22 22 22 22 22 22 Ckl ∈N1 (2,2)
(3.1)
ED
M
AN US
where f (u) = 0.5(|u + 1) − |u − 1|), σkl = 0.02, r = 1, t ≥ 1, xij (s) = ϕij (s), s ∈ [0.0025, 1], i, j = 1, 2, # # " " 0.8(1 + 0.1 cos t) 0.4(1 + 0.2 sin t) a11 (t) a12 (t) , = 0.8(1 + 0.1 cos t) 0.4(1 + 0.2 sin t) a21 (t) a22 (t) # # " " C11 (t) C12 (t) 0.02 cos(10πt) 0.02 sin(10πt) , = 0.01 cos(10πt) 0.02 sin(10πt) C21 (t) C22 (t) # # " " 0.02 sin(10πt)et 0.02 sin(10πt)et L11 (t) L12 (t) , = 0.03 sin(10πt)et 0.04 sin(10πt)et L21 (t) L22 (t) # # " " 0.0012 sin(t) 0.0012 cos(t) ζ11 (t) ζ12 (t) = , 0.0012 sin(t) 0.0012 cos(t) ζ21 (t) ζ22 (t) # # " " 0.0012 sin t 0.0012 cos t γ11 (t) γ12 (t) . = 0.0012 sin t 0.0012 cos t γ21 (t) γ22 (t)
2
e 5 , then e− e e
−(t−s)
s
,t ≥
a11 (θ)dθ + s, ζ11
=
4
≤ e 5 e−(t−s) , e− + ζ12
CE
2 5
Rt
PT
P P kl Hence M = 1. Let a∗11 (t) = a∗21 (t) = 0.8, a∗12 (t) = a∗22 (t) = 0.4, Ckl ∈N1 (1,1) |C11 (t)| ≤ 0.07, Ckl ∈N1 (1,2) P P 4 kl kl kl |C12 (t)| ≤ 0.07, Ckl ∈N1 (2,1) |C21 (t)| ≤ 0.07, Ckl ∈N1 (2,2) |C22 (t)| ≤ 0.07, κ11 = κ21 = e 5 , κ21 = κ22 = =
+ ζ21
=
Rt
+ ζ22
s
a12 (θ)dθ
4
≤ e 5 e−(t−s) , e−
Rt s
a21 (θ)dθ
2
≤ e 5 e−(t−s) , e−
Rt s
a22 (θ)dθ
≤
= 0.0012. By direct computation, we can check that all the as-
sumptions in Theorem 2.1 are fulfilled, then system (3.1) has a unique anti-periodic solution, which is
AC
globally stable. This results are illustrated in Figure 1 and Figure 2.
4.
Conclusions
During the past decades, shunting inhibitory cellular neural networks with delays attract much attention from many researchers due to the potential applications in numerous disciplines. This paper is concerned with a class of neutral shunting inhibitory cellular neural networks with time-varying delays and D operator.
Applying the differential inequality theory and Lyaunov functional method, a set of sufficient criteria which guarantee the the existence and exponential stability of anti-periodic solutions of neutral shunting inhibitory cellular neural networks with time-varying delays and D operator are obtained. The sufficient criteria can be easily tested in practice by simple algebra computation. The obtained results play an important role in designing neural networks. The obtained results also complement many previous publications (for instance,[6-8,10,13,15-16,21,25-32]). The results solved in this paper can not be applied to [6-8,10,13,1516,21,25-32]. Furthermore, the analysis method in this paper can be applied to some other similar neural
9
ACCEPTED MANUSCRIPT
80
CR IP T
60
x11(t), x12(t)
40
20
0
−40
−60
0
10
20
30
40 t
50
60
AN US
−20
70
80
M
Figure 1: Numerical solutions x11 and x12 of system (3.1). The red line stands for x11 and the
PT
ED
blue line stands for x12 .
100
CE
80 60
x21(t), x22(t)
AC
40 20
0
−20 −40 −60 −80
−100
0
10
20
30
40 t
50
60
70
80
Figure 2. Numerical solutions x21 and
x22 of system (3.1). The red line stands for x21 and the blue line stands for x22 .
10
ACCEPTED MANUSCRIPT
CR IP T
networks with D operator. For example, fuzzy cellular neural networks with the following form: n n X X 0 [x (t) − c (t)x (t − σ (t))] = −a (t)x (t) b (t)f (x (t − τ (t))) + dij (t)uj (t) i i i i i i ij j j ij j=1 j=1 n n n _ ^ ^ + αij (t)gj (xj (t)) + βij (t)gj (xj (t)) + Tij (t)uj (t) j=1 j=1 j=1 n _ + Sij (t)uj (t) + Ii (t), t ∈ R, i = 1, 2, · · · , n. j=1
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PT
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[30] Y.K. Li, J.Y. Shu, Anti-periodic solutions to impulsive shunting inhibitory cellular neural networks with distributed delays on time scales, Communications in Nonlinear Science and Numerical Simulation 16 (8) (2011) 3326-3336. [31] M. Sayli, E. Yilmaz, Anti-periodic solutions for state-dependent impulsive recurrent neural networks
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with time-varying and continuously distributed delays, Annals of Operations Research (2016) in press. [32] Z. Long, New results on anti-periodic solutions for SICNNs with oscillating coefficients in leakage terms, Neuraocomputing 171 (1) (2016) 503-509.
[33] A. Bouzerdoum, R.B. Pinter, Analysis and analog implementation of directionally sensitive shunting
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inhibitory neural networks, Proceedings of Spie 1473 (1991) 29-38.
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Changjin Xu graduated from Huaihua University, China, in 1994. He received the M. S. degree from Kunming University of Science and Technology in 2004 and the Ph.D. degree from Central South University, China, in 2010. He is currently a professor at the Guizhou Key Laboratory of Economics System Simulation at Guizhou University of Finance and Economics. He has published about 100 refereed journal papers. He is a Reviewer of Mathematical Reviews and Zentralbatt-Math. His research
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interests include nonlinear systems, neural networks, anti-periodic solution, stability and bifurcation theorey.
Peiluan Li graduated from Wuhan University, China, in 2001. He received the M. S. degree
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from Wuhan University in 2004 and the Ph.D. degree from Central South University, China, in 2010. He was a postdoctoral from 2011 to 2013 in Hunan Normal University, China. He is currently an associate
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professor at the School of Mathematics and Statistics of Henan University of Science and Technology. He has published about 70 refereed journal papers. His research interests include nonlinear systems, functional
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differential equations, boundary value problems.
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