# Almost Periodic Solutions for Neutral-Type BAM Neural Networks with Delays on Time Scales

Authors: Yongkun Li; Li Yang
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

Using the existence of the exponential dichotomy of linear dynamic
equations on time scales, a fixed point theorem and the theory of calculus on time scales,
we obtain some sufficient conditions for the existence and exponential stability of almost
periodic solutions for a class of neutral-type BAM neural networks with delays on time
scales. Finally, a numerical example illustrates the feasibility of our results and also
shows that the continuous-time neural network and its discrete-time analogue have the
same dynamical behaviors. The results of this paper are completely new even if the
time scale𝕋=ℝ or ℤ and complementary to the previously known results.


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## Body

## 1. Introduction

The bidirectional associative memory (BAM) neural network, which was introduced by Kosko (see [1]), is a special recurrent neural network that can store bipolar vector pairs and is composed of neurons arranged in two layers. The neurons in one layer are fully interconnected to the neurons in the other layer, while there are no interconnections among neurons in the same layer.Recently, due to its wide range of applications, for instance, pattern recognition, associative memory, and combinatorial optimization, BAM neural network has received much attention. For example, in [2–4], some sufficient conditions were obtained for the stability of the equilibrium points of BAM neural networks; in [5, 6], authors investigated the periodic solutions of BAM neural networks by using the continuation theorem of coincidence degree theory; in [7–9], authors studied the almost periodic solution for BAM neural networks by using the exponential dichotomy and fixed point theorems; for other results about BAM neural networks, the reader may see [10–13] and reference therein.Since it is natural and important that systems will contain some information about the derivative of the past state to further describe and model the dynamics for such complex neural reactions [14], many authors investigated the dynamical behaviors of neutral-type neural networks with delays [15–26]. For example, in [21], under the assumptions that the activation functions satisfy boundedness and Lipschitz conditions, authors discussed global asymptotic stability of neutral-type BAM neural networks with delays as follows:
(1)xi′(t)+∑j=1m‍eijxj′(t-h)=-aixi(t)-∑j=1m‍sijfj(yj(t-τ))+Ii,=-aixi(t)-∑j=1m‍sijfjsi=1,2,…,m,yj′(t)+∑i=1m‍νjiyj′(t-d)=-cjyj(t)-∑i=1m‍tjigi(xi(t-δ))+Jj,=-aixi(t)-∑j=1m‍sijfjj=1,2,…,m.Also, it is well known that the study of dynamical systems on time scales is now an active area of research. One of the reasons for this is the fact that the study on time scales unifies the study of both discrete and continuous processes, besides many others. The pioneering works in this direction are [27–30]. The theory of time scales was initiated by Stefan Hilger in his Ph.D. thesis in 1988, providing a rich theory that unifies and extends discrete and continuous analysis [31, 32]. The time scales calculus has a tremendous potential for applications in some mathematical models of real processes and phenomena studied in physics, chemical technology, population dynamics, biotechnology and economics, neural networks, and social sciences.In fact, both continuous and discrete systems are very important in implementation and applications. But, it is troublesome to study the the dynamical properties for continuous and discrete systems, respectively. Therefore, it is meaningful to study that on time scale which can unify the continuous and discrete situations (see [13, 31, 33–40]).Motivated by the above, in this paper, we propose a neutral-type BAM neural network with delays on time scales as follows:(2)xiΔ(t)=-ai(t)xi(t)+∑j=1m‍aji(t)fj(yj(t-τji(t)))+∑j=1m‍pji(t)gj(yjΔ(t-σji(t)))+Ii(t),+∑j=1m‍pji(t)gjddddt∈𝕋,i=1,2,…,n,yjΔ(t)=-bj(t)yj(t)+∑i=1n‍bij(t)hi(xi(t-ζij(t)))+∑i=1n‍qij(t)ki(xiΔ(t-ςij(t)))+Jj(t),+∑j=1m‍pji(t)gjdddt∈𝕋,j=1,2,…,m,
where 𝕋 is an almost periodic time scale; n, m are the number of neurons in layers; xi(t) and yj(t) denote the activations of the ith neuron and the jth neuron at time t; ai and bj represent the rate with which the ith neuron and jth neuron will reset their potential to the resting state in isolation when they are disconnected from the network and the external inputs at time t; fj, gj, hi, and ki are the input-output functions (the activation functions); τji, σji, ζij, and ςij are transmission delays at time t and satisfy t-τji(t)∈𝕋, t-σji(t)∈𝕋, t-ζij(t)∈𝕋, and t-ςij(t)∈𝕋 for t∈𝕋; aji,bij are elements of feedback templates at time t; pji, qij are elements of feed-forward templates at time t; and Ii, Jj denote biases of the ith neuron and the jth neuron at time t, i=1,2,…,n, j=1,2,…,m.Our main purpose of this paper is, using the exponential dichotomy of linear dynamic equations on time scales, a fixed point theorem and the theory of calculus on time scales, to study the existence and exponential stability of almost periodic solutions for (2). Our results of this paper are new and complementary to the previously known results even if the time scale 𝕋=ℝ or ℤ.For convenience, we denote[a,b]𝕋={t∣t∈[a,b]∩𝕋}. For an almost periodic function f:𝕋→ℝ, denote f+=supt∈𝕋|f(t)|, f-=inft∈𝕋|f(t)|. Set that 𝕏={ϕ=(φ1,φ2,…,φn,ψ1,ψ2,…,ψm)T|φi,ψj∈C1(𝕋,ℝ), φi,ψj are almost periodic functions on 𝕋,i=1,2,…,n,j=1,2,…,m} with the norm ∥ϕ∥=max{|φ|1,|ψ|1}, where |φ|1=max{|φ|0,|φΔ|0}, |ψ|1=max{|ψ|0,|ψΔ|0}, |φ|0=max1≤i≤nφi+, |φΔ|0=max1≤i≤n(φiΔ(t))+, |ψ|0=max1≤j≤mψj+, and |ψΔ|0=max1≤j≤m(ψjΔ(t))+, and C1(𝕋,ℝ) is the set of continuous functions with continuous derivatives on 𝕋, then 𝕏 is a Banach space.The initial condition of (2) is
(3)xi(s)=φi(s),yj(s)=ψj(s),s∈[-θ,0]𝕋,
where θ=max{max(i,j){τji+,σji+,ζij+,ςij+}}, φi, ψj∈C1([-θ,0]𝕋,ℝ), i=1,2,…,n, j=1,2,…,m.Throughout this paper, we assume that the following conditions hold:(H1)
aji,pji,bij,qij,Ii,Jj∈C(𝕋,ℝ), ai,bj∈C(𝕋,ℝ+) with -ai,-bj∈ℛ+, τji,σji,ζij, and ςij∈C(𝕋,𝕋∩ℝ+) are all almost periodic functions, where ℛ+ denotes the set of positively regressive functions from 𝕋 to ℝ, i=1,2,…,n,j=1,2,…,m;(H2)
fj,gj,hi,ki∈C(ℝ,ℝ), and there exist positive constants Hjf, Hjg, Hih, and Hik such that
(4)|fj(u)-fj(v)|≤Hjf|u-v|,|gj(u)-gj(v)|≤Hjg|u-v|,|hi(u)-hi(v)|≤Hih|u-v|,|ki(u)-ki(v)|≤Hik|u-v|,
for all u,v∈ℝ, i=1,2,…,n, j=1,2,…,m.This paper is organized as follows. In Section2, we introduce some notations and definitions and state some preliminary results which are needed in later sections. In Section 3, we establish some sufficient conditions for the existence of almost periodic solutions of (2). In Section 4, we prove that the almost periodic solution obtained in Section 3 is exponentially stable. In Section 5, we give an example to illustrate the feasibility of our results obtained in previous sections. We draw a conclusion in Section 6.


## 2. Preliminaries

In this section, we introduce some definitions and state some preliminary results.Definition 1 (see [32]).
Let𝕋 be a nonempty closed subset (time scale) of ℝ. The forward and backward jump operators σ,ρ:𝕋→𝕋 and the graininess μ:𝕋→ℝ+ are defined, respectively, by
(5)σ(t)=inf{s∈𝕋:s>t},ρ(t)=sup{s∈𝕋:s<t},μ(t)=σ(t)-t.Lemma 2 (see [32]).
Assume thatp,q:𝕋→ℝ are two regressive functions, then(i)
e0(t,s)≡1 and ep(t,t)≡1;(ii)
ep(t,s)=1/ep(s,t)=e⊖p(s,t);(iii)
ep(t,s)ep(s,r)=ep(t,r);(iv)
(ep(t,s))Δ=p(t)ep(t,s).Lemma 3 (see [32]).
Letf, g be Δ-differentiable functions on T. Then(i)
(ν1f+ν2g)Δ=ν1fΔ+ν2gΔ, for any constants ν1, ν2;(ii)
(fg)Δ(t)=fΔ(t)g(t)+f(σ(t))gΔ(t)=f(t)gΔ(t)+fΔ(t)g(σ(t)).Lemma 4 (see [32]).
Assume thatp(t)≥0 for t≥s. Then ep(t,s)≥1.Definition 5 (see [32]).
A functionf:𝕋→ℝ is positively regressive if 1+μ(t)f(t)>0 for all t∈𝕋.Lemma 6 (see [32]).
Suppose thatp∈ℛ+. Then(i)
ep(t,s)>0, for all t,s∈𝕋;(ii)
ifp(t)≤q(t) for all t≥s, t,s∈𝕋, then ep(t,s)≤eq(t,s) for all t≥s.Lemma 7 (see [32]).
Ifp∈ℛ and a,b,c∈𝕋, then
(6)[ep(c,·)]Δ=-p[ep(c,·)]σ,∫ab‍p(t)ep(c,σ(t))Δt=ep(c,a)-ep(c,b).Lemma 8 (see [32]).
Leta∈𝕋k, b∈𝕋, and assume that f:𝕋×𝕋k→ℝ is continuous at (t,t), where t∈𝕋k with t>a. Also assume that fΔ(t,·) is rd-continuous on [a,σ(t)]. Suppose that for each ε>0, there exists a neighborhood U of τ∈[a,σ(t)] such that
(7)|f(σ(t),τ)-f(s,τ)-fΔ(t,τ)(σ(t)-s)|≤ε|σ(t)-s|,dd∀s∈U,
where fΔ denotes the derivative of f with respect to the first variable. Then,(i)
g(t):=∫at‍f(t,τ)Δτ implies gΔ(t):=∫at‍fΔ(t,τ)Δτ+f(σ(t),t);(ii)
h(t):=∫tb‍f(t,τ)Δτ implies hΔ(t):=∫tb‍fΔ(t,τ)Δτ-f(σ(t),t).Definition 9 (see [34]).
A time scale𝕋 is called an almost periodic time scale if
(8)Π:={τ∈ℝ:t+τ∈𝕋,∀t∈𝕋}≠{0}.Definition 10 (see [34]).
Let𝕋 be an almost periodic time scale. A function f(t):𝕋→ℝn is said to be almost periodic on 𝕋, if for any ε>0, the set
(9)E(ε,f)={τ∈Π:|f(t+τ)-f(t)|<ε,∀t∈𝕋}
is relatively dense in 𝕋; that is, for any ε>0, there exists a constant l(ε)>0 such that each interval of length l(ε) contains at least one τ∈E(ε,f) such that
(10)|f(t+τ)-f(t)|<ε,∀t∈𝕋.
The set E(ε,f) is called the ε-translation set of f(t), τ is called the ε-translation number of f(t), and l(ε) is called the inclusion of E(ε,f).Lemma 11 (see [34]).
Iff∈C(𝕋,ℝn) is an almost periodic function, then f is bounded on 𝕋.Lemma 12 (see [34]).
Iff,g∈(𝕋,ℝn) are almost periodic functions, then f+g,fg are also almost periodic.Definition 13 (see [35]).
LettingX∈ℝn and A(t) be a n×n matrix-valued function on 𝕋, the linear system
(11)XΔ(t)=A(t)X(t),t∈𝕋
is said to admit an exponential dichotomy on 𝕋 if there exist positive constants ki, αi, i=1,2, projection P, and the fundamental solution matrix X(t) of (11) satisfying
(12)|X(t)PX-1(s)|≤k1e⊖α1(t,s),s,t∈𝕋,t≥s,|X(t)(I-P)X-1(s)|≤k2e⊖α2(s,t),s,t∈𝕋,t≤s,
where |·| is a matrix norm on 𝕋; that is, if A=(aij)n×m, then we can take |A|=(∑i=1n∑j=1m|aij|2)1/2.Lemma 14 (see [34]).
If (11) admits an exponential dichotomy, then the following almost periodic system
(13)XΔ(t)=A(t)X(t)+g(t),t∈𝕋
has an almost periodic solution as follows:
(14)X(t)=∫-∞t‍X(t)PX-1(σ(s))g(s)Δs-∫t+∞‍X(t)(I-P)X-1(σ(s))g(s)Δs,
where X(t) is the fundamental solution matrix of (11).Lemma 15 (see [35]).
IfA(t) is a uniformly bounded rd-continuous n×n matrix-valued function on 𝕋, and there is a δ>0 such that
(15)|aii(t)|-∑j≠i‍|aij(t)|-12μ(t)(∑j=1n‍|aij(t)|)2-δ2μ(t)≥2δ,|aii(t)|-∑j≠i‍|aij(t)|-12μ(t)t∈𝕋,i=1,2,…,n,
then (11) admits an exponential dichotomy on 𝕋.Definition 16.
Letz*(t)=(x1*(t),x2*(t),…,xn*(t),y1*(t), y2*(t),…,ym*(t))T be an almost periodic solution of (2) with initial value ϕ*(s)=(φ1*(s),φ2*(s),…,φn*(s),ψ1*(s),ψ2*(s),…,ψm*(s))T. If there exists a positive constant λ with -λ∈ℛ+ such that for t0∈[-θ,0]𝕋, there exists M>1 such that for an arbitrary solution z(t)=(x1(t),x2(t),…,xn(t),y1(t),y2(t),…,ym(t))T of (2) with initial value ϕ(s)=(φ1(s),φ2(s),…,φn(s),ψ1(s), ψ2(s),…,ψm(s))T satisfies
(16)|z(t)-z*(t)|1≤M∥ϕ-ϕ*∥e-λ(t,t0),t∈[-θ,∞)𝕋,t≥t0,
where |z(t)-z*(t)|1=max{max1≤i≤n{|xi(t)-xi*(t)|, |(xi(t)-xi*(t))Δ|}, max1≤j≤m{|yj(t)-yj*(t)|, |(yj(t)-yj*(t))Δ|}}, ∥ϕ-ϕ*∥=max{max1≤i≤nsups∈[-θ,0]𝕋{|φi(s)-φi*(s)|, |(φi(s)-φi*(s))Δ|}, max1≤j≤msups∈[-θ,0]𝕋{|ψj(s)-ψj*(s)|, |(ψj(s)-ψj*(s))Δ|}}. Then, the solution z*(t) is said to be exponentially stable.


## 3. Existence of Almost Periodic Solutions

In this section, we will state and prove the sufficient conditions for the existence of almost periodic solutions of (2).Letϕ0(t)=(φ10(t),φ20(t),…,φn0(t),ψ10(t),ψ20(t),…,ψm0(t))T, where φi0(t)=∫-∞t‍e-ai(t,σ(s))×Ii(s)Δs,ψj0(t)=∫-∞t‍e-bj(t,σ(s))Jj(s)Δs, i=1,2,…,n,j=1,2,…,m, and let L be a constant satisfying L≥max{∥ϕ0∥, max1≤j≤m{|fj(0)|,|gj(0)|,max1≤i≤n{|hi(0)|,ki(0)|}. We have the following theorem.Theorem 17.
Let(H1) and (H2) hold. Suppose that(H3)
there exists a positive constantδ such that
(17)ck(t)-12μ(t)ck2(t)-δ2μ(t)≥2δ,t∈𝕋,k=1,2,…,n+m,
where
(18)ck(t)={ai(t),k=i,i=1,2,…,n,bj(t),k=n+j,j=1,2,…,m;(H4)
max{max1≤i≤n{(θi/ai-), (1+(ai+/ai-))θi}, max1≤j≤m{(γj/bj-),(1+(bj+/bj-))γj}}≤(1/2), where θi=∑j=1m(aji+Hjf+pji+Hjg+aji++pji+), γj=∑i=1n(bij+Hih+qij+Hik+bij++qij+), i=1,2,…,n,j=1,2,…,m.Then, (2) has a unique almost periodic solution in 𝕏0={ϕ∈𝕏∣∥ϕ-ϕ0∥≤L}, ϕ(t)=(φ1(t), φ2(t),…,φn(t),ψ1(t),ψ2(t),…,ψm(t))T.Proof.
For any givenϕ∈𝕏, we consider the following almost periodic system:
(19)xiΔ(t)=-ai(t)xi(t)+Fi(t,ψ)+Ii(t),i=1,2,…,n,yjΔ(t)=-bj(t)yj(t)+Gj(t,φ)+Jj(t),j=1,2,…,m,
where
(20)Fi(t,ψ)=∑j=1m‍aji(t)fj(ψj(t-τji(t)))+∑j=1m‍pji(t)gj(ψjΔ(t-σji(t))),i=1,2,…,n,Gj(t,φ)=∑i=1n‍bij(t)hi(φi(t-ζij(t)))+∑i=1n‍qij(t)ki(φiΔ(t-ςij(t))),j=1,2,…,m.
Since (H3) holds, it follows from Lemma 15 that the linear system
(21)xiΔ(t)=-ai(t)xi(t),i=1,2,…,n,yjΔ(t)=-bj(t)yj(t),j=1,2,…,m
admits an exponential dichotomy on 𝕋. Thus, by Lemma 14, we obtain that (19) has an almost periodic solution, which is expressed as follows:
(22)xiφ(t)=∫-∞t‍e-ai(t,σ(s))(Fi(s,ψ)+Ii(s))Δs,dddddi=1,2,…,n,yjψ(t)=∫-∞t‍e-bj(t,σ(s))(Gj(s,φ)+Ji(s))Δs,dddddj=1,2,…,m.
For ϕ∈𝕏0, then ∥ϕ∥≤∥ϕ-ϕ0∥+∥ϕ0∥≤2L. Define the following operator:
(23)Φ:𝕏0⟶𝕏0,(φ1,φ2,…,φn,ψ1,ψ2,…,ψm)T⟶(x1φ,x2φ,…,xnφ,y1ψ,y2ψ,…,ymψ)T.
We will show that Φ is a contraction.
First, we show that for anyϕ∈𝕏0, we have Φϕ∈𝕏0. Note that
(24)|Fi(s,ψ)|=|∑j=1m‍aji(s)fj(ψj(s-τji(s)))d+∑j=1m‍pji(s)gj(ψjΔ(s-σji(s)))|≤∑j=1m‍aji+(|fj(ψj(s-τji(s)))-fj(0)|+|fj(0)|)d+∑j=1m‍pji+(|gj(ψjΔ(s-σji(s)))-gj(0)|ddddddddd+|gj(0)||gj(ψjΔ(s-σji(s)))-gj(0)|)≤∑j=1m‍(aji+Hjf+pji+Hjg)|ψ|1d+∑j=1m‍(aji+|fj(0)|+pji+|gj(0)|)≤∑j=1m‍(aji+Hjf+pji+Hjg)∥ϕ∥d+∑j=1m‍(aji+|fj(0)|+pji+|gj(0)|)≤∑j=1m‍(aji+Hjf+pji+Hjg+aji++pji+)2L:=2θiL,ddddddddddddddddddddi=1,2,…,n
and similarly,
(25)|Gj(s,φ)|≤∑i=1n‍(bij+Hih+qij+Hik+bij++qij+)2L:=2γjL,dddddddddddddddddj=1,2,…,m.
Therefore, we have
(26)|(Φ(ϕ-ϕ0))i(t)|=|∫-∞t‍e-ai(t,σ(s))Fi(s,ψ)Δs|≤∫-∞t‍e-ai(t,σ(s))|Fi(s,ψ)|Δs≤∫-∞t‍e-ai-(t,σ(s))2θiLΔs≤2θiLai-,i=1,2,…,n,|(Φ(ϕ-ϕ0))n+j(t)|=|∫-∞t‍e-bj(t,σ(s))Gj(s,φ)Δs|≤∫-∞t‍e-bj(t,σ(s))|Gj(s,φ)|Δs≤∫-∞t‍e-bj-(t,σ(s))2γjLΔs≤2γjLbj-,j=1,2,…,m.
On the other hand, for i=1,2,…,n, we have
(27)|(Φ(ϕ-ϕ0))iΔ(t)|=|(∫-∞t‍e-ai(t,σ(s))Fi(s,ψ)Δs)tΔ|=|Fi(t,ψ)-ai(t)∫-∞t‍e-ai(t,σ(s))Fi(s,ψ)Δs|≤|Fi(t,ψ)|+|ai(t)|dd×∫-∞t‍e-ai(t,σ(s))|Fi(s,ψ)|Δs≤(1+ai+ai-)2θiL
and for j=1,2,…,m,
(28)|(Φ(ϕ-ϕ0))n+jΔ(t)|≤(1+bj+bj-)2γjL.
In view of (H4), we have
(29)∥Φϕ-ϕ0∥≤max{max1≤i≤n{2θiLai-,(1+ai+ai-)2θiL},aamax1≤j≤m{2γjLbj-,(1+bj+bj-)2γjL}}≤L,
that is, Φϕ∈𝕏0. Next, we show that Φ is a contraction. For ϕ=(φ1,φ2,…,φn,ψ1,ψ2,…,ψm)T, ϑ=(ξ1,ξ2,…,ξn,η1,η2,…,ηm)T∈𝕏0, for i=1,2,…,n, denote by
(30)Fi(1)(s,ψ,η)=∑j=1m‍aji(s)(fj(ψj(s-τji(s)))-fj(ηj(s-τji(s)))),Fi(2)(s,ψ,η)=∑j=1m‍pji(s)(gj(ψjΔ(s-σji(s)))-gj(ηjΔ(s-σji(s)))).
Then, we have
(31)|(Φϕ-Φζ)i(t)|=|∫-∞t‍e-ai(t,σ(s))(Fi(1)(s,ψ,η)+Fi(2)(s,ψ,η))Δs|≤∫-∞t‍e-ai(t,σ(s))≤d×(∑j=1m‍aji+Hjf|ψj(s-τji(s))-ηj(s-τji(s))|dddddd+∑j=1m‍pji+Hjg|ψjΔ(s-σji(s))dddddddddddddddddddd-ηjΔ(s-σji(s))|∑j=1m‍)Δs≤1ai-∑j=1m‍(aji+Hjf+pji+Hjg)|ψ-η|1,i=1,2,…,n,|(Φϕ-Φζ)iΔ(t)|=|(∫-∞t‍e-ai(t,σ(s))(Fi(1)(s,ψ,η)+Fi(2)(s,ψ,η))Δs)tΔ|=|∫-∞t‍Fi(1)(t,ψ,η)+Fi(2)(t,ψ,η)-ai(t)ddddd×∫-∞t‍e-ai(t,σ(s))(Fi(1)(s,ψ,η)+Fi(2)(s,ψ,η))Δs|≤|Fi(1)(t,ψ,η)|+|Fi(2)(t,ψ,η)|+|ai(t)|ddddd×∫-∞t‍e-ai(t,σ(s))|Fi(1)(s,ψ,η)+Fi(2)(s,ψ,η)|Δs≤∑j=1m‍(aji+Hjf+pji+Hjg)|ψ-η|1+ai+ai-∑j=1m‍(aji+Hjf+pji+Hjg)|ψ-η|1=(1+ai+ai-)∑j=1m‍(aji+Hjf+pji+Hjg)|ψ-η|1,ddddddddddddddddddddddi=1,2,…,n.
In a similar way, we have
(32)|(Φϕ-Φζ)n+j(t)|≤1bj-∑i=1n‍(bij+Hih+qij+Hik)|φ-ξ|1,ddddddddddddj=1,2,…,m,|(Φϕ-Φζ)n+jΔ(t)|≤(1+bj+bj-)×∑i=1n‍(bij+Hih+qij+Hik)|φ-ξ|1,dddddddddj=1,2,…,m.
By (H4), we have
(33)max{max1≤i≤n{1ai-ϱi,(1+ai+ai-)ϱi},ddmax1≤j≤m{1bj-ρj,(1+bj+bj-)ρj}}<1,
where ϱi=∑j=1m(aji+Hjf+pji+Hjg) and ρj=∑i=1n(bij+Hih+qij+Hik), i=1,2,…,n,j=1,2,…,m. It implies that ∥Φϕ-Φζ∥<∥φ-ψ∥. Hence, Φ is a contraction. Therefore, Φ has a fixed point in 𝕏0; that is, (2) has a unique almost periodic solution in 𝕏0. This completes the proof of Theorem 17.


## 4. Exponential Stability of Almost Periodic Solution

In this section, we will study the exponential stability of almost periodic solution of (2).Theorem 18.
Let (H1)–(H4) hold. Suppose further that(H5)
ϱi≥(ai--ai+ai-)/(ai++ai-) and ρj≥(bj--bj+bj-)/(bj++bj-), where ϱi=∑j=1m(aji+Hjf+pji+Hjg) and ρj=∑i=1n(bij+Hih+qij+Hik),i=1,2,…,n,j=1,2,…,m.Then, the almost periodic solution of (2) is exponentially stable.Proof.
By Theorem17, (2) has an almost periodic solution ω(t)=(α1(t),α2(t),…,αn(t),β1(t),β2(t),…,βm(t))T with initial condition ϕ*(s)=(φ1*(s),φ2*(s),…,φn*(s),ψ1*(s),ψ2*(s),…,ψm*(s))T. Suppose that z(t)=(x1(t),x2(t),…,xn(t),y1(t),y2(t),…,ym(t))Tis an arbitrary solution of (2) with initial condition ϕ(s)=(φ1(s),φ2(s),…,φn(s),ψ1(s),ψ2(s),…,ψm(s))T. Denote v(t)=(u1(t),u2(t),…,un(t),ν1(t),ν2(t),…,νm(t))T, where ui(t)=xi(t)-αi(t), νj(t)=yj(t)-βj(t), i=1,2,…,n, j=1,2,…,m. Then, it follows from (2) that
(34)uiΔ(t)=-ai(t)ui(t)+∑j=1m‍aji(t)(fj(yj(t-τji(t)))dddddddddd-fj(βj(t-τji(t))))+∑j=1m‍pji(t)(gj(yjΔ(t-σji(t)))dddddddddd-gj(βjΔ(t-σji(t)))),dddddddddddddddi=1,2,…,n,(35)νjΔ(t)=-bj(t)νj(t)+∑i=1n‍bij(t)(hi(xi(t-ζij(t)))dddddddd-hi(αi(t-ζij(t))))+∑i=1n‍qij(t)(ki(xiΔ(t-ςij(t)))dddddddd-ki(αiΔ(t-ςij(t)))),dddddddddddddj=1,2,…,m.
The initial condition of (34) and (35) is
(36)ui(s)=φi(s)-φi*(s),νj(s)=ψj(s)-ψj*(s),s∈[-θ,0]𝕋,
where i=1,2,…,n,j=1,2,…,m.
Multiplying both sides of (34) by e-ai(t,σ(s)) and integrating on [t0,t]𝕋, where t0∈[-θ,0]𝕋, we get
(37)ui(t)=ui(t0)e-ai(t,t0)ddd+∫t0t‍e-ai(t,σ(s))dddddd×{∑j=1m‍aji(s)(fj(yj(s-τji(s)))dddddddddddddddddd-fj(βj(s-τji(s))))ddddddddd+∑j=1m‍pji(s)(gj(yjΔ(s-σji(s)))ddddddddddddddddddd-gj(βjΔ(s-σji(s))))∑j=1m‍}Δs,ddddddddddddddddddddddddddddddi=1,2,…,n.
Similarly, multiplying both sides of (35) by e-bj(t,σ(s)) and integrating on [t0,t]𝕋, we get
(38)νj(t)=νj(t0)e-bj(t,t0)dddd+∫t0t‍e-bj(t,σ(s))ddddddd×{∑i=1n‍bij(s)(hi(xi(s-ζij(s)))dddddddddddddddddd-hi(αi(s-ζij(s))))ddddddddddd+∑i=1n‍qij(s)(ki(xiΔ(s-ςij(s)))dddddddddddddddddddd-ki(αiΔ(s-ςij(s))))∑i=1n‍}Δs,ddddddddddddddddddddddddddddj=1,2,…,m.
For positive constant α<min{min1≤i≤nai-,min1≤j≤mbj-} with -α∈ℛ+, we have e⊖α(t,t0)>1, where t∈[-θ,t0]𝕋. Take
(39)M>max{ϵ1,ϵ2},
where
(40)ϵ1=max1≤i≤n{ai+ai-ai--(ai++ai-)ϱi,ai-ai--ϱi},ϵ2=max1≤j≤m{bj+bj-bj--(bj++bj-)ρj,bj-bj--ρj}.
By (H4), we have
(41)ai->(ai++ai-)ϱi,ai->ϱi,bj->(bj++bj-)ρj,bj->ρj,i=1,2,…,n,j=1,2,…,m.
In view of (H5), we have M>1. Hence, it is obvious that
(42)|v(t)|1≤Me⊖α(t,t0)∥ϕ-ϕ*∥,∀t∈[-θ,t0]𝕋.
We claim that
(43)|v(t)|1≤Me⊖α(t,t0)∥ϕ-ϕ*∥,∀t∈(t0,+∞)𝕋.
To prove this claim, we show that for any p>1, the following inequality holds:
(44)|v(t)|1<pMe⊖α(t,t0)∥ϕ-ϕ*∥,∀t∈(t0,+∞)𝕋,
which means that, for i=1,2,…,n, we have
(45)|ui(t)|<pMe⊖α(t,t0)∥ϕ-ϕ*∥,∀t∈(t0,+∞)𝕋,(46)|uiΔ(t)|<pMe⊖α(t,t0)∥ϕ-ϕ*∥,∀t∈(t0,+∞)𝕋,
and for j=1,2,…,m, we have
(47)|νj(t)|<pMe⊖α(t,t0)∥ϕ-ϕ*∥,∀t∈(t0,+∞)𝕋,(48)|νjΔ(t)|<pMe⊖α(t,t0)∥ϕ-ϕ*∥,∀t∈(t0,+∞)𝕋.
By way of contradiction, assume that (44) does not hold. Firstly, we consider the following four cases.
Case  1. Equation (45) is not true and (46)–(48) are all true. Then, there exists t1∈(t0,+∞)𝕋 and i0∈{1,2,…,n} such that
(49)|ui0(t1)|≥pMe⊖α(t1,t0)∥ϕ-ϕ*∥,|ui0(t)|<pMe⊖α(t,t0)∥ϕ-ϕ*∥,t∈(t0,t1)𝕋,dddd|ul(t)|<pMe⊖α(t,t0)∥ϕ-ϕ*∥,forl≠i0,t∈(t0,t1]𝕋,l=1,2,…,n.
Therefore, there must be a constant δ1≥1 such that
(50)|ui0(t1)|=δ1pMe⊖α(t1,t0)∥ϕ-ϕ*∥,|ui0(t)|<δ1pMe⊖α(t,t0)∥ϕ-ϕ*∥,t∈(t0,t1)𝕋,dd|ul(t)|<δ1pMe⊖α(t,t0)∥ϕ-ϕ*∥,forl≠i0,t∈(t0,t1]𝕋,l=1,2,…,n.
Note that, in view of (37), we have
(51)|ui0(t1)|=|∑j=1m‍ui0(t0)e-ai0(t1,t0)+∫t0t1‍e-ai0(t1,σ(s))ddddd×{∑j=1m‍aji0(s)(fj(yj(s-τji0(s)))ddddddddddddddddd-fj(βj(s-τji0(s))))ddddddddd+∑j=1m‍pji0(s)(gj(yjΔ(s-σji0(s)))dddddddddddddddddd-gj(βjΔ(s-σji0(s))))∑j=1m‍}Δs|≤e-ai0(t1,t0)∥ϕ-ϕ*∥+∫t0t1‍e-ai0(t1,σ(s))dddd×(∑j=1m‍aji0+Hjf|νj(s-τji0(s))|ddddddd+∑j=1n‍pji0+Hjg|νjΔ(s-σji0(s))|)Δs≤e-ai0(t1,t0)∥ϕ-ϕ*∥+∫t0t1‍e-ai0(t1,σ(s))ddd×(∑j=1m‍aji0+Hjfδ1pMe⊖α(s-τji0(s),t0)∥ϕ-ϕ*∥dddddd+∑j=1m‍pji0+Hjgδ1pMe⊖α(s-σji0(s),t0)∥ϕ-ϕ*∥)Δs=∥ϕ-ϕ*∥e⊖α(t1,t0)e-ai0⊕α(t1,t0)+δ1pMe⊖α(t1,t0)∥ϕ-ϕ*∥×∫t0t1‍e-ai0(t1,σ(s))ddd×(∑j=1m‍aji0+Hjfe⊖α(s-τji0(s),t1)dddddd+∑j=1m‍pji0+Hjge⊖α(s-σji0(s),t1))Δs≤∥ϕ-ϕ*∥e⊖α(t1,t0)e-ai0-+α(t1,t0)+δ1pMe⊖α(t1,t0)∥ϕ-ϕ*∥×∫t0t1‍e-ai0-(t1,σ(s))ddd×(∑j=1m‍aji0+Hjfexp{-ατ~}+∑j=1m‍pji0+Hjgexp{-ασ~})Δs≤∥ϕ-ϕ*∥e⊖α(t1,t0)+δ1pMe⊖α(t1,t0)∥ϕ-ϕ*∥1-ai0-×∫t0t1‍(-ai0-)e-ai0-(t1,σ(s))Δsddd×(∑j=1m‍aji0+Hjfexp{-ατ~}+∑j=1m‍pji0+Hjgexp{-ασ~})=∥ϕ-ϕ*∥e⊖α(t1,t0)-1ai0-δ1pMe⊖(-α)(t1,t0)×∥ϕ-ϕ*∥(e-ai0-(t1,t0)-1)×(∑j=1m‍aji0+Hjfexp{-ατ~}+∑j=1m‍pji0+Hjgexp{-ασ~})<δ1pM∥ϕ-ϕ*∥e⊖α(t1,t0)×[1δ1pM+1ai0-(∑j=1m‍aji0+Hjfexp{-ατ~}ddddddddddddd+∑j=1m‍pji0+Hjgexp{-ασ~})]<δ1pM∥ϕ-ϕ*∥e⊖α(t1,t0)×(1M+1ai0-∑j=1m‍(aji0+Hjf+pji0+Hjg))<δ1pMe⊖α(t1,t0)∥ϕ-ϕ*∥,
where τ~=min(i,j)τji-,σ~=min(i,j)σji-. In the proof, we use the inequality e-ai0-(t1,t0)<1. Thus, we get a contradiction.
Case  2. Equation (46) is not true and (45), (47), and (48) are all true. Then, there exists t2∈(t0,+∞)𝕋 and i1∈{1,2,…,n} such that
(52)|ui1Δ(t2)|≥pMe⊖α(t2,t0)∥ϕ-ϕ*∥,|ui1Δ(t)|<pMe⊖α(t,t0)∥ϕ-ϕ*∥,t∈(t0,t2)𝕋,dd|ulΔ(t)|<pMe⊖α(t,t0)∥ϕ-ϕ*∥,forl≠i1,t∈(t0,t2]𝕋,l=1,2,…,n.
Hence, there must be a constant δ2≥1 such that
(53)|ui1Δ(t2)|=δ2pMe⊖α(t2,t0)∥ϕ-ϕ*∥,|ui1Δ(t)|<δ2pMe⊖α(t,t0)∥ϕ-ϕ*∥,t∈(t0,t2)𝕋,dddd|ulΔ(t)|<δ2pMe⊖α(t,t0)∥ϕ-ϕ*∥,forl≠i1,t∈(t0,t2]𝕋,l=1,2,…,n.
Note that, in view of (37), we have
(54)uiΔ(t)=-ai(t)ui(t0)e-ai(t,t0)+∫t0t‍(-ai(t))e-ai(t,σ(s))ddddd×{∑j=1m‍aji(s)(fj(yj(s-τji(s)))dddddddddddddddd-fj(βj(s-τji(s))))dddddddd+∑j=1m‍pji(s)(gj(yjΔ(s-σji(s)))ddddddddddddddddd-gj(βjΔ(s-σji(s))))∑j=1m‍}Δs+e-ai(σ(t),σ(t))×{∑j=1m‍aji(t)(fj(yj(t-τji(t)))dddddddddddd-fj(βj(t-τji(t))))ddddd+∑j=1m‍pji(t)(gj(yjΔ(t-σji(t)))dddddddddddddd-gj(βjΔ(t-σji(t))))∑j=1m‍},dddddddddddddddddddddi=1,2,…,n.
Thus, we have
(55)|ui1Δ(t2)|≤ai1+e-ai1-(t2,t0)∥ϕ-ϕ*∥+ai1+∫t0t2‍e-ai1(t2,σ(s))ddddddd×(∑j=1m‍aji1+Hjf|νj(s-τji1(s))|ddddddddd+∑j=1m‍pji1+Hjg|νjΔ(s-σji1(s))|)Δs+∑j=1m‍aji1+Hjf|νj(t2-τji1(t2))|+∑j=1m‍pji1+Hjg|νjΔ(t2-σji1(t2))|≤ai1+e-ai1-(t2,t0)∥ϕ-ϕ*∥+ai1+∫t0t2‍e-ai1(t2,σ(s))ddddd×(∑j=1m‍aji1+Hjfδ2pMe⊖(-α)(s-τji1(s),t0)∥ϕ-ϕ*∥ddddddddd+∑j=1m‍pji1+Hjgδ2pMe⊖α(s-σji1(s),t0)dddddddddddddd×∥ϕ-ϕ*∥∑j=1m‍)Δs+δ2pMe⊖α(t2-τji1(t2),t0)∥ϕ-ϕ*∥∑j=1m‍aji1+Hjf+δ2pMe⊖α(t2-σji1(t2),t0)∥ϕ-ϕ*∥∑j=1m‍pji1+Hjg≤ai1+e-ai1-(t2,t0)∥ϕ-ϕ*∥+ai1+δ2pMe⊖α(t2,t0)∥ϕ-ϕ*∥×∫t0t2‍e-ai1(t2,σ(s))dddd×(∑j=1m‍aji1+Hjfe⊖(-α)(s-τji1(s),t2)dddddddd+∑j=1m‍pji1+Hjge⊖α(s-σji1(s),t2)∑j=1m‍)Δs+δ2pMe⊖α(t2,t0)∥ϕ-ϕ*∥×∑j=1m‍(aji1+Hjfe⊖α(t2-τji1(t2),t2)ddddddd+pji1+Hjge⊖α(t2-σji1(t2),t2))≤ai1+e-ai1-(t2,t0)∥ϕ-ϕ*∥+ai1+δ2pMe⊖α(t2,t0)∥ϕ-ϕ*∥×∫t0t2‍e-ai1-(t2,σ(s))ddd×(∑j=1m‍aji1+Hjfexp{-ατ~}+∑j=1m‍pji1+Hjgexp{-ασ~})Δs+δ2pMe⊖α(t2,t0)∥ϕ-ϕ*∥×∑j=1m‍(aji1+Hjfexp{-ατ~}+pji1+Hjgexp{-ασ~})≤ai1+e-ai1-(t2,t0)∥ϕ-ϕ*∥+ai1+δ2pMe⊖α(t2,t0)∥ϕ-ϕ*∥×1-ai1-∫t0t2‍(-ai1-)e-ai1-(t2,σ(s))Δs×(∑j=1m‍aji1+Hjfexp{-ατ~}+∑j=1m‍pji1+Hjgexp{-ασ~})+δ2pMe⊖α(t2,t0)∥ϕ-ϕ*∥∑j=1m‍(aji1+Hjf+pji1+Hjg)=ai1+∥ϕ-ϕ*∥e⊖α(t2,t0)e-ai1-⊕α(t2,t0)-ai1+ai1-δ2pMe⊖α(t2,t0)∥ϕ-ϕ*∥(e-ai1-(t2,t0)-1)×(∑j=1m‍aji1+Hjf×exp{-ατ~}+∑j=1m‍pji1+Hjgexp{-ασ~})+δ2pMe⊖α(t2,t0)∥ϕ-ϕ*∥∑j=1m‍(aji1+Hjf+pji1+Hjg)≤δ2pMe⊖α(t2,t0)∥ϕ-ϕ*∥×{ai1+δ2pM+∑j=1m‍(aji1+Hjf+pji1+Hjg)dddd+ai1+ai1-∑j=1m‍(aji1+Hjfexp{-ατ~}ddddddddddd+pji1+Hjgexp{-ασ~})∑j=1m‍}<δ2pMe⊖α(t2,t0)∥ϕ-ϕ*∥×{ai1+M+(1+ai1+ai1-)∑j=1m‍(aji1+Hjf+pji1+Hjg)}<δ2pMe⊖α(t2,t0)∥ϕ-ϕ*∥.
We also get a contradiction.
Case  3. Equation (47) is not true and (45), (46), and (48) are all true. Then, there exists t3∈(t0,+∞)𝕋 and j0∈{1,2,…,m} such that
(56)|νj0(t3)|≥pMe⊖α(t3,t0)∥ϕ-ϕ*∥,|νj0(t)|<pMe⊖α(t,t0)∥ϕ-ϕ*∥,t∈(t0,t3)𝕋,ddddd|νι(t)|<pMe⊖α(t,t0)∥ϕ-ϕ*∥,forι≠j0,t∈(t0,t3]𝕋,ι=1,2,…,m.
Therefore, there must be a constant δ3≥1 such that
(57)|νj0(t3)|=δ3pMe⊖α(t3,t0)∥ϕ-ϕ*∥,|νj0(t)|<δ3pMe⊖α(t,t0)∥ϕ-ϕ*∥,t∈(t0,t3)𝕋,dddd|νι(t)|<δ3pMe⊖α(t,t0)∥ϕ-ϕ*∥,forι≠j0,t∈(t0,t3]𝕋,ι=1,2,…,m.
Then, in a similar way, in view of (38), we have that
(58)|νj0(t3)|<δ3pM∥ϕ-ϕ*∥e⊖α(t3,t0)×(1M+1bj0-∑i=1n‍(bij0+Hih+qij0+Hik))<δ3pMe⊖α(t3,t0)∥ϕ-ϕ*∥,
which is also a contradiction.
Case4. Equation (48) is not true and (45), (46), and (47) are all true. Then, there exists t4∈(t0,+∞)𝕋 and j1∈{1,2,…,m} such that
(59)|νj1Δ(t4)|≥pMe⊖α(t4,t0)∥ϕ-ϕ*∥,|νj1Δ(t)|<pMe⊖α(t,t0)∥ϕ-ϕ*∥,t∈(t0,t4)𝕋,dddd|νιΔ(t)|<pMe⊖α(t,t0)∥ϕ-ϕ*∥,forι≠j1,t∈(t0,t4]𝕋,ι=1,2,…,m.
Hence, there must be a constant δ4≥1 such that
(60)|νj1Δ(t4)|≥δ4pMe⊖α(t4,t0)∥ϕ-ϕ*∥,|νj1Δ(t)|<δ4pMe⊖α(t,t0)∥ϕ-ϕ*∥,t∈(t0,t4)𝕋,ddd|νιΔ(t)|<δ4pMe⊖α(t,t0)∥ϕ-ϕ*∥,forι≠j1,t∈(t0,t4]𝕋,ι=1,2,…,m.
Similarly, in view of (38), we obtain that
(61)|νj1Δ(t4)|<δ4pMe⊖α(t4,t0)∥ϕ-ϕ*∥×{bj1+M+(1+bj1+bj1-)∑i=1n‍(bij0+Hih+qij0+Hik)}<δ4pMe⊖α(t4,t0)∥ϕ-ϕ*∥.
It is also a contradiction.
By the above four cases, for other cases of negative proposition of (44), we can obtain a contradiction. Therefore, (44) holds. Let p→1, then (43) holds. We can take -λ=⊖α, then λ>0 and -λ∈ℛ+. Hence, we have that
(62)|v(t)|1≤M∥ϕ-ϕ*∥e-λ(t,t0),t∈[-θ,∞)𝕋,t≥t0,
which means that the almost periodic solution ω(t) of (2) is exponentially stable. This completes the proof of Theorem 18.


## 5. An Example

In this section, we present an example to illustrate the feasibility of our results obtained in previous sections.Example 1.
Letn=m=2. Consider the following neutral-type BAM neural networks with delays on a time scale 𝕋(63)xiΔ(t)=-ai(t)xi(t)+∑j=12‍aji(t)fj(yj(t-τji(t)))+∑j=12‍pji(t)gj(yjΔ(t-σji(t)))+Ii(t),yjΔ(t)=-bj(t)yj(t)+∑i=12‍bij(t)hi(xi(t-ζij(t)))+∑i=12‍qij(t)ki(xiΔ(t-ςij(t)))+Jj(t),
where t∈𝕋,  i, j=1,2 and the coefficients are as follows:
(64)a1(t)=0.975+0.025sint,a2(t)=0.93+0.01cost,a11(t)=0.003sint,a12(t)=0.002cost,a21(t)=0.004sint,a22(t)=0.001sint,p11(t)=0.001+0.003cost,p12(t)=0.001sint,p21(t)=0.006+0.001sint,p22(t)=0.003cost,f1(u)=0.2sinu,f2(u)=2cosu,g1(u)=0.2cosu,g2(u)=2sinu,b1(t)=0.95+0.01sin3t,b2(t)=0.92+0.01cos2t,b11(t)=0.002sin2t,b12(t)=0.003cost,b21(t)=0.001sint,b22(t)=0.004cost,q11(t)=0.001cos3t,q12(t)=0.004sint,q21(t)=0.003sint,q22(t)=0.004cost,h1(u)=sinu,h2(u)=1.5cosu,k1(u)=cosu,k2(u)=1.5sinu,I1(t)=I2(t)=0.6sin3t,J1(t)=J2(t)=0.5cost,τji(t)=σji(t)=0.12sint,ζij(t)=ςij(t)=0.37sint,i,j=1,2.
By calculating, we have(65)a1+=1,a1-=0.95,a2+=0.94,a2-=0.92,b1+=0.96,b1-=0.94,b2+=0.93,b2-=0.91,a11+=0.003,a12+=0.002,a21+=0.004,a22+=0.001,p11+=0.004,p12+=0.001,p21+=0.007,p22+=0.003,b11+=0.002,b12+=0.003,b21+=0.001,b22+=0.004,q11+=0.001,q12+=0.004,q21+=0.003,q22+=0.004,H1f=H1g=0.2,H2f=H2g=2,H1h=H1k=1,H2h=H2k=1.5.
If 𝕋=ℝ, then μ(t)=0, and if 𝕋=ℤ, then μ(t)=1. We can verify for the above two cases, and all conditions of Theorems 17 and 18 are satisfied. Therefore, whether 𝕋=ℝ or 𝕋=ℤ, (63) has an almost periodic solution, which is exponentially stable. That is, the continuous-time neural network and its discrete-time analogue have the same dynamical behaviors.


## 6. Conclusion

In this paper, we establish some sufficient conditions ensuring the existence and exponential stability of almost periodic solutions for a class of neutral-type BAM neural networks with delays on time scales. Our results obtained in this paper are completely new even in case of the time scale𝕋=ℝ or ℤ and complementary to the previously known results. Besides, our method used in this paper may be used to study many other neutral-type neural networks with delays such as shunting inhibitory cellular neural networks, Cohen-Grossberg neural networks, and so on.



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*Source: 942309-2013-07-10.xml*