This paper is concerned with the shunting inhibitory cellular neural networks (SICNNs) with time-varying delays in the leakage (or forgetting) terms. Under proper conditions, we employ a novel argument to establish a criterion on the global exponential stability of pseudo almost periodic solutions by using Lyapunov functional method and differential inequality techniques. We also provide numerical simulations to support the theoretical result.
1. Introduction
In the last three decades, shunting inhibitory cellular neural networks (SICNNs) have been extensively applied in psychophysics, speech, perception, robotics, adaptive pattern recognition, vision, and image processing. Hence, they have been the object of intensive analysis by numerous authors in recent years. In particular, there have been extensive results on the problem of the existence and stability of the equilibrium point and periodic and almost periodic solutions of SICNNs with time-varying delays in the literature. We refer the reader to [1–7] and the references cited therein.
It is well known that SICNNs have been introduced as new cellular neural networks (CNNs) in Bouzerdoum et al. in [1, 8, 9], which can be described by
(1)xij′(t)=-aij(t)xij(t)-∑Ckl∈Nr(i,j)Cijkl(t)f(xkl(t-τkl(t)))xij(t)-∑Ckl∈Nq(i,j)Bijkl(t)·∫0∞Kij(u)g(xkl(t-u))duxij(t)+Lij(t),555555555555i=1,2,…,m,j=1,2,…,n,
where Cij denotes the cell at the (i,j) position of the lattice. The r-neighborhood Nr(i,j) of Cij is given as
(2)Nr(i,j)={Ckl:max(|k-i|,|l-j|)≤r,555551≤k≤m,1≤l≤n(|k-i|,|l-j|)},
where Nq(i,j) is similarly specified, xij is the activity of the cell Cij, Lij(t) is the external input to Cij, the function aij(t)>0 represents the passive decay rate of the cell activity, Cijkl(t) and Bijkl(t) are the connection or coupling strength of postsynaptic activity of the cell transmitted to the cell Cij, and the activity functions f(·) and g(·) are continuous functions representing the output or firing rate of the cell Ckl, and τkl(t)≥0 corresponds to the transmission delay.
Obviously, the first term in each of the right side of (1) corresponds to stabilizing negative feedback of the system which acts instantaneously without time delay; these terms are variously known as “forgettin” or leakage terms (see, for instance, Kosko [10], Haykin [11]). It is known from the literature on population dynamics and neural networks dynamics (see Gopalsamy [12]) that time delays in the stabilizing negative feedback terms will have a tendency to destabilize a system. Therefore, the authors of [13–19] dealt with the existence and stability of equilibrium and periodic solutions for neuron networks model involving leakage delays. Recently, Liu and Shao [20] considered the following SICNNs with time-varying leakage delays:
(3)xij′(t)=-aij(t)xij(t-ηij(t))-∑Ckl∈Nr(i,j)Cijkl(t)f(xkl(t-τkl(t)))xij(t)-∑Ckl∈Nq(i,j)Bijkl(t)×∫0∞Kij(u)g(xkl(t-u))duxij(t)+Lij(t),
where i=1,2,…,m, j=1,2,…,n, ηij,:ℝ→[0+∞) denotes the leakage delay. By using Lyapunov functional method and differential inequality techniques, in [20], some sufficient conditions have been established to guarantee that all solutions of (1) converge exponentially to the almost periodic solution. Moreover, it is well known that the global exponential convergence behavior of solutions plays a key role in characterizing the behavior of dynamical system since the exponential convergent rate can be unveiled (see [21–24]). However, to the best of our knowledge, few authors have considered the exponential convergence on the pseudo almost periodic solution for (1). Motivated by the above discussions, in this paper, we will establish the existence and uniqueness of pseudo almost periodic solution of (1) by using the exponential dichotomy theory and contraction mapping fixed point theorem. Meanwhile, we also will give the conditions to guarantee that all solutions and their derivatives of solutions for (1) converge exponentially to the pseudo almost periodic solution and its derivative, respectively.
For convenience, we denote by ℝp(ℝ=ℝ1) the set of all p-dimensional real vectors (real numbers). We will use
(4){xij(t)}=(x11(t),…,x1n(t),…,xi1(t),…,xin(t),…,xm1(t),…,xmn(t))∈ℝm×n.
For any x(t)={xij(t)}∈ℝm×n, we let |x| denote the absolute-value vector given by |x|={|xij|} and define ∥x(t)∥=max(i,j){|xij(t)|}. A matrix or vector A≥0 means that all entries of A are greater than or equal to zero. A>0 can be defined similarly. For matrices or vectors A1 and A2, A1≥A2 (resp. A1>A2) means that A1-A2≥0 (resp. A1-A2>0). For the convenience, we will introduce the notations:
(5)h+=supt∈ℝ|h(t)|,h-=inft∈ℝ|h(t)|,
where h(t) is a bounded continuous function.
The initial conditions associated with system (3) are of the form:
(6)xij(s)=φij(s),s∈(-∞,0],ij∈J:={11,…,1n,21,…,2n,…,m1,…,mn},
where φij(·) and φij′(·) are real-valued bounded continuous functions defined on (-∞,0].
The paper is organized as follows. Section 2 includes some lemmas and definitions, which can be used to check the existence of almost periodic solutions of (3). In Section 3, we present some new sufficient conditions for the existence of the continuously differentiable pseudo almost periodic solution of (3). In Section 4, we establish sufficient conditions on the global exponential stability of pseudo almost periodic solutions of (3). At last, an example and its numerical simulation are given to illustrate the effectiveness of the obtained results.
2. Preliminary Results
In this section, we will first recall some basic definitions and lemmas which are used in what follows.
In this paper, BC(ℝ,ℝp) denotes the set of bounded continued functions from ℝ to ℝp. Note that (BC(ℝ,ℝp),∥·∥∞) is a Banach space where ∥·∥∞ denotes the sup norm ∥f∥∞:=supt∈ℝ∥f(t)∥.
Definition 1 (see [25, 26]).
Let u(t)∈BC(ℝ,ℝp). u(t) is said to be almost periodic on ℝ if, for any ɛ>0, the set T(u,ɛ)={δ:∥u(t+δ)-u(t)∥<ɛforallt∈ℝ} is relatively dense; that is, for any ɛ>0, it is possible to find a real number l=l(ɛ)>0; for any interval with length l(ɛ), there exists a number δ=δ(ɛ) in this interval such that ∥u(t+δ)-u(t)∥<ɛ, forallt∈ℝ.
We denote by AP(ℝ,ℝn) the set of the almost periodic functions from ℝ to ℝn. Besides, the concept of pseudo almost periodicity (pap) was introduced by Zhang in the early nineties. It is a natural generalization of the classical almost periodicity. Precisely, define the class of functions PAP0(ℝ,ℝ) as follows:
(7){f∈BC(ℝ,ℝn)∣limT→+∞12T∫-TT|f(t)|dt=0}.
A function f∈BC(ℝ,ℝn) is called pseudo almost periodic if it can be expressed as
(8)f=h+φ,
where h∈AP(ℝ,ℝn) and φ∈PAP0(ℝ,ℝn). The collection of such functions will be denoted by PAP(ℝ,ℝn). The functions h and φ in the above definition are, respectively, called the almost periodic component and the ergodic perturbation of the pseudo almost periodic function f. The decomposition given in definition above is unique. Observe that (PAP(ℝ,ℝn),∥·∥∞) is a Banach space and AP(ℝ,ℝn) is a proper subspace of PAP(ℝ,ℝn) since the function ϕ(t)=cosπt+cost+e-t4sin2t is pseudo almost periodic function but not almost periodic. It should be mentioned that pseudo almost periodic functions possess many interesting properties; we shall need only a few of them and for the proofs we shall refer to [25].
Lemma 2 (see [25, page 57]).
If f∈PAP(ℝ,ℝ) and g is its almost periodic component, then we have
(9)g(ℝ)⊂f(ℝ)¯.
Therefore ∥f∥∞≥∥g∥∞≥infx∈ℝ|g(x)|≥infx∈ℝ|f(x)|.
Lemma 3 (see [25, page 140]).
Suppose that both functions f and its derivative f′ are in PAP(ℝ,ℝ). That is, f=g+φ and f′=α+β, where g,α∈AP(ℝ,ℝ) and φ,β∈PAP0(ℝ,ℝ). Then the functions g and φ are continuous differentiable so that
(10)g′=α,φ′=β.
Lemma 4.
Let B*={f∣f,f′∈PAP(ℝ,ℝ)} equipped with the induced norm defined by ∥f∥B*=max{∥f∥∞,∥f′∥∞}=max{supt∈ℝ|f(t)|,supt∈ℝ|f′(t)|}, and then B* is a Banach space.
Proof.
Suppose that {fp}p=1+∞ is a Cauchy sequence in B*, and then for any ɛ>0, there exists N(ɛ)>0, such that
(11)∥fp-fq∥B*=max{supt∈ℝ|fp(t)-fq(t)|,supt∈ℝ|fp′(t)-fq′(t)|}<ɛ,555555555555555555555555555555555555∀p,q≥N(ɛ).
By the definition of pseudo almost periodic function, let
(12)fp=gp+φp,wheregp∈AP(ℝ,ℝ),5555φp∈PAP0(ℝ,ℝ),p=1,2,….
From Lemma 3, we obtain
(13)fp′=gp′+φp′,wheregp′∈AP(ℝ,ℝ),555555φp′∈PAP0(ℝ,ℝ),p=1,2,….
On combining (11) with Lemma 2, we deduce that, {gp}p=1+∞,{gp′}p=1+∞⊂AP(ℝ,ℝ) are Cauchy sequence, so that {φp}p=1+∞,{φp′}p=1+∞⊂PAP0(ℝ,ℝ) are also Cauchy sequence.
Firstly, we show that there exists g∈AP(ℝ,ℝ) such that gp uniformly converges to g, as p→+∞.
Note that {gp} is Cauchy sequence in AP(ℝ,ℝ). forallɛ>0, ∃N(ɛ), such that forallp,q≥N(ɛ)(14)|gp(t)-gq(t)|<ɛ,∀t∈ℝ.
So for fixed t∈ℝ, it is easy to see {gp(t)}p=1+∞ is Cauchy number sequence. Thus, the limits of gp(t) exist as p→+∞ and let g(t)=limp→+∞gp(t). In (14), let q→+∞, and we have
(15)|g(t)-gp(t)|≤ɛ,∀t∈ℝ,p≥N(ɛ).
Thus, gn uniformly converges to g, as p→+∞. Moreover, from the Theorem 1.9 [26, page 5], we obtain g∈AP(ℝ,ℝ). Similarly, we also obtain that there exist g*∈AP(ℝ,ℝ) and φ,φ*∈BC(ℝ,ℝ), such that
(16)|g*(t)-gp′(t)|≤ɛ,|φ(t)-φp(t)|≤ɛ,|φ*(t)-φp′(t)|≤ɛ,∀t∈ℝ,p≥N(ɛ),
which lead to
(17)gp′⟹g*,φp⟹φ,φp′⟹φ*,
where p→+∞ and “⇒” means uniform convergence.
Next, we claim that φ,φ*∈PAP0(ℝ). Together with (16) and the facts that
(18)limr→+∞12r∫-rr|φp(s)|ds=0,limr→+∞12r∫-rr|φp′(s)|ds=0,p=1,2,…,12r∫-rr|φ(s)|ds≤12r∫-rr|φ(s)-φp(s)|ds+12r∫-rr|φp(s)|ds,r>0,n=1,2,…,12r∫-rr|φ*(s)|ds≤12r∫-rr|φ*(s)-φp′(s)|ds+12r∫-rr|φp′(s)|ds,r>0,p=1,2,…,
we have
(19)limr→+∞12r∫-rr|φ(s)|ds=0,limr→+∞12r∫-rr|φ*(s)|ds=0.
Hence φ,φ*∈PAP0(ℝ). Let f=g+φ, f*=g*+φ*, then f=g+φ∈PAP(ℝ), f*=g*+φ*∈PAP(ℝ) and fp⇒f, fp′⇒f* as p→+∞.
Finally, we reveal f′=f*. For t,Δt∈ℝ, it follows that
(20)fp(t+Δt)-fp(t)=∫tt+Δtfp′(s)ds.
In view of the uniform convergence of fp and fp′, let p→+∞ for (20), and we get
(21)f(t+Δt)-f(t)=∫tt+Δtf*(s)ds,
which implies that
(22)f*(t)=limΔt→0∫tt+Δtf*(s)dsΔt=limΔt→0f(t+Δt)-f(t)Δt=f′(t).
In summary, in view of (15), (16), and (22), we obtain that the Cauchy sequence {fp}p=1+∞⊂B* satisfies
(23)∥fp-f∥B*⟶0(p⟶+∞),
and f∈B*. This yields that B* is a Banach space. The proof is completed.
Remark 5.
Let B={f∣f,f′∈PAP(ℝ,ℝn×m)} equipped with the induced norm defined by ∥f∥B=max{∥f∥∞,∥f′∥∞}=max{supt∈ℝ∥f(t)∥,supt∈ℝ∥f′(t)∥}. It follows from Lemma 4 that B is a Banach space.
Definition 6 (see [19, 20]).
Let x∈ℝp and Q(t) be a p×p continuous matrix defined on ℝ. The linear system
(24)x′(t)=Q(t)x(t)
is said to admit an exponential dichotomy on ℝ if there exist positive constants k, α, and projection P and the fundamental solution matrix X(t) of (24) satisfying
(25)∥X(t)PX-1(s)∥≤ke-α(t-s),fort≥s,∥X(t)(I-P)X-1(s)∥≤ke-α(s-t),fort≤s.
Lemma 7 (see [19]).
Assume that Q(t) is an almost periodic matrix function and g(t)∈PAP(ℝ,ℝp). If the linear system (24) admits an exponential dichotomy, then pseudo almost periodic system
(26)x′(t)=Q(t)x(t)+g(t)
has a unique pseudo almost periodic solution x(t), and
(27)x(t)=∫-∞tX(t)PX-1(s)g(s)ds-∫t+∞X(t)(I-P)X-1(s)g(s)ds.
Lemma 8 (see [19, 20]).
Let ci(t) be an almost periodic function on ℝ and
(28)M[ci]=limT→+∞1T∫tt+Tci(s)ds>0,i=1,2,…,p.
Then the linear system
(29)x′(t)=diag(-c1(t),-c2(t),…,-cp(t))x(t)
admits an exponential dichotomy on ℝ.
3. Existence of Pseudo Almost Periodic Solutions
In this section, we establish sufficient conditions on the existence of pseudo almost periodic solutions of (3).
For ij,kl∈J, aij:ℝ→(0,+∞) is an almost periodic function, ηij,τkl:ℝ→[0,+∞), and Lij,Cijkl,Bijkl:ℝ→ℝ are pseudo almost periodic functions. We also make the following assumptions which will be used later.
We also make the following assumptions.
There exist constants Mf, Mg, Lf, and Lg such that
(30)|f(u)-f(v)|≤Lf|u-v|,|f(u)|≤Mf,|g(u)-g(v)|≤Lg|u-v|,|g(u)|≤Mg,5555555555555555555555555∀u,v∈ℝ.
For ij∈J, the delay kernels Kij:[0,∞)→ℝ are continuous, and |Kij(t)|eβt are integrable on [0,∞) for a certain positive constant β.
Let
(31)L=max{max(i,j){supt∈ℝ|∫-∞te-∫staij(u)duLij(s)ds|},555555555max(i,j){supt∈ℝ|∫-∞tLij(t)-aij(t)555555555555555555×∫-∞te-∫staij(u)duLij(s)ds|supt∈ℝ|∫-∞tLij(t)-aij(t)}{supt∈ℝ|∫-∞te-∫staij(u)duLij(s)ds|}}>0.
Moreover, there exists a constant κ such that
(32)0<κ≤L,max(i,j){1aij-Eij,(1+aij+aij-)Eij}≤κ,max(i,j){1aij-Fij,(1+aij+aij-)Fij}<1,
where
(33)Eij=[aij+ηij++∑Ckl∈Nr(i,j)Cijkl+(Lf(κ+L)+|f(0)|)5555+∑Ckl∈Nq(i,j)Bijkl+5555×∫0∞|Kij(u)|du(Lg(κ+L)+|g(0)|)∑Ckl∈Nr(i,j)](κ+L),555555555555555555555555555555555555555ij∈J,Fij=[aij+ηij++∑Ckl∈Nr(i,j)Cijkl+(Mf+Lf(κ+L))5555+∑Ckl∈Nq(i,j)Bijkl+(∫0∞|Kij(u)|duMg555555555555555555+∫0∞|Kij(u)|duLg(κ+L))∑Ckl∈Nr(i,j)],55555555555555555555555555555555555555ij∈J.
Lemma 9.
Assume that assumptions (S1) and (S2) hold. Then, for φ(·)∈PAP(ℝ,ℝ), the function ∫0∞Kij(u)g(φ(t-u))du belongs to PAP(ℝ,ℝ), where ij∈J.
Proof.
Let φ∈PAP(ℝ,ℝ). Obviously, (S1) implies that g is a uniformly continuous function on ℝ. By using Corollary 5.4 in [25, page 58], we immediately obtain the following:
(34)g(φ(t))=χ1(t)+χ2(t)∈PAP(ℝ,ℝ),
whereχ1∈AP(ℝ,ℝ) and χ2∈PAP0(ℝ,ℝ). Then, for any ɛ>0, it is possible to find a real number l=l(ɛ)>0; for any interval with length l, there exists a number τ=τ(ɛ) in this interval such that
(35)|χ1(t+τ)-χ1(t)|<ɛ1+∫0∞|Kij(u)|du,∀t∈ℝ,ij∈J,limr→+∞12r∫-rr|χ2(v)|dv=0.
It follows that
(36)|∫0∞Kij(u)χ1(t+τ-u)du-∫0∞Kij(u)χ1(t-u)du|≤∫0∞|Kij(u)||χ1(t+τ-u)-χ1(t-u)|du<∫0∞|Kij(u)|duɛ1+∫0∞|Kij(u)|du<ɛ,∀t∈ℝ,ij∈J,limr→+∞12r∫-rr|∫0∞Kij(u)χ2(v-u)du|dv≤limr→+∞12r∫-rr∫0∞|Kij(u)||χ2(v-u)|dudv=limr→+∞12r∫0∞|Kij(u)|∫-rr|χ2(v-u)|dvdu=limr→+∞12r∫0∞|Kij(u)|∫-r-ur-u|χ2(z)|dzdu≤limr→+∞12r∫0∞|Kij(u)|∫-r-ur+u|χ2(z)|dzdu≤limr→+∞∫0∞|Kij(u)|(1+1ru)12(r+u)×∫-r-ur+u|χ2(z)|dzdu≤limr→+∞∫0∞|Kij(u)|e(1/r)u12(r+u)∫-r-ur+u|χ2(z)|dzdu≤limr→+∞∫0∞|Kij(u)|eβu12(r+u)∫-r-ur+u|χ2(z)|dzdu=0,wherer>1β,ij∈J.
Thus,
(37)∫0∞Kij(u)χ1(t-u)du∈AP(ℝ,ℝ),∫0∞Kij(u)χ2(t-u)du∈PAP0(ℝ,ℝ),
which yield
(38)∫0∞Kij(u)gj(φ(t-u))du=∫0∞Kij(u)χ1(t-u)du+∫0∞Kij(u)χ2(t-u)du∈PAP(ℝ,ℝ),ij∈J.
The proof of Lemma 9 is completed.
Theorem 10.
Let (S1), (S2), and (S3) hold. Then, there exists at least one continuously differentiable pseudo almost periodic solution of system (3).
Proof.
Let φ∈B. Obviously, the boundedness of φ′ and (S1) imply that f and φij are uniformly continuous functions on ℝ for ij∈J. Set f~(t,z)=φij(t-z)(ij∈J). By Theorem 5.3 in [25, page 58] and Definition 5.7 in [25, page 59], we can obtain that f~∈PAP(ℝ×Ω) and f~ is continuous in z∈K and uniformly in t∈ℝ for all compact subset K of Ω. This, together with τij,ηij∈PAP(ℝ,ℝ) and Theorem 5.11 in [25, page 60], implies that
(39)φij(t-τij(t))∈PAP(ℝ,ℝ),φij(t-ηij(t))∈PAP(ℝ,ℝ),555555555555555555ij∈J.
Again from Corollary 5.4 in [25, page 58], we have
(40)f(φij(t-τij(t)))∈PAP(ℝ,ℝ),ij∈J,
which, together with Lemma 9, implies
(41)aij(t)∫t-ηij(t)tφij′(s)ds=aij(t)φij(t)-aij(t)φij(t-ηij(t))∈PAP(ℝ,ℝ),5555555555555555555555555555555555555555ij∈J,-∑Ckl∈Nr(i,j)Cijkl(t)f(φkl(t-τkl(t)))φij(t)-∑Ckl∈Nq(i,j)Bijkl(t)×∫0∞Kij(u)g(φkl(t-u))duφij(t)+Lij(t)∈PAP(ℝ,ℝ),555555555555555555555555555555555555555555ij∈J.
For any φ∈B, we consider the pseudo almost periodic solution xφ(t) of nonlinear pseudo almost periodic differential equations
(42)xij′(t)=-aij(t)xij(t)+aij(t)∫t-ηij(t)tφij′(s)ds-∑Ckl∈Nr(i,j)Cijkl(t)f(φkl(t-τkl(t)))φij(t)-∑Ckl∈Nq(i,j)Bijkl(t)×∫0∞Kij(u)g(φkl(t-u))duφij(t)+Lij(t),55555555555555555555555555555555ij∈J.
Then, notice that M[aij]>0, ij∈J, and it follows from Lemma 8 that the linear system,
(43)xij′(t)=-aij(t)xij(t),ij∈J,
admits an exponential dichotomy on ℝ. Thus, by Lemma 7, we obtain that the system (42) has exactly one pseudo almost periodic solution:
(44)xφ(t)={xijφ(t)}={∫-∞te-∫staij(u)du×[aij(s)∫s-ηij(s)sφij′(u)du-∑Ckl∈Nr(i,j)Cijkl(s)×f(φkl(s-τkl(s)))φij(s)-∑Ckl∈Nq(i,j)Bijkl(s)×∫0∞Kij(u)g(φkl(s-u))duφij(s)+Lij(s)∫s-ηij(s)s]ds∫-∞t}.
From (S1), (S2), and the Corollary 5.6 in [25, page 59], we get
(45)(xφ(t))′={xijφ′(t)}={[aij(t)∫t-ηij(t)tφij′(s)ds55555-∑Ckl∈Nr(i,j)Cijkl(t)f(φkl(t-τkl(t)))φij(t)55555-∑Ckl∈Nq(i,j)Bijkl(t)5555×∫0∞Kij(u)g(φkl(t-u))duφij(t)+Lij(t)∫t-ηij(t)t]5555-aij(t)5555×∫-∞te-∫staij(u)du55×[aij(s)∫s-ηij(s)sφij′(u)du55-∑Ckl∈Nr(i,j)Cijkl(s)5×f(φkl(s-τkl(s)))φij(s)5-∑Ckl∈Nq(i,j)Bijkl(s)5×∫0∞Kij(u)g(φkl(s-u))duφij(s)555+Lij(s)∫s-ηij(s)s]ds}
which is a pseudo almost periodic function. Therefore, xφ∈B. Let φ0(t)=x0(t). Then,
(46)φ0(t)={φij0(t)}={∫-∞te-∫staij(u)duLij(s)ds}∈B,L=∥φ0∥B.
Set
(47)B**={φ∣φ∈B,∥φ-φ0∥B≤κ}.
If φ∈B**, then
(48)∥φ∥B≤∥φ-φ0∥B+∥φ0∥B≤κ+L.
Now, we define a mapping T:B**→B** by setting
(49)T(φ)(t)=xφ(t),∀φ∈B**.
We next prove that the mapping T is a contraction mapping of the B**.
First we show that, for any φ∈B**, T(φ)=xφ∈B**.
Note that
(50)|T(φ)(t)-φ0(t)|={|∫-∞te-∫staij(u)du×[aij(s)∫s-ηij(s)sφij′(u)du5-∑Ckl∈Nr(i,j)Cijkl(s)f(φkl(s-τkl(s)))φij(s)5-∑Ckl∈Nq(i,j)Bijkl(s)5×∫0∞Kij(u)g(φkl(s-u))duφij(s)∫s-ηij(s)s]ds|}≤{∫-∞te-∫staij-du×[aij+ηij+∥φ∥B5+∑Ckl∈Nr(i,j)Cijkl+(|f(φkl(s-τkl(s)))-f(0)(φkl(s-τkl(s)))|+|f(0)|(s-τkl(s)))∥φ∥B5+∑Ckl∈Nq(i,j)Bijkl+5×∫0∞|Kij(u)|55×(|g(φkl(s-u))-g(0)|+|g(0)|)du55×∥φ∥Baij+ηij+∥φ∥B]ds∫-∞t}≤{1aij-[aij+ηij++∑Ckl∈Nr(i,j)Cijkl+(Lf∥φ∥B+|f(0)|)+∑Ckl∈Nq(i,j)Bijkl+∫0∞|Kij(u)|du(Lg∥φ∥B+|g(0)|)]×∥φ∥B[∑Ckl∈Nr(i,j)]}≤{1aij-[aij+ηij++∑Ckl∈Nr(i,j)Cijkl+(Lf(κ+L)+|f(0)|)+∑Ckl∈Nq(i,j)Bijkl+×∫0∞|Kij(u)|du(Lg(κ+L)+|g(0)|)∑Ckl∈Nr(i,j)](κ+L)[∑Ckl∈Nr(i,j)]},|(T(φ)(t)-φ0(t))′|={|[aij(t)∫t-ηij(t)tφij′(s)ds5-∑Ckl∈Nr(i,j)Cijkl(t)f(φkl(t-τkl(t)))φij(t)5-∑Ckl∈Nq(i,j)Bijkl(t)5×∫0∞Kij(u)g(φkl(t-u))duφij(t)[∫t-ηij(t)t]]-aij(t)∫-∞te-∫staij(u)du×[aij(s)∫s-ηij(s)sφij′(u)du-∑Ckl∈Nr(i,j)Cijkl(s)f(φkl(s-τkl(s)))φij(s)-∑Ckl∈Nq(i,j)Bijkl(s)×∫0∞Kij(u)g(φkl(s-u))duφij(s)]ds[∫t-ηij(t)t]|}≤{(1+aij+aij-)×[aij+ηij++∑Ckl∈Nr(i,j)Cijkl+(Lf∥φ∥B+|f(0)|)+∑Ckl∈Nq(i,j)Bijkl+×∫0∞|Kij(u)|du(Lg∥φ∥B+|g(0)|)∑Ckl∈Nr(i,j)]∥φ∥B}≤{(1+aij+aij-)×[aij+ηij++∑Ckl∈Nr(i,j)Cijkl+(Lf(κ+L)+|f(0)|)+∑Ckl∈Nq(i,j)Bijkl+×∫0∞|Kij(u)|du(Lg(κ+L)+|g(0)|)∑Ckl∈Nr(i,j)]×(κ+L)(1+aij+aij-)}.
It follows that
(51)∥T(φ)-φ0∥B≤max(i,j){1aij-Eij,(1+aij+aij-)Eij}≤κ;
that is, T(φ)=xφ∈B**.
Second, we show that T is a contract operator.
In fact, in view of (44), (48), (S1), (S2), and (S3), for φ,ψ∈B**, we have
(52)|T(φ(t))-T(ψ(t))|={|(T(φ(t))-T(ψ(t)))ij|}={|(ψkl(s-u))duψij(s)∫0∞)∫s-ηij(s)s]∫-∞te-∫staij(u)du×[aij(s)∫s-ηij(s)s(φij′(u)-ψij′(u))du-∑Ckl∈Nr(i,j)Cijkl(s)×(f(φkl(s-τkl(s)))φij(s)-f(ψkl(s-τkl(s)))ψij(s))-∑Ckl∈Nq(i,j)Bijkl(s)×(∫0∞Kij(u)g(φkl(s-u))duφij(s)-∫0∞Kij(u)g×(ψkl(s-u))duψij(s)∫0∞)∫s-ηij(s)s]ds|∫-∞t}≤{∫-∞te-∫staij(u)du×[(∫0∞)aij+ηij+∥φ-ψ∥B+∑Ckl∈Nr(i,j)Cijkl+(-f(ψkl(s-τkl(s)))||f(φkl(s-τkl(s)))|555555555555555555555555555×|φij(s)-ψij(s)|555555555555555555555555555+|f(φkl(s-τkl(s)))555555555555555555555555555555-f(ψkl(s-τkl(s)))||ψij(s)|)+∑Ckl∈Nq(i,j)Bijkl+(∫0∞|Kij(u)|×|g(φkl(s-u))|du|φij(s)-ψij(s)|+∫0∞|Kij(u)|×|g(φkl(s-u))-g(ψkl(s-u))|du×|ψij(s)|∫0∞)]ds}≤{∫0∞|Kij(u)|duLg∥φ-ψ∥B∥ψ∥B)]1aij-[∫0∞|Kij(u)|duLg∥φ-ψ∥B∥ψ∥B)aij+ηij+∥φ-ψ∥B+∑Ckl∈Nr(i,j)Cijkl+(Mf∥φ-ψ∥B+Lf∥φ-ψ∥B∥ψ∥B)+∑Ckl∈Nq(i,j)Bijkl+×(∫0∞|Kij(u)|duMg∥φ-ψ∥B+∫0∞|Kij(u)|duLg∥φ-ψ∥B∥ψ∥B)]}={1aij-[(∫0∞|Kij(u)|duLg∥ψ∥B)aij+ηij++∑Ckl∈Nr(i,j)Cijkl+(Mf+Lf∥ψ∥B)+∑Ckl∈Nq(i,j)Bijkl+×(∫0∞|Kij(u)|duMg+∫0∞|Kij(u)|duLg∥ψ∥B)]∥φ-ψ∥B}≤{1aij-[∫0∞|Kij(u)|duLg(κ+L))aij+ηij++∑Ckl∈Nr(i,j)Cijkl+(Mf+Lf(κ+L))+∑Ckl∈Nq(i,j)Bijkl+×(∫0∞|Kij(u)|duMg+∫0∞|Kij(u)|duLg(κ+L))]×∥φ-ψ∥B[∫0∞|Kij(u)|duLg(κ+L))aij+ηij+},|(T(φ(t))-T(ψ(t)))′|={|(T′(φ(t))-T′(ψ(t)))ij|}={|[aij(t)∫t-ηij(t)t(φij′(s)-ψij′(s))ds-∑Ckl∈Nr(i,j)Cijkl(t)×(f(φkl(t-τkl(t)))φij(t)-f(ψkl(t-τkl(t)))ψij(t))-∑Ckl∈Nq(i,j)Bijkl(t)×(∫0∞Kij(u)g(φkl×(t-u))duφij(t)-∫0∞Kij(u)g(ψkl(t-u))du×ψij(t)∫0∞)∫t-ηij(t)t]-aij(t)∫-∞te-∫staij(u)du×[aij(s)∫s-ηij(s)s(φij′(u)-ψij′(u))du-∑Ckl∈Nr(i,j)Cijkl(s)×(f(φkl(s-τkl(s)))φij(s)-f(ψkl(s-τkl(s)))×ψij(s))-∑Ckl∈Nq(i,j)Bijkl(s)×(∫0∞Kij(u)g(φkl(s-u))duφij(s)-∫0∞Kij(u)g×(ψkl(s-u))duψij(s)∫0∞Kij(u)g(φkl(s-u))duφij(s))∫s-ηij(s)s]ds|}≤{(1+aij+aij-)[∫0∞aij+ηij+5+∑Ckl∈Nr(i,j)Cijkl+(Mf+Lf(κ+L))5+∑Ckl∈Nq(i,j)Bijkl+(∫0∞|Kij(u)|duMg5+∫0∞|Kij(u)|du5×Lg(κ+L)∫0∞)]5×∥φ-ψ∥B(1+aij+aij-)},
which yields
(53)∥T(φ)-T(ψ)∥B≤max(i,j){1aij-Fij,(1+aij+aij-)Fij}∥φ-ψ∥B,
which implies that the mapping T:B**→B** is a contraction mapping. Therefore, using Theorem 0.3.1 of [27], we obtain that the mapping T possesses a unique fixed point
(54)x*={xij*(t)}∈B**,Tx*=x*.
By (42) and (44), x* satisfies (42). So (3) has at least one continuously differentiable pseudo almost periodic solution x*. The proof of Theorem 10 is now completed.
4. Exponential Stability of the Pseudo Almost Periodic Solution
In this section, we will discuss the exponential stability of the pseudo almost periodic solution of system (3).
Definition 11.
Let x*(t)={xij*(t)} be the pseudo almost periodic solution of system (3). If there exist constants α>0 and M>1 such that, for every solution x(t)={xij(t)} of system (3) with any initial value φ(t)={φij(t)} satisfying (6),(55)∥x(t)-x*(t)∥1=max(i,j){max{|xij(t)-xij*(t)|,|xij′(t)-xij*′(t)|}}≤M∥φ-x*∥0e-αt,∀t>0,
where ∥φ-x*∥0=max{supt≤0max(i,j)|φij(t)-xij*(t)|,supt≤0max(i,j)|φij′(t)-xij*′(t)|}. Then x*(t) is said to be globally exponentially stable.
Theorem 12.
Suppose that all conditions in Theorem 10 are satisfied. Then system (3) has at least one pseudo almost periodic solution x*(t). Moreover, x*(t) is globally exponentially stable.
Proof.
By Theorem 10, (3) has at least one continuously differentiable pseudo almost periodic solution x*(t)={xij*(t)} such that
(56)∥x*∥B≤κ+L.
Suppose that x(t)={xij(t)} is an arbitrary solution of (1) associated with initial value φ(t)={φij(t)} satisfying (6). Let y(t)={yij(t)}={xij(t)-xij*(t)}. Then
(57)yij′(t)=-aij(t)yij(t-ηij(t))-∑Ckl∈Nr(i,j)Cijkl(t)×[f(xkl(t-τkl(t)))xij(t)-f(xkl*(t-τkl(t)))xij*(t)]-∑Ckl∈Nq(i,j)Bijkl(t)×[∫0∞Kij(u)g(xkl(t-u))duxij(t)-∫0∞Kij(u)g(xkl*(t-u))duxij*(t)]=-aij(t)yij(t)+aij(t)∫t-ηij(t)tyij′(u)du-∑Ckl∈Nr(i,j)Cijkl(t)[f(xkl(t-τkl(t)))xij(t)555-f(xkl*(t-τkl(t)))555×xij*(t)]-∑Ckl∈Nq(i,j)Bijkl(t)×[∫0∞Kij(u)g(xkl(t-u))duxij(t)-∫0∞Kij(u)g(xkl*(t-u))duxij*(t)].
Define continuous functions Γi(ω) and Πi(ω) by setting
(58)Γij(ω)=-aij-+ω+aij+ηij+eωηij++∑Ckl∈Nr(i,j)Cijkl+(Mf+Lfeωτkl+(κ+L))+∑Ckl∈Nq(i,j)Bijkl+×(∫0∞|Kij(u)|duMg+∫0∞|Kij(u)|Lgeωudu(κ+L)),Πij(ω)=(1+aij+aij--ω)×[(∫0∞)aij+ηij+eωηij++∑Ckl∈Nr(i,j)Cijkl+(Mf+Lfeωτkl+(κ+L))+∑Ckl∈Nq(i,j)Bijkl+(∫0∞|Kij(u)|duMg55+∫0∞|Kij(u)|Lgeωudu55×(κ+L)∫0∞)],
where t>0, ω∈[0,β], ij∈J. Then, from (S3), we have
(59)Γij(0)=-aij-+aij+ηij++∑Ckl∈Nr(i,j)Cijkl+(Mf+Lf(κ+L))+∑Ckl∈Nq(i,j)Bijkl+×(∫0∞|Kij(u)|duMg+∫0∞|Kij(u)|Lgdu(κ+L))=-aij-(1-1aij-Fij)<0,ij∈J,Πij(0)=(1+aij+aij-)×[aij+ηij++∑Ckl∈Nr(i,j)Cijkl+(Mf+Lf(κ+L))+∑Ckl∈Nq(i,j)Bijkl+×(∫0∞|Kij(u)|duMg+∫0∞|Kij(u)|Lgdu(κ+L))[∑Ckl∈Nr(i,j)]]=(1+aij+aij-)Fij<1,ij∈J,
which, together with the continuity of Γij(ω) and Πij(ω), implies that we can choose a constant λ∈(0,min{β,min(i,j)aij-}) such that
(60)Γij(λ)=-aij-+λ+aij+ηij+eληij++∑Ckl∈Nr(i,j)Cijkl+(Mf+Lfeλτkl+(κ+L))+∑Ckl∈Nq(i,j)Bijkl+×(∫0∞|Kij(u)|duMg+∫0∞|Kij(u)|Lgeλudu(κ+L))=(aij--λ)(βijaij--λ-1)<0,(61)Πij(λ)=(1+aij+aij--λ)×[aij+ηij+eληij++∑Ckl∈Nr(i,j)Cijkl+(Mf+Lfeλτkl+(κ+L))+∑Ckl∈Nq(i,j)Bijkl+(∫0∞|Kij(u)|duMg+∫0∞|Kij(u)|Lgeλudu×(κ+L)∫0∞)∑Ckl∈Nr(i,j)]=(1+aij+aij--λ)βij<1,
where
(62)βij=aij+ηij+eληij++∑Ckl∈Nr(i,j)Cijkl+(Mf+Lfeλτkl+(κ+L))+∑Ckl∈Nq(i,j)Bijkl+×(∫0∞|Kij(u)|duMg+∫0∞|Kij(u)|Lgeλudu(κ+L)),ij∈J.
Let M be a constant such that
(63)M>aij--λβij>1,∀ij∈J,
which, together with (60), yields
(64)1M-βijaij--λ<0,βijaij--λ<1,∀ij∈J.
Consequently, for any ɛ>0, it is obvious that
(65)∥y(t)∥1<(∥φ-x*∥0+ε)e-λt<M(∥φ-x*∥0+ε)e-λt555t∈(-∞,0].
In the following, we will show that
(66)∥y(t)∥1<M(∥φ-x*∥0+ε)e-λt,∀t>0.
Otherwise, there must exist ij∈J and θ>0 such that
(67)∥y(θ)∥1=max{|yij(θ)|,|yij′(θ)|}=M(∥φ-x*∥0+ɛ)e-λθ,∥y(t)∥1<M(∥φ-x*∥0+ɛ)e-λt,∀t∈(-∞,θ).
Note that
(68)yij′(s)+aij(s)yij(s)=aij(s)∫s-ηij(s)syij′(u)du-∑Ckl∈Nr(i,j)Cijkl(s)×[f(xkl(s-τkl(s)))xij(s)-f(xkl*(s-τkl(s)))xij*(s)]-∑Ckl∈Nq(i,j)Bijkl(s)×[∫0∞Kij(u)g(xkl(s-u))duxij(s)-∫0∞Kij(u)g(xkl*(s-u))duxij*(s)],55555555555555555555s∈[0,t],t∈[0,θ].
Multiplying both sides of (68) by e∫0saij(u)du and integrating on [0,t], we get
(69)yij(t)=yij(0)e-∫0taij(u)du+∫0te-∫staij(u)du×[aij(s)∫s-ηij(s)syij′(u)du-∑Ckl∈Nr(i,j)Cijkl(s)×(f(xkl(s-τkl(s)))xij(s)-f(xkl*(s-τkl(s)))xij*(s))-∑Ckl∈Nq(i,j)Bijkl(s)×(∫0∞Kij(u)g(xkl(s-u))duxij(s)-∫0∞Kij(u)g(xkl*(s-u))duxij*(s))]ds,55555555555555555555555555555555t∈[0,θ].
Thus, with the help of (67), we have
(70)|yij(θ)|=|[(∫0∞)]yij(0)e-∫0θaij(u)du+∫0θe-∫sθaij(u)du×[aij(s)∫s-ηij(s)syij′(u)du-∑Ckl∈Nr(i,j)Cijkl(s)(f(xkl(s-τkl(s)))xij(s)-f(xkl*(s-τkl(s)))xij*(s))-∑Ckl∈Nq(i,j)Bijkl(s)×(∫0∞Kij(u)g(xkl(s-u))duxij(s)-∫0∞Kij(u)g(xkl*(s-u))duxij*(s))∫s-ηij(s)s]ds|≤(∥φ-x*∥0+ɛ)e-aij-θ+∫0θe-∫sθaij(u)du×[(∫0∞)aij+ηij+M(∥φ-x*∥0+ɛ)e-λ(s-ηij(s))+∑Ckl∈Nr(i,j)Cijkl+(|f(xkl(s-τkl(s)))||xij(s)-xij*(s)|+|f(xkl(s-τkl(s)))-f(xkl*(s-τkl(s)))|×|xij*(s)||f(xkl(s-τkl(s)))|)+∑Ckl∈Nq(i,j)Bijkl+×(∫0∞|Kij(u)||g(xkl(s-u))|du|xij(s)-xij*(s)|+∫0∞|Kij(u)||g(xkl(s-u))-g(xkl*(s-u))|du×|xij*(s)|∫0∞)]ds≤(∥φ-x*∥0+ɛ)e-aij-θ+∫0θe-∫sθaij(u)du×[∫0∞aij+ηij+M(∥φ-x*∥0+ɛ)e-λ(s-ηij(s))+∑Ckl∈Nr(i,j)Cijkl+(Mf|yij(s)|+Lf|ykl(s-τkl(s))||xij*(s)|)+∑Ckl∈Nq(i,j)Bijkl+×(∫0∞|Kij(u)|duMg|yij(s)|+∫0∞|Kij(u)|Lg|ykl(s-u)|du×|xij*(s)|∫0∞)]ds≤(∥φ-x*∥0+ɛ)e-aij-θ+∫0θe-∫sθaij(u)du×[∫0∞aij+ηij+M(∥φ-x*∥0+ɛ)e-λ(s-ηij(s))+∑Ckl∈Nr(i,j)Cijkl+×(MfM(∥φ-x*∥0+ɛ)e-λs+LfM(∥φ-x*∥0+ɛ)e-λ(s-τkl(s))|xij*(s)|)+∑Ckl∈Nq(i,j)Bijkl+×(∫0∞|Kij(u)|duMgM(∥φ-x*∥0+ɛ)e-λs+∫0∞|Kij(u)|LgM(∥φ-x*∥0+ɛ)×e-λ(s-u)du|xij*(s)|∫0∞)]ds≤M(∥φ-x*∥0+ɛ)×{∫0∞|Kij(u)|Lgeλudu(κ+L))]e-aij-θM+∫0θe-∫sθaij(u)due-λs×[(∫0∞)aij+ηij+eληij++∑Ckl∈Nr(i,j)Cijkl+(Mf+Lfeλτkl+(κ+L))+∑Ckl∈Nq(i,j)Bijkl+×(∫0∞|Kij(u)|duMg+∫0∞|Kij(u)|Lgeλudu(κ+L))]ds}≤M(∥φ-x*∥0+ɛ)×{e-aij-θM+e-aij-θ∫0θe(aij--λ)sdsβij}≤M(∥φ-x*∥0+ɛ)e-λθ×[e(λ-aij-)θM+βijaij--λ(1-e(λ-aij-)θ)]=M(∥φ-x*∥0+ɛ)e-λθ×[(1M-βijaij--λ)e(λ-aij-)θ+βijaij--λ],
which, together with (64) and (67), implies that
(71)|yij(θ)|<M(∥φ-x*∥0+ɛ)e-λθ,(72)∥y(θ)∥1=max{|yij(θ)|,|yij′(θ)|}=|yij′(θ)|=M(∥φ-x*∥ξ+ɛ)e-λθ.
From (60), (61) and (67)–(72) yield
(73)|yij′(θ)|≤aij(θ)|yij(θ)|+|aij(θ)∫θ-ηij(θ)θyij′(u)du-∑Ckl∈Nr(i,j)Cijkl(θ)×[f(xkl(θ-τkl(θ)))xij(θ)-f(xkl*(θ-τkl(θ)))xij*(θ)]-∑Ckl∈Nq(i,j)Bijkl(θ)×[∫0∞Kij(u)g(xkl(θ-u))duxij(θ)-∫0∞Kij(u)g(xkl*(θ-u))duxij*(θ)]|≤aij+|yij(θ)|+[aij+ηij+M(∥φ-x*∥ξ+ɛ)e-λ(θ-ηij(θ))+∑Ckl∈Nr(i,j)Cijkl+×(|f(xkl(θ-τkl(θ)))||xij(θ)-xij*(θ)|+|f(xkl(θ-τkl(θ)))-f(xkl*(θ-τkl(θ)))|×|xij*(θ)||f(xkl(θ-τkl(θ)))|)+∑Ckl∈Nq(i,j)Bijkl+(∫0∞|Kij(u)||g(xkl(θ-u))|du×|xij(θ)-xij*(θ)|+∫0∞|Kij(u)|×|g(xkl(θ-u))-g(xkl*(θ-u))|du×|xij*(θ)|∫0∞)≤{aij+[(1M-βijaij--λ)e(λ-aij-)θ+βijaij--λ]+aij+ηij+eληij++∑Ckl∈Nr(i,j)Cijkl+×(Mf+Lfeλτkl+(κ+L))+∑Ckl∈Nq(i,j)Bijkl+×(∫0∞|Kij(u)|duMg+∫0∞|Kij(u)|Lgeλudu(κ+L))[(1M-βijaij--λ)e(λ-aij-)θ+βijaij--λ]}M×(∥φ-x*∥0+ɛ)e-λθ≤M(∥φ-x*∥0+ɛ)e-λθ×[aij+(1M-βijaij--λ)e(λ-aij-)θ+βij(aij+aij--λ+1)]<M(∥φ-x*∥0+ɛ)e-λθ,
which contradicts (72). Hence, (66) holds. Letting ɛ→0+, we have from (66) that
(74)∥y(t)∥1≤M∥φ-x*∥0e-λt,∀t>0,
which implies
(75)∥x(t)-x*(t)∥1≤M∥φ-x*∥0e-λt,∀t>0.
This completes the proof.
5. An Example
In this section, we give an example with numerical simulation to demonstrate the results obtained in previous sections.
Example 13.
Consider the following SICNNs with time-varying delays in the leakage terms:(76)dxijdt=-aij(t)xij(t-ηij(t))-∑ckl∈Nr(i,j)Cijklf(xkl(t-sin2t))xij(t)-∑Ckl∈Nq(i,j)Bijkl∫0∞Kij(u)g(xkl(t-u))duxij+Lij(t),i,j=1,2,3,(77)[a11a12a13a21a22a23a31a32a33]=[113313313][B11B12B13B21B22B23B31B32B33]=[C11C12C13C21C22C23C31C32C33]=[0.10.20.10.200.20.10.20.1],[η11η12η13η21η22η23η31η32η33]=0.01[sin23t+0.11+t2cos23t+0.11+t2sin22t+0.11+t2cos25t+0.11+t2sin25t+0.11+t2cos22t+0.11+t2sin22t+0.11+t2cos23t+0.11+t2sin22t+0.11+t2][L11L12L13L21L22L23L31L32L33]=[0.7+0.24sin22t-11+t20.41+0.5cos2t10.61+0.2cos2t-11+t20.67+0.2sin2t10.59+0.4cos4t-11+t20.5+0.41sin2t1].Set
(78)κ=0.7,r=q=1,Kij(u)=|sinu|e-u,5555555555555555555i=1,2,3,j=1,2,3,f(x)=g(x)=150(|x-1|-|x+1|),
clearly,
(79)Mf=Mg=0.04,Lf=Lg=0.04,∑Ckl∈N1(1,1)C11kl=∑Ckl∈N1(1,1)B11kl=0.5,∑Ckl∈N1(1,2)C12kl=∑Ckl∈N1(1,2)B12kl=0.8,∑Ckl∈N1(1,3)C13kl=∑Ckl∈N1(1,3)B13kl=0.5,∑Ckl∈N1(2,1)C21kl=∑Ckl∈N1(2,1)B21kl=0.8,∑Ckl∈N1(2,2)C22kl=∑Ckl∈N1(2,2)B22kl=1.2,∑Ckl∈N1(2,3)C23kl=∑Ckl∈N1(2,3)B23kl=0.8,∑Ckl∈N1(3,1)C31kl=∑Ckl∈N1(3,1)B31kl=0.5,∑Ckl∈N1(3,2)C32kl=∑Ckl∈N1(3,2)B32kl=0.8,∑Ckl∈N1(3,3)C33kl=∑Ckl∈N1(3,3)B33kl=0.5,
where ij∈J={11,12,13,21,22,23,31,32,33}. Then,
(80)L=max{max(i,j){supt∈ℝ|∫-∞te-∫staij(u)duLij(s)ds|},max(i,j){supt∈ℝ|∫-∞te-∫staij(u)duLij(s)dsLij(t)-aij(t)×∫-∞te-∫staij(u)duLij(s)ds|supt∈ℝ|∫-∞te-∫staij(u)duLij(s)dsLij(t)-aij(t)}{supt∈ℝ|∫-∞te-∫staij(u)duLij(s)ds|}}=1>0,0.7=κ≤L=1,max(i,j){1aij-Eij,(1+aij+aij-)Eij}=0.6603≤κ,max(i,j){1aij-Fij,(1+aij+aij-)Fij}=0.5804<1.
It follows that system (56) satisfies all the conditions in Theorems 10 and 12. Hence, system (76) has exactly one pseudo almost periodic solution. Moreover, the pseudo almost periodic solution is globally exponentially stable. The fact is verified by the numerical simulation in Figures 1, 2, and 3 and there are three different initial values which are φ11≡1, φ12≡-3, φ13≡4, φ21≡2, φ22≡5, φ23≡3, φ33≡-1, φ32≡-2, φ33≡-5; φ11≡2, φ12≡-1, φ13≡5, φ21≡4, φ22≡2, φ23≡1, φ33≡-3, φ32≡-4, φ33≡3 and φ11≡-2, φ12≡1, φ13≡-5, φ21≡-4, φ22≡-2, φ23≡-1, φ33≡3, φ32≡4, φ33≡-3, respectively.
Numerical solutions of system (76) for different initial values.
Numerical solutions of system (76) for different initial values.
Numerical solutions of system (76) for different initial values.
Remark 14.
By using the inequality analysis technique, in [19, 20], the authors obtained the existence of almost periodic solution of SICNNs with leakage delays, but they did not give the existence and global exponential convergence for the pseudo almost periodic solution. Since [1–9] only dealt with SICNNs without leakage delays, [14–18, 21–24] give no opinions about the problem of pseudo almost periodic solutions for SICNNs with leakage delays. One can observe that all the results in these literatures and the references therein cannot be applicable to prove the existence and exponential stability of pseudo almost periodic solutions for SICNNs (56).
Conflict of Interests
The authors declare no conflict of interests. They also declare that they have no financial and personal relationships with other people or organizations that can inappropriately influence their work; there is no professional or other personal interest of any nature or kind in any product, service, and/or company that could be construed as influencing the position presented in, or the review of, this present paper.
Acknowledgments
The authors would like to express the sincere appreciation to the reviewers for their helpful comments in improving the presentation and quality of the paper. This work was supported by the National Natural Science Foundation of China (Grant no. 11201184), the Natural Scientific Research Fund of Zhejiang Provincial of P. R. China (Grant no. LY12A01018), and the Natural Scientific Research Fund of Zhejiang Provincial Education Department of P. R. China (Grant no. Z201122436).
BouzerdoumA.PinterR. B.Shunting inhibitory cellular neural networks: derivation and stability analysis199340321522110.1109/81.222804MR1232563ZBL0825.93681ChenA.CaoJ.Almost periodic solution of shunting inhibitory CNNs with delays20022982-316117010.1016/S0375-9601(02)00469-3MR1917000ZBL0995.92003ChérifF.Existence and global exponential stability of pseudo almost periodic solution for SICNNs with mixed delays2012391-223525110.1007/s12190-011-0520-1MR2914474CaiM.ZhangH.YuanZ.Positive almost periodic solutions for shunting inhibitory cellular neural networks with time-varying delays200878454855810.1016/j.matcom.2007.08.001MR2424562ZBL1147.34053ShaoJ.Anti-periodic solutions for shunting inhibitory cellular neural networks with time-varying delays200837230501150162-s2.0-4644908778210.1016/j.physleta.2008.05.064FanQ.ShaoJ.Positive almost periodic solutions for shunting inhibitory cellular neural networks with time-varying and continuously distributed delays20101561655166310.1016/j.cnsns.2009.06.026MR2576792ZBL1221.37182ZhaoC.FanQ.WangW.Anti-periodic solutions for shunting inhibitory cellular neural networks with time-varying coefficients201031259267BouzerdoumA.NabetB.PinterR. B.Analysis and analog implementation of directionally sensitive shunting inhibitory neural networks1473Visual Information Processing: From Neurons to ChipsApril 19912938Proceedings of the SPIE2-s2.0-0025568612BouzerdoumA.PinterR. B.PinterR. B.NabetB.Nonlinear lateral inhibition applied to motion detection in the fly visual system1992Boca Raton, Fla, USACRC Press423450KoskoB.1992New Delhi, IndiaPrentice HallMR1123937HaykinS.1999New Jersey, NJ, USAPrentice HallGopalsamyK.199274Dordrecht, The NetherlandsKluwer AcademicMathematics and its ApplicationsMR1163190GopalsamyK.Leakage delays in BAM200732521117113210.1016/j.jmaa.2006.02.039MR2270073ZBL1116.34058ZhangH.ShaoJ.Almost periodic solutions for cellular neural networks with time-varying delays in leakage terms201321924114711148210.1016/j.amc.2013.05.046MR3073298LiX.RakkiyappanR.BalasubramaniamP.Existence and global stability analysis of equilibrium of fuzzy cellular neural networks with time delay in the leakage term under impulsive perturbations2011348213515510.1016/j.jfranklin.2010.10.009MR2771833ZBL1241.92006BalasubramaniamP.VembarasanV.RakkiyappanR.Leakage delays in T-S fuzzy cellular neural networks20113321111362-s2.0-7995609755910.1007/s11063-010-9168-3LiuB.Global exponential stability for BAM neural networks with time-varying delays in the leakage terms201314155956610.1016/j.nonrwa.2012.07.016MR2969855ZBL1260.34138ChenZ.A shunting inhibitory cellular neural network with leakage delays and continuously distributed delays of neutral type20132372429243410.1007/s00521-012-1200-2ZhangH.YangM.Global exponential stability of almost periodic solutions for SICNNs with continuously distributed leakage delays2013201314307981MR3035317ZBL1277.3410310.1155/2013/307981LiuB.ShaoJ.Almost periodic solutions for SICNNs with time-varying delays in the leakage terms20132013article 49410.1186/1029-242X-2013-494LiuB.Global exponential stability of positive periodic solutions for a delayed Nicholson's blowflies model2014412121222110.1016/j.jmaa.2013.10.049MR3145795MengJ.Global exponential stability of positive pseudo almost periodic solutions for a model of hematopoiesis20132013746307610.1155/2013/463076OuC.Almost periodic solutions for shunting inhibitory cellular neural networks20091052652265810.1016/j.nonrwa.2008.07.004MR2523228ZBL1205.34087LiL.FangZ.YangY.A shunting inhibitory cellular neural network with continuously distributed delays of neutral type20121331186119610.1016/j.nonrwa.2011.09.011MR2863947ZBL1239.34085ZhangC.2003Beijing, ChinaScience Press10.1007/978-94-007-1073-3MR2000981FinkA. M.1974377Berlin, GermanySpringerLecture Notes in MathematicsMR0460799HaleJ. K.19802ndHuntington, NY, USARobert E. KriegerMR587488