This paper is concerned with piecewise pseudo almost periodic solutions of a class of interval general BAM neural networks with mixed time-varying delays and impulsive perturbations. By adopting the exponential dichotomy of linear differential equations and the fixed point theory of contraction mapping. The sufficient conditions for the existence of piecewise pseudo almost periodic solutions of the interval general BAM neural networks with mixed time-varying delays and impulsive perturbations are obtained. By adopting differential inequality techniques and mathematical methods of induction, the global exponential stability for the piecewise pseudo almost periodic solutions of the interval general BAM neural networks with mixed time-varying delays and impulsive perturbations is discussed. An example is given to illustrate the effectiveness of the results obtained in the paper.
Citation: Yanshou Dong, Junfang Zhao, Xu Miao, Ming Kang. Piecewise pseudo almost periodic solutions of interval general BAM neural networks with mixed time-varying delays and impulsive perturbations[J]. AIMS Mathematics, 2023, 8(9): 21828-21855. doi: 10.3934/math.20231113
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This paper is concerned with piecewise pseudo almost periodic solutions of a class of interval general BAM neural networks with mixed time-varying delays and impulsive perturbations. By adopting the exponential dichotomy of linear differential equations and the fixed point theory of contraction mapping. The sufficient conditions for the existence of piecewise pseudo almost periodic solutions of the interval general BAM neural networks with mixed time-varying delays and impulsive perturbations are obtained. By adopting differential inequality techniques and mathematical methods of induction, the global exponential stability for the piecewise pseudo almost periodic solutions of the interval general BAM neural networks with mixed time-varying delays and impulsive perturbations is discussed. An example is given to illustrate the effectiveness of the results obtained in the paper.
This paper considers the existence and global exponential stability of piecewise pseudo almost periodic solutions for interval general BAM neural networks with mixed time-varying delays and impulsive perturbations. The mixed time-varying delays include leakage delays and time-varying delays:
{x′i(t)=−ai(t)xi(t−αi(t))+m∑j=1sji(t)fj(xj(t),yj(t−τji(t)))+ci(t),t>0,t≠tk,i=1,2,⋯,m,Δxi(t)=xi(t+)−xi(t−)=Ik(xi(t)),t=tk,k∈Z+,y′j(t)=−bj(t)yj(t−βj(t))+m∑i=1tij(t)gi(xi(t−δij(t)),yi(t))+dj(t),t>0,t≠tk,j=1,2,⋯,m,Δyj(t)=yj(t+)−yj(t−)=Jk(yj(t)),t=tk,k∈Z+, | (1.1) |
where Z+ is the set of nonnegative integers; the numbers of neurons in layers X and Y are denoted by m. xi(t) represents the state variables of the i-th neurons at time t. yj(t) represents the state variables of the j-th neurons at time t. ai(t)>0, bj(t)>0 are continuous functions and represent the decay rates of neurons in different layers, respectively. sji(t), tij(t) are the connection weights. fj(⋅,⋅), gi(⋅,⋅) represent the activation functions of the j-th and i-th units, respectively. ci(t), dj(t) represent the external inputs of the i-th neuron and the j-th neuron acting on different layers at time t, respectively. αi(t)>0, βj(t)>0 are time-varying leakage delays. τji(t)>0, δij(t)>0 are time-varying delays respectively satisfying 1−τ′ji(t)>0, 1−δ′ij(t)>0. The sequence {tk} has no finite accumulation point. Ik(⋅):Rm→R and Jk(⋅):Rm→R, k∈Z+.
The initial value conditions of the system (1.1) are as follows:
{xi(θ)=φi(θ),θ∈[−r1,0],i=1,2,⋯,m,yj(θ)=ψj(θ),θ∈[−r2,0],j=1,2,⋯,m, | (1.2) |
where r1=max1≤i,j≤m{supt∈R+|αi(t)|,supt∈R+|δij(t)|}, r2=max1≤i,j≤m{supt∈R+|βj(t)|,supt∈R+|τji(t)|}, φi(θ)∈C1([−r1,0],R), i=1,2,⋯,m, ψj(θ)∈C1([−r2,0],R) and j=1,2,⋯,m.
BAM neural networks have more complex dynamic behavior. They are composed of two-layer nonlinear feedback networks. Neurons in different layers are interconnected, and there is no connection between neurons in the same layer [1]. It extends the single-layer self-association learning rule to the two-layer hetero-association mode. Time delays in neural networks not only decrease the transmission speed of the networks, but may also change the stability of the networks. There have been many excellent results on the stability and bifurcation behavior of delayed neural networks [2,3,4,5,6,7,8,9]. The existence and stability of solutions for neural networks with time delays have been widely explored [10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29]. In particular, the dynamic behavior of neural networks with leakage delays [10,12,13,24,25,27,29] is still a hot research topic.
Impulses are ubiquitous in the actual neural networks and the dynamic behavior of the neural networks are often affected by impulsive perturbations. And the study of impulsive control theory for neural networks has made a lot of advancements [11,15,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40].
Compared with the activation functions fj(⋅), gi(⋅) in BAM neural networks [10,11,13,14,15,17,19,21,22,23,24], the activation functions fj(xj,yj), gi(xj,yj) in the interval general BAM neural networks [16,17,20,21,22,24] more clearly show the connections between different layers of neurons in the networks. Ding and Huang [16] studied the existence and global robust exponential stability of equilibrium points for the following a class of interval general BAM neural networks with constant delays by using fixed-point theory and constructing suitable Lyapunov functions:
{x′i(t)=−aixi(t)+m∑j=1sjifj(xj(t),yj(t−τji))+ci,y′j(t)=−bjyj(t)+m∑i=1tijgi(xi(t−δij),yi(t))+dj. | (1.3) |
By variable transformations, fixed point theory and constructing suitable delay differential inequalities, Xu et al. [17] studied the existence, uniqueness and global exponential stability of the equilibrium point of system (1.3) with proportional delays. Given ai=ai(t), bj=bj(t), sji=sji(t) and tij=tij(t) in the system (1.3), the authors of [20] studied the existence and global exponential stability of periodic solutions for the system (1.3) on time scales by using fixed point theory and constructing suitable Lyapunov functions. Given ai=ai(t), bj=bj(t), sji=sji(t), tij=tij(t), ci=ci(t) and dj=dj(t) in the system (1.3), Duan [22] studied the existence and global exponential stability of pseudo almost periodic solutions for system (1.3) by using exponential dichotomy, fixed point theory and inequality techniques. Given ai=ai(t), bj=bj(t), sji=sji(t), tij=tij(t), ci=ci(t) and dj=dj(t) in the system (1.3), the authors of [24] studied the existence and exponential stability of almost periodic solutions for system (1.3) with leakage delays by using exponential dichotomy, fixed point theory and inequality techniques.
However, in the existing literature, we have not found any research on the dynamic behavior of interval general BAM neural networks with impulses and leakage delays. The stability of the interval general BAM neural networks may be destroyed by external perturbations and leakage delays. Therefore, the effects of leakage delays and impulsive perturbations on the dynamic behavior of interval general BAM neural networks are worth exploring.
Motivated by the above discussions, this paper studies the existence and global exponential stability of piecewise pseudo almost periodic solutions for the interval general BAM neural network described by (1.1) with mixed time-varying delays and impulsive perturbations. The mixed time-varying delays include leakage delays and time-varying delays.
Throughout this paper, let e+i=supt∈R+|ei(t)|, e−i=inft∈R+|ei(t)|, e+ij=supt∈R+|eij(t)| and e−ij=inft∈R+|eij(t)|. Let T be the set of the real sequences {tk}k∈Z+ such that σ_=infk∈Z+t1k=infk∈Z+(tk+1−tk)>0, so limk→+∞tk=+∞. For {tk}k∈Z+⊂T, assume that {tjk:tjk=tk+j−tk,k,j∈Z+} are equipotentially almost periodic and t0=0; it can be easily proved that the sequence {tjk} satisfies tjk+i−tjk=tik+j−tik and tjk−tik=tj−ik+i. And suppose that the following conditions hold.
(H1) For u1, v1, u2, v2∈R, there are nonnegative constants Pj, Qj, Hi and Ki such that |fj(u1,v1)−fj(u2,v2)|≤Pj|u1−u2|+Qj|v1−v2| and |gi(u1,v1)−gi(u2,v2)|≤Hi|u1−u2|+Ki|v1−v2|.
(H2) ai(t), bj(t)∈C(R+,R+) are almost periodic functions, and supt∈R+|ai(t)|=a+i, inft∈R+|ai(t)|=a−i, supt∈R+|bj(t)|=b+j, inft∈R+|bj(t)|=b−j. sji(t), tij(t), ci(t), dj(t):R+→R are piecewise pseudo almost periodic functions, and supt∈R+|sji(t)|=s+ji, supt∈R+|tij(t)|=t+ij, supt∈R+|ci(t)|=c+i and supt∈R+|dj(t)|=d+j.
(H3) Ik, Jk∈PAP(Z+,Rn) and there are two positive constants I, J satisfying 0<I, J<1 such that |Ik(u1)−Ik(u2)|≤I|u1−u2|, |Jk(u1)−Jk(u2)|≤J|u1−u2| for u1, u1∈R, k∈Z+.
Remark 1.1. In the existing literature on BAM neural networks, in addition to the condition (H1), the activation functions fj(⋅) and gi(⋅) are required to satisfy fj(0)=0 and gi(0)=0. In this paper, the activation functions fj(0,0)=0, gi(0,0)=0 are not required, as it only needs to meet the condition (H1).
In this paper, first, by using the exponential dichotomy of linear differential equations and the fixed point theory for contraction mapping, the existence of piecewise pseudo almost periodic solutions for system (1.1) satisfying the initial value conditions given by (1.2) is studied. And then by using mathematical methods of induction and inequality techniques, the global exponential stability of piecewise pseudo-almost periodic solutions for system (1.1) satisfying the initial value conditions given by (1.2) will be discussed.
Remark 1.2. Compared with the systems studied in [11,16,17,20,21,22,24,25], the system (1.1) studied in this paper is more general, as some classical Hopfield neural networks and BAM neural networks are special cases of the system (1.1). Duan [22] studied the effects of constant delays in the activation functions on the dynamic behavior of the system, but they did not consider the effects of leakage delays and impulsive perturbations on the dynamic behavior of the system. The mixed time-varying delays studied in this paper include not only the time-varying delays in the activation functions but also the leakage delays; the influences of impulsive perturbations on the dynamic behavior of the system are also considered. This paper not only considers the norm ‖⋅‖∞ but also ‖(⋅)′‖∞. Therefore, the results obtained in [21,22,24] are special cases of the research results in this paper.
This section mainly gives the necessary definitions, lemmas and notations to prove the existence and global exponential stability of the piecewise pseudo almost periodic solutions of the system described by (1.1)–(1.2).
C1(R,Rn) is the set of continuous functions with a continuous derivative, and
‖ϕ‖1=max{max{‖φ‖∞,‖φ′‖∞},max{‖ψ‖∞,‖ψ′‖∞}}, |
where ‖⋅‖∞=max1<i<msupt∈R|⋅|.
BC(R,Rn) is the set of bounded continuous functions, and (BC(R,Rn),‖⋅‖∞) is a Banach space.
Let PC(R,Rn) be the set of all piecewise continuous functions ϕ:R→Rn such that ϕ is countinuous at t for t∉{tk:k∈Z+} and that ϕ(t+k), ϕ(t−k) exist and ϕ(t−k)=ϕ(tk) for k∈Z+.
Define L∞(Z+,Rn)={u:Z+→Rn,‖u‖=supn∈Z+|u(n)|} and PAP0(Z+,Rn)={u∈L∞(Z+,Rn):limn→∞12nn∑k=−n‖u(k)‖=0}.
Definition 2.1. [34] A function f∈C(R,Rn) is called almost periodic if for ∀ϵ>0, there is L(ϵ)>0 such that every interval of length L(ϵ) contains at least a number δ∈T(f,ϵ) such that |f(t+δ)−f(t)|<ϵ for any t∈R, where T(f,ϵ)={δ∣|f(t+δ)−f(t)|<ϵ,∀t∈R}. The set of all almost periodic functions is represented by AP(R,Rn).
Definition 2.2. [35] A sequence {un} is called almost periodic if for ∀ϵ>0, there is a natural number L(ϵ) such that for k∈Z, there is at least one number q in [k,k+L(ϵ)] for which the inequality ‖un+q−un‖<ϵ holds for all n∈N. The set of all almost periodic sequences is represented by AP(Z,Rn).
Definition 2.3. [36] A function ϕ∈PC(R,Rn) is called piecewise almost periodic if the following conditions are satisfied.
● (1) {tjk}, tjk=tk+j−tk, k, j∈Z are equipotentially almost periodic, that is to say, for any ϵ>0, there is a relatively dense set in R of ϵ-almost periods that are common for any of the sequences {tjk}.
● (2) For any ϵ>0, there is a constant δ(ϵ)>0 such that, if t′ and t″ belong to the same interval of continuity of ϕ and |t′−t″|<δ, then ‖ϕ(t′)−ϕ(t″)‖<ϵ.
● (3) For any ϵ>0, there is a relatively dense set Ω(ϵ) in R such that, if ρ∈Ω(ϵ), then ‖ϕ(t+ρ)−ϕ(t)‖<ϵ for t∈R, which satisfies |t−ti|>ϵ, i∈Z.
Denote by APT(R,Rn) the set of piecewise almost periodic functions; UPC(R,Rn) is the set of the function f∈PC(R,Rn) such that f satisfies the condition (2) in Definition 2.3. Let
PC0T(R,Rn)={ϕ∈PC(R,Rn),limt→∞‖ϕ(t)‖=0}, |
PAP0T(R,Rn)={ϕ∈PC(R,Rn),limr→∞12r∫r−r‖ϕ‖dt=0}. |
Obviously, PC0T(R,Rn)⊂PAP0T(R,Rn).
Definition 2.4. f∈PC(R×Ω1×Ω2,Rn) is said to be piecewise almost periodic in t uniformly in (φ,ψ)∈Ω1×Ω2 if for every compact set K1×K2⊆Ω1×Ω2, {f(⋅,φ,ψ):(φ,ψ)∈K1×K2} is uniformly bounded, and for ∀ϵ>0, there exists a relatively dense set Ω(ϵ) such that ‖f(t+τ,φ,ψ)−f(t,φ,ψ)‖≤ϵ for (φ,ψ)∈K1×K2, τ∈Ω(ϵ), and given t∈R, |t−ti|>ϵ.
Definition 2.5. [37] Sequence {un}n∈Z∈L∞(Z,Rn) is called pseudo almost periodic if there exist u1n∈AP(Z,Rn) and u2n∈PAP0(Z,Rn) such that un=u1n+u2n. The set of pseudo almost periodic sequences is denoted by PAP(Z,Rn).
Definition 2.6. [38] A function ϕ∈PC(R,Rn) is known as piecewise pseudo almost periodic if it can be disintegrated into ϕ=ϕ1+ϕ2, where ϕ1∈APT(R,Rn) and ϕ2∈PAP0T(R,Rn). The set of piecewise pseudo almost periodic functions is denoted by PAPT(R,Rn).
Lemma 2.1. [38] Let the sequence of vector-valued functions {Ii}i∈Z be pseudo almost periodic and there be a constant L>0 such that ‖Ii(u)−Ii(v)‖≤L‖u−v‖ for all u,v∈Ω,i∈Z if ϕ∈PAPT(R,X)∩UPC(R,X) such that R(ϕ)⊂Ω; then Ii(ϕ(ti)) is pseudo almost periodic.
Lemma 2.2. [39] If φ(⋅)∈PAPT(R,Rn), then φ(⋅−h)∈PAPT(R,Rn).
Lemma 2.3. [39] If φ(⋅),ψ(⋅)∈PAPT(R,R), then φ×ψ∈PAPT(R,R).
Lemma 2.4. [39] If fi(⋅)∈C(R,Rn) satisfies the Lipschitz condition, ψ(⋅)∈PAPT(R,R), ψ′(⋅)∈PAPT(R,R) and δ(⋅)∈APT(R,R) such that for all t∈R, 1−δ′(t)>0, fi(ψ(⋅−δ(⋅)))∈PAPT(R,R).
Lemma 2.5. [40] Assume that f∈PAPT(R×Ω1×Ω2,X) and the following conditions hold:
● (1) {f(t,u,v):t∈R,u∈K1,v∈K2} is bounded for every bounded subset K1×K2⊆Ω1×Ω2.
● (2) {f(t,u,v)} is uniformly continuous in each bounded subset of Ω1×Ω2 uniformly in t∈R.
If φ∈PAPT(R,X), ψ∈PAPT(R,X) such that R(φ)×R(ψ)⊂Ω1×Ω2, then f(t,φ(⋅),ψ(⋅))∈PAPT(R,X), where R(φ), R(ψ) are the ranges of φ, ψ, respectively.
Lemma 2.6. [40] Let f∈PAPT(R×Ω1×Ω2,X), φ∈PAPT(R,X), ψ∈PAPT(R,X) and R(φ)×R(ψ)⊂Ω1×Ω2. Assume that there exists a constant Lf>0 such that ‖f(t,u1,v1)−f(t,u2,v2)‖≤Lf(‖u1−u2‖+‖v1−v2‖), t∈R, u1, u2∈Ω1 and v1, v2∈Ω2. Then, f(⋅,φ(⋅),ψ(⋅))∈PAPT(R,X).
Definition 2.7. [34] Let x∈Rn and A(⋅) be an n×n continuous matrix defined on R. The linear system x′(t)=A(t)x(t) is said to admit an exponential dichotomy on R if there exist positive constants k and α and a projection P such that the fundamental solution matrix X(t) of x′(t)=A(t)x(t) satisfies the following:
‖X(t)PX−1(s)‖≤ke−α(t−s),t≥s,‖X(t)(I−P)X−1(s)‖≤ke−α(s−t),t≤s. |
Lemma 2.7. [34] Let ai(⋅) be an almost periodic function on R and M[ai]=limT→+∞∫t+Ttai(s)ds>0, i=1,2,...,n. Then, the system x′(t)=diag(−a1(t),−a2(t),...,−an(t))x(t) admits an exponential dichotomy on R.
Lemma 2.8. [41] Let A(⋅) be an almost periodic matrix function and f(⋅)∈PAP(R,Rn). If x′(t)=A(t)x(t) admits an exponential dichotomy, then the pseudo almost periodic system x′(t)=A(t)x(t)+f(t) has a unique pseudo almost periodic solution x(t), and
x(t)=∫t−∞X(t)PX−1(s)f(s)ds−∫∞tX(t)(I−P)X−1(s)f(s)ds. |
Definition 2.8. Let ¯z(t)=(¯x(t),¯y(t))T be the solution of system (1.1) satisfying the initial value conditions ¯ϕ=(¯φ(θ),¯ψ(θ))T. If there exist constants M≥1, η>0 such that for any solution z(t)=(x(t),y(t))T of the system (1.1) satisfying the initial value conditions ϕ=(φ(θ),ψ(θ))T, the following inequality holds
‖z(t)−¯z(t)‖1≤M‖ϕ−¯ϕ‖1e−ηt,t≥0, |
where ‖ϕ−¯ϕ‖1=max{max{‖φ‖∞,‖φ′‖∞},max{‖ψ‖∞,‖ψ′‖∞}}. Then, ¯z(t) is said to be globally exponentially stable.
This section establishes the conditions of the existence for piecewise pseudo almost periodic solutions of the system described by (1.1)–(1.2).
Lemma 3.1. Let f∈PAPT(R×Ω1×Ω2,X), φ, ψ∈PAPT(R,Rn) and R(φ)×R(ψ)⊂Ω1×Ω2. There are constants P>0, Q>0 such that for t∈R, φ, ¯φ∈Ω1, ψ, ¯ψ∈Ω2, |f(t,φ,ψ)−f(t,¯φ,¯ψ)|≤P|φ−¯φ|+Q|ψ−¯ψ| and θ(⋅)∈C(R,R). Then, f(t,φ(t−θ(⋅)),ψ), f(t,φ,ψ(t−θ(⋅)))∈PAPT(R,X) and f(t,φ(t−θ(⋅)),ψ(t−θ(⋅)))∈PAPT(R,X).
Proof. Since φ, ψ∈PAPT(R,R), then φ=φ1+φ2, ψ=ψ1+ψ2, where φ1, ψ1∈APT(R,R) and φ2, ψ2∈PAP0T(R,R). We have
f(t,φ(t−θ(⋅)),ψ)=f(t,φ1(t−θ(⋅))+φ2(t−θ(⋅)),ψ1+ψ2)=f(t,φ1(t−θ(⋅)),ψ1)+f(t,φ1(t−θ(⋅))+φ2(t−θ(⋅)),ψ1+ψ2)−f(t,φ1(t−θ(⋅)),ψ1)=f1+f2, |
where f1=f(t,φ1(t−θ(⋅)),ψ1), f2=f(t,φ1(t−θ(⋅))+φ2(t−θ(⋅)),ψ1+ψ2)−f(t,φ1(t−θ(⋅)),ψ1). From Lemmas 2.2–2.5, f1=f(t,φ1(t−θ(⋅)),ψ1)∈APT(R,R) holds. Now f2∈PAP0T(R,R) will be proved. Since
limr→∞12r∫r−r|f2|dt≤limr→∞12r∫r−r(P|φ2(t−θ(t)|+Q|ψ2|)dt=Plimr→∞12r∫r−r|φ2(t−θ(t)|dt+Qlimr→∞12r∫r−r|ψ2|dt=0. |
Thus f2∈PAP0T(R,R). So, f(t,φ(t−θ(⋅)),ψ)∈PAPT(R,X). In a similar way, f(t,φ,ψ(t−θ(⋅)))∈PAPT(R,X) and f(t,φ(t−θ(⋅)),ψ(t−θ(⋅)))∈PAPT(R,X). The proof is complete.
Theorem 3.1. If conditions (H1)–(H3) are true. For any ϕ=(φ,ψ)T∈PAPT(R+,R2m) and ¯ϕ=(¯φ,¯ψ)T∈PAPT(R+,R2m), define the operator
Φ1ϕ(t)=(∫t−∞e−∫tsa1(u)du¯F1(s,φ,ψ)ds...∫t−∞e−∫tsam(u)du¯Fm(s,φ,ψ)ds∫t−∞e−∫tsb1(u)du¯G1(s,φ,ψ)ds...∫t−∞e−∫tsbm(u)du¯Gm(s,φ,ψ)ds), |
where
¯Fi(t,φ,ψ)=ai(t)∫tt−αi(t)φ′i(s)ds+m∑j=1sji(t)fj[φj(t),ψj(t−τji(t))]+ci(t),i=1,2,⋯,m,¯Gj(t,φ,ψ)=bj(t)∫tt−βj(t)ψ′j(s)ds+m∑i=1tij(t)gi[φi(t−δij(t)),ψi(t)]+dj(t),j=1,2,⋯,m. |
Then, Φ1ϕ(⋅) is a mapping from PAPT(R+,R2m) to PAPT(R+,R2m).
Proof. By Lemmas 2.2–2.5 and Lemma 3.1, ¯Fi(t,φ,ψ)∈PAPT(R+,R); thus,
¯Fi(t,φ,ψ)=¯F1i(t,φ,ψ)+¯F2i(t,φ,ψ), |
where ¯F1i(t,φ,ψ)∈APT(R+,R), ¯F2i(t,φ,ψ)∈PAP0T(R+,R).
For i=1,2,...,m, we have
∫t−∞e−∫tsai(u)du¯Fi(s,φ,ψ)ds=∫t−∞e−∫tsai(u)du¯F1i(s,φ,ψ)ds+∫t−∞e−∫tsai(u)du¯F2i(s,φ,ψ)ds=Ψ1(t)+Ψ2(t). |
Step 1: Prove that Ψ1(t)∈UPC(R+,R). Obviously, Ψ1(t)∈PC(R+,R). For t′,t″∈(tk,tk+1),k∈Z+,t″<t′,
|Ψ1(t′)−Ψ1(t″)|=|∫t′−∞e−∫t′sai(u)du¯F1i(s,φ,ψ)ds−∫t″−∞e−∫t″sai(u)du¯F1i(s,φ,ψ)ds|≤|∫t″−∞(e−∫t′sai(u)du−e−∫t″sai(u)du)¯F1i(s,φ,ψ)ds|+|∫t′t″e−∫t′sai(u)du¯F1i(s,φ,ψ)ds|≤|e−∫t′t″ai(u)du−1|∫t″−∞e−∫t″sai(u)du|¯F1i(s,φ,ψ)|ds+∫t′t″e−∫t′sai(u)du|¯F1i(s,φ,ψ)|ds≤a+i(t′−t″)∫t″−∞e−a−i(t″−s)|¯F1i(s,φ,ψ)|ds+∫t′t″e−a−i(t′−s)|¯F1i(s,φ,ψ)|ds≤(a+ia−i(t′−t″)+∫t′t″e−a−i(t′−s)ds)‖¯F1i(s,φ,ψ)‖∞. |
For ∀ϵ>0, there is 0<δ<min{a−iϵ2‖¯F1i‖∞a+i,ϵ2‖¯F1i‖∞} such that for t′, t′ satisfying 0<|t′−t″|<δ, |Ψ1(t′)−Ψ1(t″)|<ϵ holds. So, Ψ1(t)∈UPC(R+,R).
Step 2: Prove that Ψ1(⋅)∈APT(R+,R). Since ¯F1i(t,φ,ψ)∈APT(R+,R), for ∀ϵ>0, there is a relatively dense set Ω(ϵ); for τ∈Ω(ϵ), t∈R+, |t−tk|>ϵ, i∈Z+ such that |¯F1i(s+τ,φ(s+τ),ψ(s+τ))−¯F1i(s,φ(s),ψ(s))|<a−iϵ. Then
|Ψ1(t+τ)−Ψ1(t)|=|∫t+τ−∞e−∫t+τsai(u)du¯F1i(s,φ,ψ)ds−∫t−∞e−∫tsai(u)du¯F1i(s,φ,ψ)ds|≤|∫t−∞e−∫t+τs+τai(u)du¯F1i(s+τ,φ(s+τ),ψ(s+τ))ds−∫t−∞e−∫tsai(u)du¯F1i(s,φ(s),ψ(s))ds|≤|∫t−∞e−a−i(t−s)¯F1i(s+τ,φ(s+τ),ψ(s+τ))ds−∫t−∞e−a−i(t−s)¯F1i(s,φ(s),ψ(s))ds|≤∫t−∞e−a−i(t−s)|¯F1i(s+τ,φ(s+τ),ψ(s+τ))−¯F1i(s,φ(s),ψ(s))|ds<ϵ. |
Consequently, Ψ1(⋅)∈APT(R,R).
Step 3: Prove that Ψ2(⋅)∈PAP0T(R+,R). Since ¯F2i(s−u,φ(s−u),ψ(s−u)),¯F2i(s,φ(s),ψ(s))∈PAP0T(R+,R), then
limr→∞12r∫r−r|Ψ2(⋅)|dt=limr→∞12r∫r−r|∫t−∞e−∫tsai(u)du¯F2i(s,φ,ψ)ds|dt≤limr→∞12r∫r−r|∫t−∞e−a−i(t−s)¯F2i(s,φ,ψ)ds|dt≤limr→∞12r∫r−re−a−itdt∫−r−∞ea−is‖¯F2i(s,φ,ψ)‖∞ds+limr→∞12r∫r−r∫t−re−a−i(t−s)|¯F2i(s,φ,ψ)|dsdt≤limr→∞‖¯F2i(s,φ,ψ)‖∞2ra−i∫r−re−a−i(t+r)dt+limr→∞12r∫r−r∫t+r0e−a−iu|¯F2i(t−u,φ(t−u),ψ(t−u))|dudt≤limr→∞‖¯F2i(s,φ,ψ)‖∞2ra−i∫r−re−a−i(t+r)dt+∫∞0e−a−iu(limr→∞12r∫r−r|¯F2i(t−u,φ(t−u),ψ(t−u))|dt)du=0. |
Consequently, Ψ2(⋅)∈PAP0T(R+,R). So, ∫t−∞e−∫tsai(u)du¯Fi(s,φ,ψ)ds∈PAPT(R+,R). Similarly, for j=1,2,...,m, we have that ∫t−∞e−∫tsbj(u)du¯Gj(s,φ,ψ)ds∈PAPT(R+,R).
Therefore Φ1ϕ(⋅)∈PAPT(R+,R2m).
Theorem 3.2. If conditions (H1)–(H3) are true, for any ϕ=(φ,ψ)T∈PAPT(R+,R2m), define the operator
Φ2ϕ(t)=(∑tk<te−∫ttka1(u)duIk(φ1(tk))...∑tk<te−∫ttkam(u)duIk(φm(tk))∑tk<te−∫ttkb1(u)duJk(ψ1(tk))...∑tk<te−∫ttkbm(u)duJk(ψm(tk))). |
Then, Φ2ϕ(⋅) is a mapping from PAPT(R+,R2m) to PAPT(R+,R2m).
Proof. Assume that t∈(tl,tl+1], l∈Z+; therefore, t−tk=t−tl+tl−tk≥(l−k)σ_, and
∑tk<te−a−i(t−tk)≤∑0≤j=l−k<∞e−a−ijσ_≤11−e−a−iσ_. |
For any ϵ>0, take the positive constant δ<ϵ(1−e−a−iσ_)a+i‖Ik(φi(tk))‖∞, for t′,t″∈(tk,tk+1),k∈Z+,t′<t″, provided that |t′−t″|<δ; there is ζ∈(t′,t″) such that
|∑tk<t′e−∫t′tkai(u)duIk(φi(tk))−∑tk<t″e−∫t″tkai(u)duIk(φi(tk))|≤a+i∑tk<ζe−a−i(ζ−tk)‖Ik(φi(tk))‖∞|t′−t″|≤a+i‖Ik(φi(tk))‖∞1−e−a−iσ_|t′−t″|<ϵ. |
That is, ∑tk<te−∫ttkai(u)duIk(φi(tk))∈UPC(R+,R). By Lemma 2.1 and (H3), Ik(φi(tk))∈PAP(Z+,R). Let Ik(φi(tk))=I1k+I2k, where I1k∈AP(Z+,R), I2k∈PAP0(Z+,R); then,
∑tk<te−∫ttkai(u)duIk(φi(tk))=∑tk<te−∫ttkai(u)duI1k(φi(tk))+∑tk<te−∫ttkai(u)duI2k(φi(tk))=¯I1(⋅)+¯I2(⋅). |
First, we show that ¯I1(⋅)∈APT(R+,R). Since {tjk:k,j∈Z+} are equipotentially almost periodic, for any ϵ>0, there is a relatively dense set Ω(ϵ) of R+ and Q(ϵ) of Z+ such that for t∈(tk,tk+1], |t−tk|>ϵ, |t−tk+1|>ϵ, k∈Z+, τ∈Ω(ϵ), q∈Q(ϵ), from Lemma 35 (see [35]), we have that t+τ>tk+τ+ϵ>tk+q and tk+q+1>tk+1−ϵ+τ>t+τ; so, tk+q<t+τ<tk+q+1. Then
|¯I1(t+τ)−¯I1(t)|=|∑tk<t+τe−∫t+τtkai(u)duI1k(φi(tk))−∑tk<te−∫ttkai(u)duI1k(φi(tk))|≤∑tk<te−∫ttkai(u)du|I1k+q(φi(tk))−I1k(φi(tk))|≤ϵ∑tk<te−a−i(t−tk)≤ϵ1−e−a−iσ_. |
So, ¯I1(⋅)∈APT(R+,R).
Next, we show that ¯I2(⋅)∈PAP0T(R+,R). For k, n∈Z+ and t∈(tk,tk+1], let
χn(t)={e−∫ttk−nai(u)duI2k−n(φi(tk−n)),0≤n≤k,0,n>k; |
since
limt→∞|e−∫ttk−nai(u)duI2k−n(φi(tk−n))|≤limt→∞e−a−i(t−tk−n)‖I2k(φi(tk))‖∞=0, |
limt→∞|χn(t)|=0; there is χn(t)∈PC0T(R,Rn)⊂PAP0T(R,Rn).
Let Sn(t)=n∑l=0χl(t), because
|sn(t)|≤n∑l=0e−a−i(t−tk−l)‖I2k(φi(tk))‖∞≤n∑l=0e−a−i(t−tk)e−lσ_a−i‖I2k(φi(tk))‖∞. |
Therefore, limt→∞|sn(t)|=0. So, sn(t)∈PAP0T(R,Rn).
And because limn→∞sn(t)=¯I2(⋅) uniformly holds on R+, from Lemma 2.2 in [40], ¯I2(⋅)∈PAP0T(R+,R) holds. Therefore, ∑tk<te−∫ttkai(u)duIk(φi(tk))∈PAPT(R+,R).
In the same way, ∑tk<te−∫ttkbj(u)duJk(ψj(tk))∈PAPT(R+,R) can be obtained. Thus Φ2ϕ(⋅)∈PAPT(R+,R2m).
From Theorems 3.1 and 3.2, the following theorem is obtained.
Theorem 3.3. If conditions (H1)–(H3) are true, for any ϕ=(φ,ψ)T∈PAPT(R+,R2m) and ϕ′=(φ′,ψ′)T∈PAPT(R+,R2m), define the operator
Φϕ(t)=Φ1ϕ(t)+Φ2ϕ(t). |
Then, Φϕ(t) maps PAPT(R+,R2m) into PAPT(R+,R2m), and Φ′ϕ(t) maps PAPT(R+,R2m) into PAPT(R+,R2m).
Theorem 3.4. Assume that conditions (H1)–(H3) are true and the following condition holds:
(H4) pij=max{max{θia−i+I1−e−a−iσ_,θia−i(a+i+a−i)+Ia+i1−e−a−iσ_},max{γjb−j+J1−e−b−jσ_,γib−j(b+j+b−j)+Ib+j1−e−b−jσ_}}<1,
where θi=a+iα+i+m∑j=1s+ji(Pj+Qj), γj=b+jβ+j+m∑i=1t+ij(Hi+Ki), i,j=1,2,⋯,m.
Then, there is a unique piecewise differentiable pseudo almost periodic solution of system (1.1) in the region
X0={ϕ=(φ1,φ2,⋯,φm,ψ1,ψ2,⋯,ψm)T∣ϕ∈PAPT(R+,R2m),‖ϕ−ϕ0‖1≤pL1−p}, |
where ϕ0=(φ01,φ02,⋯,φ0m,ψ01,ψ02,⋯,ψ0m)T, and
L=max{max1≤i≤m{lia−i+l′i,lia−i(a+i+a−i)+a+il′i},max1≤i≤m{ljb−j+l′j,ljb−j(b+j+b−j)+b+jl′j}},φ0i(t)=∫t−∞e−∫tsai(u)du(ci(s)+m∑j=1sji(s)fj(0,0))ds+∑tk<te−∫ttkai(u)duIk(0),ψ0j(t)=∫t−∞e−∫tsbj(u)du(dj(s)+m∑i=1tij(s)gi(0,0))ds+∑tk<te−∫ttkbj(u)duJk(0). |
Proof. It is easy to show that X0⊂PAPT(R+,R2m) is a closed convex set. Rewrite (1.1) into the following form
{x′i(t)=−ai(t)xi(t)+ai(t)∫tt−αi(t)x′i(s)ds+m∑j=1sji(t)fj(xj(t),yj(t−τji(t)))+ci(t),t≠tk,i=1,2,⋯,m,Δxi(t)=xi(t+k)−xi(t−k)=Ik(xi(tk)),t=tk,y′j(t)=−bj(t)yj(t)+bj(t)∫tt−βj(t)y′j(s)ds+m∑i=1tij(t)gi(xi(t−δij(t)),yi(t))+dj(t),t≠tk,j=1,2,⋯,m,Δyj(t)=yj(t+k)−yj(t−k)=Jk(yj(tk)),t=tk. | (3.1) |
We consider the following system
{x′i(t)=−ai(t)xi(t)+¯Fi(t,φ,ψ)+Ik(φi),i=1,2,⋯,m,y′j(t)=−bj(t)yj(t)+¯Gj(t,φ,ψ)+Jk(ψj),j=1,2,⋯,m, | (3.2) |
where
¯Fi(t,φ,ψ)=ai(t)∫tt−αi(t)φ′i(s)ds+m∑j=1sji(t)fj(φj(t),ψj(t−τji(t)))+ci(t),i=1,2,⋯,m,¯Gj(t,φ,ψ)=bj(t)∫tt−βj(t)ψ′j(s)ds+m∑i=1tij(t)gi(φi(t−δij(t)),ψi(t))+dj(t),j=1,2,⋯,m. |
Since M[ai]>0, M[bi]>0, from Lemma 2.7, we have that the linear system
{x′i(t)=−ai(t)xi(t),i=1,2,⋯,m,y′j(t)=−bj(t)yj(t),j=1,2,⋯,m | (3.3) |
admits an exponential dichotomy on R+. Thus, by Lemma 2.8, we obtain that system (3.2) has a unique piecewise pseudo almost periodic solution, which is expressed as follows:
Φϕ(t)=(∫t−∞e−∫tsa1(u)du¯F1(s,φ,ψ)ds+∑tk<te−∫ttka1(u)duIk(φ1(tk))...∫t−∞e−∫tsam(u)du¯Fm(s,φ,ψ)ds+∑tk<te−∫ttkam(u)duIk(φm(tk))∫t−∞e−∫tsb1(u)du¯G1(s,φ,ψ)ds+∑tk<te−∫ttkb1(u)duJk(ψ1(tk))...∫t−∞e−∫tsbm(u)du¯Gm(s,φ,ψ)ds+∑tk<te−∫ttkbm(u)duJk(ψm(tk))). | (3.4) |
Let li=c+i+m∑j=1s+ji|fj(0,0)|, l′i=supk∈Z+|Ik(0)|1−e−a−iσ_, lj=d+j+m∑i=1t+ij|gi(0,0)| and l′j=supk∈Z+|Jk(0)|1−e−b−jσ_. Then,
|φ0i(t)|=|∫t−∞e−∫tsai(u)du(ci(s)+m∑j=1sji(s)fj(0,0))ds+∑tk<te−∫ttkai(u)duIk(0)|≤∫t−∞e−a−i(t−s)(c+i+m∑j=1s+ji|fj(0,0)|)ds+∑tk<te−a−i(t−tk)|Ik(0)|≤lia−i+l′i. |
Similarly, ψ0j(t)≤ljb−j+l′j.
In addition,
|(φ0i)′(t)|=|ci(t)+m∑j=1sji(t)fj[0,0]−ai(t)∫t−∞e−∫tsai(u)du(ci(s)+m∑j=1sji(s)fj[0,0])ds−∑tk<tai(t)e−∫ttkai(u)duIk(0)|≤li(1+a+i∫t−∞e−a−i(t−s)ds)+a+i∑tk<te−a−i(t−tk)Ik(0)≤lia−i(a+i+a−i)+a+il′i. |
Similarly, |(ψ0j)′(t)|≤ljb−j(b+j+b−j)+b+jl′j.
Therefore, ‖ϕ0‖1≤L. For ϕ∈X0, ‖ϕ‖1≤‖ϕ−ϕ0‖1+‖ϕ0‖1≤L1−p.
We will prove that Φϕ:X0→X0 is a contraction.
First, we prove that for any ϕ∈X0 and Φϕ∈X0, let
Fi(t,φ,ψ)=ai(t)∫tt−αi(t)φ′i(s)ds+m∑j=1sji(t)(fj(φj(t),ψj(t−τji(t)))−fj(0,0)),Gj(t,φ,ψ)=bj(t)∫tt−βj(t)ψ′j(s)ds+m∑i=1tij(t)(gi(φi(t−δij(t)),ψi(t))−gi(0,0)). |
Then,
|Fi(t,φ,ψ)|≤a+i‖φ′‖∞α+i+m∑j=1s+ji(|fj(φj(t),ψj(t−τji(t)))−fj(0,0)|)≤a+i‖φ′‖∞α+i+m∑j=1s+ji(Pj|φj(t)|+Qj|ψj(t−τji(t))|)≤a+i‖φ′‖∞α+i+m∑j=1s+ji(Pj‖φ‖∞+Qj‖ψ‖∞)≤(a+iα+i+m∑j=1s+ji(Pj+Qj))‖ϕ‖1=θi‖ϕ‖1,i=1,2,⋯,m. |
Similarly, |Gj(t,φ,ψ)|≤(b+jβ+j+m∑i=1t+ij(Hi+Ki))‖ϕ‖1=γj‖ϕ‖1,j=1,2,⋯,m, where θi=a+iα+i+m∑j=1s+ji(Pj+Qj),γj=b+jβ+j+m∑i=1t+ij(Hi+Ki),i,j=1,2,⋯,m.
For i=1,2,⋯,m, we have
|(Φϕ−ϕ0)i(t)|=|∫t−∞e−∫tsai(u)duFi(s,φ,ψ)ds+∑tk<te−∫ttkai(u)du(Ik(φi(tk))−Ik(0))|≤θi‖ϕ‖1∫t−∞e−(t−s)a−ids+∑tk<te−(t−tk)a−iI‖ϕ‖1=(θia−i+I1−e−a−iσ_)‖ϕ‖1; |
similarly, for j=1,2,⋯,m, |(Φϕ−ϕ0)j(t)|≤(γjb−j+J1−e−b−jσ_)‖ϕ‖1.
Furthermore, for i=1,2,⋯,m, we have
|(Φϕ−ϕ0)′i(t)|=|(∫t−∞e−∫tsai(u)duFi(s,φ,ψ)ds+∑tk<te−∫ttkai(u)du(Ik(φi(tk))−Ik(0)))′|=|Fi(t,φ,ψ)−ai(t)∫t−∞e−∫tsai(u)duFi(s,φ,ψ)ds−∑tk<tai(t)e−∫ttkai(u)du(Ik(φi(tk))−Ik(0))|≤(θia−i(a+i+a−i)+a+iI1−e−a−iσ_)‖ϕ‖1. |
Similarly, for j=1,2,⋯,m, we have
|(Φϕ−ϕ0)′j(t)|≤(γjb−j(b+j+b−j)+b+jJ1−e−b−jσ_)‖ϕ‖1. |
In the light of (H4), we have that ‖Φϕ−ϕ0‖1≤p‖ϕ‖1<pL1−p, that is Φϕ∈X0.
Next, we prove that Φϕ is a contraction. For ϕ=(φ1,φ2,⋯,φm,ψ1,ψ2,⋯,ψm)T,¯ϕ=(¯φ1,¯φ2,⋯,¯φm,¯ψ1,¯ψ2,⋯,¯ψm)T∈X0, it follows that
|¯Fi(t,φ,ψ)−¯Fi(t,¯φ,¯ψ)|≤ai(t)∫tt−αi(t)|φ′i(s)−¯φ′i(s)|ds+m∑j=1sji(t)|fj[φj(t),ψj(t−τji(t))]−fj[¯φj(t),¯ψj(t−τji(t))]|≤a+i|φ′−¯φ′|∞α+i+m∑j=1s+ji(Pj|φ−¯φ|∞+Qj|ψ−¯ψ|∞)≤(a+iα+i+m∑j=1s+ji(Pj+Qj))‖ϕ−¯ϕ‖1=θi‖ϕ−¯ϕ‖1,i=1,2,⋯,m, |
and |¯Gj(t,φ,ψ)−¯Gj(t,¯φ,¯ψ)|≤(b+jβ+j+m∑i=1t+ij(Pi+Qi))‖ϕ−¯ϕ‖1=γj‖ϕ−¯ϕ‖1,j=1,2,⋯,m.
And because
|∑tk<te−∫ttkai(u)duIk(φi(tk))−∑tk<te−∫ttkai(u)duIk(¯φi(tk))|≤∑tk<te−∫ttkai(u)du|Ik(φi(tk))−Ik(¯φi(tk))|≤I1−e−a−iσ_‖ϕ−¯ϕ‖1, |
and
|(∑tk<te−∫ttkai(u)duIk(φi(tk))−∑tk<te−∫ttkai(u)duIk(¯φi(tk)))′|≤ai(t)∑tk<te−∫ttkai(u)du|Ik(¯φi(tk))−Ik(φi(tk))|≤a+iI1−e−a−iσ_‖ϕ−¯ϕ‖1. |
Similarly,
|∑tk<te−∫ttkbj(u)duJk(ψj(tk))−∑tk<te−∫ttkbj(u)duJk(¯ψj(tk))|≤I1−e−b−jσ_‖ϕ−¯ϕ‖1|(∑tk<te−∫ttkbj(u)duJk(ψj(tk))−∑tk<te−∫ttkbj(u)duJk(¯ψj(tk)))′|≤b+jI1−e−b−jσ_‖ϕ−¯ϕ‖1. |
So, for i=1,2,⋯,m, we have
|Φϕ−Φ¯ϕ|i≤∫t−∞e−∫tsai(u)du|¯Fi(s,φ,ψ)−¯Fi(s,¯φ,¯ψ)|ds+∑tk<te−∫ttkai(u)du|Ik(φi(tk))−Ik(¯φi(tk))|≤θia−i‖ϕ−¯ϕ‖1+I1−e−a−iσ_‖ϕ−¯ϕ‖1=(θia−i+I1−e−a−iσ_)‖ϕ−¯ϕ‖1, |
and
|(Φϕ−Φ¯ϕ)′|i=(∫t−∞e−∫tsai(u)du(¯Fi(s,φ,ψ)−¯Fi(s,¯φ,¯ψ))ds+∑tk<te−∫ttkai(u)du(Ik(φi(tk))−Ik(¯φi(tk))))′≤|Fi(t,φ,ψ)−Fi(t,¯φ,¯ψ)|+a+i∫t−∞e−∫tsai(u)du|Fi(s,φ,ψ)−Fi(s,¯φ,¯ψ)|ds+a+iI1−e−a−iσ_‖ϕ−¯ϕ‖1≤(θia−i(a+i+a−i)+a+iI1−e−a−iσ_)‖ϕ−¯ϕ‖1. |
Similarly, for j=1,2,⋯,m, we have
|Φϕ−Φ¯ϕ|j≤(γjb−j+J1−e−b−jσ_)‖ϕ−¯ϕ‖1,|(Φϕ−Φ¯ϕ)′|j≤(γjb−j(b+j+b−j)+b+jJ1−e−b−jσ_)‖ϕ−¯ϕ‖1. |
By (H4), we have
‖Φϕ−Φ¯ϕ‖1≤pij‖ϕ−¯ϕ‖1, |
which implies that Φϕ is a contraction. Therefore, Φϕ has a fixed point in X0, that is, (1.1) has a unique piecewise pseudo almost periodic solution in X0. This completes the proof.
This section states and proves the sufficient conditions for the global exponential stability of piecewise pseudo almost periodic solutions of the system described by (1.1)–(1.2).
Theorem 4.1. Assume that conditions (H1)–(H4) are true and the following condition holds.
(H5) For i, j=1,2,⋯,m,
a−i>max{a+iα+i+m∑j=1s+ji(Pj+Qj),(a+i+a−i)[a+iα+i+m∑j=1s+ji(Pj+Qj)]},b−j>max{b+jβ+j+m∑i=1t+ij(Hi+Ki),(b+j+b−j)[b+jβ+j+m∑i=1t+ij(Hi+Ki)]}. |
Then, the piecewise pseudo almost periodic solutions of the system described by (1.1)–(1.2) is globally exponentially stable.
Proof. According to Theorem 3.4, the system (1.1) has at least one piecewise differentiable pseudo-almost periodic solution (¯x1(t),¯x2(t),⋯,¯xm(t),¯y1(t),¯y2(t),⋯,¯ym(t))T∈X0 with the initial value condition ¯ϕ=(¯φ1(θ),¯φ2(θ),⋯,¯φm(θ),¯ψ1(θ),¯ψ2(θ),⋯,¯ψm(θ))T. Let
(x1(t),x2(t),⋯,xm(t),y1(t),y2(t),⋯,ym(t))T |
be an arbitrary solution of system (1.1) with the initial value condition
ϕ=(φ1(θ),φ2(θ),⋯,φm(θ),ψ1(θ),ψ2(θ),⋯,ψm(θ))T. |
Let ui(⋅)=xi(⋅)−¯xi(⋅), vj(⋅)=yj(⋅)−¯yj(⋅); for t≠tk and t>0, we have
u′i(t)=−ai(t)ui(t)+ai(t)∫tt−αi(t)u′i(s)ds+m∑j=1sji(t)[fj(xj(t),yj(t−τji(t)))−fj(¯xj(t),¯yj(t−τji(t)))], | (4.1) |
v′j(t)=−bj(t)vj(t)+bj(t)∫tt−βj(t)vj(s)ds+m∑i=1tij(t)[gi(xi(t−δij(t)),yi(t)−gi(xi(t−δij(t)),yi(t)]. | (4.2) |
The initial value conditions are as follows:
{ui(θ)=φi(θ)−¯φi(θ),θ∈[−r1,0],i=1,2,⋯,m,vj(θ)=ψj(θ)−¯ψj(θ),θ∈[−r2,0],j=1,2,⋯,m. |
Multiplying e∫stkai(τ)dτ on both sides of (4.1) and e∫stkbj(τ)dτ on both sides of (4.2), and integrating over [tk,t], we derive
ui(t)=ui(tk)e−∫ttkai(τ)dτ+∫ttke−∫tsai(τ)dτ{ai(s)∫ss−αi(s)u′i(τ)dτ+m∑j=1sji(s)[fj(xj(s),yj(s−τji(s)))−fj(¯xj(s),¯yj(s−τji(s)))]}ds, | (4.3) |
vj(t)=vj(tk)e−∫ttkbj(τ)dτ+∫ttke−∫tsbj(τ)dτ{bj(s)∫ss−βj(s)v′j(τ)dτ+m∑i=1tij(s)[fi(xi(s−δij(s)),yi(s))−fj(¯xj(s−δij(s)),¯yj(s))]}ds. | (4.4) |
Consider the following functions:
Ci(s)=a−i−s−hi(s),s∈(0,a−i),s∈(0,a−i),Di(s)=a−i−s−(a+i+a−i)hi(s)+shi(s),s∈(0,a−i),¯Cj(s)=b−j−s−¯hj(s),s∈(0,b−j),¯Dj(s)=b−j−s−(b+j+b−j)¯hj(s)+s¯hj(s),s∈(0,b−j), |
where hi(s)=a+iesα+i−1s+m∑j=1s+ji(Pj+Qjesτ+ji), ¯hj(s)=b+jesβ+j−1s+m∑i=1t+ij(Hiesδ+ij+Ki).
For i,j=1,2,⋯,m, since Ci(s), Di(s) are continuous on (0,a−i), ¯Cj(s), ¯Dj(s) are continuous on (0,b−j), lims→0Ci(s)>0, lims→0¯Cj(s)>0, lims→0Di(s)>0, lims→0¯Dj(s)>0, lims→a−iCi(s)<0, lims→a−iDi(s)<0, lims→b−j¯Cj(s)<0 and lims→b−j¯Dj(s)<0. There exist constants si1, si2∈(0,a−i) and sj3, sj4∈(0,b−j) such that Ci(si1)=0, Di(si2)=0, ¯Cj(sj3)=0 and ¯Dj(sj4)=0. There exists a constant η∈(0,min1≤i,j≤m{si1,si2,sj3,sj4}) such that Ci(η)>0, Di(η)>0, ¯Cj(η)>0 and ¯Dj(η)>0. Obviously, there is η∈(0,min{a−i,b−j}).
Take a positive sequence {Mk}∞k=1 satisfying
1M1<1N0min{hi(η)a−i−η,¯hj(η)b−j−η}, |
1Mk<1Nk−1min{hi(η)a−i−η,¯hj(η)b−j−η}, |
where Nk=Mkmax{1+I,1+J}, k≥1, N0=max{1+I,1+J}.
Let
w(t)={wi(t)=ui(t),i=1,2,⋯,m,wm+j(t)=vj(t),j=1,2,⋯,m. |
Let Υ=‖ϕ−¯ϕ‖1; for i,j=1,2,⋯,m, it is obvious that there exists a positive constant M0>1 such that the following inequalities hold
‖wi(θ)‖1≤M0Υe−ηt,θ∈[−r1,0],‖wm+j(θ)‖1≤M0Υe−ηt,θ∈[−r2,0]. |
For t=t+0, we have
|ui(t+0)|=|xi(t+0)−¯xi(t+0)|≤|xi(t−0)−¯xi(t−0)|+|I1(xi(t0))−I1(¯xi(t0))|=|ui(t−0)|+I|ui(t−0)|≤(1+I)Υ,|vj(t+0)|=|yj(t+0)−¯yj(t+0)|≤|yj(t−0)−y∗j(t−0)|+|J1(yj(t0))−J1(y∗j(t0))|=|vj(t−0)|+J|vj(t−0)|≤(1+J)Υ. |
Therefore,
‖wl(t+0)‖∞≤N0Υ. | (4.5) |
Now we will prove, for t∈(0,t1], l=1,2,⋯,2m, that
‖wl(t)‖1≤M1Υe−ηt. | (4.6) |
In order to prove that (4.6) is true, we first prove that for any ρ>1, t∈(0,t1], the following inequality holds:
‖wl(t)‖1<M1ρΥe−ηt. | (4.7) |
If (4.7) is not established, there are l0∈{1,2,⋯,2m} and t′∈(0,t1] such that
‖wl0(t′)‖1≥M1ρΥe−ηt′,‖wl(t)‖1<M1ρΥe−ηt,t∈(0,t′), | (4.8) |
where t′∈(0,t1] is the smallest variable such that (4.8) holds. There is a constant σ≥1 such that
‖wl0(t′)‖1=σM1ρΥe−ηt′ | (4.9) |
and ‖wl(t)‖1<M1ρΥe−ηt,t∈(0,t′).
For l0∈{1,2,⋯,m}, let i0=l0; from (H5) and (4.3), we derive
|ui0(t′)|≤N0Υe−a−i0t′+∫t′0e−∫t′sai0(τ)dτ{a+i0∫ss−αi0(s)M1ρΥe−ητdτ+m∑j=1s+ji0(Pj|uj(s)|+Qj|vj(s−τji0(s))|)}ds≤N0Υe−a−i0t′+∫t′0e−∫t′sai0(τ)dτ{a+i0∫ss−αi(s)M1ρΥe−ητdτ+m∑j=1s+ji0(PjM1ρΥe−ηs+QjM1ρΥe−η(s−τ+ji0))}ds≤N0Υe−a−i0t′+e−a−i0t′M1ρΥe(a−i0−η)t′−1a−i0−ηhi0(η)=N0Υe−a−i0t′−M1ρΥe−a−i0t′a−i0−ηhi0(η)+M1ρΥe−ηt′a−i0−ηhi0(η)≤{N0M1−1a−i0−ηhi0(η)}e−a−i0t′M1ρΥ+1a−i0−ηhi0(η)M1ρΥe−ηt′<M1ρΥe−ηt′, | (4.10) |
|u′i0(t′)|=|−ui0(0)ai0(t′)e−∫t′0ai0(τ)dτ−ai0(t′)∫t′0e−∫t′sai0(τ)dτ{ai0(s)∫ss−αi0(s)u′i0(τ)dτ+m∑j=1sji0(s)[fj(xj(s),yj(s−τji0(s)))−fj(¯xj(s),¯yj(s−τji0(s)))]}ds+e−∫t′0ai0(τ)dτe∫t′0ai0(τ)dτ{ai0(t′)∫t′t′−αi0(t′)u′i0(τ)dτ+m∑j=1sji0(t′)[fj(xj(t′),yj(t′−τji0(t′)))−fj(¯xj(t′),¯yj(t′−τji0(t′)))]}|≤|ui0(0)|a+i0e−a−i0t′+a+i0e−a−i0t′∫t′0ea−i0s{a+i0∫ss−αi0(s)|u′i0(τ)|dτ+m∑j=1s+ji0(Pj|uj(s)|+Qj|vj(s−τji0(s))|)}ds+a+i∫t′t′−αi0(t′)|u′i0(τ)|dτ+m∑j=1s+ji0(Pj|uj(t′)|+Qj|vj(t′−τji0(t′))|)≤N0Υa+i0e−a−i0t′+a+i0e−a−i0t′∫t′0ea−i0s{a+i0∫ss−αi0(s)M1ρΥe−ητdτ+m∑j=1s+ji0(PjM1ρΥe−ηs+QjM1ρΥe−ηseητ+ji0)}ds+a+i0∫t′t′−αi0(t′)M1ρΥe−ητdτ+m∑j=1s+ji0(PjM1ρΥe−ηt′+QjM1ρΥe−ηt′eητ+ji0)≤{N0M1a+i0e−a−i0t′+a+i0e−ηt′−e−a−i0t′a−i0−ηhi0(η)}M1ρΥ+hi0(η)M1ρΥe−ηt′={N0M1−hi0(η)a−i0−η}a+i0e−a−i0t′M1ρΥ+{a+i0a−i0−ηhi0(η)+hi0(η)}M1ρΥe−ηt′<M1ρΥe−ηt′, | (4.11) |
where hi0(η)=a+i0eηα+i0−1η+m∑j=1s+ji0(Pj+Qjeητ+ji0). Equations (4.10) and (4.11) are both in conflict with (4.9).
For l0∈{m+1,m+2,⋯,2m}, let j0=l0−m; similarly, from (H5) and (4.4), we derive
|vj0(t′)|≤{N0M1−1b−j0−η¯hj0(η)}e−b−j0t′M1ρΥ+1b−j0−η¯hj0(η)M1ρΥe−ηt′<M1ρΥe−ηt′, | (4.12) |
|v′j0(t′)|={N0M1−¯hj0(η)b−j0−η}b+j0e−b−j0t′M1ρΥ+{b+j0b−j0−η¯hj0(η)+¯hj0(η)}M1ρΥe−ηt′<M1ρΥe−ηt′, | (4.13) |
where ¯hj0(η)=b+j0eηβ+j0−1η+m∑i=1t+ij0(Hieηδ+ij0+Ki). Equations (4.12) and (4.13) are both in conflict with (4.9).
Therefore, (4.8) is not valid. For t∈(0,t1], we have that (4.7) holds. Let ρ→1; for t∈(0,t1], (4.6) holds.
For t=t+1, we have
|ui(t+1)|=|xi(t+1)−¯xi(t+1)|≤|xi(t−1)−¯xi(t−1)|+|I1(xi(t1))−I1(¯xi(t1))|=|ui(t−1)|+I|ui(t−1)|≤(1+I)M1Υe−ηt1,|vj(t+1)|=|yj(t+1)−¯yj(t+1)|≤|yj(t−1)−y∗j(t−1)|+|J1(yj(t1))−J1(y∗j(t1))|=|vj(t−1)|+J|vj(t−1)|≤(1+J)M1Υe−ηt1. |
Therefore,
‖wl(t+1)‖∞≤N1Υe−ηt1. | (4.14) |
For t∈(t1,t2], l=1,2,⋯,2m, we certify that
‖wl(t)‖1≤M2Υe−ηt. | (4.15) |
In order to prove that (4.15) is true, we first prove that for any ρ>1 and t∈(t1,t2], the following inequality holds:
‖wl(t)‖1<M2ρΥe−ηt. | (4.16) |
If (4.16) is not established, there are l1∈{1,2,⋯,2m} and t″∈(t1,t2] such that
‖wl1(t″)‖1≥M2ρΥe−ηt′,‖wl(t)‖1<M2ρΥe−ηt,t∈(t1,t″), | (4.17) |
where t″∈(t1,t2] is the smallest variable such that (4.17) holds. There is a constant σ1≥1 such that
‖wl1(t′)‖1=σ1M2ρΥe−ηt′ | (4.18) |
and ‖wl(t)‖1<M2ρΥe−ηt,t∈(t1,t″).
For l1∈{1,2,⋯,m}, let i1=l1; from (H5) and (4.3), we derive
|ui1(t″)|≤N1ρΥe−ηt1e−a−i1(t″−t+1)+∫t″t+1e−∫t″sai1(τ)dτ{a+i1∫ss−αi1(s)M2ρΥe−ητdτ+m∑j=1s+ji1(Pj|uj(s)|+Qj|vj(s−τji1(s))|)}ds≤N1ρΥe−ηt1e−a−i1(t″−t+1)+e−a−i1t″M2ρΥe(a−i1−η)t″−e(a−i1−η)t1a−i1−ηhi1(η)=N1ρΥe−ηt1e−a−i1(t″−t+1)+M2ρΥe−ηt″−e−a−i1(t″−t1)e−ηt1a−i1−ηhi1(η)≤{N1M2−hi1(η)a−i1−η}e−a−i1(t″−t1)M2ρΥe−ηt1+1a−i1−ηhi1(η)M2ρΥe−ηt″<M2ρΥe−ηt″, | (4.19) |
|u′i1(t″)|≤|ui1(t+1)|a+i1e−a−i1(t″−t+1)+a+i1e−a−i1t″∫t″t+1ea−i1s{a+i1∫ss−αi1(s)|u′i1(τ)|dτ+m∑j=1s+ji1(Pj|uj(s)|+Qj|vj(s−τji1(s))|)}ds+a+i∫t″t″−αi1(t″)|u′i1(τ)|dτ+m∑j=1s+ji1(Pj|uj(t″)|+Qj|vj(t″−τji1(t″))|)≤N1ρΥe−ηt1a+i1e−a−i1(t″−t1)+a+i1e−a−i1t″∫t″t+1ea−i1s{a+i1∫ss−αi1(s)M2ρΥe−ητdτ+m∑j=1s+ji1(PjM2ρΥe−ηs+QjM2ρΥe−ηseητ+ji1)}ds+a+i1∫t″t″−αi1(t″)M2ρΥe−ητdτ+m∑j=1s+ji1(PjM2ρΥe−ηt″+QjM2ρΥe−ηt″eητ+ji1)≤{N1M2−hi1(η)a−i1−η}a+i1e−a−i1(t″−t1)M2ρΥe−ηt1+{a+i1a−i1−ηhi1(η)+hi1(η)}M2ρΥe−ηt″<M2ρΥe−ηt″, | (4.20) |
where hi1(η)=a+i1eηα+i1−1η+m∑j=1s+ji1(Pj+Qjeητ+ji1). Equations (4.19) and (4.20) are both in conflict with (4.18).
For l1∈{m+1,m+2,⋯,2m}, let j1=l1−m; similarly, from (H5) and (4.4), we derive
|vj1(t″)|≤{N1M2−¯hj1(η)b−j1−η}e−b−j1(t″−t1)M2ρΥe−ηt1+1b−j1−η¯hj1(η)M2ρΥe−ηt″<M2ρΥe−ηt″, | (4.21) |
|v′j1(t″)|≤{N1M2−¯hj1(η)b−j1−η}b+j1e−b−j1(t″−t1)M2ρΥe−ηt1+{b+j1b−j1−η¯hj1(η)+¯hj1(η)}M2ρΥe−ηt″<M2ρΥe−ηt″, | (4.22) |
where ¯hj1(η)=b+j1eηβ+j1−1η+m∑i=1t+ij1(Hieηδ+ij1+Ki). Equations (4.21) and (4.22) are both in conflict with (4.18).
Therefore, (4.17) is not valid. For t∈(t1,t2], we have that (4.16) holds. Let ρ→1; for t∈(t1,t2], (4.15) holds.
Similar to the proof of (4.14) and (4.15), we deduce that
‖wl(t+k−1)‖∞≤Nk−1Υe−ηtk−1,‖wl(t)‖1≤MkΥe−ηt,t∈(tk−1,tk]. | (4.23) |
Let M=supk≥1max{Mk,Nk−1}; obviously, M>1; we have
‖wl(t)‖1≤MΥe−ηt,t>0. | (4.24) |
Therefore, for i,j=1,2,⋯,m,
‖xi(t)−¯xi(t)‖1≤MΥe−ηt,t>0,‖yj(t)−¯yj(t)‖1≤MΥe−ηt,t>0. | (4.25) |
From Definition 2.8, the piecewise pseudo almost periodic solution
(¯x1(t),¯x2(t),⋯,¯xm(t),¯y1(t),¯y2(t),⋯,¯ym(t))T |
of system (1.1) with the initial value conditions
¯ϕ=(¯φ1(θ),¯φ2(θ),⋯,¯φm(θ),¯ψ1(θ),¯ψ2(θ),⋯,¯ψm(θ))T |
is said to be globally exponentially stable.
This section presents an example to illustrate the effectiveness of our results obtained in previous sections. Consider the following interval general BAM neural network with mixed time-varying delays and impulsive perturbations:
{x′i(t)=−ai(t)xi(t−αi(t))+2∑j=1sji(t)fj(xj(t),yj(t−τji(t)))+ci(t),t>0,t≠2k,i=1,2,Δxi(2k)=Ik(xi(2k)),k∈Z+,y′j(t)=−bj(t)yj(t−βj(t))+2∑i=1tij(t)gi(xi(t−δij(t)),yi(t))+dj(t),t>0,t≠2k,j=1,2,Δyj(2k)=Jk(yj(2k)),k∈Z+, | (5.1) |
where fj(xj(t),yj(t−τji(t)))=|xj(t)|+|yj(t−τji(t)|,gi(xi(t−δij(t)),yi(t))=|xi(t−δij(t))|+|yi(t)|,i,j=1,2, and
a(t)=(0.2+0.1cos2t0.2+0.1sin4t),b(t)=(0.2+0.1cos2t0.2+0.1sin4t),s(t)=(180+180sint180+180cost180+180sint180+180cost); |
t(t)=(180+180sint180+180cost180+180sint180+180cost),c(t)=(0.5+0.1sint10(1+t2)0.2+0.1cost10(1+t2)),d(t)=(0.3+0.1sint10(1+t2)0.4+0.1cost10(1+t2)); |
α(t)=(18|sint|18|cost|),β(t)=(18|cost|18|sint|),δ(t)=(0.4|sint|0.2|cost|0.3|cost|0.1|sint|); |
τ(t)=(0.5|cost|0.3|sint|0.2|sint|0.4|cost|),(Δx1(2k)Δx2(2k)Δy1(2k)Δy2(2k))=(−180x1(2k)+180sin(x1(2k))+4−180x2(2k)+180cos(x2(2k))+3−180y1(2k)+180sin(y1(2k))+2−180y2(2k)+180cos(y2(2k))+1.5). |
Obviously,
a+=(0.30.3),a−=(0.20.2),b+=(0.30.3),b−=(0.20.2),α+=(1818),s+=(280280280280); |
β+=(1818),t+=(280280280280),δ+=(0.40.20.30.1),τ+=(0.50.30.20.4). |
By calculation, it is easy to acquire σ_=2, Pj=Qj=Hi=Ki=1, i,j=1,2,⋯,m, I=J=140 and
(θ1θ2γ1γ2)=(1180118011801180),(I1−e−a−1σ_I1−e−a−2σ_J1−e−b−1σ_J1−e−b−2σ_)=(0.07580.07580.07580.0758). |
For i,j=1,2,⋯,m, pij=0.7633<1 and 0.2>1180. That is, the conditions (H1)–(H5) are valid. From Theorems 3.4 and 4.1, system (5.1) has a piecewise pseudo almost periodic solution, which is globally exponentially stable. Computer simulations of the existence and stable behavior of the pseudo-almost periodic solutions are given in Figures 1 and 2, respectively.
The pseudo almost periodic behaviors have been applied to the qualitative theory of differential equations. Piecewise pseudo almost periodic solutions of BAM neural networks are generalizations of almost periodic solutions, which have obvious scientific significance and application value in many fields such as signal processing, pattern recognition, associative memory, image processing and optimization problems.
This paper investigated the piecewise pseudo almost periodic solutions of the interval general BAM neural networks with mixed time-varying delays and impulsive perturbations. By employing the exponential dichotomy of linear differential equations, fixed-point theory for contraction mapping, differential inequality techniques and mathematical methods of induction, the effective conditions for the existence and global exponential stability of piecewise pseudo almost periodic solutions of system (1.1) have been established. From Theorems 3.4 and 4.1, the delays contained in the activation functions do not affect the existence and global exponential stability of piecewise pseudo almost periodic solutions of system (1.1). The existence and global exponential stability of the piecewise pseudo almost periodic solutions of the system (1.1) are determined by the negative feedback terms, leakage delays, connection weights, impulsive perturbations and the Lipschitz constants of the activation functions.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
The authors declare that they have no competing interests concerning the publication of this article.
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