The following theorem is about the existence and global attractivity of positive periodic solutions of system (4).
Proof.
We firstly consider the existence of positive periodic solutions of system (4). For system (4), we introduce new variables ui(t) (i=1,2,…,n) such that
(13)xi(t)=exp{ui(t)}, i=1,2,…,n.
Then, system (4) is rewritten in the following form:
(14)u˙i(t)=ri(t)-αi(t)exp{ui(t)} +∑l=1maiil(t)∫-τ0kiil(s)exp{ui(t+s)}ds +∑j≠in ∑l=1maijl(t)∫-τ0kijl(s)exp{uj(t+s)}ds, i=1,2,…,n.
In order to apply Lemma 1 to system (14), we introduce the normed vector spaces X and Z as follows. Let C(R,Rn) denote the space of all continuous function u(t)=(u1(t),u2(t),…,un(t)):R→Rn. We take
(15)X=Z={u(t)∈C(R,Rn):u(t) zis an ω-periodic function}
with norm
(16)∥u∥=∑i=1nmaxt∈[0,ω]|ui(t)|.
It is obvious that X and Z are the Banach spaces.
We define a linear operator L:DomL⊂X→Z and a continuous operator N:X→Z as follows:
(17)Lu(t)=u˙(t),Nu(t)=(Nu1(t),Nu2(t),…,Nun(t)),
where
(18)Nui(t)=ri(t)-αi(t)exp{ui(t)} +∑l=1maiil(t)∫-τ0kiil(s)exp{ui(t+s)}ds +∑j≠in ∑l=1maijl(t)∫-τ0kijl(s)exp{uj(t+s)}ds, i=1,2,…,n.
Further, we define continuous projectors P:X→X and Q:Z→Z as follows:
(19)Pu(t)=1ω∫0ωu(t)dt, Qv(t)=1ω∫0ωv(t)dt.
We easily see that ImL={v∈Z:∫0ωv(t)dt=0} and KerL=Rn. It is obvious that ImL is closed in Z and dimKerL=n. Since for any v∈Z there are unique v1∈Rn and v2∈ImL with
(20)v1=1ω∫0ωv(t)dt, v2(t)=v(t)-v1,
such that v(t)=v1+v2(t), we have co dim ImL=n. Therefore, L is a Fredholm mapping of index zero. Furthermore, the generalized inverse (to L) Kp:ImL→KerP∩DomL is given in the following form:
(21)Kpv(t)=∫0tv(s)ds-1ω∫0ω∫0tv(s)ds dt.
For convenience, we denote F(t)=(F1(t),F2(t),…,Fn(t)) as follows:
(22)Fi(t)=ri(t)-αi(t)exp{ui(t)} +∑l=1maiil(t)∫-τ0kiil(s)exp{ui(t+s)}ds +∑j≠in ∑l=1maijl(t)∫-τ0kijl(s)exp{uj(t+s)}ds, i=1,2,…n.
Thus, we have
(23)QNu(t)=1ω∫0ωF(t)dt,Kp(I-Q)Nu(t)=KpINu(t)-KpQNu(t)=∫0tF(s)ds-1ω∫0ω∫0tF(s)ds dt+(12-tω)∫0ωF(s)ds.
From formulas (23), we easily see that QN and Kp(I-Q)N are continuous operators. Furthermore, it can be verified that Kp(I-Q)N(Ω¯)¯ is compact for any open bounded set Ω⊂X by using Arzela-Ascoli theorem and QN(Ω¯) is bounded. Therefore, N is L-compact on Ω¯ for any open bounded subset Ω⊂X.
Now, we reach the position to search for an appropriate open bounded subset Ω for the application of the continuation theorem (Lemma 1) to system (4).
Corresponding to the operator equation Lu(t)=λNu(t) with parameter λ∈(0,1), we have
(24)u˙i(t)=λFi(t), i=1,2,…,n,
where Fi(t) (i=1,2,…,n) is given in (22).
Assume that u(t)=(u1(t),u2(t),…,un(t))∈X is a solution of system (24) for some parameter λ∈(0,1). By integrating system (24) with the interval [0,ω], we obtain the following:
(25)∫0ω[ri(t)-αi(t)exp{ui(t)}∑j≠in∫-τ0 +∑l=1maiil(t)∫-τ0kiil(s)exp{ui(t+s)}ds +∑j≠in ∑l=1maijl(t)∫-τ0kijl(s)exp{uj(t+s)}ds]dt=0, i=1,2,…n.
Consequently,
(26)∫0ω[αi(t)exp{ui(t)}∑j≠in∫-τ0 -∑l=1maiil(t)∫-τ0kiil(s)exp{ui(t+s)}ds -∑j≠in ∑l=1maijl(t)∫-τ0kijl(s)exp{uj(t+s)}ds]dt =r-iω, i=1,2,…n.
From the continuity of u(t)=(u1(t),u2(t),…,un(t)), there exist constants ξi,ηi∈[0,ω] (i=1,2,…,n) such that
(27)ui(ξi)=maxt∈[0,ω]ui(t), ui(ηi)=mint∈[0,ω]ui(t), i=1,2,…,n.
From (26) and (27), we obtain
(28)∫0ωαi(t)exp{ui(ξi)}dt≥r-iω, i=1,2,…,n.
Therefore, we further have
(29)ui(ξi)≥ln(r-iα-i), i=1,2,…,n.
For each i,j=1,2,…,n and l=1,2,…,m, we have
(30)∫0ωaijl(t)∫-τ0kijl(s)exp{uj(t+s)}ds dt =∫-τ0∫0ωaijl(t)kijl(s)exp{uj(t+s)}dt ds =∫-τ0∫ss+ωaijl(v-s)kijl(s)exp{uj(v)}dv ds =∫-τ0∫0ωaijl(v-s)kijl(s)exp{uj(v)}dv ds =∫0ω∫-τ0aijl(v-s)kijl(s)exp{uj(v)}ds dv =∫0ω(∫-τ0aijl(t-s)kijl(s)ds)exp{uj(t)}dt.
Hence, from (26) we further obtain
(31)∫0ω[(αi(t)-∑l=1m(∫-τ0aiil(t-s)kiil(s)ds))exp{ui(t)}∑j≠in -∑j≠in ∑l=1m(∫-τ0aijl(t-s)kijl(s)ds)exp{uj(t)}]dt =r-iω, i=1,2,…,n.
Consequently,
(32)∫0ω[∑j≠1n ∑l=1m(∫-τ0a1jl(t-s)k1jl(s)ds)(α1(t)-∑l=1m(∫-τ0a11l(t-s)k11l(s)ds))exp{u1(t)} -∑j≠1n ∑l=1m(∫-τ0a1jl(t-s)k1jl(s)ds)exp{uj(t)}]dt +∫0ω[(α2(t)-∑l=1m(∫-τ0a22l(t-s)k22l(s)ds)) ×exp{u2(t)}-∑j≠2n ∑l=1m(∫-τ0a2jl(t-s)k2jl(s)ds) × exp{uj(t)}(∑l=1m(∫-τ0a22l(t-s)))]dt +⋯+∫0ω[(αn(t)∫-τ0 -∑l=1m(∫-τ0annl(t-s)knnl(s))ds∑l=1m) ×exp{un(t)} -∑j≠nn ∑l=1m(∫-τ0anjl(t-s)knjl(s)ds) ×exp{uj(t)}(αn(t)∫-τ0]dt =∫0ω[α1(t)-∑l=1m(∫-τ0a11l(t-s)k11l(s)ds∑j≠1n +∑j≠1n∫-τ0aj1l(t-s)kj1l(s)ds)] ×exp{u1(t)}dt +∫0ω[α2(t)-∑l=1m(∫-τ0a22l(t-s)k22l(s)ds +∑j≠2n∫-τ0aj2l(t-s) × kj2l(s)ds∫-τ0)∑l=1m] ×exp{u2(t)}dt +⋯+∫0ω[αn(t)×kjnl(s)ds∫-τ0)∑l=1m(∫-τ0annl(t-s)knnl(s)ds -∑l=1m(∫-τ0annl(t-s)knnl(s)ds +∑j≠nn∫-τ0ajnl(t-s) × kjnl(s)ds∫-τ0)∑l=1m(∫-τ0annl(t-s)knnl(s)ds] ×exp{un(t)}dt =∫0ω[×kj1l(s)∑j≠1n](∫-τ0ds)α1(t) -∑l=1m(∫-τ0[a11l(t-s)k11l(s)+∑j≠1naj1l(t-s) × kj1l(s)∑j≠1n]ds∫-τ0)] ×exp{u1(t)}dt +∫0ω[α2(t)-∑l=1m(∫-τ0[a22l(t-s)k22l(s)kj2l(∫-τ0) +∑j≠2naj2l(t-s) × kj2l(s)∫-τ0]ds∫-τ0)∑l=1m] ×exp{u2(t)}dt +⋯+∫0ω[∫-τ0∑j≠nnαn(t)×kjnl(s)]∫-τ0)ds] -∑l=1m(∫-τ0[∑j≠nnannl(t-s)knnl(s)∑j≠nnkjnl +∑j≠nnajnl(t-s) × kjnl(s)∑j≠nn]ds∫-τ0)] ×exp{un(t)}dt=∑i=1nr-iω.
From the assumptions of Theorem 2, we can obtain
(33)∫0ω[αi(t)-∑l=1m(∫-τ0[aiil(t-s)kiil(s)kjil+∑j≠inajil(t-s)kjil(s)] +∑j≠inajil(t-s) × kjil(s)]ds∫-τ0aiil(t-s)kiil(s)kjil)+∑j≠inajil(t-s)kjil(s)] ×exp{ui(t)}dt ≤∑i=1nr-iω, i=1,2,…,n.
Hence,
(34)δi∫0ωexp{ui(t)}dt≤∑i=1nr-iω, i=1,2,…,n.
Consequently,
(35)∫0ωexp{ui(t)}dt≤∑i=1nr-iωδi, i=1,2,…,n.
From (35), we further obtain
(36)ui(ηi)≤ln(∑i=1nr-iδi), i=1,2,…,n.
On the other hand, directly from system (14) we have
(37)∫0ω|u˙i(t)|dt ≤∫0ω[|ri(t)|+αi(t)exp{ui(t)}∑j≠1n∫-τ0 +∑l=1maiil(t)∫-τ0kiil(s)exp{ui(t+s)}ds +∑j≠1n ∑l=1maijl(t)∫-τ0kijl(s) ×exp{uj(t+s)}ds∑j≠1n∫-τ0]dt =∫0ω|ri(t)|dt +∫0ω[αi(t)+∑l=1m∫-τ0aiil(t-s)kiil(s)ds] ×exp{ui(t)}dt +∫0ω(∑j≠in ∑l=1m∫-τ0aijl(t-s)kijl(s)ds) ×exp{uj(t)}dt ≤∫0ω|ri(t)|dt+∫0ωαi(t)exp{ui(t)}dt +∑l=1maiilM∫0ωexp{ui(t)}dt +∑j≠in ∑l=1maijlM∫0ωexp{uj(t)}dt ≤|ri|-ω+∑j=1n ∑l=1maijlM ∑i=1nr-iωδi +αiM ∑i=1nr-iωδi, i=1,2,…,n.
From (36) and (37), we have, for any t∈[0,ω],
(38)ui(t)≤ui(ηi)+∫0ω|u˙i(t)|dt≤ln(∑i=1nr-iδi)+|ri|-ω +∑j=1n ∑l=1maijlM ∑i=1nr-iωδi+αiM ∑i=1nr-iωδi=:Mi, i=1,2,…,n.
Further, from (29) and (37), we have, for any t∈[0,ω],
(39)ui(t)≥ui(ξi)-∫0ω|u˙i(t)|dt≥ln(r-iα-i)-|ri|-ω-∑j=1n ∑l=1maijlM∑i=1nr-iωδi -αiM∑i=1nr-iωδi=:Ni, i=1,2,…,n.
Therefore, from (38) and (39) we have
(40)maxt∈[0,ω]|ui(t)|≤max{|Mi|,|Ni|}=:Bi, i=1,2,…,n.
It can be seen that the constants Bi (i=1,2,…,n) are independent of parameter λ∈(0,1).
For any u=(u1,u2,…,un)∈Rn, from (18) we can obtain
(41)QNu=(QNu1,QNu2,…,QNun),
where
(42)QNu=r-i-(α-i-a-ii)exp{ui}+∑j≠ina-ijexp{uj}, i=1,2,…,n.
We consider the following algebraic equation:
(43)r-i-(α-i-a-ii)vi+∑j≠ina-ijvj=0, i=1,2,…,n.
From the assumption of Theorem 2, the equation has a unique positive solution v*=(v1*,v2*,…,vn*). Hence, the equation QNu=0 has a unique solution u*=(u1*,u2*,…,un*)=(lnv1*,lnv2*,…,lnvn*)∈Rn.
Choosing constant B>0 large enough such that |u1*|+|u2*|+⋯+|un*|<B and B>B1+B2+⋯+Bn, we define a bounded open set Ω⊂X as follows:
(44)Ω={u∈X:∥u∥<B}.
It is clear that Ω satisfies conditions (a) and (b) of Lemma 1. On the other hand, by direct calculating we can obtain
(45)deg{JQN,Ω∩KerL,(0,0,…,0)} =sgn|fu11fu21⋯fun1fu12fu22⋯fun2⋯⋯⋯⋯fu1nfu2n⋯funn|,
where
(46)fuji=-(α-i-a-ij)exp{uj*}, i=j,fuji=a-ijexp{uj*}, i≠j, i,j=1,2,…,n.
From the assumption of Theorem 2, we have
(47)|fu11fu21⋯fun1fu12fu22⋯fun2⋯⋯⋯⋯fu1nfu2n⋯funn|≠0.
From this, we finally have
(48)deg{JQN,Ω∩KerL,(0,0,…,0)}≠0.
This shows that Ω satisfies condition (c) of Lemma 1. Therefore, system (14) has a ω-periodic solution u*(t)=(u1*(t),u2*(t),…,un*(t))∈Ω-. Further, from (13), system (4) has a positive ω-periodic solution x*(t)=(x1*(t),x2*(t),…,xn*(t)).
Next, we will consider the global attractivity of positive periodic solutions x*(t)=(x1*(t),x2*(t),…,xn*(t)) of system (4). Choose positive constants mi>0, Mi>0 such that
(49)mi≤xi*(t)≤Mi, i=1,2,…,n.
From the assumption of Theorem 2, there exists constant β>0 such that for all t≥0 we have
(50)δi≥β>0, i=1,2,…,n.
Let (x1(t),x2(t),…,xn(t)) be any solution of system (4), we define Lyapunov function as follows:
(51)Vi(t)=μi|lnxi*(t)-lnxi(t)|+∑j=1n ∑l=1mμj∫-τ0kijl(s)∫t+staijl(θ-s)×|xj*(θ)-xj(θ)|dθ ds.
Calculating the upper right derivation of Vi(t) along system (4) for i=1,2,…,n, we have
(52)D+Vi(t)=sign(xi*(t)-xi(t)) ×[-μiαi(t)(xi*(t)-xi(t))(xj*(t+θ)-xj(t+θ))ds +∑j=1n ∑l=1mμjaijl(t)∫-τ0kijl(s) ×(xj*(t+θ)-xj(t+θ))ds] +∑j=1n ∑l=1mμj∫-τ0aijl(t-s)kijl(s)ds|xj*(t)-xj(t)| -∑j=1n ∑l=1mμjaijl(t)∫-τ0kijl(s) ×|xj*(t+θ)-xj(t+θ)|ds≤-μiαi(t)|xi*(t)-xi(t)| +∑j=1n ∑l=1mμj∫-τ0aijl(t-s)kijl(s)ds|xj*(t)-xj(t)|.
Further, we define a Lyapunov function as follows:
(53)V(t)=∑i=1nVi(t).
Calculating the upper right derivation of V(t), from (52) we finally can obtain, for all t≥0,
(54)D+V(t)≤-∑i=1nδi|xi*(t)-xi(t)|.
Integrating from 0 to t on both sides of (54) and by (50) produces
(55)V(t)+β∫0t(∑i=1n|xi*(s)-xi(s)|)ds≤V(0), t≥0,
then
(56)∫0t(∑i=1n|xi*(s)-xi(s)|)ds≤V(0)β, t≥0.
By the definition of V(t) and (53), we have
(57)∑i=1nμi|lnxi*(t)-lnxi(t)|≤V(t)≤V(0), t≥0.
Therefore, for i=1,2,…,n we have
(58)μi|lnxi*(t)-lnxi(t)|≤V(0), t≥0,
which, together with (49), leads to
(59)miexp{-V(0)μi}≤xi(t)≤Miexp{V(0)μi}, i=1,2,…,n,
and, hence, ∑i=1n|xi*(t)-xi(t)|∈L1[0,+∞). From the boundedness of xi*(t) and (58), it follows that xi(t) (i=1,2,…,n) are bounded for t≥0. It is obvious that both xi(t) and xi*(t) satisfy the equations of system (4), then by system (4) and the boundedness of xi(t) and xi*(t), we know that the derivatives x˙i(t) and x˙i*(t) are bounded. Furthermore, we can obtain that x˙i*(t)-x˙i(t) (i=1,2,…,n) and their derivatives remain bounded on [0,+∞). Therefore ∑i=1n|xi*(t)-xi(t)| is uniformly continuous on [0,+∞). Thus, from (56), we have
(60)limt→+∞ ∑i=1n|xi*(t)-xi(t)|=0.
Therefor,
(61)limt→+∞(xi*(t)-xi(t))=0, i=1,2,…,n.
This completes the proof of Theorem 2.