# Positive Periodic Solutions for Impulsive Functional Differential Equations with Infinite Delay and Two Parameters

Authors: Zhenguo Luo; Liping Luo; Yunhui Zeng
Journal: Journal of Applied Mathematics
 (2014)

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## Abstract

We apply the Krasnoselskii’s fixed point theorem to study the existence of multiple positive periodic solutions for a class of impulsive functional differential equations with infinite delay and two parameters. In particular, the presented criteria improve and generalize some related results in the literature. As an application, we study some special cases of systems, which have been studied extensively in the literature.


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## Body

## 1. Introduction

First, we give the following definitions. LetJ⊂R denote by PC(J,Rn) the set of operators φ:J→Rn which are continuous for t∈J, t≠tk and have discontinuities of the first kind at the points tk∈J(k∈Z+) but are continuous from the left at these points. For each y=(y1,y2,…,yn)T∈Rn, the norm of y is defined as |y|=∑i=1n|yi|. The matrix A>B(A≤B) means that each pair of corresponding elements of A and B satisfies the inequality “>”(“≤”). In particular, A is called a positive matrix if A>0.Impulsive differential equations are suitable for the mathematical simulation of evolutionary process whose states are subject to sudden changes at certain moments. Equations of this kind are found in almost every domain of applied sciences; numerous examples are given in [1–3]. In recent years, in [4–11], many researchers have obtained some properties of impulsive differential equations, such as oscillation, asymptotic behavior, stability, and existence of solutions. However, to this day, still no scholars investigate the existence of multiple positive periodic solutions for impulsive functional differential equations with infinite delay and two parameters. Motivated by this, in this paper, we mainly consider the following impulsive functional differential equations with two parameters:
(1)y′(t)=-A(t,y(t))y(t)+λB(t,y(t))F(t,u(t)),t∈R,t≠tk,Δy(tk)=μIk(tk,y(tk)),k∈Z+,(2)y′(t)=A(t,y(t))y(t)-λB(t,y(t))F(t,u(t)),t∈R,t≠tk,Δy(tk)=μIk(tk,y(tk)),k∈Z+,
where
(3)u(t)=(y(c1(t)),…,y(cn-1(t)),∫-∞t‍k(t-ξ)y(ξ)dξ∫-∞t‍k(t-ξ)y(ξ)dξ),
and λ>0, μ>0 are two parameters, A(t,y(t))=diag[a1(t,y(t)),a2(t,y(t)),…,an(t,y(t))],  B(t,y(t))=diag[b1(t,y(t)),b2(t,y(t)),…,bn(t,y(t))], ai,bi∈C(R×R+,R+)  (i=1,2,…,n) are ω-periodic, that is, ai(t+ω,y(t+ω))=ai(t,y(t)), bi(t+ω,y(t+ω))=bi(t,y(t)), F=(f1,…,fn)T,  F(t,u(t)) is an operator on R×BC(R,Rn) (here BC(R,Rn) denotes the Banach space of bounded continuous operator ϕ:R→Rn with the norm ∥ϕ∥=∑i+1nsupθ∈R|ϕ(θ)|, where ϕ=(ϕ1,…,ϕn)T),  fi(t+ω,u(t+ω))=fi(t,u(t)), Δy(tk)=y(tk+)-y(tk) (here y(tk+) represents the right limit of y(t) at the point tk), Ik=(I1k,I2k,…,Ink)∈C(R+n,R-n), that is, y(t) changes decreasingly suddenly at times tk. ω>0 is a constant, Z+={1,2,3,…}, R=(-∞,+∞), R+=[0,+∞), and R-=(-∞,0]. We assume that there exists an integer q>0 such that tk+q=tk+ω, Ii(k+q)=Iik, i=1,2,…,n, where 0<t1<t2<⋯<tq<ω.Models of forms (1) and (2) have been proposed for population dynamics (single species growth models), physiological processes (such as production of blood cells, respiration, and cardiac arrhythmias), and other practical problems. Equations (1) and (2) are very general and incorporate many famous mathematical models extensively studied in the literature [12–21]. In this paper, we will study the existence of positive periodic solutions in more cases than the previously mentioned papers and obtain some easily verifiable sufficient criteria.Throughout the paper, we make the following assumptions.(H1)
ai,bi:R×R+→R+ satisfy Caratheodory conditions; that is, ai(t,y) and bi(t,y) are locally Lebesgue measurable in t for each fixed y and are continuous in y for each fixed t are ω-periodic functions in t. Moreover, there exist ω-periodic functions a1i,a2i,b1i,b2i:R→R+, which are locally bounded Lebesgue measurable so that a1i(t)≤ai(t,y(t))≤a2i(t),  b1i(t)≤b1i(t,y(t))≤b2i(t) and ∫0ω‍a1i(t)dt>0, ∫0ω‍b1i(t)dt>0.   λ>0, μ>0 are two parameters.(H2)
fi(t,u(t)) is ω-periodic with respect to the first variable, that is, fi(t+ω,u(t+ω))=fi(t,u(t)) such that fi(t,u(t))≢0, i=1,2,…,n.(H3)
The delay kernelk:R+→R+ is integrable and is normalized such that ∫0+∞‍k(t)dt=1, cj(t):R→R such that cj(t)≤t, j=1,2,…,n-1.(H4)
{tk}, k∈Z+ satisfies 0<t1<t2<⋯<tk<⋯, and limk→+∞tk=+∞; Ik:R×R+→R, k∈Z+ satisfy Caratheodory conditions and are ω-periodic functions in t. Moreover, Ik(t,0)=0 for all k∈Z+. There exists a positive constant q such that tk+q=tk+ω, Ik+q(tk+q,y(tk+q))=Ik(tk,y(tk)), k∈Z+. Without loss of generality, we can assume that tk≠0 and [0,ω]∩{tk,k∈Z+}={t1,t2,…,tq}.In addition, the parameters in this paper are assumed to be not identically equal to zero.To conclude this section, we summarize in the following a few concepts and results that will be needed in our arguments.Definition 1 (see [22]).
LetX be a real Banach space and let E be a closed, nonempty subset of X. E is said to be a cone if(1)
αx+βy∈E for all x,y∈E, and α,β>0;(2)
x,-x∈E imply x=0.Lemma 2 (see Krasnoselskii’s fixed point theorem [23–26]).
LetE be a cone in a real Banach space X. Assume that Ω1 and Ω2 are open subsets of X with 0∈Ω1⊂Ω1¯⊂Ω2, where Ωi={x∈X:∥x∥<ri}(i=1,2). Let T:E∩(Ω2∖Ω1-)→E be a completely continuous operator and satisfies either(1)  ∥Tx∥≥∥x∥, for any x∈E∩∂Ω1 and ∥Tx∥≤∥x∥, for any x∈E∩∂Ω2,or(2)  ∥Tx∥≤∥x∥, for any x∈E∩∂Ω1 and ∥Tx∥≥∥x∥, for any x∈E∩∂Ω2.
Then,T has a fixed point in E∩(Ω2∖Ω1-).For convenience in the following discussion, we introduce the following notations:(4)Fa=limsupu∈E,∥u∥→amaxt∈[0,ω]∫0ω‍|F(t,u(t))|dt∥u∥,Fa=liminfu∈E,∥u∥→amint∈[0,ω]∫0ω‍|f(t,u(t))|dt∥u∥,Ia=limsupy∈E,∥y∥→amaxt∈[0,ω]∑t≤tk<t+ω|Ik(t,y)|∥y∥,Ia=liminfy∈E,∥y∥→amint∈[0,ω]∑t≤tk<t+ω|Ik(t,y)|∥y∥,
where a denotes either 0 or ∞,  ∥u∥=max{|u1|,|u2|,…,|un|}, and ∥y∥=max{|y1|,|y2|,…,|yn|}.The paper is organized as follows. In Section2, firstly, we give some definitions and lemmas. Secondly, we derive some existence theorems for one or two positive periodic solutions of (1) which are established by using Krasnoselskii’s fixed point theorem under some conditions. In Section 3, existence theorems for one or two positive periodic solutions of (2) are also established by using Krasnoselskii’s fixed point theorem under some conditions. As applications in Section 4, we study some particular cases of systems (1) and (2) which have been investigated extensively in the references mentioned earlier.


## 2. Existence of Periodic Solution of (1)

We establish the existence of positive periodic solutions of (1) by applying the Krasnoselskii’s fixed point theorem on cones. We will first make some preparations and list below a few preliminary results. For (t,s)∈R2, 1≤i≤n, we define
(5)Gi(t,s)=e∫ts‍ai(ξ,y(ξ))dξe∫0ω‍ai(ξ,y(ξ))dξ-1,G(t,s)=diag[G1(t,s),G2(t,s),…,Gn(t,s)].
It is clear that Gi(t+ω,s+ω)=Gi(t,s), (∂Gi(t,s)/∂t)=ai(t,y(t))Gi(t,s), Gi(t,t+ω)-Gi(t,t)=1. In view of (H1), we also define for 1≤i≤n(6)αi:=min0≤t≤s≤ω|Gi(t,s)|=1e∫0ω‍a2i(ξ)dξ-1,βi:=max0≤t≤s≤ω|Gi(t,s)|=e∫0ω‍a2i(ξ)dξe∫0ω‍a1i(ξ)dξ-1,α=min1≤i≤nαi,β=max1≤i≤nβi,σ=αβ∈(0,1),Bi(t)=max{|b1i(t)|,|b2i(t)|},Bi′(t)=min{|b1i(t)|,|b2i(t)|},B(t)=max1≤i≤n{Bi(t)},B′(t)=min1≤i≤n{Bi′(t)}.
Let X={y=(y1(t),y2(t),…,yn(t))T∈PC(R,Rn)∣y(t+ω)=y(t)} with the norm ∥y∥=∑i=1n|yi|0, |yi|0=supt∈[0,ω]|yi(t)|. It is easy to verify that (X,∥·∥) is a Banach space. Define E as a cone in X by
(7)E={y=(y1(t),y2(t),…,yn(t))T∈X:yi(t)≥σ∥yi∥0,1111111111111111111111111111111111(y1(t),y2(t),…,yn(t))Tt∈[0,ω]}.
We easily verify that E is a cone in X. We define an operator T:X→X as follows:
(8)(Ty)(t)=((T1y)(t),(T2y)(t),…,(Tny)(t))T,
where
(9)(Tiy)(t)=λ∫tt+ω‍Gi(t,s)bi(s,y(s))fi(s,u(s))ds+μ∑t≤tk<t+ω‍Gi(t,tk)Iik(tk,y(tk)).
The proofs of the main results in this paper are based on an application of Krasnoselskii’s fixed point theorem in cones. To make use of the fixed point theorem in cone, firstly, we need to introduce some definitions and lemmas.Definition 3 (see [1]).
A functiony:R→(0,+∞) is said to be a positive solution of (1) if the following conditions are satisfied:(a)
y(t) is absolutely continuous on each (tk,tk+1);(b)
for eachk∈Z+, y(tk+) and y(tk-) exist, and y(tk-)=y(tk);(c)
y(t) satisfies the first equation of (1) for almost everywhere in R and y(tk) satisfies the second equation of (1) at impulsive point tk, k∈Z+.Definition 4 (see [22]).
LetX be a real Banach space; E is a cone of X. The semiorder induced by the cone E is denoted by “≤”. That is, x≤y if and only if y-x∈P for any x,y∈E.Lemma 5 (see [27]).
Assume thatf(t) and g(t) are continuous nonnegative functions defined on the interval [α,β]; then there exists ξ∈[α,β] such that
(10)∫αβ‍f(t)g(t)dt=f(ξ)∫αβ‍g(t)dt.Lemma 6.
Assume that(H1)–(H4) hold. The existence of positive ω-periodic solution of (1) is equivalent to that of nonzero fixed point of T in E.Proof.
Assume thaty=(y1(t),y2(t),…,yn(t))T∈X is a periodic solution of (1). Then, we have
(11)[yi(t)e∫0t‍ai(s,y(s))ds]′=λe∫0t‍ai(s,y(s))dsbi(t,y(t))fi(t,u(t)),1111111111111111t≠tk,i=1,2,…,n.
Integrating the above equation over [t,t+ω], we can have
(12)yi(v)e∫0v‍ai(s,y(s))ds|ttm1+nω+yi(v)e∫0v‍ai(s,y(s))ds|tm1+nωtm2+nω+⋯+yi(v)e∫0v‍ai(s,y(s))ds|tmq+nωt+ω=λ∫tt+ω‍e∫0v‍ai(s,y(s))dsbi(v,x(v))fi(v,u(v))dv,
where tmk+nω∈(t,t+ω), mk∈{1,2,…,q}, k=1,2,…,q, n∈Z+.
Therefore,(13)yi(t)e∫0t‍ai(s,y(s))ds[e∫tt+ω‍ai(s,y(s))ds-1]-∑t≤tk<t+ω‍Δyi(tmk)e∫0tmk+nω‍ai(s,y(s))ds=λ∫tt+ω‍e∫0v‍ai(s,y(s))dsbi(v,y(v))fi(v,u(v))dv,
which can be transformed into
(14)yi(t)=λ∫tt+ω‍Gi(t,s)bi(s,y(s))fi(s,u(s))ds11+μ∑t≤tk<t+ω‍Gi(t,tk)Iik(tk,y(tk))=(Tiy)(t).
Thus, yi is a periodic solution for (9).
Ify=(y1(t),y2(t),…,yn(t))T∈X and Ty=(T1y,T2y,…,Tny)T=y with y≠0, then for any t=tk, derivative the two sides of (9) about t,(15)(Tiy)′(t)=ddt[λ∫tt+ω‍Gi(t,s)bi(s,y(s))fi(s,u(s))ds]=λ[Gi(t,t+ω)bi(t+ω,y(t+ω))fi(t+ω,u(t+ω))1111-Gi(t,t)bi(t,y(t))fi(t,u(t))]-ai(t,y(t))yi(t)=-ai(t,y(t))yi(t)+λbi(t,y(t))fi(t,u(t))=yi′(t).
For any t=tj, j∈Z+, we have from (9) that
(16)yi(tj+)-yi(tj)=λ∫tjtj+ω‍[Gi(tj+,s)-Gi(tj,s)]1111×bi(s,y(s))fi(s,u(s))ds11+μ∑tj+≤tk<tj+ω‍Gi(tj+,tk)Iik(tk,y(tk))11-μ∑tj≤tk<tj+ω‍Gi(tj,tk)Iik(tk,y(tk))=μIik(tk,y(tk)).
Hence y(t)=(y1(t),y2(t),…,yn(t))T∈X is a positive ω-periodic solution of (1). Thus we complete the proof of Lemma 6.Lemma 7.
Assume that(H1)–(H4) hold. Then T:E→E is well defined.Proof.
From (9), it is easy to verify that (Ty)(t) is continuous in (tk,tk+1), (Ty)(tk+) and (Ty)(tk-) exist, and (Ty)(tk-)=(Ty)(tk) for each k∈Z+. Moreover, for any y∈E(17)(Ty)(t+ω)=λ∫t+ωt+2ω‍G(t+ω,s)b(s,y(s))F(s,u(s))ds11+μ∑t+ω≤tk<t+2ω‍G(t+ω,tk)Ik(tk,y(tk))=λ∫tt+ω‍G(t+ω,v+ω)b(v+ω,y(v+ω))1111×F(v+ω,u(v+ω))dv11+μ∑t≤tk<t+ω‍G(t,tk)Ik(tk,y(tk))=λ∫tt+ω‍G(t,s)b(s,y(s))F(s,u(s))ds11+μ∑t≤tk<t+ω‍G(t,tk)Ik(tk,y(tk))=(Ty)(t).
Therefore, (Ty)∈X. From (9), we have
(18)|Tiy|0≤βi[λ∫0ω‍|bi(s,y(s))fi(s,u(s))|dsμ∑t≤tk<t+ω‍Iik(tk,y(tk))11111+μ∑t≤tk<t+ω‍Iik(tk,y(tk))].
On the other hand, we obtain
(19)(Tiy)(t)≥αi[λ∫0ω‍|bi(s,y(s))fi(s,u(s))|ds∑t≤tk<t+ω‍Iik(tk,y(tk))111111+μ∑t≤tk<t+ω‍Iik(tk,y(tk))]≥αiβi|Tiy|0≥σ|Tiy|0.
Therefore, Ty∈E. This completes the proof of Lemma 7.Lemma 8.
Assume that(H1)–(H4) hold. Then T:E→E is completely continuous.Proof.
We first show thatT is continuous. By (H3)-(H4), f and Hk are continuous in y; it follows that for any ϵ>0, let δ>0 be small enough to satisfy that if y,y*∈E, with |y-y*|<δ,
(20)|f(s,u(s))-f(s,u*(s))|<ϵ2B¯λβω,s∈R,|Hk(tk,y(tk))-Ik(tk,y*(tk))|<ϵ2βμq,k∈Z+.
Therefore,
(21)∥(Ty)(t)-(Ty*)(t)∥11=∑i=1n‍|Tiy-Tiy*|011≤β∑i=1n‍λ∫tt+ω‍|bi(s,y(s))fi(s,u(s))1111111111111111-bi(s,y(s))fi(s,u*(s))|ds1111+βμ∑i=1n‍∑t≤tk<t+ω‍|Iik(tk,y(tk))11111111111111111-hik(tk,y*(tk))|11<βλB¯ωϵ2B¯λβω+βμqϵ2βμq11=ϵ,
which implies that T is continuous on E.
Next we show thatT maps a bounded set into a bounded set. Indeed, let C∈E be a bounded set. For any t∈R and x∈C, by (9), we have
(22)∥(Ty)(t)∥=∑i=1n‍|Tiy|0≤β[λ∑i=1n‍∫tt+ω‍|bi(s,y(s))fi(s,u(s))|ds∑t≤tk<t+ω‍|Ik(tk,y(tk))|11111+μ∑i=1n‍∑t≤tk<t+ω‍|Iik(tk,y(tk))|]=β[∫0ω‍b2i(s)F(s,u(s))ds∑t≤tk<t+ω‍|Ik(tk,y(tk))|11111+∑t≤tk<t+ω‍|Ik(tk,y(tk))|].
Since C is bounded, in view of the continuity of T, it follows from (21) that Ty is bounded and {Ty:y∈C} is uniformly bounded. Finally, we show that the family of functions {Ty:y∈C} is equicontinuous on [0,ω]. Let θ1,θ2∈[0,ω] with θ1<θ2. From (9), for any y∈C, we have
(23)∥(Ty)(θ2)-(Ty)(θ1)∥11≤λ∑i=1n‍[∫θ1θ2‍(Gi(θ2,s)-Gi(θ1,s))11111111111×bi(s,y(s))fi(s,u(s))ds11111111111+∫θ2θ1+ω‍(Gi(θ2,s)-Gi(θ1,s))111111111111111×bi(s,y(s))fi(s,u(s))ds11111111111+∑θ1≤tk<θ2‍(Gi(θ2,tk)-Gi(θ1,tk))111111111111111111×|Iik(tk,y(tk))|11111111111+∑θ2≤tk<θ1+ω‍(Gi(θ2,tk)-Gi(θ1,tk))1111111111111111111×∫θ1θ2‍|Iik(tk,y(tk))|].
Since for y∈C, t∈[0,ω], 0≤k≤q, bi(t,y(t)),fi(t,u(t)), and Iik(tk,y(tk)) are uniformly bounded in X, in view of (23), it is easy to see that when θ2-θ1 tends to zero, |(Ty)(θ2)-(Ty)(θ1)| tends uniformly to zero in X. Hence, {Ty:y∈C} is a family of uniformly bounded and equicontinuous functions on [0,ω]. By Arzela-Ascoli theorem, the operator T is completely continuous. The proof of Lemma 8 is complete.Our main results of this paper are as follows.Theorem 9.
Assume that(H1)–(H4) and(H5)
there exists aR>0 such that F(t,u)≥(R/(2ασB′(ξ)λω)), ∑t≤tk<t+ωIk(tk,y(tk))≥(R/(2ασμ)), for ∥u∥≤∥y∥∈[σR,R];(H6)
F0=I0=F∞=I∞=0hold. Then, (1) has two positive ω-periodic solutions.Proof.
First, we defineΩR={y∈X:∥y∥<R}; then ΩR is an open subset of X. Then, for any y∈E∩∂Ωr, we have y(t)≥σR. Consequently,
(24)uj(t)=y(cj(t))≥σ∥y∥≥σ∥u(t)∥,un(t)=∫-∞t‍k(t-ξ)y(ξ)dξ≥σ∥y∥≥σ∥u(t)∥;
then
(25)∥u(t)∥=max1≤j≤n-1{y(cj(t)),∫-∞t‍k(t-ξ)y(ξ)dξ}≥σR,
and from the definition of u(t), we know ∥u(t)∥≤∥y(t)∥. From (9), (H5), and Lemma 5, we get
(26)∥(Ty)(t)∥=∑i=1n‍|Tiy|0=∑i=1n‍[λ∫tt+ω‍|Gi(t,s)bi(s,y(s))fi(s,u(s))|ds∑t≤tk<t+ω‍111111+μ∑t≤tk<t+ω‍|Gi(t,tk)Iik(tk,y(tk))|]≥λB′(ξ)∑i=1n‍∫tt+ω‍|Gi(t,s)fi(s,u(s))|ds11+μ∑i=1n‍∑t≤tk<t+ω‍|Gi(t,tk)Iik(tk,y(tk))|=α[λB′(ξ)∫0ω‍|F(s,u(s))|ds∑t≤tk<t+ω‍11111+μ∑t≤tk<t+ω‍|Ik(tk,y(tk))|]≥ασ(λB′(ξ)ωR2ασB′(ξ)λω+Rμ2ασμ)=R>∥y∥.
This yields
(27)∥Ty∥>∥y∥,foranyy∈E∩∂ΩR.
On the other hand, if F0=I0=0 holds, then we can choose 0<R1<R, so that ∫0ω‍|f(t,u(t))|dt≤ϵ∥u∥; from the definition of u(t), we know ∥u∥≤∥y∥. Thus, we have ∫0ω‍|f(t,u(t))|dt≤ϵ∥y∥, ∑t≤tk<t+ω|Ik(tk,y(tk))|≤ϵ∥y∥ for y∈[0,R1], t∈[0,ω], 1≤k<q, where constant ϵ>0 satisfies ϵβ(λB(ξ)+μ)≤1. By (9) and Lemma 5, we can obtain
(28)(Ty)(t)=∑i=1n‍(Tiy)=∑i=1n‍[λ∫tt+ω‍Gi(t,s)bi(s,y(s))fi(s,u(s))ds∑t≤tk<t+ω‍111111+μ∑t≤tk<t+ω‍Gi(t,tk)Iik(tk,y(tk))]≤λB(ξ)∑i=1n‍∫tt+ω‍|Gi(t,s)fi(s,u(s))|ds11+μ∑i=1n‍∑t≤tk<t+ω‍|Gi(t,tk)Iik(tk,y(tk))|=β[λB(ξ)∫0ω‍|F(s,u(s))|ds∑t≤tk<t+ω‍11111+μ∑t≤tk<t+ω‍|Ik(tk,y(tk))|]≤ϵβ(λB(ξ)+μ)∥y∥≤∥y∥.
This yields
(29)∥(Ty)(t)∥≤∥y∥,foranyy∈E∩∂ΩR1.
In view of (27) and (29), by Lemma 5, it follows that T has a fixed point y*∈E∩(ΩR∖ΩR1-) with R1<∥y*∥<R, which is a positive ω-periodic solution of (1).
Likewise, in view ofF∞=I∞=0, for any 0<ϵ<(1/(4βB(ξ)λω)), there is N>0 such that
(30)F(t,u(t))≤ϵ∥u∥,for∥u∥>N.
Let ΩR2={y∈X:∥y∥<R2}, where
(31)R2≥4N+4B(ξ)λωmaxt∈[0,ω],∥u∥≤NF(t,u(t))+2βμ∑t≤tk<t+ω‍|Ik(tk,y(tk))|.
Then for any y∈E⋂‍∂ΩR2, from (9), (30), and (31), we have
(32)∥(Ty)(t)∥=∑i=1n‍|Tiy|0=∑i=1n‍[λ∫tt+ω‍|Gi(t,s)bi(s,y(s))fi(s,u(s))|ds∑t≤tk<t+ω‍111111+μ∑t≤tk<t+ω‍|Gi(t,tk)Iik(tk,y(tk))|]≤βB(ξ)∑i=1n‍∫tt+ω‍|Gi(t,s)fi(s,u(s))|ds11+μ∑i=1n‍∑t≤tk<t+ω‍|Gi(t,tk)Iik(tk,y(tk))|=β[λB(ξ)∫tt+ω‍|F(s,u(s))|ds∑t≤tk<t+ω‍111111+μ∑t≤tk<t+ω‍|Ik(tk,y(tk))|]=β{λB(ξ)[∫I1‍|F(s,u(s))|ds∑t≤tk<t+ω‍1111111111+∫I2‍|F(s,u(s))|ds]111111+μ∑t≤tk<t+ω‍|Ik(tk,y(tk))|}≤R24+∥y∥4+R22=∥y∥,
where I1={s∈[0,ω],∥u(s)∥≤N}, I2={s∈[0,ω],∥u(s)∥>N}. This yields
(33)∥Ty∥<∥y∥,foranyy∈E∩∂ΩR2.
In view of (27) and (33), by Lemma 5, it follows that T has a fixed point y¯∈E∩(ΩR2∖ΩR-) with R<∥y¯∥<R2, which is a positive ω-periodic solution of (1). Therefore (1) has at least two positive periodic solutions; that is, R1<∥y*∥<R<∥y¯∥<R2. This proves Theorem 9.Corollary 10.
Assume that(H1)–(H4) and(H5)
There exists aR>0 such that F(t,u)≥(R/2ασB′(ξ)λω), ∑t≤tk<t+ωIk(tk,y(tk))≥(R/2ασμ), for ∥u∥≤∥y∥∈[σR,R];(H7)
F0=I0=0, or F∞=I∞=0hold. Then, (1) has a positive ω-periodic solution.Theorem 11.
Assume that(H1)–(H4) and(H8)
there exists ar>0 such that F(t,u)≤(r/2βB(ξ)λω), ∑t≤tk<t+ωIk(tk,y(tk))≤(r/2βμ), for ∥u∥≤∥y∥≤r;(H9)
F0=I0=F∞=I∞=∞hold. Then, (1) has two positive ω-periodic solutions.Proof.
We defineΩr={y∈X:∥y∥<r}, where R satisfied (H8); then Ωr is an open subset of X and 0∈Ωr. For any y∈E∩∂Ωr, by the definition of u(t), we get ∥u(t)∥≤∥y(t)∥=r. Furthermore, by (9), (H8), and Lemma 5, we have
(34)∥(Ty)(t)∥=∑i=1n‍|Tiy|0=∑i=1n‍[λ∫tt+ω‍|Gi(t,s)bi(s,y(s))fi(s,u(s))|∑t≤tk<t+ω‍111111+μ∑t≤tk<t+ω‍|Gi(t,tk)Iik(tk,y(tk))|]≤λB(ξ)∑i=1n‍∫tt+ω‍|Gi(t,s)fi(s,u(s))|ds11+μ∑i=1n‍∑t≤tk<t+ω‍|Gi(t,tk)Iik(tk,y(tk))|=β[λB(ξ)∫0ω‍|F(s,u(s))|ds∑t≤tk<t+ω‍11111+μ∑t≤tk<t+ω‍|Ik(tk,y(tk))|]≤β(λB(ξ)ωr2βB(ξ)λω+r2βμ)≤r=∥y∥.
This implies for any y∈E∩∂Ωr(35)∥Ty∥≤∥y∥.
On the one hand, since F0=I0=∞, then for any M1>1/(2αB′(ξ)σλω) there exists 0<r1<r such that
(36)F(t,u(t))≥M1∥u∥,for∥u∥<r1,uj≥σ∥u∥,j=1,2,…,n.
Letting Ωr1={y∈X:∥y∥<r1}, then for any y∈E⋂‍∂Ωr1, one has ∥y(t)∥=r1, ∑t≤tk<t+ω|Hk(tk,y(tk))|≥(1/(2αμ))∥y(t)∥=(r1/(2αμ)), y(t)≥σ∥y(t)∥=σr1. Consequently,
(37)uj(t)=y(cj(t))≥σ∥y∥≥σ∥u(t)∥,un(t)=∫-∞t‍k(t-ξ)y(ξ)dξ≥σ∥y∥≥σ∥u(t)∥,σr1≤∥u(t)∥=max1≤j≤n-1{y(cj(t)),∫-∞t‍k(t-ξ)y(ξ)dξ}≤∥y(t)∥=r1,
where, together with (42) and Lemma 5, we have
(38)∥(Ty)(t)∥=∑i=1n‍|Tiy|0=∑i=1n‍[λ∫tt+ω‍|Gi(t,s)bi(s,y(s))fi(s,u(s))|ds∑t≤tk<t+ω‍111111+μ∑t≤tk<t+ω‍|Gi(t,tk)Iik(tk,y(tk))|]≥αB′(ξ)∑i=1n‍∫tt+ω‍|Gi(t,s)fi(s,u(s))|ds11+μ∑i=1n‍∑t≤tk<t+ω‍|Gi(t,tk)Iik(tk,y(tk))|=α[λB′(ξ)∫tt+ω‍|F(s,u(s))|ds∑t≤tk<t+ω‍11111+μ∑t≤tk<t+ω‍|Ik(tk,y(tk))|]≥αλB′(ξ)M1σ∥y(t)∥ω+∥y(t)∥2=∥y∥.
This implies for any y∈E∩∂Ωr1(39)∥Ty∥≥∥y∥.
In view of (35) and (39), by Lemma 2, it follows that T has a fixed point y*∈E∩(ΩR∖Ωr1-) with r1<∥y*∥<r, which is a positive ω-periodic solution of (1). On the other hand, if F∞=I∞=∞, then for any M2>(1/2αB′(ξ)σλω) there exists r2>r such that
(40)F(t,u(t))≥M2∥u∥,for∥u∥>r2,uj≥σ∥u∥,j=1,2,…,n.
Letting Ωr2={y∈X:∥y∥<r2}, then for any y∈E∩∂Ωr2, one has ∥y(t)∥=r2, ∑t≤tk<t+ω|Ik(tk,y(tk))|≥(1/2αμ)∥y(t)∥=(r2/2αμ), y(t)≥σ∥y(t)∥=σr2. Consequently,
(41)uj(t)=y(cj(t))≥σ∥y∥≥σ∥u(t)∥,un(t)=∫-∞t‍k(t-ξ)y(ξ)dξ≥σ∥y∥≥σ∥u(t)∥,∥u(t)∥=max1≤j≤n-1{y(cj(t)),∫-∞t‍k(t-ξ)y(ξ)dξ}≥σ∥y(t)∥=σr2,
where, together with (41) and Lemma 5, we have
(42)∥(Ty)(t)∥=∑i=1n‍|Tiy|0=∑i=1n‍[λ∫tt+ω‍|Gi(t,s)bi(s,y(s))fi(s,u(s))|ds∑t≤tk<t+ω‍11111+μ∑t≤tk<t+ω‍|Gi(t,tk)Iik(tk,y(tk))|]≥αB′(ξ)∑i=1n‍∫tt+ω‍|Gi(t,s)fi(s,u(s))|ds11+μ∑i=1n‍∑t≤tk<t+ω‍|Gi(t,tk)Iik(tk,y(tk))|=α[λB′(ξ)∫tt+ω‍|F(s,u(s))|ds∑t≤tk<t+ω‍11111+μ∑t≤tk<t+ω‍|Ik(tk,y(tk))|]≥αλB′(ξ)M2σr2ω+∥y(t)∥2≥∥y∥.
This yields
(43)∥(Ty)(t)∥≥∥y∥,foranyy∈E∩∂Ωr2.
In view of (35) and (43), by Lemma 2, it follows that T has a fixed point y¯∈E∩(Ωr2∖Ωr-) with r<∥y¯∥<r2, which is a positive ω-periodic solution of (1). Therefore, (1) has at least two positive periodic solutions; that is r1<∥y*∥<r<∥y¯∥<r2. This proves Theorem 11.Corollary 12.
Assume that(H1)–(H4) and(H8)
there exists ar>0 such that F(t,u)≤(r/2βB(ξ)λω), ∑t≤tk<t+ωHk(tk,y(tk))≤(r/2βμ), for ∥u∥≤∥y∥≤r;(H10)
F0=I0=∞, or F∞=I∞=∞;hold. Then, (1) has a positive ω-periodic solution.Theorem 13.
Assume that(H1)–(H5), and (H8) hold. Then, (1) has a positive ω-periodic solution y with ∥y∥ lying between R and r, which are defined in (H5) and (H8), respectively.Proof.
Without loss of generality, we may assume thatr<R; then for any y∈E∩∂Ωr, by the definition of u(t), we get ∥u(t)∥≤∥y(t)∥=r. Furthermore, by (9), (H8), and Lemma 5, we have
(44)∥(Ty)(t)∥=∑i=1n‍|Tiy|0=∑i=1n‍[λ∫tt+ω‍|Gi(t,s)bi(s,y(s))fi(s,u(s))|∑t≤tk<t+ω‍111111+μ∑t≤tk<t+ω‍|Gi(t,tk)Iik(tk,y(tk))|]≤λB(ξ)∑i=1n‍∫tt+ω‍|Gi(t,s)fi(s,u(s))|ds11+μ∑i=1n‍∑t≤tk<t+ω‍|Gi(t,tk)Iik(tk,y(tk))|=β[λB(ξ)∫0ω‍|F(s,u(s))|ds∑t≤tk<t+ω‍11111+μ∑t≤tk<t+ω‍|Ik(tk,y(tk))|]≤β(λB(ξ)ωr2βB(ξ)λω+r2βμ)≤r=∥y∥.
This implies for any y∈E∩∂Ωr(45)∥Ty∥≤∥y∥.
Now, we let ΩR={y∈X:∥y∥<R}; then ΩR is an open subset of X. Then, for any y∈E∩∂ΩR, we have y(t)≥σR. Consequently,
(46)uj(t)=y(cj(t))≥σ∥y∥≥σ∥u(t)∥,un(t)=∫-∞t‍k(t-ξ)y(ξ)dξ≥σ∥y∥≥σ∥u(t)∥,∥u(t)∥=max1≤j≤n-1{y(cj(t)),∫-∞t‍k(t-ξ)y(ξ)dξ}≥σR,
and from the definition of u(t), we know ∥u(t)∥≤∥y(t)∥. From (9), (H5), and Lemma 5, we get
(47)∥(Ty)(t)∥=∑i=1n‍|Tiy|0=∑i=1n‍[λ∫tt+ω‍|Gi(t,s)bi(s,y(s))fi(s,u(s))|ds∑t≤tk<t+ω‍111111+μ∑t≤tk<t+ω‍|Gi(t,tk)Iik(tk,y(tk))|]≥λB′(ξ)∑i=1n‍∫tt+ω‍|Gi(t,s)fi(s,u(s))|ds11+μ∑i=1n‍∑t≤tk<t+ω‍|Gi(t,tk)Iik(tk,y(tk))|=α[λB′(ξ)∫0ω‍|F(s,u(s))|ds∑t≤tk<t+ω‍11111+μ∑t≤tk<t+ω‍|Ik(tk,y(tk))|]≥ασ(λB′(ξ)ωR2ασB′(ξ)λω+Rμ2ασμ)=R>∥y∥.
This yields
(48)∥Ty∥>∥y∥,foranyy∈E∩∂ΩR.
In view of (45) and (48), by Lemma 2, it follows that T has a fixed point y*∈E∩(ΩR∖Ωr-) with r<∥y*∥<R, which is a positive ω-periodic solution of (1). This proves Theorem 13.Theorem 14.
In addition to(H1)–(H4), suppose the following conditions hold:(H11)
0<F0≤(1/2βλB(ξ)ω), 0<I0≤(1/2βμ);(H12)
F∞≥(1/2ασλB′(ξ)ω), I∞≥(1/2ασμ);then, (1) has a positive ω-periodic solution.Proof.
In view of0<F0≤(1/2βλB(ξ)ω), 0<I0≤1/(2βμ), there exists a sufficiently small r>0 such that
(49)∫0ω‍|F(t,u(t))|dt∥u∥≤12βλB(ξ)ω,forany∥u(t)∥≤r,∑t≤tk<t+ω|Ik(t,y)|∥y∥≤12βμ,forany∥y(t)∥≤r,
which yields
(50)∫0ω‍|F(t,u(t))|dt≤12βλB(ξ)ω∥u∥≤r2βλB(ξ)ω,forany∥u(t)∥≤r,∑t≤tk<t+ω‍|Ik(t,y)|≤12βμ∥y∥≤r2βμ,forany∥y(t)∥≤r.
Therefore, condition (H8) is satisfied. On the other hand, since F∞≥(1/2ασλB′(ξ)ω), I∞≥(1/2ασμ), there exists a sufficiently large R>0(R>r) such that
(51)∫0ω‍|F(t,u(t))|dt∥u∥≥12ασλB′(ξ)ω,forany∥u(t)∥∈[σR,R],∑t≤tk<t+ω|Ik(t,y)|∥y∥≥12ασμ,forany∥y(t)∥∈[σR,R],
which yields
(52)∫0ω‍|F(t,u(t))|dt11≥12ασλB′(ξ)ω∥u∥11≥R2ασλB′(ξ)ω,forany∥u(t)∥∈[σR,R],∑t≤tk<t+ω‍|Ik(t,y)|≥12ασμ∥y∥∑t≤tk<t+ω‍|Ik(t,y)|≥R2ασμ,forany∥y(t)∥∈[σR,R].
Thus, condition (H5) is satisfied. By Theorem 13, we complete the proof.Theorem 15.
Assume that(H1)–(H4) and(H13)
F0>(1/2ασλB′(ξ)ω), I0>1/2ασμ;(H14)
0<F∞≤(1/2βλB(ξ)ω), 0<I∞≤(1/(2βσμ))hold. Then, (1) has a positive ω-periodic solution.Proof.
In view ofF0>(1/2ασλB′(ξ)ω), I0>(1/2ασμ), for ϵ=min{F0-(1/2ασλB′(ξ)ω),I0-(1/2ασμ)}, there exists a sufficiently small r>0 such that
(53)∫0ω‍|F(t,u(t))|dt∥u∥≥F0-ϵ≥12ασλB′(ξ)ω,forany0≤∥u(t)∥≤r,uj(t)≥σ∥u(t)∥,∑t≤tk<t+ω|Ik(t,y)|∥y∥≥I0-ϵ≥12ασμ,forany∥y(t)∥∈[σr,r],
which yields
(54)∫0ω‍|F(t,u(t))|dt11≥12ασλB′(ξ)ω∥u∥11≥r2ασλB′(ξ)ω,forany∥u(t)∥∈[σr,r],∑t≤tk<t+ω‍|Ik(t,y)|≥12ασμ∥y∥∑t≤tk<t+ω‍|Ik(t,y)|≥r2ασμ,forany∥y(t)∥∈[σr,r].
Therefore, condition (H5) is satisfied. On the other hand, since 0<F∞≤(1/2βλB(ξ)ω), 0<I∞≤(1/2βσμ), for ϵ=max{(1/2βλB(ξ)ω)-F∞,(1/2βσμ)-I∞}, there exists a sufficiently large R(R>r) such that
(55)∫0ω‍|F(t,u(t))|dt∥u∥≤F∞+ϵ≤12βλB(ξ)ω,forany∥u(t)∥≥R,∑t≤tk<t+ω|Ik(t,y)|∥y∥≤I∞+ϵ≤12βμ,forany∥y(t)∥≥R.
In the following, we consider two cases to prove (H8) to be satisfied: ∫0ω‍|F(t,u(t))|dt, ∑t≤tk<t+ω|Ik(t,y)| are bounded or unbounded. The bounded case is clear. If ∫0ω‍|F(t,u(t))|dt, and ∑t≤tk<t+ω|Ik(t,y)| are unbounded, then there exist u*,y*∈R+n, R1=∥y*∥≥∥u*∥≥R and t0∈[0,ω] such that
(56)∫0ω‍|F(t,u(t))|dt≤∫0ω‍|F(t,u*(t))|dt,forany∥u(t)∥≤∥u*∥≤R1,∑t≤tk<t+ω‍|Ik(t,y)|≤∑t≤tk<t+ω‍|Ik(t,y*)|,forany∥y(t)∥≤∥y*∥+R1.
Since R1=∥y*∥≥∥u*∥≥R, then we get
(57)∫0ω‍|F(t,u(t))|dt≤∫0ω‍|F(t,u*(t))|dt≤∥u*∥2βλB(ξ)ω≤∥y*∥2βλB(ξ)ω=R12βλB(ξ)ω,forany∥u(t)∥≤∥u*∥≤R1,∑t≤tk<t+ω‍|Ik(t,y)|≤∑t≤tk<t+ω‍|Ik(t,y*)|≤∥y*∥2βμ=R12βμ,1111111111111forany∥y(t)∥≤∥y*∥+R1.
Thus, condition (H8) is satisfied. By Theorem 14, we complete the proof.Theorem 16.
Assume that(H1)–(H5), (H11), and (H14) hold. Then, (1) has at least two positive ω-periodic solutions y*, y¯ satisfying 0<∥y*∥<R<∥y¯∥, where R is defined in (H5).Proof.
From(H5) and the proof of Theorem 9, we know that there exists a sufficiently large R>0 such that
(58)∥Ty∥>∥y∥,foranyy∈E∩∂ΩR.
On the one hand, from (H11) and the proof of Theorem 14, we know that there exists a sufficiently small r>0(r<R) such that
(59)∥Ty∥<∥y∥,foranyy∈E∩∂Ωr.
In view of (58) and (59), by Lemma 2, it follows that T has a fixed point y*∈E∩(ΩR∖Ωr-) with r<∥y*∥<R, which is a positive ω-periodic solution of (1).
On the other hand, from(H14) and the proof of Theorem 15, we know that there exists a sufficiently large R1>0(R1>R) such that
(60)∥Ty∥<∥y∥,foranyy∈E∩∂ΩR1.
In view of (58) and (60), by Lemma 2, it follows that T has a fixed point y¯∈E∩(ΩR1∖ΩR-) with R<∥y¯∥<R1, which is a positive ω-periodic solution of (1). Thus, (1) has at least two positive ω-periodic solutions y*, y¯ satisfying r<∥y*∥<R<∥y¯∥<R1. The proof is completed.Corollary 17.
In addition to(H1)–(H5), suppose (H11) or (H14) holds. Then, (1) has at least one positive ω-periodic solution.Theorem 18.
Assume that(H1)–(H4), (H8), (H12), and (H13) hold. Then, (1) has at least two positive ω-periodic solutions y*, y¯ satisfying 0<∥y*∥<r<∥y¯∥, where r is defined in (H8).Proof.
From(H8) and the proof of Theorem 9, we know that there exists a sufficiently small r>0 such that
(61)∥Ty∥<∥y∥,foranyy∈E∩∂Ωr.
On the one hand, from (H12) and the proof of Theorem 14, we know that there exists a sufficiently large r1>0(r1>r) such that
(62)∥Ty∥>∥y∥,foranyy∈E∩∂Ωr1.
In view of (61) and (62), by Lemma 2, it follows that T has a fixed point y*∈E∩(Ωr1∖Ωr-) with r<∥y*∥<r1, which is a positive ω-periodic solution of (1).
On the other hand, from(H13) and the proof of Theorem 15, we know that there exists a sufficiently small r*>0(r*<r) such that
(63)∥Ty∥>∥y∥,foranyy∈E∩∂Ωr*.
In view of (61) and (63), by Lemma 2, it follows that T has a fixed point y*∈E∩(Ωr∖Ωr*-) with r*<∥y*∥<r, which is a positive ω-periodic solution of (1). Thus, (1) has at least two positive ω-periodic solutions y*, y¯ satisfying r*<∥y*∥<r<∥y¯∥<r1. The proof is completed.Corollary 19.
In addition to(H1)–(H4), and (H8), suppose (H12) or (H13) holds. Then, (1) has at least one positive ω-periodic solution.Theorem 20.
Assume that(H1)–(H4), (H6), (H12) hold. Then, (1) has at least two positive ω-periodic solutions y*, y¯ satisfying 0<∥y*∥<∥y¯∥.Proof.
From(H6) and the proof of Theorem 9, if F0=I0=0, we know that there exists a sufficiently small r>0 such that
(64)∥Ty∥<∥y∥,foranyy∈E∩∂Ωr.
In view of F∞=I∞=0, we know that there exists a sufficiently large R>0(R>r) such that
(65)∥Ty∥<∥y∥,foranyy∈E∩∂ΩR.
On the one hand, from (H12) and the proof of Theorem 14, we know that there exists a sufficiently large r1>0(R>r1>r) such that
(66)∥Ty∥>∥y∥,foranyy∈E∩∂Ωr1.
In view of (64) and (66), by Lemma 2, it follows that T has a fixed point y*∈E∩(Ωr1∖Ωr-) with r<∥y*∥<r1, which is a positive ω-periodic solution of (1). In view of (65) and (66), by Lemma 2, it follows that T has a fixed point y¯∈E∩(ΩR∖Ωr1-) with r1<∥y¯∥<R, which is a positive ω-periodic solution of (1). Thus, (1) has at least two positive ω-periodic solutions y*, y¯ satisfying r<∥y*∥<r1<∥y¯∥<R. The proof is completed.Corollary 21.
Assume that(H1)–(H4), (H7), and (H12) hold. Then, (1) has at least one positive ω-periodic solution.Theorem 22.
Assume that(H1)–(H4), (H6), and (H13) hold. Then, (1) has at least two positive ω-periodic solutions y*, y¯ satisfying 0<∥y*∥<∥y¯∥.Proof.
From(H13) and the proof of Theorem 15, we know that there exists a sufficiently small r>0 such that
(67)∥Ty∥>∥y∥,foranyy∈E∩∂Ωr.
From (H6) and the proof of Theorem 9, on the one hand, if F0=I0=0, we know that there exists a sufficiently small r*>0(r*<r) such that
(68)∥Ty∥<∥y∥,foranyy∈E∩∂Ωr*.
In view of (67) and (68), by Lemma 2, it follows that T has a fixed point y*∈E∩(Ωr∖Ωr*-) with r*<∥y*∥<r, which is a positive ω-periodic solution of (1).
On the other hand, ifF∞=I∞=0, we know that there exists a sufficiently large R>0(R>r) such that
(69)∥Ty∥<∥y∥,foranyy∈E∩∂ΩR.
In view of (67) and (69), by Lemma 2, it follows that T has a fixed point y¯∈E∩(ΩR∖Ωr-) with r<∥y¯∥<R, which is a positive ω-periodic solution of (1). Thus, (1) has at least two positive ω-periodic solutions y*, y¯ satisfying r*<∥y*∥<r<∥y¯∥<R. The proof is completed.Corollary 23.
Assume that(H1)–(H4), (H7), and (H13) hold. Then, (1) has at least one positive ω-periodic solution.
Similarly, one can prove the following theorems and corollaries.Theorem 24.
Assume that(H1)–(H4), (H9), and (H11) hold. Then, (1) has at least two positive ω-periodic solutions y*, y¯ satisfying 0<∥y*∥<∥y¯∥.Corollary 25.
Assume that(H1)–(H4), (H10), and (H11) hold. Then, (1) has at least one positive ω-periodic solution.Theorem 26.
Assume that(H1)–(H4), (H9), and (H14) hold. Then, (1) has at least two positive ω-periodic solutions y*, y¯ satisfying 0<∥y*∥<∥y¯∥.Corollary 27.
Assume that(H1)–(H4), (H10), and (H14) hold. Then, (1) has at least one positive ω-periodic solution.


## 3. Existence of Periodic Solution of (2)

Now, we are in a position to attack the existence of positive periodic solutions of (2). By carrying out similar arguments as in Section 2, it is not difficult to establish sufficient criteria for the existence of positive periodic solutions of (2). For simplicity, we prefer to list below the corresponding criteria for (2) without proof, since the proofs are very similar to those in Section 2.For(t,s)∈R2, 1≤i≤n, we define
(70)Gi*(t,s)=e-∫ts‍ai(ξ,y(ξ))dξ1-e-∫0ω‍ai(ξ,y(ξ))dξ111111=e∫st+ω‍ai(ξ,y(ξ))dξe∫0ω‍ai(ξ,y(ξ))dξ-1,G*(t,s)=diag[G1*(t,s),G2*(t,s),…,Gn*(t,s)].
It is clear that Gi*(t,t)≥Gi*(t,s)≥G*(t,t+ω), ∂Gi*(t,s)/(∂t)=ai(t,y(t))Gi*(t,s), Gi*(t,t)-Gi*(t,t+ω)=1. In view of (H1), we also define for 1≤i≤n(71)αi*:=min0≤t≤s≤ω|Gi*(t,s)|=1e∫0ω‍a2i(ξ)dξ-1=αi,βi*:=max0≤t≤s≤ω|Gi*(t,s)|=e∫0ω‍a2i(ξ)dξe∫0ω‍a1i(ξ)dξ-1=βi,α*=min1≤i≤nαi*=α,β*=max1≤i≤nβi*=β,δ=α*β*∈(0,1)=σ,Bi(t)=max{|b1i(t)|,|b2i(t)|},Bi′(t)=min{|b1i(t)|,|b2i(t)|},B(t)=max1≤i≤n{Bi(t)},B′(t)=min1≤i≤n{Bi′(t)}.
Let X={y=(y1(t),y2(t),…,yn(t))T∈PC(R,Rn)∣y(t+ω)=y(t)} with the norm ∥y∥=∑i=1n|yi|0, |yi|0=supt∈[0,ω]|yi(t)|. It is easy to verify that (X,∥·∥) is a Banach space. Define P as a cone in X by
(72)P={y=(y1(t),y2(t),…,yn(t))T∈X:yi(t)11(y1(t),y2(t),…,yn(t))T≥δ∥yi∥0,t∈[0,ω]}.
We easily verify that P is a cone in X. We define an operator ϕ:X→X as follows:
(73)(ϕy)(t)=((ϕ1y)(t),(ϕ2y)(t),…,(ϕny)(t))T,
where
(74)(ϕiy)(t)=λ∫tt+ω‍Gi*(t,s)bi(s,y(s))fi(s,u(s))ds111+μ∑t≤tk<t+ω‍Gi*(t,tk)Iik(tk,y(tk)).
The proof of the following lemmas, theorems, and corollaries is similar to those in Section 2; we omit all the details here.Lemma 28.
Assume that(H1)–(H4) hold. The existence of positive ω-periodic solution of (2) is equivalent to that of nonzero fixed point of ϕ in P.Lemma 29.
Assume that(H1)–(H4) hold. Then ϕ:P→P is well defined.Lemma 30.
Assume that(H1)–(H4) hold. Then ϕ:P→P is completely continuous.Theorem 31.
Assume(H1)–(H4) and (H5) hold. Moreover, if one of the following conditions holds:
(75)(H6);(H11),(H14),
then, (2) has two positive ω-periodic solutions y* and y¯ satisfying 0<∥y*∥<R<∥y¯∥, where R is defined in (H5).Theorem 32.
Assume(H1)–(H4) and (H8) hold. Moreover, if one of the following conditions holds:
(76)(H9);(H12),(H13),
then, (2) has two positive ω-periodic solutions y* and y¯ satisfying 0<∥y*∥<r<∥y¯∥, where r is defined in (H8).Theorem 33.
Assume(H1)–(H4) hold. Moreover, if one of the following conditions holds:
(77)(H6),(H12);(H6),(H13);(H9),(H11);(H9),(H14),
then, (2) has two positive ω-periodic solutions y* and y¯ satisfying 0<∥y*∥<∥y¯∥.Theorem 34.
Assume(H1)–(H5) hold. Moreover, if one of the following conditions holds:
(78)(H7);(H8);(H11);(H14),
then, (2) has at least one positive ω-periodic solution.Theorem 35.
Assume(H1)–(H4) and (H8) hold. Moreover, if one of the following conditions holds:
(79)(H10);(H12);(H13),
then, (2) has at least one positive ω-periodic solution.Theorem 36.
Assume(H1)–(H4) hold. Moreover, if one of the following conditions holds:
(80)(H7),(H10);(H7),(H12);(H7),(H13);(H10),(H11);(H10),(H14);(H11),(H12);(H13),(H14),
then, (2) has at least one positive ω-periodic solution.


## 4. Applications

In this section, as some applications of our main results, we will consider some special cases of systems (1) and (2), which have been investigated extensively in the literature.Application1. Consider the following equations:
(81)y′(t)=-A(t,y(t))y(t)+λB(t,y(t))F(t,u(t)),t∈R,(82)y′(t)=A(t,y(t))y(t)-λB(t,y(t))F(t,u(t)),t∈R,
where u(t)=(y(c1(t)),…,y(cn-1(t)),∫-∞t‍k(t-ξ)y(ξ)dξ), which are special cases of systems (1) and (2) without impulse, respectively. First, we list several assumptions:(P1)
=(H1);(P2)
=(H2);(P3)
=(H3);(P4)
there exists aR>0 such that F(t,u)≥(R/(ασB′(ξ)λω)) for ∥u∥≤∥y∥∈[σR,R];(P5)
there exists ar>0 such that F(t,u)≤(r/(2βB(ξ)λω)) for ∥u∥≤∥y∥≤r;(P6)
F0=0;(P7)
F∞=0;(P8)
F0=∞;(P9)
F∞=∞;(P10)
F0=η1∈(0,1/βλB(ξ)ω);(P11)
F∞=θ1∈(0,1/βλB(ξ)ω);(P12)
F0=η2∈(1/ασλB′(ξ)ω,∞);(P13)
F∞=θ2∈(1/ασλB′(ξ)ω,∞).By applying theorems in Sections2 and 3 and to systems (81) and (82), respectively, we obtain the following theorems.Theorem 37.
Assume(P1)–(P4) hold. Moreover, if one of the following conditions holds:
(83)(P6),(P7);(P10),(P11),
then, (81) and (82) have at least two positive ω-periodic solutions.Theorem 38.
Assume(P1)–(P3) and (P5) hold. Moreover, if one of the following conditions holds:
(84)(P8),(P9);(P12),(P13),
then, (81) and (82) have at least two positive ω-periodic solutions.Theorem 39.
Assume(P1)–(P3) hold. Moreover, if one of the following conditions holds:
(85)(P6),(P7),(P12);(P6),(P7),(P13);(P8),(P9),(P10);(P8),(P9),(P11),
then, (81) and (82) have at least two positive ω-periodic solutions.Theorem 40.
Assume(P1)–(P4) hold. Moreover, if one of the following conditions holds:
(86)(P5);(P8);(P9);(P10);(P11),
then, (81) and (82) have at least one positive ω-periodic solution.Theorem 41.
Assume(P1)–(P3) and (P5) hold. Moreover, if one of the following conditions holds:
(87)(P6);(P7);(P12);(P13),
then (81) and (82) have at least one positive ω-periodic solution.Theorem 42.
Assume(P1)–(P3), hold. Moreover, if one of the following conditions holds:
(88)(P6),(P9);(P6),(P13);(P7),(P8);(P7),(P12);(P8),(P11);(P9),(P10);(P10),(P13);(P11),(P12),
then, (81) and (82) have at least one positive ω-periodic solution.Application2. Consider the following equations:
(89)y′(t)=-a(t)y(t)+f(t,u(t)),t∈R,t≠tk,Δy(tk)=μIk(tk,y(tk)),k∈Z+,(90)y′(t)=a(t)y(t)-f(t,u(t)),t∈R,t≠tk,Δy(tk)=μIk(tk,y(tk)),k∈Z+,
where
(91)u(t)=(y(g1(t)),…,y(gn-1(t)),∫-∞t‍1111111∫-∞tk(t-ξ)y(ξ)dξ)
which are special cases of systems (1) and (2), respectively. For convenience in the following discussion, on the one hand, we introduce the following notations:
(92)fa=limsupu∈E,∥u∥→amaxt∈[0,ω]∫0ω‍|f(t,u(t))|dt∥u∥,fa=liminfu∈E,∥u∥→amint∈[0,ω]∫0ω‍|f(t,u(t))|dt∥u∥,Ia=limsupy∈E,∥y∥→amaxt∈[0,ω]∑t≤tk<t+ω|Ik(t,y)|∥y∥,Ia=liminfy∈E,∥y∥→amint∈[0,ω]∑t≤tk<t+ω|Ik(t,y)|∥y∥,
where a denotes either 0 or ∞, ∥u∥=max{|u1|,|u2|,…,|un|}. On the other hand, we list several assumptions:(A1)
=(H1);(A2)
=(H2);(A3)
=(H3);(A4)
=(H4);(A5)
there exists aR>0 such that f(t,u)≥(R/2ασλω), ∑t≤tk<t+ωHk(tk,y(tk))≥(R/2ασμ), for ∥u∥≤∥y∥∈[σR,R];(A6)
there exists ar>0 such that f(t,u)≤(r/2βλω), ∑t≤tk<t+ωI(tk,y(tk))≤(r/2βμ), for ∥u∥≤∥y∥≤r;(A7)
f0=I0=0;(A8)
f∞=I∞=0;(A9)
f0=I0=∞;(A10)
f∞=I∞=∞;(A11)
f0=ρ1∈(0,1/2βλω), I0=ξ1∈(0,1/2βμ);(A12)
f∞=ϱ1∈(0,1/2βλω), I∞=ς1∈(0,1/2βμ);(A13)
f0=ρ2∈(1/2ασλω,∞), I0=ξ2∈(1/2ασμ,∞);(A14)
f∞=ϱ2∈(1/2ασλω,∞), I∞=ς2∈(1/2ασμ,∞).By applying theorems in Sections2 and 3 and to systems (89) and (90), respectively, we obtain the following theorems.Theorem 43.
Assume(A1)–(A5) hold. Moreover, if one of the following conditions holds:
(93)(A7),(A8);(A11),(A12),
then, (89) and (90) have at least two positive ω-periodic solutions.Theorem 44.
Assume(A1)–(A4) and (A6) hold. Moreover, if one of the following conditions holds:
(94)(A9),(A10);(A13),(A14),
then, (89) and (90) have at least two positive ω-periodic solutions.Theorem 45.
Assume(A1)–(A4) hold. Moreover, if one of the following conditions holds:
(95)(A7),(A8),(A13);(A7),(A8),(A14);(A9),(A10),(A11);(A9),(A10),(A12),
then, (89) and (90) have at least two positive ω-periodic solutions.Theorem 46.
Assume(A1)–(A5) hold. Moreover, if one of the following conditions holds:
(96)(A5);(A9);(A10);(A11);(A12),
then, (89) and (90) have at least one positive ω-periodic solution.Theorem 47.
Assume(A1)–(A4) and (A6) hold. Moreover, if one of the following conditions holds:
(97)(A7);(A8);(A13);(A14),
then, (89) and (90) have at least one positive ω-periodic solution.Theorem 48.
Assume(A1)–(A4) hold. Moreover, if one of the following conditions holds:
(98)(A7),(A10);(A7),(A14);(A8),(A9);(A8),(A13);(A9),(A12);(A10),(A11);(A11),(A14);(A12),(A13),
then, (89) and (90) have at least one positive ω-periodic solution.Application3. Consider the following equations:
(99)y′(t)=-a(t)y(t)+f(t,u(t)),t∈R,(100)y′(t)=a(t)y(t)-f(t,u(t)),t∈R,
where
(101)u(t)=(y(c1(t)),…,y(cn-1(t)),∫-∞t‍1111111∫-∞tk(t-ξ)y(ξ)dξ),
which are special cases of systems (89) and (90) without impulse, respectively, that is, I(tk,y(tk))=0. For convenience in the following discussion, we list several assumptions:(A1*)
=(H1);(A2*)
=(H2);(A3*)
=(H3);(A4*)
there exists aR>0 such that f(t,u)≥(R/2ασλω), for ∥u∥≤∥y∥∈[σR,R];(A5*)
there exists ar>0 such that f(t,u)≤(r/2βλω), for ∥u∥≤∥y∥≤r;(A6*)
f0=0; (A7*)f∞=0;(A8*)
f0=∞; (A9*)f∞=∞;(A10*)
f0=ρ1∈(0,1/(2βλω));(A11*)
f∞=ϱ1∈(0,1/(2βλω));(A12*)
f0=ρ2∈(1/(2ασλω,∞));(A13*)
f∞=ϱ2∈(1/(2ασλω,∞)).We obtain the following theorems.Corollary 49.
Assume(A1*)–(A4*) hold. Moreover, if one of the following conditions holds:
(102)(A6*),(A7*);(A10*),(A11*),
then, (99) and (100) have at least two positive ω-periodic solutions.Corollary 50.
Assume(A1*)–(A3*) and (A5*) hold. Moreover, if one of the following conditions holds:
(103)(A8*),(A9*);(A12*),(A13*),
then, (99) and (100) have at least two positive ω-periodic solutions.Corollary 51.
Assume(A1*)–(A3*) hold. Moreover, if one of the following conditions holds:
(104)(A6*),(A7*),(A12*);(A6*),(A7*),(A13*);(A8*),(A9*),(A10*);(A8*),(A9*),(A11*),
then, (99) and (100) have at least two positive ω-periodic solutions.Corollary 52.
Assume(A1*)–(A4*) hold. Moreover, if one of the following conditions holds:
(105)(A5*);(A8*);(A9*);(A10*);(A11*),
then, (99) and (100) have at least one positive ω-periodic solution.Corollary 53.
Assume(A1*)–(A3*), (A5*) hold. Moreover, if one of the following conditions holds:
(106)(A6*);(A7*);(A12*);(A13*),
then, (99) and (100) have at least one positive ω-periodic solution.Corollary 54.
Assume(A1*)–(A3*) hold. Moreover, if one of the following conditions holds:
(107)(A6*),(A9*);(A6*),(A13*);(A7*),(A8*);(A7*),(A12*);(A8*),(A11*);(A9*),(A10*);(A10*),(A13*);(A11*),(A12*),
then, (99) and (100) have at least one positive ω-periodic solution.
Hence, our results generalize and improve the corresponding results of [16].Application4. Consider the generalized logistic model of single species [16, 21] with impulse and two parameters:
(108)y′(t)=y(t)a(t)-λy(t)[∑i=1n‍bi(t)y(t-τi(t))11111111+∑i=1n‍c(t)∫-∞t‍k(t-s)y(s)ds],t∈R,t≠tk,Δy(tk)=μIk(tk,y(tk)),k∈Z+,
where a(t),bi(t),τi(t),c(t)∈C(R,R+) are ω-periodic, λ>0, μ>0 are two parameters, and k:R+→R+ is integrable such that ∫0ω‍k(t)dt=1.Theorem 55.
Assume thatIk∈C(R,R-) and the following conditions hold:(1)
I0=0, I∞=∞;(2)
∑i=1nbiL+cL>0;then, (108) has at least one positive ω-periodic solution, where
(109)α=1e∫0ω‍a(t)dt-1,β=e∫0ω‍a(t)dte∫0ω‍a(t)dt-1,σ=αβ,gM=maxt∈[0,ω]g(t),gL=mint∈[0,ω]g(t).Proof.
We let(110)f(t,u(t))=u0[∑i=1n‍bi(t)ui(t)+c(t)un+1],u=(u0,u1,…,un,un+1)∈Rn+2.
Then, (108) can be seen as a special form of (2) satisfying (H1)–(H4). We can construct the same Banach space X and cones E as in Section 2. Then for any y∈E,
(111)∫0ω‍|f(t,u(t))|dt=∫0ω‍|u0[∑i=1n‍bi(t)ui+c(t)un+1]|dt≥σ∥u∥2[∑i=1n‍biL+cL]ω,
so we have
(112)∫0ω‍|f(t,u(t))|dt∥u∥⟶∞,∥u∥⟶∞;
that is, f∞=∞. On the other hand,
(113)∫0ω‍|f(t,u(t))|dt=∫0ω‍|u0[∑i=1n‍bi(t)ui+c(t)un+1]|dt≤∥u∥2[∑i=1n‍biM+cM]ω,
which can lead to
(114)∫0ω‍|f(t,u(t))|dt∥u∥⟶0,∥u∥⟶0;
that is, f0=0. Then in view of (2), (112), and (114), we can obtain that (108) has at least one positive ω-periodic solution. The proof is complete.Application5. Consider the generalized so-called Nicholson’s Blowflies model [19, 20] with impulse and two parameters:
(115)y′(t)=-y(t)a(t)+λb(t)y(t-τ(t))e-c(t)y(t-τ(t)),t∈R,t≠tk,Δy(tk)=μIk(tk,y(tk)),k∈Z+,
where a(t),b(t),c(t)τ(t)∈C(R,R+) are ω-periodic and λ>0, μ>0 are two parameters.Theorem 56.
Assume thatIk∈C(R,R-) and the following conditions hold:(3)
I0=∞, I∞=0;
then, (115) has at least one positive ω-periodic solution, where
(116)η=1e∫0ω‍a(t)dt-1,θ=e∫0ω‍a(t)dte∫0ω‍a(t)dt-1,δ=ηθ,gM=maxt∈[0,ω]g(t),gL=mint∈[0,ω]g(t).Proof.
We letf(t,u(t))=b(t)u(t)e-c(t)u(t). Then, (115) can be seen as a special form of (1) satisfying (H1)–(H4). We can construct the same Banach space X and cones P as in Section 2. Then for any y∈P,
(117)∫0ω‍|f(t,u(t))|dt=∫0ω‍|b(t)u(t)e-c(t)u(t)|dt≥∥u∥bLecM∥u∥ω.
So we have
(118)∫0ω‍|f(t,u(t))|dt∥u∥⟶0,∥u∥⟶∞;
that is, f∞=0. On the other hand,
(119)∫0ω‍|f(t,u(t))|dt∫0ω‍|b(t)u(t)e-c(t)u(t)|dt≤∥u∥bMecL∥u∥ω,
which can lead to
(120)∫0ω‍|f(t,u(t))|dt∥u∥⟶∞,∥u∥⟶0;
that is, f0=∞. Then in view of (1), (93), and (120), we can obtain that (115) has at least one positive ω-periodic solution. The proof is complete.



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