# Three Positive Periodic Solutions to Nonlinear Neutral Functional Differential Equations with Parameters on Variable Time Scales

Authors: Yongkun Li; Chao Wang
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

Using two successive reductions: B-equivalence of the system on a variable time scale to a system on a time scale and a reduction to an impulsive differential equation and by Leggett-Williams fixed point theorem, we investigate the existence of three positive periodic solutions to the nonlinear neutral functional differential equation on variable time scales with a transition condition between two consecutive parts of the scale(d/dt)(x(t)+c(t)x(t-α))=a(t)g(x(t))x(t)-∑j=1nλjfj(t,x(t-vj(t))), (t,x)∈𝕋0(x),Δt|(t,x)∈𝒮2i=Πi1(t,x)-t, Δx|(t,x)∈𝒮2i=Πi2(t,x)-x, where Πi1(t,x)=t2i+1+τ2i+1(Πi2(t,x)) and Πi2(t,x)=Bix+Ji(x)+x,i=1,2,….λj(j=1,2,…,n) are parameters, 𝕋0(x) is a variable time scale with (ω,p)-property, c(t),a(t), vj(t), and fj(t,x)(j=1,2,…,n) are ω-periodic functions of t, Bi+p=Bi,Ji+p(x)=Ji(x) uniformly with respect to i∈ℤ.


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## Body

## 1. Introduction

In the last several decades, the theory of dynamic equations on time scales (DETS) has been developed very intensively. For the full description of the equations we refer to the nicely written books [1, 2] and papers [3, 4]. The equations have a very special transition condition for adjoint elements of time scales. To enlarge the field of applications of the DETS, Akhmet and Turan proposed to generalize the transition operator [5], correspondingly to investigate differential equations on variable time scales with transition condition (DETC). In [6], Akhmet and Turan proposed some basic theory of dynamic equations on variable time scales; the method of investigation is by means of two successive reductions: B-equivalence of the system [7–9] on a variable time scale to a system on a time scale and a reduction to an impulsive differential equation [5, 7]. Consequently, these results are very effective to develop methods of investigation of mechanical models with impacts.Also, neutral differential equations arise in many areas of applied mathematics, and for this reason these equations have received much attention in the last few decades; they are not only an extension of functional differential equations but also provide good models in many fields including biology, mechanics, and economics. In particular, qualitative analysis such as periodicity and stability of solutions of neutral functional differential equations has been studied extensively by many authors. We refer to [10–19] for some recent work on the subject of periodicity and stability of neutral equations. In [20], the authors discussed a class of neutral functional differential equations with impulses and parameters on nonvariable time scales(1.1)(x(t)+c(t)x(t-r1))Δ=a(t)g(x(t))x(t)-∑i=1nλifi(t,x(t-τi(t))),t≠tj,t∈T,j=1,2,…,q,x(tj-)-x(tj+)=Ij(x(tj)),t=tj,j=1,2,…,q,
where λi, i=1,2,…,n are parameters, 𝕋 is an ω-periodic nonvariable time scale, a∈C(𝕋,ℝ+),c∈C(𝕋,[0,1)) and both of them are ω-periodic functions, τi∈C(𝕋,ℝ),i=1,2,…,n are ω-periodic functions, fi∈C(𝕋×ℝ+,ℝ+),i=1,2,…,n are nondecreasing with respect to their second arguments and ω-periodic with respect to their first arguments, respectively; g∈C(ℝ,ℝ+) and there exist two positive constants l,L such that 0<l≤g(x)≤L<∞ for all x>0,Ij∈C(ℝ,ℝ+)(j=1,2,…,q) and is bounded, r1 is a constant.To the best of authors’ knowledge, there has been no paper published on the existence of solutions to neutral functional differential equations on variable time scales. Our main purpose of this paper is by using theory of dynamic equations on variable time scales to investigate the existence of three positive periodic solutions to the nonlinear neutral functional differential equations on variable time scales with a transition condition between two consecutive parts of the scale(1.2)ddt(x(t)+c(t)x(t-α))=a(t)g(x(t))x(t)-∑j=1nλjfj(t,x(t-vj(t))),(t,x)∈T0(x),Δt|(t,x)∈S2i=Πi1(t,x)-t,Δx∣(t,x)∈S2i=Πi2(t,x)-x,
where Πi1(t,x)=t2i+1+τ2i+1(Πi2(t,x)) and Πi2(t,x)=Bix+Ji(x)+x,i=1,2,…λj(j=1,2,…,n) are parameters, 𝕋0(x) is a variable time scale with (ω,p)-property, c(t),a(t), vj(t), and fj(t,x)(j=1,2,…,n) are ω-periodic functions of t, Bi+p=Bi,Ji+p(x)=Ji(x) uniformly with respect to i∈ℤ.For convenience, we introduce the notation(1.3)a̅=max⁡t∈[0,ω]a(t),a̲=min⁡t∈[0,ω]a(t),c̲=min⁡t∈[0,ω]c(t),c̅=max⁡t∈[0,ω]c(t),r0=exp⁡{∫ω0a(s)ds}.Throughout this paper, we assume the following.(H1)
𝕋0(x) is a variable time scale with (ω,p)-property, c∈C(ℝ,[0,1)),a∈C(ℝ,ℝ+),vj∈C(ℝ,ℝ), and fj(t,x)∈C(𝕋0(x),ℝ+),(j=1,2,…,n) are ω-periodic functions of t, Ji∈C(ℝ,ℝ),Bi+p=Bi,Ji+p(x)=Ji(x) uniformly with respect to i∈ℤ,Ji(0)=0,Bi≥0 for each i∈ℤ.(H2)
fj(t,x)∈C(𝕋0(x),ℝ+) is nondecreasing with respect to x and fj(t,0)=0 for each j∈{1,2,…,n}.(H3)
g∈C(ℝ,ℝ+) and there exist two positive constants l, L such that 0<l≤g(x)≤L<∞ for all x>0.(H4)
There exists a numberδ>0 such that (c̲+c¯)/2≤δ<1,0≤((2δ-c̲)(1-c̲2))/(1-(2δ-c̲)2)<r0L(1-r0l)/(1-r0L).


## 2. Preliminaries

LetE be a real Banach space and P be a cone in E. A map ρ is said to be a nonnegative continuous concave functional on P if ρ:P→[0,∞) is continuous and
(2.1)ρ(tx+(1-t)y)≥tρ(x)+(1-t)ρ(y)∀x,y∈Pandt∈[0,1].For numbersβ1,β4 such that 0<β1<β4 and ρ is a nonnegative continuous concave function on P, we define the following sets: Pβ1={x∈P:∥x∥<β1},Pβ1¯={x∈P:∥x∥≤β1},P(ρ,β1,β4)={x∈P:β1≤ρ(x),∥x∥≤β4}.Now, we state the following Leggett-Williams fixed-point theorem, which is critical to the proof of our main results.Lemma 2.1 (see [21]).
LetT:Pβ4¯→Pβ4¯ be completely continuous and ρ nonnegative continuous concave functional on P such that ρ(u)≤∥u∥ for all u∈Pβ4¯. Suppose that there exist positive constants β1,β2,β3,β4 with 0<β1<β2<β3≤β4 such that(1)
{u∈P(ρ,β2,β3):ρ(x)>β2}≠ϕ and ρ(Tu)>β2 for u∈P(ρ,β2,β3);(2)
∥Tu∥<β1 for u∈Pβ1¯;(3)
ρ(Tu)>β2 for u∈P(ρ,β2,β4) with ∥Tu∥>β3.ThenT has at least three fixed points u1,u2,u3 satisfying(2.2)u1∈Pβ1,u2∈{u∈P(ρ,β2,β4):ρ(u)>β2},u3∈Pβ4¯∖(P(ρ,β2,β4)∪Pβ1¯).Let𝕋 be a periodic time scale, and let  E={x∈C(𝕋,ℝ):x(t)=x(t+ω)} be a Banach space with the norm ∥x∥=supt∈[0,ω]∩𝕋{|x(t)|:x∈E}, and let  Φ:E→E be defined by(2.3)(Φx)(t)=x(t)+c(t)x(t-τ).Lemma 2.2 (see [20]).
If0≤c(t)<1 and E is a Banach space, then Φ has a bounded inverse Φ-1 on E, and for all x∈E,
(2.4)(Φ-1x)(t)=∑j≥0∏0≤i≤j-1(-1)kc(t-iτ)x(t-jτ)
and ∥Φ-1x∥≤∥x∥/(1-c¯).Definition 2.3 (see [6]).
A nonempty closed set𝕋0(x) in ℝ×ℝn is said to be a variable time scale if for any x0∈ℝn the projection of 𝕋0(x0) on time axis, that is, the set {t∈ℝ:(t,x0)∈𝕋0(x0)} is a time scale in Hilger sense.
Fix a sequence{ti}⊂ℝ such that ti<ti+1 for all i∈ℤ, and |ti|→∞ as |i|→∞. Denote δi=t2i-1-t2i, κi=t2i-t2i-1 and take a sequence of functions {τi(x)}⊂C(ℝn,ℝ). Assume that(C1)
for some positive numbersθ′,θ∈ℝ,θ′≤ti+1-ti≤θ;(C2)
there existsl0, 0<2l0<θ′ such that ∥τi(x)∥≤l0 for all x∈ℝn,i∈ℤ.
Denote(2.5)li:=inf⁡x∈Rn{ti+τi(x)},ri:=sup⁡x∈Rn{ti+τi(x)}.
We set(2.6)Ei={(t,x)∈R×Rn:t2i+τ2i(x)<t<t2i+1+τ2i+1(x)},Si={(t,x)∈R×Rn:t=ti+τi(x)},Di={(t,x)∈R×Rn:t2i-1+τ2i-1(x)≤t≤t2i+τ2i(x)}.
Following [6], we denote 𝕋0(x):=⋃i=-∞∞𝒟i and 𝕋c=⋃i=-∞∞[t2i-1,t2i].A transition operatorΠi:𝒮2i→𝒮2i+1, for all i∈ℤ, such that Πi(t,y)=(Πi1(t,y),Πi2(t,y)) where Πi1:𝒮2i→ℝ and Πi2:𝒮2i→ℝn, and(2.7)Πi1(t,y)=t2i+1+τ2i+1(Πi2(t,y)),Πi2(t,y)=Ii(y)+y,
where Ii:ℝn→ℝn is a function. One can easily see that Πi1(t,y) is the time coordinate of (t+,y+):=Πi(t,y), the image of (t,y)∈𝒮2i under the operator Πi, and Πi2(t,y) is the space coordinate of the image.Lett=αi and t=βi be the moments that the graph of y=φ(t) intersects the surface 𝒮2i-1 and 𝒮2i, respectively, where the surfaces are defined previously. Then, we set the nonvariable time scale(2.8)Tcφ:=⋃i=-∞∞[αi,βi],
which is the domain of φ, and define the Δ-derivative as in the introduction. That is, for t=βi, we have(2.9)φΔ(βi)=φ(αi+1)-φ(βi)αi+1-βi,φΔ(t)=lim⁡s→tφ(s)-φ(t)s-t,
for any other t∈𝕋cφ, whenever the limit exists.Consider the system on variable time scales:(2.10)y′=A(t)y+f(t,y),(t,y)∈T0(y),Δt|(t,y)∈S2i=Πi1(t,y)-t,Δy|(t,y)∈S2i=Πi2(t,y)-y,
where A(t):ℝ→ℝn×n is an n×n continuous real-valued matrix function, Bi is an n×n matrix, functions f(t,y):𝕋0(y)→ℝn and Ji(y):ℝn→ℝn are continuous, Πi1(t,y)=t2i+1+τ2i+1(Πi2(t,y)), and Πi2(t,y)=Biy+Ji(y)+y.For anyα,β∈ℝ we define the oriented interval [α,β̂] as(2.11)[α,β̂]={[α,β],ifα≤β,[β,α],  otherwise.Consider the nonvariable time scale(2.12)Tc0=⋃i=-∞∞[l2i-1,r2i],
where li,ri,i∈ℤ are defined by (2.5) for the variable time scale 𝕋0(y), and take a continuation f̃:𝕋c0×ℝn→ℝn of f:𝕋0(y)→ℝn which is Lipschitzian with the same Lipschitz constant l; furthermore, if f is a monotone function, a continuation f̃ can also have the same monotony with f. Set 𝕋c:=⋃i=-∞∞[t2i-1,t2i].(C3)
∥τi(x)-τi(y)∥+∥Ji(x)-Ji(y)∥+∥f(t,x)-f(t,y)∥≤l∥x-y∥ for arbitrary x,y∈ℝn, where l is a Lipschitz constant.By (C3), in [6], one can see the following important lemma.Lemma 2.4 (see [6]).
Assume (C3) is satisfied. Then there are mappings Wi(z):ℝn→ℝn,i∈ℤ, such that, corresponding to each solution y(t) of (2.10), there is a solution z(t) of the system
(2.13)z′=A(t)z+f̃(t,z),t≠t2i,z(t2i+1)=Biz(t2i)+Wi(z(t2i))+z(t2i),
such that y(t)=z(t) for all t∈𝕋c except possibly on [t2i-1,αi]̂ and [βi,t2î], where αi and βi are the moments that y(t) meets the surfaces 𝒮2i-1 and 𝒮2i, respectively.
Furthermore, the functionsWi satisfy the inequality
(2.14)‖Wi(z)-Wi(y)‖≤k(l)l‖z-y‖,
uniformly with respect to i∈ℤ for all z,y∈ℝn such that ∥z∥≤h and ∥y∥≤h; here k(l0)=k(l0,h) is a bounded function. Under the sense of Lemma 2.4, we say that systems (2.10) and (2.13) are B-equivalent.Proof.
Fixi∈ℤ. Let z(t) be the solution of (2.10) such that z(t2i)=z, and assume that αi and βi are solutions of α=t2i-1+τ2i-1(z(α)) and β=t2i+τ2i(z(β)), respectively. Let z1(t) be the solution of the system
(2.15)z′=A(t)z+f̃(t,z)
with the initial condition z1(αi+1)=Πi2(βi,z(βi)).
We first note thatz1(αi+1)=(I+Bi)z(βi)+Ji(z(βi)). Moreover, for t∈[t2i,βî],(2.16)z(t)=z(t2i)+∫t2it[A(s)z(s)+f̃(s,z(s))]ds,
and for t∈[αi+1,t2i+1̂],
(2.17)z1(t)=z1(αi+1)+∫αi+1t[A(s)z1(s)+f(s,z1(s))]ds=(I+Bi)z(βi)+Ji(z(βi))+∫αi+1t[A(s)z1(s)+f̃(s,z1(s))]ds=(I+Bi)[z(t2i)+∫t2iβi[A(s)z(s)+f̃(s,z(s))]ds]+Ji(z(βi))+∫αi+1t[A(s)z1(s)+f̃(s,z1(s))]ds.
Thus, we have
(2.18)Wi(z)=(I+Bi)∫t2iβi[A(s)z(s)+f̃(s,z(s))]ds+Ji(z(βi))+∫αi+1t2i+1[A(s)z1(s)+f̃(s,z1(s))]ds.
Substituting (2.18) in (2.13), we see that Wi(z) satisfies the first conclusion of the lemma.
Next, we prove (2.14). Let ∥z(t2i)∥≤h. By employ integrals (2.16) and (2.17), we find that the solutions z(t) and z1(t) determined above satisfy the inequalities ∥z(t)∥≤H and ∥z1(t)∥≤H on [βi,t2î] and[αi+1,t2i+1̂], where
(2.19)H=[M(1+l)+(1+N+l)(h+Ml)eNl+l2]eNl+l2.
Let y(t) be the solution of (2.10) such that y(t2i)=y, and assume thatα̅i andβ̅i are solutions ofα̅=t2i-1+τ2i-1(y(α̅)) andβ̅=t2i+τ2i(y(β̅)), respectively. Let y1(t) be the solution of (2.15) with initial condition y1(α̅i+1)=Πi2(β̅i,y(β̅i)). Without loss of any generality, we assume thatβ̅i≥βi and α̅i+1≤αi+1. Application of the Gronwall-Bellman lemma shows that, for t∈[βi,t2î],
(2.20)‖z(t)-y(t)‖≤e(N+l)l‖z-y‖.
The equation
(2.21)y(β̅i)=y(βi)+∫βiβ̅i[A(s)y(s)+f̃(s,y(s))]ds
gives us
(2.22)‖y(β̅i)-y(βi)‖≤(NH+lH+M)(β̅i-βi).
Thus, we obtain
(2.23)‖z(βi)-y(β̅i)‖≤e(N+l)l‖z-y‖+(NH+lH+M)(β̅i-βi).
Now condition (C3) together with (2.23) leads to
(2.24)β̅i-βi≤le(N+l)l1-l(NH+lH+M)‖z-y‖.
Hence (2.23) becomes
(2.25)‖z(βi)-y(β̅i)‖≤e(N+l)l1-l(NH+lH+M)‖z-y‖.
On the other hand,
(2.26)y1(αi+1)=y1(α̅i+1)+∫α̅i+1αi+1[A(s)y1(s)+f̃(s,y1(s))]ds
gives us
(2.27)‖y1(αi+1)-y1(α̅i+1)‖≤(NH+lH+M)(αi+1-α̅i+1).
Using the transition operators and (2.25) we get
(2.28)‖z1(αi+1)-y1(α̅i+1)‖≤(1+N+l)e(N+l)l1-l(NH+lH+M)‖z-y‖.
Condition (C3) and (2.28) imply that
(2.29)αi+1-α̅i+1≤l(1+N+l)e(N+l)l1-l(NH+lH+M)‖z-y‖.
From (2.27)–(2.29) we obtain
(2.30)‖z1(αi+1)-y1(α̅i+1)‖≤H1e(N+l)l‖z-y‖,
where H1=(1+N+l)[1+l(NH+lH+M)]/[1-l(NH+lH+M)]. Solutions z1(t) and y1(t) on [αi+1,t2i+1̂] satisfy the inequality
(2.31)‖z1(t)-y1(t)‖≤H1e2(N+l)l‖z-y‖.
Now subtracting the expression
(2.32)Wi(y)=(I+Bi)∫t2iβ̅i[A(s)y(s)+f̃(s,y(s))]ds+Ji(y(β̅i))+∫α̅i+1t2i+1[A(s)y1(s)+f̃(s,y1(s))]ds
from (2.18) and using (2.20), (2.24), (2.29), and (2.31), we conclude that (2.14) holds. The proof is complete.A special transformation calledψ-substitution [5], which is change of the independent variable and defined for t∈⋃i=-∞∞(t2i-1,t2i] as
(2.33)ψ(t)={t-∑0<t2k<tδk,t≥0,t+∑t≤t2k<0δk,t<0,
where δk=t2k+1-t2k. Setting si=ψ(t2i), we see that this transformation has an inverse given by(2.34)ψ-1(s)={s+∑0<sk<sδk,s≥0,s-∑s≤sk<0δk,s<0.Lemma 2.5 (see [5, 6]).
ψ′(t)=1 if t∈⋃i=-∞∞(t2i-1,t2i].Proof.
Assume thatt≥0. Then,
(2.35)ψ′(t)=lim⁡h→0ψ(t+h)-ψ(t)h=lim⁡h→01h[(t+h-∑0<t2k<t+hδk)-(t-∑0<t2k<tδk)]=1.
The assertion for t<0 can be proved in the same way. The proof is complete.Definition 2.6 (see [5]).
The time scale𝕋0 is said to have an ω-property if there exists a number ω∈ℝ+ such that t+ω∈𝕋0 whenever t∈𝕋0.Definition 2.7 (see [5]).
A sequence{ai}⊂ℝ is said to satisfy an (ω,p)-property if there exist numbers ω∈ℝ+ and p∈ℕ such that ai+p=ai+ω for all i∈ℤ.Definition 2.8 (see [6]).
The variable time scale𝕋0(y) is said to satisfy an (ω,p)-property if (t±ω,y) is in 𝕋0(y) whenever (t,y) is. In this case, there exists p∈ℕ such that the sequences {t2i-1} and {t2i} satisfy the (ω,p)-property and τi+p(y)=τi(y) for all i∈ℤ.
Suppose now that (2.10) is ω-periodic; that is, 𝕋0(y) satisfies the (ω,p)-property, A(t) and f(t,y) are ω-periodic functions of t, and Bi+p=Bi,Ji+p(y)=Ji(y) uniformly with respect to i∈ℤ.Lemma 2.9 (see [6]).
If (2.10) is ω-periodic, then the sequence Wi(z) is p-periodic uniformly with respect to z∈ℝn.Proof.
Since the variable time scale𝕋0(y) satisfies an (ω,p)-property, by (2.18), one can easily see that Wi(z) is p-periodic uniformly with respect to z∈ℝn. The proof is complete.Lemma 2.10 (see [5]).
If𝕋0 has an ω-property, then the sequence {si},si=ψ(t2i), is (ω̃,p0)-periodic with
(2.36)ω̃=ω-∑0<t2k<ωδk=ψ(ω).Proof.
In order to prove this lemma, we only need to verify thatsi+p0=si+ω̃ for all i. Assume that i≥0,i=np0+j for some n∈ℤ,0≤j<p0 and 0<t0<···<t2(p0-1)<ω. Then
(2.37)si+p0=ψ(t2(i+p0))=t2(i+p0)-∑0<t2k<t2(i+p0)δk=t2i+ω-∑0<t2k<t2iδk-∑t2i≤t2k<t2(i+p0)δk=ψ(t2i)+ω-∑k=ii+p0-1δk=si+ω-∑k=jj+p0-1δk+np0=si+ω-∑k=jj+p0-1δk=si+ω-∑k=0p0-1δk=si+ω-∑0<t2k<ωδk=si+ω̃,
where we have used the fact that
(2.38)∑k=jj+p0-1δk=∑k=jp0-1δk+∑k=p0j+p0-1δk=∑k=jp0-1δk+∑k=0j-1δk+p0=∑k=jp0-1δk+∑k=0j-1δk=∑k=0p0-1δk.
All other cases can be verified similarly. The proof is complete.Lemma 2.11 (see [5, 6]).
If𝕋0(y) satisfies an (ω,p)-property, then ψ(t+ω)=ψ(t)+ψ(ω).Proof.
Assume thatt≥0. By Lemma 2.10, we have
(2.39)ψ(t+ω)=t+ω-∑0<t2k<t+ωδk=t+ω-∑0<t2k<ωδk-∑ω≤t2k<t+ωδk=t-∑ω≤t2k<t+ωδk+ψ(ω)=ψ(t)+ψ(ω).
The assertion for t<0 can be proved in the same way. The proof is complete.Lemma 2.12 (see [5, 6]).
A functionϕ(t) is an ω-periodic function on 𝕋c, if and only if ϕ(ψ-1(s)) is an ω̃-periodic function on ℝ, where ω̃=ψ(ω).Proof.
By Lemma2.11, s+ω̃=ψ(t+ω). Then the equality
(2.40)ϕ(ψ-1(s+ω̃))=ϕ(t+ω)=ϕ(t)=ϕ(ψ-1(s))
completes the proof.For any fixedx0∈ℝ, we set (2.41)PC(T(x0))={x∈C((t2i-1+τ2i-1(x0),t2i+τ2i(x0)),R):Πi2(t,x)=Bix+Ji(x)+x,i=1,2,…}
and consider the Banach space (2.42)E={x:x∈PC(T(x0)):x(t)=x(t+ω)}.
Let Φ:E→E be defined by(2.43)(Φx)(t)=x(t)+c(t)x(t-α):=y(t).Using the inverse transformation ofΦ, we can obtain(2.44)Πi1(t,Φ-1y)=t2i+1+τ2i+1(Πi2(t,Φ-1y)),Πi2(t,Φ-1y)=BiΦ-1y+Ji(Φ-1y)+Φ-1y.
From the second equation, it is easy to get(2.45)Φ(Πi2(t,Φ-1y))=(ΦBiΦ-1)y+(ΦJiΦ-1)y+y:=Π̃i2(t,y),
that is, Πi2(t,Φ-1y)=Φ-1(Π̃i2(t,y)). Hence(2.46)Πi1(t,Φ-1y)=t2i+1+τ2i+1(Φ-1Π̃i2(t,y))=t2i+1+τ2i+1Φ-1(Π̃i2(t,y)):=t2i+1+τ̃2i+1(Π̃i2(t,y)):=Π̃i1(t,y).Therefore, we can obtain the other variable time scale planeT0(y) by the inverse transformation of Φ and(2.47)Ei′={(t,y)∈R×R:t2i+τ̃2i(y)<t<t2i+1+τ̃2i+1(y)},Si′={(t,y)∈R×R:t=ti+τ̃i(y)},Di′={(t,y)∈R×R:t2i-1+τ̃2i-1(y)≤t≤t2i+τ̃2i(y)}.Hence, (1.2) can be changed into the following form:(2.48)y′=a(t)g((Φ-1y)(t))y(t)-a(t)H(y(t))-∑j=1nλjfj(t,(Φ-1y)(t-vj(t))),(t,y)∈T0(y),Δt|(t,y)∈S2i′=Π̃i1(t,y)-t,Δy|(t,y)∈S2i′=Π̃i2(t,y)-y,
where H(y(t))=c(t)g((Φ-1y)(t))(Φ-1y)(t-α).Define a cone inE by(2.49)P0={y(t)∈E:y(t)≥k‖y‖},
where k∈(((2δ-c̲)(1-c̲2))/(1-(2δ-c̲)2),r0L(1-r0l)/(1-r0L)].Lemma 2.13 (see [20]).
Suppose that conditions (H1)–(H4) hold and 0≤c(t)<1 and y∈P0, then
(2.50)α̃‖y‖≤(Φ-1y)(t)≤11-c̅‖y‖,(2.51)lc̲α̃‖y‖≤H(y(t))≤Lc̅1-c̅‖y‖,
where α̃=(k/(1-c̲2))-(2δ-c̲)/(1-(2δ-c̲)2).(H5)
∥τi(x)-τi(y)∥+∥Ji(x)-Ji(y)∥+∑j=1n∥fj(t,x)-fj(t,y)∥≤l0∥x-y∥ for arbitrary x,y∈ℝn, where l0 is a Lipschitz constant.In view of (H5) by Lemma 2.4, it is easy to get the following lemma.Lemma 2.14.
Assume that (H5) is satisfied. Then there are mappings Wi(z):ℝ→ℝ,i∈ℤ such that, corresponding to each solution y(t) of (2.48), there is a solution z(t) of the system
(2.52)z′=a(t)g((Φ-1z)(t))z(t)-a(t)H(z(t))-∑j=1nλjf̃j(t,(Φ-1z)(t-vj(t))),t≠t2i,z(t2i+1)=(ΦBiΦ-1)(z(t2i))+(ΦWiΦ-1)(z(t2i))+z(t2i),t=t2i
such that y(t)=z(t) for all t∈𝕋c except possibly on [t2i-1,αî] and [βi,t2î] where αi and βi are the moments that y(t) meets the surfaces 𝒮2i-1′ and 𝒮2i′, respectively.
Furthermore, the functionsWi satisfy the inequality
(2.53)‖Wi(z)-Wi(y)‖≤k(l0)l0‖z-y‖,
uniformly with respect to i∈ℤ for all z,y∈ℝn such that ∥z∥≤h and ∥y∥≤h; here k(l0)=k(l0,h) is a bounded function. Under the sense of lemma 2.14, we say that systems (2.48) and (2.52) are B-equivalent.Proof.
For fixedi∈ℤ. Let z(t) be the solution of (2.48) such that z(t2i)=z, and assume that αi and βi are solutions of α=t2i-1+τ̃2i-1(z(α)) and β=t2i+τ̃2i(z(β)), respectively. Let z1(t) be the solution of the system
(2.54)z′=a(t)g((Φ-1z)(t))z(t)-a(t)H(z(t))-∑j=1nλjf̃j(t,(Φ-1z)(t-vj(t)))
with the initial condition z1(αi+1)=Π̃i2(βi,z(βi)).
We first note thatz1(αi+1)=(ΦBiΦ-1)(z(βi))+(ΦJiΦ-1)(z(βi))+z(βi). Moreover, for t∈[t2i,βî],
(2.55)z(t)=z(t2i)+∫2it<?leftdot?>-∑j=1nλjf̃j(s,(Φ-1z)(s-vj(s)))[∑j=1nλjf̃j(s,(Φ-1z)(s-vj(s)))a(s)g((Φ-1z)(s))z(s)-a(s)H(z(s))-∑j=1nλjf̃j(s,(Φ-1z)(s-vj(s)))]ds,
and for t∈[αi+1,t2i+1̂],
(2.56)z1(t)=z1(αi+1)+∫αi+1t[∑j=1na(s)g((Φ-1z1)(s))z1(s)-a(s)H(z1(s))-∑j=1nλjf̃j(s,(Φ-1z1)(s-vj(s)))]ds=(ΦBiΦ-1)(z(βi))+(ΦJiΦ-1)(z(βi))+z(βi)+∫αi+1t[∑j=1na(s)g((Φ-1z1)(s))z1(s)-a(s)H(z1(s))-∑j=1nλjf̃j(s,(Φ-1z1)(s-vj(s)))]ds=(Φ(Bi+I)Φ-1)[z(t2i)+∫t2iβi[∑j=1nλjf̃ja(s)g((Φ-1z)(s))z(s)-a(s)H(z(s))-∑j=1nλjf̃j(s,(Φ-1z)(s-vj(s)))]ds]+(ΦJiΦ-1)(z(βi))+∫αi+1t[∑j=1na(s)g((Φ-1z1)(s))z1(s)-a(s)H(z1(s))-∑j=1nλjf̃j(s,(Φ-1z1)(s-vj(s)))]ds.
Thus, we set
(2.57)Wi(z)=(Φ(Bi+I)Φ-1)∫t2iβi[-∑j=1nλjf̃j(s,(Φ-1z)(s-vj(s)))]a(s)g((Φ-1z)(s))z(s)-a(s)H(z(s))-∑j=1nλjf̃j(s,(Φ-1z)(s-vj(s)))]ds+(ΦJiΦ-1)(z(βi))+∫αi+1t2i+1[<?leftdot?>-a(s)H(z1(s))-∑j=1nλjf̃j(s,(Φ-1z1)(s-vj(s)))a(s)g((Φ-1z1)(s))z1(s)-a(s)H(z1(s))-∑j=1nλjf̃j(s,(Φ-1z1)(s-vj(s)))]ds.
Substituting (2.57) in (2.52), we see that Wi(z) satisfies the first conclusion of the lemma.
The rest of the proof is similar to that of Lemma2.4, and we can use Gronwall-Bellman lemma to show that Wi(z) satisfies (2.53) and it will be omitted here. This completes the proof.Next, we will useψ-substitution, reducing (2.52) to an impulsive differential equation. Letting m(s)=z(ψ-1(s)), we obtain, for t≠t2i,
(2.58)(Φ-1z)(t-vj(t))=(Φ-1z)(ψ-1(s)-vj(ψ-1(s)))=(Φ-1z)(ψ-1(ψ(ψ-1(s)-vj(ψ-1(s)))))=(Φ-1m)(ψ(ψ-1(s)-vj(ψ-1(s)))):=ν(s),
hence
(2.59)m′=a(ψ-1(s))g((Φ-1m)(s))m(s)-a(ψ-1(s))H(m(s))-∑j=1nλjf̃j(ψ-1(s),ν(s)),
and for t=t2i, we get
(2.60)m(si-)=z(t2i+1)=(ΦBiΦ-1)(m(si))+(ΦWiΦ-1)(m(si))+m(si).
Thus, the second equation in (2.23) leads to
(2.61)Δm|s=si=(ΦBiΦ-1)(m(si))+(ΦWiΦ-1)(m(si)),
where Δm|s=si=m(si-)-m(si). Hence, m(s) is a solution of the impulsive differential equation:
(2.62)m′=a(ψ-1(s))g((Φ-1m)(s))m(s)-a(ψ-1(s))H(m(s))-∑j=1nλjf̃j(ψ-1(s),ν(s)),s≠si,Δm∣s=si=(ΦBiΦ-1)(m(si))+(ΦWiΦ-1)(m(si)),s=si.In the following, we set(2.63)PC̅={m:m∣(si,si+1)∈C((si,si+1),R),m(si+)=m(si),i=1,2,…}
and consider the Banach space
(2.64)E̅={m:m∈PC̅,m(s)=m(s+ω̃)}
with the norm ∥m∥=sups∈[0,ω̃]{|m(s)|:m∈E̅}, where ω̃=ψ(ω). Define a cone inE̅ by
(2.65)P={m(s)∈E̅:m(s)≥k‖m‖},
where k∈((2δ-c̲)(1-c̲2)/(1-(2δ-c̲)2),r0L(1-r0l)/(1-r0L)].Let the operatorΨ:P→E̅ be defined by
(2.66)(Ψm)(s)=∫ss+ω̃G(s,θ){a(ψ-1(θ))H(m(θ))+∑j=1nλjf̃j(ψ-1(θ),ν(θ))}dθ+∑i:si∈[s,s+ω̃]G(s,si)((ΦBiΦ-1)(m(si))+(ΦWiΦ-1)(m(si))),
where
(2.67)G(s,θ)=e∫θsa(ψ-1(r))g((Φ-1m)(r))dr1-e∫ω̃0a(ψ-1(r))g((Φ-1m)(r))dr,θ∈[s,s+ω̃].
By the assumptions, we have
(2.68)r0L1-r0L≤G(s,θ)≤11-r0l.Lemma 2.15.
m is an ω̃-periodic solution of (2.62) if and only if m is a fixed point of the operator Ψ.Proof.
Ifm(s) is an ω̃-periodic solution of (2.62), for any s∈ℝ, there exists i∈ℤ such that si is the first impulsive point after s. Hence, for θ∈[s,si], we have
(2.69)m(θ)=e∫sθa(ψ-1(τ))g((Φ-1m)(τ))dτ-∫sθe∫rθa(ψ-1(τ))g((Φ-1m)(τ))dτ{a(ψ-1(r))H(m(r))+∑j=1nλjf̃j(ψ-1(r),ν(r))}dr,
then
(2.70)m(si)=e∫ssia(ψ-1(τ))g((Φ-1m)(τ))dτm(s)-∫ssie∫rsia(ψ-1(τ))g((Φ-1m)(τ))dτ{a(ψ-1(r))H(m(r))+∑j=1nλjf̃j(ψ-1(r),ν(r))}dr.
Again, for θ∈(si,si+1], then
(2.71)m(s)=e∫siθa(ψ-1(τ))g((Φ-1m)(τ))dτm(si+)-∫siθe∫rθa(ψ-1(τ))g((Φ-1m)(τ))dτ{a(ψ-1(r))H(m(r))+∑j=1nλjf̃j(ψ-1(r),ν(r))}dr=e∫siθa(ψ-1(τ))g((Φ-1m)(τ))dτ{∑j=1nλjf̃j(ψ-1(r),ν(r))(ΦBiΦ-1)(m(si))+(ΦWiΦ-1)(m(si))+m(si)}-∫siθe∫rθa(ψ-1(τ))g((Φ-1m)(τ))dτ{a(ψ-1(r))H(m(r))+∑j=1nλjf̃j(ψ-1(r),ν(r))}dr=e∫siθa(ψ-1(τ))g((Φ-1m)(τ))dτ((ΦBiΦ-1)(m(si))+(ΦWiΦ-1)(m(si)))-∫siθe∫rθa(ψ-1(τ))g((Φ-1m)(τ))dτ{a(ψ-1(r))H(m(r))+∑j=1nλjf̃j(ψ-1(r),ν(r))}dr+e∫siθa(ψ-1(τ))g((Φ-1m)(τ))dτm(si),e∫siθa(ψ-1(τ))g((Φ-1m)(τ))dτm(si)=e∫siθa(ψ-1(τ))g((Φ-1m)(τ))dτ{∑j=1nλjf̃j(ψ-1(r),ν(r))e∫ssia(ψ-1(τ))g((Φ-1m)(τ))dτm(s)-∫ssie∫rsia(ψ-1(τ))g((Φ-1m)(τ))dτ×{∑j=1nλjf̃j(ψ-1(r),ν(r))a(ψ-1(r))H(m(r))+∑j=1nλjf̃j(ψ-1(r),ν(r))}dr}=e∫sθa(ψ-1(τ))g((Φ-1m)(τ))dτm(s)-∫ssie∫rθa(ψ-1(τ))g((Φ-1m)(τ))dτ∑j=1nλjf̃j(ψ-1(r),ν(r)){a(ψ-1(r))H(m(r))+∑j=1nλjf̃j(ψ-1(r),ν(r))}dr.
So we can obtain
(2.72)m(s)=e∫sθa(ψ-1(τ))g((Φ-1m)(τ))dτm(s)-∫sθe∫rθa(ψ-1(τ))g((Φ-1m)(τ))dτ{a(ψ-1(r))H(m(r))+∑j=1nλjf̃j(ψ-1(r),ν(r))}dr+e∫siθa(ψ-1(τ))g((Φ-1m)(τ))dτ[(ΦBiΦ-1)(m(si))+(ΦWiΦ-1)(m(si))].
Repeating the above process for θ∈[s,s+ω̃], we obtain
(2.73)m(s)=e∫sθa(ψ-1(τ))g((Φ-1m)(τ))dτm(s)-∫sθe∫rθa(ψ-1(τ))g((Φ-1m)(τ))dτ{a(ψ-1(r))H(m(r))+∑j=1nλjf̃j(ψ-1(r),ν(r))}dr+∑i:si∈[s,θ)e∫siθa(ψ-1(τ))g((Φ-1m)(τ))dτ[(Φ-1BiΦ-1)(m(si))+(ΦWiΦ-1)(m(si))].
Noticing that m(s)=m(s+ω̃) and e∫s+ω̃sa(ψ-1(τ))g((Φ-1m)(τ))dτ=e∫ω̃0a(ψ-1(τ))g((Φ-1m)(τ))dτ, we find that m is a fixed point of Ψ.
Letm be a fixed point of Ψ. If s≠si,i∈ℤ, we have
(2.74)m′(s)=G(s,s+ω̃){a(ψ-1(s+ω̃))H(m(s+ω̃))+∑j=1nλjf̃j(ψ-1(s+ω̃),ν(s+ω̃))}-G(s,s){a(ψ-1(s))H(m(s))+∑j=1nλjf̃j(ψ-1(s),ν(s))}+a(ψ-1(s))g((Φ-1m)(s))m(s).
By Lemmas 2.11 and 2.12, we have ψ(t+ω)=ψ(t)+ψ(ω)=ψ(t)+ω̃, so it is easy to have t+ω=ψ-1(ψ(t)+ω̃)=ψ-1(s+ω̃) and ν(s+ω̃)=ν(s). Therefore, we can obtain
(2.75)m′(s)=a(ψ-1(s))g((Φ-1m)(s))m(s)-{a(ψ-1(s))H(m(s))+∑j=1nλjf̃j(ψ-1(s),ν(s))}.
If s=si,i∈ℤ, we can get
(2.76)m(si+)-m(si-)=∑j:sj∈[si+,si++ω̃)G(si,sj)((ΦBiΦ-1)(m(sj))+(ΦWiΦ-1)(m(sj)))-∑j:sj∈[si-,si-+ω̃)G(si,sj)((ΦBiΦ-1)(m(sj))+(ΦWiΦ-1)(m(sj)))=G(si,si+ω̃)((ΦBiΦ-1)(m(sj)+ω̃)+(ΦWiΦ-1)(m(sj)+ω̃))-G(si,si)((ΦBiΦ-1)(m(sj))+(ΦWiΦ-1)(m(sj)))=-((ΦBiΦ-1)(m(sj))+(ΦWiΦ-1)(m(sj))).
Therefore, m is ω̃-periodic solution of (2.62). The proof is complete.Lemma 2.16.
Assume that (H1)–(H5) hold, then Ψ(P)⊂P, and Ψ:P→P is compact and continuous.Proof.
By the definition ofP, for m∈P, we have
(2.77)(Ψm)(s+ω̃)=∫s+ω̃s+2ω̃G(s+ω̃,θ){a(ψ-1(θ))H(m(θ))+∑j=1nλjf̃j(ψ-1(θ),ν(θ))}dθ+∑i:si∈[s+ω̃,s+2ω̃]G(s+ω̃,si)((ΦBiΦ-1)(m(si))+(ΦWiΦ-1)(m(si)))=∫ss+ω̃G(s+ω̃,θ+ω̃){∑j=1nλjf̃j(ψ-1(θ+ω̃),ν(θ+ω̃))a(ψ-1(θ+ω̃))H(m(θ+ω̃))+∑j=1nλjf̃j(ψ-1(θ+ω̃),ν(θ+ω̃))}dθ+∑i:si∈[s,s+ω̃]G(s+ω̃,si)((ΦBiΦ-1)(m(si))+(ΦWiΦ-1)(m(si)))=∫ss+ω̃G(s,θ){a(ψ-1(θ))H(m(θ))+∑j=1nλjf̃j(ψ-1(θ),ν(θ))}dθ+∑i:si∈[s,s+ω̃]G(s,si)((ΦBiΦ-1)(m(si))+(ΦWiΦ-1)(m(si)))=(Ψm)(s).
Thus, (Ψm)(s+ω̃)=(Ψm)(s),s∈ℝ. So in view of (2.66), (2.68), for m∈P,s∈[0,ω̃], we have
(2.78)(Ψm)(s)=∫ss+ω̃G(s,θ){a(ψ-1(θ))H(m(θ))+∑j=1nλjf̃j(ψ-1(θ),ν(θ))}dθ+∑i:si∈[s,s+ω̃]G(s,si)((ΦBiΦ-1)(m(si))+(ΦWiΦ-1)(m(si)))≥r0L1-r0L{∫ss+ω̃{a(ψ-1(θ))H(m(θ))+∑j=1nλjf̃j(ψ-1(θ),ν(θ))}dθ+∑i:si∈[s,s+ω̃]((ΦBiΦ-1)(m(si))+(ΦWiΦ-1)(m(si)))}≥k11-r0l{∫ss+ω̃{a(ψ-1(θ))H(m(θ))+∑j=1nλjf̃j(ψ-1(θ),ν(θ))}dθ+∑i:si∈[s,s+ω̃]((ΦBiΦ-1)(m(si))+(ΦWiΦ-1)(m(si)))}≥k‖Ψm‖.
Therefore, Ψm⊂P. Next, we will show that Ψ is continuous and compact. Firstly, we will consider the continuity of Ψ. Let mn∈P and ∥mn-m∥→0 as n→+∞, then m∈P and |mn(s)-m(s)|→0 as n→+∞ for any s∈[0,ω̃]. By the continuity of fj(j=1,2,…,n),g,Φ,Φ-1,Wi(i=1,2,…,p), for any s∈[0,ω̃] and ɛ>0, we have
(2.79)|H(mn(s))-H(m(s))|≤1-r0l3ω̃a̅ɛ,
and denote νn(s):=(Φ-1mn)(ψ(ψ-1(s)-vj(ψ-1(s)))), and it is easy to see that ∥mn-m∥→0 as n→+∞ implies ∥νn-ν∥→0 as n→+∞, thus
(2.80)|f̃j(ψ-1(s),νn(s))-f̃j(ψ-1(s),ν(s))|≤1-r0l3ω̃λjnɛ,j=1,2,…,n,|((ΦBiΦ-1)(mn(si))+(ΦWiΦ-1)(mn(si)))-((ΦBiΦ-1)(m(si))+(ΦWiΦ-1)(m(si)))|≤1-r0l3ω̃pɛ,
where n is sufficiently large. For s∈[0,ω̃], we have
(2.81)‖Ψmn-Ψm‖=sup⁡s∈[0,ω̃]{|{∑i:si∈[s,s+ω̃]G(s,si)((ΦBiΦ-1)(m(si))+(ΦWiΦ-1)(m(si)))}∫ss+ω̃G(s,θ){a(ψ-1(θ))H(mn(θ))+∑j=1nλjf̃j(ψ-1(θ),νn(θ))}dθ+∑i:si∈[s,s+ω̃]G(s,si)((ΦBiΦ-1)(mn(si))+(ΦWiΦ-1)(mn(si)))-∫ss+ω̃G(s,θ){a(ψ-1(θ))H(m(θ))+∑j=1nλjf̃j(ψ-1(θ),ν(θ))}dθ-∑i:si∈[s,s+ω̃]G(s,si)((ΦBiΦ-1)(m(si))+(ΦWiΦ-1)(m(si)))|}=sup⁡s∈[0,ω̃]{|∫ss+ω̃G(s,θ){∑j=1nλj(f̃j(ψ-1(θ),νn(θ))-f̃j(ψ-1(θ),ν(θ)))a(ψ-1(s))(H(mn(s))-H(m(s)))+∑j=1nλj(f̃j(ψ-1(θ),νn(θ))-f̃j(ψ-1(θ),ν(θ)))}dθ+∑i:si∈[s,s+ω̃)G(s,si)(((ΦBiΦ-1)(mn(si))+(ΦWiΦ-1)(mn(si)))∫ss+ω̃G(s,θ){a(ψ-1(s))(H(mn(s))-H(m(s)))-((ΦBiΦ-1)(m(si))+(ΦWiΦ-1)(m(si))))G(s,θ){∑j=1nλj(f̃j(ψ-1(θ),νn(θ))-f̃j(ψ-1(θ),ν(θ)))a(ψ-1(s))(H(mn(s))-H(m(s)))∫ss+ω̃G(s,θ){a(ψ-1(s))(H(mn(s))-H(m(s)))|}≤ω̃1-r0l{a̅1-r0l3ω̃a̅ɛ+∑j=1nλj1-r0l3ω̃λjnɛ+p1-r0l3ω̃pɛ}=ɛ.
Therefore, Ψ is continuous on P.
Next, we prove thatΨ is a compact operator. Let S⊂P be an arbitrary bounded set in P, then there exists a number L0>0 such that ∥m∥<L0 for any m∈S. We prove that ΨS̅ is compact. In fact, from (H1), one has Wi(0)=0,i=1,2,…; by (2.53), it is easy to see that for any z∈ℝ, one has the following:
(2.82)‖Wi(z)‖≤k(l0)l0‖z‖+‖Wi(0)‖=k(l0)l0‖z‖,i=1,2,….
So for any {mn}n∈ℕ⊂S and s∈[0,ω̃], we have
(2.83)‖Ψmn‖=sup⁡s∈[0,ω̃]{|∫ss+ω̃(s,θ){a(ψ-1(θ))H(mn(θ))+∑j=1nλjf̃j(ψ-1(θ),νn(θ))}dθ+∑i:si∈[s,s+ω̃]G(s,si)((ΦBiΦ-1)(mn(si))+(ΦWiΦ-1)(mn(si)))|}≤ω̃1-r0l{a̅Lc̅1-c̅+∑j=1nmax⁡ν∈[-L0′,L0′]s∈[0,ω̃]λjf̃j(ψ-1(s),ν(s))+L0∑i=1p(Bi+k(l0)l0)}:=K,‖(Ψmn)′‖=sup⁡s∈[0,ω̃]{|∑j=1nλjf̃j(ψ-1(s),νn(s))a(ψ-1(s))g((Φ-1mn)(s))mn(s)-{a(ψ-1(s))H(mn(s))+∑j=1nλjf̃j(ψ-1(s),νn(s))}|}≤a̅L0‖Ψmn‖+a̅Lc̅1-c̅+∑j=1nmax⁡ν∈[-L0′,L0′]s∈[0,ω̃]λjf̃j(ψ-1(s),ν(s))≤a̅L0K+a̅Lc̅1-c̅+∑j=1nmax⁡ν∈[-L0′,L0′]s∈[0,ω̃]λjf̃j(ψ-1(s),ν(s)):=Q,
where L0′=L0/(1-c̅), which implies that {Ψmn}n∈ℕ and {(Ψmn)′}n∈ℕ are uniformly bounded on[0,ω̃]. Therefore, there exists a subsequence of {Ψmn}n∈ℕ which converges uniformly on [0,ω̃]; namely,ΨS̅ is compact. The proof is complete.


## 3. Main Results

Our main results of this paper are as follows.Theorem 3.1.
Assume that (H1)–(H5) hold, 0≤c(t)<1, for a sufficiently small Lipschitz constant l0; suppose that the following conditions hold:(H6)
α̃0=(1-c̅)(1-r0l)-ω̃a̅Lc̅-(1-c̅)∑i=1p(Bi+k(l0)l0)>0.(H7)
There exist positive constantsβ1,β2, and β4 with 0<β1<β2<β4 such that
(3.1)sup⁡s∈[0,ω̃]⁡fj(ψ-1(s),β1/(1-c̅))(β1/(1-c̅))α̃0<sup⁡s∈[0,ω̃]⁡fj(ψ-1(s),β4/(1-c̅))(β4/(1-c̅))α̃0<inf⁡s∈[0,ω̃]⁡fj(ψ-1(s),α̃β2)α̃β2β̃0,
where β̃0=((1-r0L)/α̃r0L)-ω̃a̲lc̲ and ω̃=ψ(ω).Then for all j=1,2,…,n,λj∈(λj1,λj2], (1.2) has at least three positive ω-periodic solutions, where
(3.2)λj1=α̃β2β̃0ω̃ninf⁡s∈[0,ω̃]⁡fj(t,α̃β2),λj2=(β4/(1-c̅))α̃0ω̃nsup⁡s∈[0,ω̃]⁡fj(t,β4/(1-c̅)),j=1,2,…,n.Proof.
First of all, since0<α̃<1/(1-c̅) and 0<r0<1, we have α̃0>0, so
(3.3)β̃0=1-r0Lα̃r0L-ω̃a̲lc̲>1-r0Lα̃r0L-ω̃a̅lc̅>1-r0Lα̃-(1-c̅)(1-r0l)+(1-c̅)∑i=1p(Bi+k(l0)l0)>(1-c̅)(r0l-r0L)+(1-c̅)∑i=1p(Bi+k(l0)l0)>0.
Furthermore, 0<λj1<λj2 in view of (3.1).
Now, define for eachλj∈(λj1,λj2] and m∈P a mapping Ψ:P→P by
(3.4)(Ψm)(s)=∫ss+ω̃G(s,θ){a(ψ-1(θ))H(m(θ))+∑j=1nλjf̃j(ψ-1(θ),ν(θ))}dθ+∑i:si∈[s,s+ω̃]G(s,si)((ΦBiΦ-1)(m(si))+(ΦWiΦ-1)(m(si))),
and a function ρ:P→[0,∞) by
(3.5)ρ(m)=min⁡s∈[0,ω̃]m(s).
For m∈Pβ4̅, by Lemma 2.13, we have
(3.6)0<(Φ-1m)(s)<β41-c̅.
It follows from (2.51), (2.68), (3.6), and (H2), for all j=1,2,…,n,λj∈(λj1,λj2] and m∈Pβ4̅ that
(3.7)(Ψm)(s)=∫ss+ω̃G(s,θ){a(ψ-1(θ))H(m(θ))+∑j=1nλjf̃j(ψ-1(θ),ν(θ))}dθ+∑i:si∈[s,s+ω̃]G(s,si)((ΦBiΦ-1)(m(si))+(ΦWiΦ-1)(m(si)))≤11-r0l{∫ss+ω̃{a(ψ-1(θ))H(m(θ))+∑j=1nλjf̃j(ψ-1(θ),ν(θ))}dθ+∑i:si∈[s,s+ω̃]((ΦBiΦ-1)(m(si))+(ΦWiΦ-1)(m(si)))}≤11-r0l{a̅ω̃Lc̅1-c̅β4+ω̃∑j=1nλj2sup⁡s∈[0,ω̃]f̃j(ψ-1(s),β41-c̅)+β4∑i=1p(bi+k(l0)l0)}=11-r0l{a̅ω̃Lc̅1-c̅β4+∑j=1n(β4/(1-c̅))α0ω̃nsup⁡s∈[0,ω̃]⁡f̃j(ψ-1(s),β4/(1-c̅))sup⁡s∈[0,ω̃]f̃j(ψ-1(s),β41-c̅)+β4∑i=1p(Bi+k(l0)l0)∑j=1n(β4/(1-c̅))α0ω̃nsup⁡s∈[0,ω̃]⁡f̃j(ψ-1(s),β4/(1-c̅))}≤β4.
By Lemma 2.16, we know that Ψ is completely continuous onPβ4̅.
We now assert that the condition (2) of Lemma2.1 holds. Indeed, if m∈Pβ1̅, then similar to above argument, by (3.1), we have
(3.8)(Ψm)(s)=∫ss+ω̃G(s,θ){a(ψ-1(θ))H(m(θ))+∑j=1nλjf̃j(ψ-1(θ),ν(θ))}dθ+∑i:si∈[s,s+ω̃]G(s,si)((ΦBiΦ-1)(m(si))+(ΦWiΦ-1)(m(si)))≤11-r0l{∫ss+ω̃{a(ψ-1(θ))H(m(θ))+∑j=1nλjf̃j(ψ-1(θ),ν(θ))}dθ+∑i:si∈[s,s+ω̃]((ΦBiΦ-1)(m(si))+(ΦWiΦ-1)(m(si)))}≤11-r0l{{∑i:si∈[s,s+ω̃]((ΦBiΦ-1)(m(si))+(ΦWiΦ-1)(m(si)))}∫ss+ω̃{a(ψ-1(θ))H(m(θ))+∑j=1nλjf̃j(ψ-1(θ),ν(θ))}dθ+∑i:si∈[s,s+ω̃]((ΦBiΦ-1)(m(si))+(ΦWiΦ-1)(m(si)))}≤11-r0l{a̅ω̃Lc̅1-c̅β1+ω̃∑j=1nλj2sup⁡s∈[0,ω̃]f̃j(ψ-1(s),β11-c̅)+β1∑i=1p(Bi+k(l0)l0)}=11-r0l{a̅ω̃Lc̅1-c̅β1+∑j=1n(β4/(1-c̅))α0ω̃nsup⁡s∈[0,ω̃]⁡f̃j(ψ-1(s),β4/(1-c̅))sup⁡s∈[0,ω̃]f̃j(ψ-1(s),β11-c̅)+β1∑i=1p(Bi+k(l0)l0)ω̃Lc̅1-c̅β1+∑j=1n(β4/(1-c̅))α0ω̃nsup⁡s∈[0,ω̃]⁡f̃j(ψ-1(s),β4/(1-c̅))}<11-r0l{a̅ω̃Lc̅1-c̅β1+β41-c̅α0β1β4+β1∑i=1p(Bi+k(l0)l0)}=β1.
Hence, ∥Ψm∥<β1 holds.
Choose a positive constantβ3 such that 0<β2<kβ3<β3≤β4. Next, we show that the condition (1) of Lemma 2.1 holds. Obviously, ρ is a concave continuous function on P with ρ(m)≤∥m∥ for m∈Pβ4̅. We notice that if m(s)=(2/5)β2+(3/5)β3 for s∈[0,ω̃], then m∈{m∈P(ρ,β2,β3):ρ(m)>β2} which implies {m∈P(ρ,β2,β3):ρ(m)>β2}≠∅. For m∈P(ρ,β2,β3), we have
(3.9)β2≤ρ(m)=min⁡s∈[0,ω̃]m(s)≤‖m‖≤β3,
which implies, from (2.50), that
(3.10)(Φ-1m)(s)≥α̃‖m‖≥αβ2.
And it is also clear that Φ(x) is nondecreasing for x>0 and Bi,Wi∈C(ℝ,ℝ+), and we can easily have ΦBiΦ-1,ΦWiΦ-1∈C(ℝ,ℝ+). Hence
(3.11)ρ(Ψm)=min⁡s∈[0,ω̃](Ψm)(s)=min⁡s∈[0,ω̃]{∫ss+ω̃G(s,θ){a(ψ-1(θ))H(m(θ))+∑j=1nλjf̃j(ψ-1(θ),ν(θ))}dθ+∑i:si∈[s,s+ω̃]G(s,si)((ΦBiΦ-1)(m(si))+(ΦWiΦ-1)(m(si)))}≥r0L1-r0Lmin⁡s∈[0,ω̃]{∫ss+ω̃{a(ψ-1(θ))H(m(θ))+∑j=1nλjf̃j(ψ-1(θ),α̃β2)}dθ+∑i:si∈[s,s+ω̃]((ΦBiΦ-1)(m(si))+(ΦWiΦ-1)(m(si)))}>r0L1-r0L{ω̃a̲lc̲α̃β2+ω̃∑j=1nλj1inf⁡s∈[0,ω̃]f̃j(ψ-1(s),α̃β2)+min⁡β2≤m≤β3∑i=1p((ΦBiΦ-1)(m(si))+(ΦWiΦ-1)(m(si)))}≥r0L1-r0L{{∑i=1p((ΦBiΦ-1)(m(si))+(ΦWiΦ-1)(m(si)))∑j=1nα̃β2β̃0ω̃ninf⁡s∈[0,ω̃]f̃j(ψ-1(s),α̃β2)}ω̃a̲lc̲α̃β2+ω̃∑j=1nα̃β2β̃0ω̃ninf⁡s∈[0,ω̃]⁡f̃j(ψ-1(s),α̃β2)inf⁡s∈[0,ω̃]f̃j(ψ-1(s),α̃β2)+min⁡β2≤m≤β3∑i=1p((ΦBiΦ-1)(m(si))+(ΦWiΦ-1)(m(si)))∑j=1nα̃β2β̃0ω̃ninf⁡s∈[0,ω̃]f̃j(ψ-1(s),α̃β2)12}>β2
for all m∈P(ρ,β2,β3).
Finally, we prove that the condition (3) of Lemma2.1 holds. Let m∈P(ρ,β2,β4) and ∥Ψm∥>β3, then ρ(Ψm)>β2. We notice that (3.4) implies that
(3.12)‖Ψm‖≤11-r0l{{∑i=1p((ΦBiΦ-1)(m(si))+(ΦWiΦ-1)(m(si)))∑j=1nα̃β2β̃0ω̃ninf⁡s∈[0,ω̃]f̃j(ψ-1(s),α̃β2)}∫ss+ω̃{a(ψ-1(θ))H(m(θ))+∑j=1nλjf̃j(ψ-1(θ),α̃β2)}dθ+∑i:si∈[s,s+ω̃]((ΦBiΦ-1)(m(si))+(ΦWiΦ-1)(m(si)))}.
Thus
(3.13)ρ(Ψm)=min⁡s∈[0,ω̃]{∫ss+ω̃G(s,θ){a(ψ-1(θ))H(m(θ))+∑j=1nλjf̃j(ψ-1(θ),ν(θ))}dθ+∑i:si∈[s,s+ω̃]G(s,si)((ΦBiΦ-1)(m(si))+(ΦWiΦ-1)(m(si)))}≥r0L1-r0Lmin⁡s∈[0,ω̃]{{∑i=1p((ΦBiΦ-1)(m(si))+(ΦWiΦ-1)(m(si)))∑j=1nα̃β2β̃0ω̃ninf⁡s∈[0,ω̃]f̃j(ψ-1(s),α̃β2)}∫ss+ω̃{a(ψ-1(θ))H(m(θ))+∑j=1nλjf̃j(ψ-1(θ),ν(θ))}dθ+∑i:si∈[s,s+ω̃]((ΦBiΦ-1)(m(si))+(ΦWiΦ-1)(m(si)))}≥r0L(1-r0l)1-r0l‖Ψm‖≥kβ3>β2.
To sum up, all the hypotheses of Lemma 2.1 are satisfied. Hence Ψ has at least three positive fixed points. That is, (1.2) has at least three positive ω-periodic solutions. This completes the proof.Corollary 3.2.
Suppose (H1)–(H6) hold. If
(3.14)lim⁡x→∞sup⁡s∈[0,ω̃]⁡fj(ψ-1(s),x)x=0,(3.15)lim⁡x→0sup⁡s∈[0,ω̃]⁡fj(ψ-1(s),x)x=0,
where ω̃=ψ(ω); then (1.2) has at least three positive ω-periodic solutions.Proof.
In view of (3.14), we can choose β4>β2>0 such that the second inequality in (3.1) holds, and in view of (3.15), we can choose β1∈(0,β2) such that the first inequality in (3.1) holds. Therefore, the conclusion of Theorem 3.1 holds. This completes the proof.


## 4. An Example

Let us consider the variable time scale𝕋0(x) constructed by ti=i,τi(x)=(-1)il0sinx, where |x|>1 for all t∈𝕋0(x), 0<l0<(1/2) and consider π-periodic system:(4.1)(x(t)+19|cos⁡t|x(t-α))′=1π|sin⁡t|(13+13e-x)x(t)-l03n∑j=1nλjx1/2(t)ln⁡(x(t-e(1/j)|sin⁡t|)+1),(t,x)∈T0(x),x+=0.03(23)ix+0.02l0sin⁡x+x,t+=2i+1-l0sin⁡x,
where α is a constant, λj,j=1,2,…,n are nonnegative parameters. In this case, c(t)=(1/9)|cost|,a(t)=1/π,g(x(t))=(1/3)+(1/3)e-x,Bi=0.03(2/3)i,Ji(x)=0.02l0sinx,vj(t)=e(1/j)|sint| and fj(t,x(t-vj(t)))=(l0/3n)x1/2(t)ln(x(t-vj(t))+1),j=1,2,…,n,i∈ℕ. Obviously, (H1)–(H3) are satisfied, and it is easy to see that (3.14) and (3.15) hold.By the formula ofψ-substitution and δk=1, one can find(4.2)ω̃=ψ(ω)=ω-∑0<2k<ωδk=π-1.Clearly,L=(2/3),l=(1/3),0≤c(t)≤(1/9)<1,r0=e-(π-1)(1/π)≈0.5058 and c̅=1/9,c̲=0, so we can find δ=(c̲+c̅)/2=1/18, and it is easy to check that ((2δ-c̲)(1-c̲2))/(1-(2δ-c̲)2)=9/80≈0.1125<(r0L(1-r0l))/(1-r0L)=0.3532. Thus, (H4) holds. Furthermore, we also have(4.3)‖∂fj∂x‖=l03n‖ln⁡(x+1)2x1/2+x1/2x+1‖≤l03n(‖ln⁡(x+1)1/2x1/2‖+‖x1/2x+1‖)≤l03n(‖(x+1)1/2x1/2‖+‖x1/2x+1‖)=l03n(‖(1+1x)1/2‖+‖x1/2x+1‖)<l03n(2+1)=l0n.
Hence, for any x1,x2∈ℝ, one can get(4.4)∑j=1n|fj(t,x1)-fj(t,x2)|≤|∂fj∂x||x1-x2|<n⋅l0n|x1-x2|=l0|x1-x2|.
So (H5) is satisfied. For a sufficiently small l0, one can also have(4.5)α̃0=(1-c̅)(1-r0l)-ω̃a̅Lc̅-(1-c̅)∑i=1p(Bi+k(l0)l0)=0.0768-89∑i=1pk(l0)l0,
since k(l0) is a bounded function; for a sufficiently small l0, one can have α̃0>0 such that (H6) holds. Therefore, according to Corollary 3.2, (4.1) has at least three positive π-periodic solutions.



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*Source: 516476-2012-03-07.xml*