趙 昕,王淑玲,白 杰

(1.吉林農(nóng)業(yè)大學(xué) 信息技術(shù)學(xué)院,長(zhǎng)春 130118;2.東北師范大學(xué)人文學(xué)院,長(zhǎng)春 130117)

脈沖時(shí)滯微分方程理論已成為微分方程理論的重要組成部分[1].具有脈沖效應(yīng)的泛函微分方程邊界問(wèn)題解的存在性研究目前已受到人們廣泛關(guān)注,并得到了許多有意義的結(jié)果[2-9].多參數(shù)泛函系統(tǒng)的研究對(duì)生物種群的研究具有重要意義,對(duì)于參數(shù)化邊界值問(wèn)題解的存在性研究近年已取得一些進(jìn)展[10-17].但關(guān)于帶參數(shù)二階脈沖時(shí)滯微分方程邊值問(wèn)題的研究目前報(bào)道較少.本文考慮如下邊界值問(wèn)題(BVP):

(1)

其中:f∈C(J××D×n,);gi∈C(×,);D=L1([-τ,0],×n,);0≤t1≤t2≤…≤tm≤T;J0=[-τ,T];ΔΔk=1,2,…,m;常數(shù)τ>0.

定義1如果(α0,γ0)滿足下列條件,則(α0,γ0)稱為問(wèn)題(1)的上解:

定義2如果(β0,ζ0)滿足下列條件,則(β0,ζ0)稱為問(wèn)題(1)的下解:

令J-=J{t1,t2,…,tm},PC(J0,)={u:J0;當(dāng)t≠tk時(shí)u(t)連續(xù),存在,且(J,)={u:J;當(dāng)t≠tk時(shí)u′(t)連續(xù),u′(存在,且k=1,2,…,m}.顯然E={u∈PC′(J,)}為Banach空間,其范數(shù)為‖u‖PC′=max{‖u(t)‖PC,‖u′(t)‖PC′},其中‖u(t)‖PC=max{|u(t)|:t∈J0},‖u′(t)‖PC=max{|u′(t)|:t∈J}.

先考慮線性邊值問(wèn)題(BVP):

(2)

仿照文獻(xiàn)[12,14]主要結(jié)果的證明， 可得下列3個(gè)引理.

引理1u∈E∩C2(J-,)是方程(2)的解,當(dāng)且僅當(dāng)u∈PC(J,)滿足脈沖積分方程

則方程(2)有唯一解u∈E.

定義J0=(t0,t1],J1=(t1,t2],…,Jm=(tm,tm+1],a=max{tk-tk+1,k=0,1,…,m},t0=0,tm+1=T.

引理3假設(shè)p∈E∩C2(J-,)滿足:

(4)

假設(shè)條件:

(H1) (α0,γ0),(β0,γ1)是方程(1)的上下解,且(α0,γ0)≤(β0,γ1);

其中α0≤v(tk)≤u(tk)≤β(tk),k=1,2,…,m;

定理1假設(shè)條件(H1)～(H6)成立,且

(5)

則在[α0,β0]×[γ0,ζ0]上必存在單調(diào)序列{(αn(t),γn)}和{(βn(t),ζn)}?(E∩C2(J-,))×n,并分別收斂于邊值問(wèn)題(1)的極值解.

證明: 對(duì)于任意的(η,γ)∈[α0,β0]×[γ0,ζ0],考慮如下線性邊值問(wèn)題:

(6)

其中σ(t)=f(t,η(t),ηt,γ)+Mη(t)+Nηt.

由引理2可知,線性邊值問(wèn)題(6)有唯一解(u,μ)∈E×n.定義算子A,使得A(η,γ)=(u,μ).易證: 1) (α0,γ0)≤A(α0,γ0),(β0,ζ0)≥A(β0,ζ0);2)A在[α0,β0]×[γ0,ζ0]是增算子.

為證明1),設(shè)A(α0,γ0)=(α1,a),A(β0,ζ0)=(β1,b).先證明(α0,γ0)≤(α1,a).

又由引理3可知,p0(t)≤0,t∈J0,i.e.,α(t)≤α1(t).再根據(jù)式(6),得

由引理3可知,p1(t)≤0,t∈J0,i.e.,u1(t)≤u2(t).再根據(jù)式(6)和假設(shè)條件(H4),有

令(αn,γn)=A(αn-1,γn-1),(βn,ζn)=A(βn-1,ζn-1),n=1,2,…,可構(gòu)造序列{(αn(t),γn)}和{(βn(t),ζn)},使得

顯然,(αi,γi),(βi,ζi)(i=1,2,…)滿足:

根據(jù)引理3,有p(t)≤0,i.e.,αn+1(t)≤u(t),?t∈J.由條件(H4)可知

同理可以證明u(t)≤βn+1(t)(t∈J),μ≤ζn+1(t∈J).從而(αn+1,γn+1)≤(u,μ)≤(βn+1,ζn+1),所以(α*,γ*)≤(u,μ)≤(β*,ζ*).證畢.

注1實(shí)際上,同理也可以證明下列線性邊值問(wèn)題極值解的存在性:

(7)

其中:fi∈C(J××D×n,);gi∈C(×,);D=L1([-τ,0],×n,);0≤t1≤t2≤…≤tm≤T;J0=[-τ,T];ΔΔk=1,2,…,m;τ>0;cj(j=1,2,…,n1)為常數(shù).

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