多参数二阶脉冲时滞系统极值解的存在性

赵 昕,王淑玲,白 杰

(1.吉林农业大学 信息技术学院,长春 130118;2.东北师范大学人文学院,长春 130117)

脉冲时滞微分方程理论已成为微分方程理论的重要组成部分[1].具有脉冲效应的泛函微分方程边界问题解的存在性研究目前已受到人们广泛关注,并得到了许多有意义的结果[2-9].多参数泛函系统的研究对生物种群的研究具有重要意义,对于参数化边界值问题解的存在性研究近年已取得一些进展[10-17].但关于带参数二阶脉冲时滞微分方程边值问题的研究目前报道较少.本文考虑如下边界值问题(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;常数τ>0.

定义1如果(α0,γ0)满足下列条件,则(α0,γ0)称为问题(1)的上解:

定义2如果(β0,ζ0)满足下列条件,则(β0,ζ0)称为问题(1)的下解:

令J-=J{t1,t2,…,tm},PC(J0,)={u:J0;当t≠tk时u(t)连续,存在,且(J,)={u:J;当t≠tk时u′(t)连续,u′(存在,且k=1,2,…,m}.显然E={u∈PC′(J,)}为Banach空间,其范数为‖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}.

先考虑线性边值问题(BVP):

(2)

仿照文献[12,14]主要结果的证明， 可得下列3个引理.

引理1u∈E∩C2(J-,)是方程(2)的解,当且仅当u∈PC(J,)满足脉冲积分方程

则方程(2)有唯一解u∈E.

定义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假设p∈E∩C2(J-,)满足:

(4)

假设条件:

(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假设条件(H1)～(H6)成立,且

(5)

则在[α0,β0]×[γ0,ζ0]上必存在单调序列{(αn(t),γn)}和{(βn(t),ζn)}⊂(E∩C2(J-,))×n,并分别收敛于边值问题(1)的极值解.

证明: 对于任意的(η,γ)∈[α0,β0]×[γ0,ζ0],考虑如下线性边值问题:

(6)

其中σ(t)=f(t,η(t),ηt,γ)+Mη(t)+Nηt.

由引理2可知,线性边值问题(6)有唯一解(u,μ)∈E×n.定义算子A,使得A(η,γ)=(u,μ).易证: 1) (α0,γ0)≤A(α0,γ0),(β0,ζ0)≥A(β0,ζ0);2)A在[α0,β0]×[γ0,ζ0]是增算子.

为证明1),设A(α0,γ0)=(α1,a),A(β0,ζ0)=(β1,b).先证明(α0,γ0)≤(α1,a).

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

由引理3可知,p1(t)≤0,t∈J0,i.e.,u1(t)≤u2(t).再根据式(6)和假设条件(H4),有

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

显然,(αi,γi),(βi,ζi)(i=1,2,…)满足:

根据引理3,有p(t)≤0,i.e.,αn+1(t)≤u(t),∀t∈J.由条件(H4)可知

同理可以证明u(t)≤βn+1(t)(t∈J),μ≤ζn+1(t∈J).从而(αn+1,γn+1)≤(u,μ)≤(βn+1,ζn+1),所以(α*,γ*)≤(u,μ)≤(β*,ζ*).证毕.

注1实际上,同理也可以证明下列线性边值问题极值解的存在性:

(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)为常数.

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