郭彥平，苗素榮，禹長(zhǎng)龍

(河北科技大學(xué)理學(xué)院，河北石家莊 050018)

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無(wú)窮區(qū)間上二階三點(diǎn)差分方程邊值問(wèn)題正解的存在性

郭彥平，苗素榮，禹長(zhǎng)龍

(河北科技大學(xué)理學(xué)院，河北石家莊 050018)

為了將差分方程應(yīng)用到解無(wú)窮區(qū)間邊值問(wèn)題，借助于相應(yīng)線性邊值問(wèn)題Green函數(shù)的性質(zhì)，研究了無(wú)窮區(qū)間上的二階三點(diǎn)差分方程邊值問(wèn)題。通過(guò)Banach壓縮映像原理和Leray-Schauder不動(dòng)點(diǎn)定理獲得了該問(wèn)題正解的存在性和唯一性定理，推廣了已有結(jié)論。

常微分方程其他學(xué)科；差分方程；Green 函數(shù)；Leray-Schauder不動(dòng)點(diǎn)定理；無(wú)窮區(qū)間

無(wú)窮區(qū)間邊值問(wèn)題起源于應(yīng)用數(shù)學(xué)和物理領(lǐng)域，具有廣泛的應(yīng)用背景[1-2]。近年來(lái)，由于無(wú)窮區(qū)間邊值問(wèn)題的廣泛應(yīng)用，引起了學(xué)者的關(guān)注，尤其是在利用各種不動(dòng)點(diǎn)理論、拓?fù)涠壤碚摗⒅睾隙壤碚撘约吧舷陆夥椒ǖ确蔷€性泛函的工具對(duì)無(wú)窮區(qū)間上的2階、n階等邊值問(wèn)題(Dirichlet問(wèn)題、周期、脈沖、時(shí)滯等邊值問(wèn)題)的研究中獲得了重要研究成果，參見(jiàn)文獻(xiàn)[3—10]。

隨著現(xiàn)代科學(xué)技術(shù)的進(jìn)步，許多生產(chǎn)實(shí)際和科學(xué)研究中所遇到的微分方程極其復(fù)雜，很多情況下很難得到解甚至根本得不到解析表達(dá)式。為解決問(wèn)題，自然需要考慮其近似解或研究解的性質(zhì)，這就需要將微分方程離散化，這便得到了差分方程。由于差分方程在現(xiàn)代醫(yī)學(xué)、生物數(shù)學(xué)、生態(tài)學(xué)、物理學(xué)、化學(xué)等方面的廣泛應(yīng)用，已有很多學(xué)者利用各種方法和技巧對(duì)差分方程邊值問(wèn)題進(jìn)行了研究，但是相對(duì)微分方程邊值問(wèn)題則缺乏足夠的理論和方法。

對(duì)無(wú)窮區(qū)間上的差分方程邊值問(wèn)題的研究已有一些成果[11-18]，但整體上，無(wú)窮區(qū)間上的差分方程邊值問(wèn)題的結(jié)果尚少，理論尚不完善，還需要更多的研究和討論。

在文獻(xiàn)[19]中，AGARWAL等研究了無(wú)窮區(qū)間上的二階差分方程邊值問(wèn)題：

非負(fù)解的存在性。

在文獻(xiàn)[20]中，LIAN等利用Schauder不動(dòng)點(diǎn)定理和上下解的技術(shù)研究了無(wú)窮區(qū)間上的二階差分方程邊值問(wèn)題:

-Δ2xk-1=f(k,xk,Δxk-1),k∈N,

x0-aΔx0=B, Δx∞=C

1個(gè)和3個(gè)無(wú)界解的存在性。

在本文中，運(yùn)用Leray-Schauder不動(dòng)點(diǎn)定理研究無(wú)窮區(qū)間上的二階差分方程三點(diǎn)邊值問(wèn)題:

(1)

解的存在性，其中N={1,2,…,∞}，f:N×R2→R是連續(xù)函數(shù)，α∈R,α≠1，η∈N且Δx(k)=x(k+1)-x(k)。

C1)f:N×R2→R是連續(xù)函數(shù)，且?r>0，?k∈N；?φr(k)，kφr(k)∈l1，φr(k)>0；使得當(dāng)max{|u|,|v|}≤r時(shí)，?k∈N有|f(k,u,v)|≤φr(k)；

C2)f:N×R2→R是連續(xù)函數(shù)，且?p(k),q(k),r(k)∈l1；且kp(k),kq(k),kr(k)∈l1，使得?k∈N,(u,v)∈R2有|f(k,u,v)|≤p(k)|u|+q(k)|v|+r(k)。

1 預(yù)備知識(shí)

引理1 若{v(k)}k∈N和{kv(k)}k∈N∈l1，則線性邊值問(wèn)題：

由x(k)=αx(η)和式(4)可得：

,

命題得證。

附注1 顯然G(k,i)滿足Green函數(shù)的性質(zhì)，所以稱G(k,i)為無(wú)窮區(qū)間三點(diǎn)邊值問(wèn)題(2)對(duì)應(yīng)的齊次方程的Green函數(shù)。

引理2 ?k,i∈N，則

證明 對(duì)于?i∈N，G(k,i)關(guān)于k為非減函數(shù)。于是有

故而有

因此命題得證。

引理3Green函數(shù)G(k,i)滿足：

2 主要結(jié)論

引理4 設(shè)條件C1)成立，則?λ∈[0,1]，T(x,λ)在X上是全連續(xù)的。

證明 首先，證明Tx∈X。對(duì)?x∈X，則?r>0,使得‖x‖

由T的定義得：

|Δ(T(x,λ)(k))|= |(T(x,λ)(k+1)-T(x,λ)(k))|=

其次，證明T(x,λ)在X連續(xù)。設(shè)?xn∈X,當(dāng)n→∞時(shí)，xn→x。下面證明?λ∈[0,1],當(dāng)n→∞時(shí)，T(xn,λ)→T(x,λ)。由條件C1)知

其中r0>0是一實(shí)數(shù)。且maxn∈N{‖xn‖,‖x‖}≤r0，于是有

|T(xn,λ) (k)-T(x,λ)(k)|≤

且

|ΔT(xn,λ) (k)-ΔT(x,λ)(k)|≤

因此，T是連續(xù)的。

最后，證明T是緊的，即T映X中的有界集為相對(duì)緊集。設(shè)B?X為有界子集，則?r>0,?x∈B,‖x‖

且

因此，TB是有界的。又

且

|ΔT(x,λ)(k)|≤

因此，TB在無(wú)窮遠(yuǎn)處一致收斂，故TB為相對(duì)緊的。由定理1知，T(·,λ):X×[0,1]→X是全連續(xù)的。

證明 由引理1，顯然當(dāng)且僅當(dāng)x是T(·,λ)的不動(dòng)點(diǎn)時(shí)，x∈X是邊值問(wèn)題(1)的解。顯然，?x∈X有T(x,0)=0。由Leray-Schauder連續(xù)定理可知定理的證明只需證明?λ∈[0,1]，T(·,λ)在閉集X中的不動(dòng)點(diǎn)不依賴于λ即可。

下面證明T(·,λ)的不動(dòng)點(diǎn)有一個(gè)不依賴λ的先驗(yàn)界M。假設(shè)x=T(x,λ)且記

情形1α<0，?x∈X,?k∈N有

因此，

‖Δx‖∞≤ ‖λf(i,x(i),Δx(i-1))‖l1≤‖f(i,x(i),Δx(i-1))‖l1≤

‖p(i)|x(i)|+q(i)|Δx(i-1)|+r(i)‖l1≤(ηP+P1+Q)‖Δx‖∞+R,

即

同理，得：

P1‖x‖∞+Q1ΔM1+R1,k∈N,

即

記M=max{M1,ΔM1},M與λ無(wú)關(guān),則‖x‖≤M。

情形2 0≤α<1，對(duì)任意?x∈X,?k∈N有

記M=max{M2,ΔM2},M與λ無(wú)關(guān)，則‖x‖≤M。

情形3α>1，對(duì)于?x∈X，?k∈N，得：

同理，有：

且

即

記M=max{M3,ΔM3},與λ無(wú)關(guān)。所以邊值問(wèn)題(1)至少有1個(gè)解。

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Existence of positive solutions to boundary value problem of second-order three-point difference equations on infinite intervals

GUO Yanping, MIAO Surong, YU Changlong

(School of Science, Hebei University of Science and Technology, Shijiazhuang, Hebei 050018, China)

In order to apply difference equation in boundary value problem on infinite intervals, with the help of the properties of Green function in linear boundary value problems，the boundary value problem of second-order three-point difference equations on infinite intervals is studied. The theory of existence and uniqueness of positive solutions to this problem is are obtained by Banach's contraction mapping principle and Leray-Schauder fixed point theorem, which generalizes the known results.

ordinary differential equation； difference equation； Green function； Leray-Schauder fixed point theorem； infinite interval

1008-1542(2016)06-0556-06

10.7535/hbkd.2016yx06006

2016-05-12；

2016-09-06；責(zé)任編輯：張 軍

國(guó)家自然科學(xué)基金(11201112)；河北省自然科學(xué)基金(A2013208147, A2014208152, A2015208114)；河北省高等學(xué)?？茖W(xué)技術(shù)研究項(xiàng)目(QN2015175，QN2016165)

郭彥平(1965—)，男，河北張家口人，教授，博士，主要從事微分方程邊值問(wèn)題方面的研究。

E-mail： guoyanping65@sohu.com

O175 MSC(2010)主題分類:34B40

A

郭彥平，苗素榮，禹長(zhǎng)龍.無(wú)窮區(qū)間上二階三點(diǎn)差分方程邊值問(wèn)題正解的存在性 [J].河北科技大學(xué)學(xué)報(bào),2016,37(6):556-561. GUO Yanping, MIAO Surong, YU Changlong.Existence of positive solutions to boundary value problem of second-order three-point difference equations on infinite intervals[J].Journal of Hebei University of Science and Technology,2016,37(6):556-561.