具有多項(xiàng)差分算子的三階q-差分方程邊值問題

楊小輝, 李杰民(. 廣東警官學(xué)院 計(jì)算機(jī)系, 廣東 廣州 5030; . 嶺南師范學(xué)院 數(shù)學(xué)與計(jì)算科學(xué)學(xué)院, 廣東 湛江 54048)

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具有多項(xiàng)差分算子的三階q-差分方程邊值問題

楊小輝1, 李杰民2
(1. 廣東警官學(xué)院 計(jì)算機(jī)系, 廣東 廣州 510230; 2. 嶺南師范學(xué)院 數(shù)學(xué)與計(jì)算科學(xué)學(xué)院, 廣東 湛江 524048)

q-差分方程邊值問題解的存在性已經(jīng)引起國內(nèi)外數(shù)學(xué)工作者的研究興趣,并且得到許多有價(jià)值的結(jié)果.研究一類三階q-差分方程邊值問題,該問題是由一個(gè)三階q-差分方程和3個(gè)具有多項(xiàng)q-差分算子為邊界條件構(gòu)成.這種邊界條件可以看成是Sturm-Liouville邊界條件的推廣.利用Banach壓縮映射原理和Krasnoselskii不動(dòng)點(diǎn)定理,獲得了該類邊值問題解的存在性和唯一性的充分條件.所得條件簡潔,便于驗(yàn)證.結(jié)果推廣和改進(jìn)了已有文獻(xiàn)中的定理.最后,舉2個(gè)例子來演示所得結(jié)論的應(yīng)用.

q-差分方程;q-微分;q-積分; 邊值問題

1 引言及預(yù)備知識(shí)

q-差分方程歷史悠久[1-4],q-差分方程在多個(gè)學(xué)科中已得到應(yīng)用[5-8].近年來q-差分方程解的存在性問題是數(shù)學(xué)工作者研究的中心問題之一[9-17].Sturm-Liouville型邊值問題一直是大家關(guān)注的問題[18-21].B. Ahmad等[12]研究了三階q-差分方程兩點(diǎn)邊值問題

u(0)=0,Dqu(0)=0,u(1)=0

(1)

C. L. Yu等[15]研究了三階q-差分方程兩點(diǎn)邊值問題

u(0)=0,Dqu(0)=0,

(2)

正解的存在性,其中,00且(α-β)/(α+β)≤q,α、β、q都是常數(shù),

注意到邊值問題(1)和(2)僅涉及到一個(gè)q-差分算子Dq,而涉及多項(xiàng)q-差分算子的三階q-差分方程邊值問題的研究較少.受到文獻(xiàn)[12-13]的啟發(fā),本文研究具有4個(gè)q-差分算子的三階q-差分方程兩點(diǎn)邊值問題

u(0)-αDp1u(0)=0,

u(1)+βDp2u(1)=0,

(3)

其中,0

Δ=1+γ(1+p3)(1+α)+

β(1+p2)+γβ(1+p3)>0.

2 引理

首先介紹相關(guān)概念,然后給出2個(gè)引理.

定義 2.1[8]設(shè)0

定義 2.2[8]設(shè)0

定義 2.3[8]設(shè)00,函數(shù)g(t):[0,t]→R在區(qū)間[0,t]上的q-積分記為Iqg(t),定義為

若g在[a,b]上有定義,函數(shù)g(t)定義在區(qū)間[a,b]上的q-積分定義為

注意到

DqIqg(t)=g(t),

IqDqg(t)=g(t)-g(0)(g(t)在t=0處連續(xù)).

引理 2.4[8]q-差分算子有如下性質(zhì):

Dq(gh)(t)=(Dq(g(t))h(t))+g(qt)Dqh(t),

引理 2.5 設(shè)y(t)∈C[0,1],則u為邊值問題

u(0)-αDp1u(0)=0,

u(1)+βDp2u(1)=0,

(4)

的解當(dāng)且僅當(dāng)

β(1+p2)]y(s)dqs+

(1-p2)dqs}.

(5)

(6)

對(6)式在[0,t]上進(jìn)行q-積分得到

(7)

對(7)式在[0,t]上進(jìn)行q-積分得到

(8)

其中,a0、a1、a2是常數(shù).

當(dāng)t≠0時(shí),

(9)

注意到

又有

(10)

還有

(11)

同理

此時(shí),可知Dpiu(0)=a1.

當(dāng)t≠0時(shí),

于是有

類似(10)和(11)式可得

(12)

利用(4)式的邊值條件

u(0)-αDp1u(0)=0,

可以得到

(13)

把(13)式代入(8)式,并令t=1得到

由(9)式知

所以

(14)

利用u(1)+βDp2u(1)=0得到

(15)

(15)式兩邊通乘以1+q,左邊等于

a2[1+γ(1+p3)(1+α)+

β(1+p2)+γβ(1+p3)]=a2Δ,

右邊等于

β(1+p2)]y(s)dqs+

整理得

β(1+p2)]y(s)dqs+

(1-p2)dqs}.

(16)

把(16)式代入(13)式,可得a0和a1,把a(bǔ)0、a1和a2代入(8)式得到(5)式,所以u滿足(5)式.

反之,設(shè)u滿足(5)式,容易驗(yàn)證u滿足(4)式.證畢.

為了進(jìn)一步的分析,設(shè)X=C[I,R]表示從I到R的所有連續(xù)函數(shù)集合,定義范數(shù)

‖X‖=sup{|x(t)|,t∈I}.

這時(shí)X為Banach空間.記

(1+p2)q](1-p2+β)}.

(17)

3 主要結(jié)論

定理 3.1 設(shè)f∈C(I×R,R),I=[0,1],且滿足Lipschitz條件

|f(t,u)-f(t,v)|≤L|u-v|,

?t∈I,u,v∈R,

L為Lipschitz常數(shù),則當(dāng)LH<1時(shí),(3)式有唯一解,其中H為(17)式定義.

證明 構(gòu)造X上的非線性算子F為

β(1+p2)]f(s,u(s))dqs+

(1-p2)dqs},u∈X.

(18)

由f的連續(xù)性容易證明F:X→X是全連續(xù)算子,u為(18)式的解當(dāng)且僅當(dāng)u∈X為F的不動(dòng)點(diǎn).

設(shè)Br={u∈X:‖u‖≤r},當(dāng)u∈Br時(shí),有|u(t)|≤r,t∈[0,1],所以

β(1+p2)]f(s,u(s))dqs+

(1-p2)dqs}|≤

f(s,0)|+|f(s,0)|)dqs+

(1+β)(1+q)qs+β(1+p2)]×

(|f(s,u(s))-f(s,0)|+|f(s,0)|)dqs+

(|f(s,u(s))-f(s,0)|+|f(s,0)|)dqs]}≤

β(1+p2)dqs+

(1+p2)q](1-p2+β)]}≤

(1+p2)q](1-p2+β)]}≤

(Lr+M)H=LHr+MH≤

LHr+(1-δ)r=(LH+1-δ)r≤r.

(H為(17)式所定義.)這表明FBr?Br.

設(shè)u,v∈X有

[f(s,u(s))-f(s,v(s))]dqs+

β(1+p2)][f(s,u(s))-f(s,v(s))]dqs+

[f(s,u(s))-f(s,v(s))])dqs}|≤

β(1+p2)dqs+

(1+p2)q](1-p2+β)]}=LH‖u-v‖.

當(dāng)LH<1時(shí),F是壓縮映射.由Banach壓縮映射原理,F在Br內(nèi)有唯一不動(dòng)點(diǎn)u.利用引理2.5,u是(3)式的唯一解.

引理 3.2[18](Krasnoselskii不動(dòng)點(diǎn)定理) 假設(shè)K是Banach空間X的一個(gè)非空有界閉凸子集.若算子F1和F2是滿足條件:

(i)F1x+F2y∈K,x,y∈K;

(ii)F1是全連續(xù)算子;

(iii)F2是壓縮算子,

那么存在z∈K使得z=F1z+F2z.

定理 3.3 設(shè)f∈C(I×R,R),且滿足條件:

(A1) |f(t,u)-f(t,v)|≤L|u-v|,L為Lipschitz常數(shù);

(A2) 存在φ∈C(I,R+)使得

|f(t,u)|≤φ(t), ?(t,u)∈I×R,

若Lh<1,其中

(19)

則(3)式至少有一解.

證明 設(shè)Banach空間X如第二節(jié)定義.算子F1和F2分別如下定義:

(F1u)(t)=

(1+β)(1+q)qs+β(1+p2)]f(s,u(s))dqs+

(1-p2)dqs},u∈X.

由引理3.2知u為(3)式的解當(dāng)且僅當(dāng)u滿足u=F1u+F2u.設(shè)r≥‖φ‖H且固定,取K={u∈X:‖u‖≤r}.證明分3步完成.

第1步:證當(dāng)u,v∈K時(shí),F1u+F2v∈K.

‖F(xiàn)2u+F2v‖=

β(1+p2)]f(s,v(s))dqs+

(1-p2)dqs}|≤

β(1+p2)]|f(s,v(s))|dqs+

(1-p2+β)|f(s,v(s))|/(1-p2)dqs}|≤

(1+p2)q](1-p2+β)]}≤

(1+p2)q](1-p2+β)]}≤‖φ‖H≤r.

因此F1u+F2v∈K,這表明引理3.2的(i)成立.

第2步:證F1是全連續(xù)算子.由條件(A2)知F1是連續(xù),又K有界,于是可設(shè)

且t1

‖(F1u)(t2)-(F1u)(t1)‖=

qs(t1-t2)]f(s,u(s))dqs|=

上式表明F1(K)是相對緊的.由Arzelá-Ascoli定理知F1在K上是緊的,所以F1是全連續(xù)算子.因此引理3.2的(ii)成立.

第3步:證F2是壓縮算子.設(shè)u,v∈K時(shí)有

‖F(xiàn)2u-F2v‖=

[f(s,u(s))-f(s,v(s))]dqs+

[f(s,u(s))-f(s,v(s))]dqs}|≤

|f(s,u(s))-f(s,v(s))|dqs+

|f(s,u(s))-f(s,v(s))|dqs]}≤

(1+p2)q](1-p2+β)]}≤Lh‖u-v‖.

結(jié)合(19)式知F2是壓縮算子.因此引理3.2的所有條件都成立.由引理3.2知存在u∈K滿足u=F1u+F2u.所以(3)式至少有一解,即定理3.3成立.證畢.

4 例子

例 4.1 考查如下邊值問題

t∈[0,1],L>0,

u(1)+D1/4u(1)=0,

(20)

則當(dāng)0

證明 與(3)式對應(yīng),易知q=1/2,p1=1/3,p2=1/4,p3=1/5,α=1/4,β=1,γ=1,容易驗(yàn)證Δ=99/20,H≈1.604 5,

f=L[t3+cost+1+sinu(t)],

且

|f(t,u)-f(t,v)|=

|Lsinu-Lsinv|≤L|u-v|,

當(dāng)0

例 4.2 考查如下邊值問題

t∈[0,1],L>0,

u(1)+D1/4u(1)=0,

(21)

則當(dāng)0

證明 與(3)式對應(yīng),易知q=1/2,p1=1/3,p2=1/4,p3=1/5,α=1/4,β=1,γ=2,易算得Δ=153/20,h=4 070/3 213,

且

|f(t,u)-f(t,v)|=

當(dāng)0

注 4.3 文獻(xiàn)[12-13]中的定理不能應(yīng)用到(20)和(21)式.

致謝 劉玉記教授對本文提供了指導(dǎo),廣東警官學(xué)院青年項(xiàng)目(2013-Q01)和湛江師范學(xué)院自然科學(xué)研究項(xiàng)目(QL1101)對本文給予了資助,謹(jǐn)致謝意.

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(編輯 李德華)

Boundary Value Problem of Third-orderq-Difference Equations with Multi-termq-Difference Operators

YANG Xiaohui1, LI Jiemin2
(1.DepartmentofComputer,GuangdongPoliceCollege,Guangzhou510230,Guangdong;2.SchoolofMathematicsandComputationalScience,LingnanNormalCollege,Zhanjiang524048,Guangdong)

In recent years, an increasing interest in studying the existence of solutions for boundary value problems ofq-difference equations has been observed by the domestic and foreign mathematics workers. And many valuable results have been obtained. In this thesis, a class of third-orderq-difference equations with boundary value conditions is concerned. The boundary value problem is constituted of a third-orderq-difference equation and three boundary conditions containing multi-termq-difference operators. The conditions can be regarded as extension to Sturm-Liouville boundary conditions. By using Banach’s contraction mapping principle and Krasnoselskii’fixed point theorem, sufficient conditions for the existence and uniqueness of solutions of this problem are established. The present conditions are concise and are verified easily. The conclusions in this paper essentially extend and improve known results in references. Finally, two examples are given to demonstrate the use of the main result in this paper.

q-difference equation;q-derivative;q-integral; boundary value problem

2014-06-13

公安部應(yīng)用創(chuàng)新項(xiàng)目(2013YYCXGDST015)

楊小輝(1979—),男,講師,主要從事微分方程及其應(yīng)用的研究,E-mail:yxhljq@126.com

O175.7

A

1001-8395(2015)06-0875-09

10.3969/j.issn.1001-8395.2015.06.017