脈沖微分方程非局部奇異邊值問(wèn)題

苗春梅,葛渭高

(1. 長(zhǎng)春大學(xué) 理學(xué)院,長(zhǎng)春 130022;2. 吉林大學(xué) 數(shù)學(xué)學(xué)院,長(zhǎng)春 130012;3. 北京理工大學(xué) 數(shù)學(xué)學(xué)院,北京 100081)

0 引 言

脈沖微分方程應(yīng)用廣泛,關(guān)于其穩(wěn)定性、 振動(dòng)性、 邊值問(wèn)題等研究目前已有許多結(jié)果[1-12]. 但關(guān)于脈沖微分方程非局部奇異邊值問(wèn)題的研究結(jié)果較少,多數(shù)都是兩點(diǎn)邊值問(wèn)題[13-16].

Dai等[14]運(yùn)用上下解方法研究了如下奇異Emden-Fowler邊值問(wèn)題:

其中:λ,m,a,b,c,d≥0;p(t),q(t)在t=0和t=1處具有奇性;非線性項(xiàng)f(t,x)=p(t)xλ+q(t)x-m在x=0處具有奇性,但該問(wèn)題中的非線性項(xiàng)是具體的多項(xiàng)式函數(shù),不具有一般性.

在非局部邊值問(wèn)題中,帶有積分邊界條件的問(wèn)題在熱傳導(dǎo)問(wèn)題[17]、 水力問(wèn)題[18]和半導(dǎo)體問(wèn)題[19]中應(yīng)用廣泛,兩點(diǎn)邊界條件實(shí)質(zhì)上是積分邊界條件的特殊情形.

基于此,本文研究如下帶有積分邊界條件的脈沖微分方程奇異邊值問(wèn)題:

(1)

其中: 0

假設(shè)如下條件成立:

1 預(yù)備知識(shí)

PC[J]={u:J→R|u(t)在J0上連續(xù)且u(tk-0)=u(tk),u(tk+0)(k=1,2,…,p)存在};

PC1[J]={u:J→R|u′(t)在J0上連續(xù)且u′(tk-0)=u′(tk),u′(tk+0)(k=1,2,…,p)存在}.

定義1如果函數(shù)u∈PC1[J]∩C2[J′]滿足式(1)且u(t)>0,t∈(0,1),則稱u為邊值問(wèn)題(1)的正解.

定義2[4]如果對(duì)任意的u∈S,ε>0,存在δ>0,使得s,t∈Jk(k=1,2,…,p)且|s-t|<δ,有|u(s)-u(t)|<ε,則稱集合S?PC[J]是擬等度連續(xù)的.

定義3[4]如果對(duì)任意的u∈S,ε>0,存在δ>0,使得s,t∈Jk(k=1,2,…,p)且|s-t|<δ,有|u(s)-u(t)|<ε和|u′(s)-u′(t)|<ε,則稱集合S?PC1[J]是擬等度連續(xù)的.

引理1[4]集合S?PC[J](S?PC1[J])在PC[J](PC1[J])上相對(duì)緊當(dāng)且僅當(dāng)S是有界的且擬等度連續(xù).

引理2對(duì)常數(shù)ak,bk≥0(k=1,2,…,p)和y∈L1[J],邊值問(wèn)題

(2)

存在唯一解:

證明: 由式(2)知

u″(t)=-y(t),t∈J′.

(4)

(5)

再對(duì)式(5)從0到t積分得

(6)

將式(6)兩邊同時(shí)乘以g(t),再?gòu)?到1積分,由邊界條件可得

結(jié)合式(6)可得

從而,對(duì)任意的t∈[0,1],有

證畢.

為研究奇異邊值問(wèn)題(1)解的存在性,先考慮如下邊值問(wèn)題:

(7)

定義算子T:PC[J] →PC[J]為

引理3算子T:PC[J] →PC[J]全連續(xù).

因此,T(B)是一致有界的.

對(duì)任意的ε>0,t,s∈Jk(k=0,1,…,p),

引理4假設(shè)存在與λ無(wú)關(guān)的常數(shù)R>a≥0,使得對(duì)任意的λ∈(0,1),邊值問(wèn)題

(9)

的解u(t)都有‖u‖≠R,則當(dāng)λ=1時(shí),邊值問(wèn)題(9)至少有一個(gè)解u∈PC1[J]∩C2[J′],且‖u‖≤R.

證明: 對(duì)任意的λ∈[0,1],u∈PC[J],定義

由引理3知,Nλ:PC[J] →PC[J]全連續(xù). 易證u(t)是邊值問(wèn)題(9)的解當(dāng)且僅當(dāng)u是Nλ在PC[J]中的不動(dòng)點(diǎn). 令Ω={u∈PC[J]|‖u‖a,因此(I-N0)u(t)=u(t)-N0u(t)=u(t)-a≠0,t∈J. 對(duì)λ∈(0,1),若存在u∈?Ω,使得(I-Nλ)u(t)=0,t∈J,則u(t)是問(wèn)題(9)的解,由引理的條件可得‖u‖≠R,與u∈?Ω矛盾. 因此,對(duì)任意的u∈?Ω和λ∈[0,1],有Nλu≠u. 又由Leray-Schauder度的同倫不變性得,Deg{I-N1,Ω,θ}=Deg{I-N0,Ω,θ}=1. 因此,N1在Ω中存在不動(dòng)點(diǎn)u,即λ=1時(shí),邊值問(wèn)題(9)至少存在一個(gè)解u∈PC1[J]∩C2[J′],且‖u‖≤R. 證畢.

引理5如果u(t)是邊值問(wèn)題(7)的解,則:

1)u(t)在Jk(k=0,1,…,p)上是凹的;

2)u′(t)≥ 0,t∈J0,u′(tk-0)≥u′(tk+0)≥0,Δu(tk)≥0,k=1,2,…,p;

3)u(t)≥a,t∈[0,1].

2 主要結(jié)果

定理1設(shè)(H1)成立,再假設(shè)下列條件成立,則邊值問(wèn)題(1)至少存在一個(gè)正解:

(H3) 對(duì)任意的l>0,存在函數(shù)ψl: [0,1] → (0,∞),使得f(t,u)≥ψl(t),(t,u)∈J′×(0,l];

證明: 由(H4)知,存在M>0和0<ε<(1-σ)M,使得

(11)

(12)

有解.

為了證明對(duì)任意的m∈N0,邊值問(wèn)題(12)都有解,先考慮邊值問(wèn)題:

(13)

其中:

下面應(yīng)用引理4證明邊值問(wèn)題(13)有解. 為此,先考慮如下一族邊值問(wèn)題:

(14)

由(H2)知,對(duì)任意的x∈J′,

-u″(x)=λq(x)f*(x,u(x))=λq(x)f(x,u(x))≤q(x)[f1(u(x))+f2(u(x))],

(15)

對(duì)式(15)從t(t∈J0)到1積分并由u′(t)的單調(diào)性可得

(16)

將式(16)兩邊同時(shí)除以f1(u(t)),再由0到1積分可得

進(jìn)而有

結(jié)合式(11)可得‖u‖=u(1)≠M(fèi). 又由引理4知,對(duì)任意固定的m∈N0,邊值問(wèn)題(13)至少有一個(gè)解um∈PC1[J]∩C2[J′],滿足‖um‖≤M. 由引理5知,um(t)≥1/m>0,從而

因此um(t)也是邊值問(wèn)題(12)的解.

下面將獲得um(t)(?m∈N0)的擬下界,即存在常數(shù)L>0,L0≥0(與m無(wú)關(guān)),使得

um(t)≥Lt+L0,t∈J.

(17)

由于

0<1/m≤um(t)≤M,t∈J.

(18)

故由(H3)知,存在連續(xù)函數(shù)ψM: [0,1] → (0,∞),使得:f(t,um(t))≥ψM(t),t∈J′. 又由引理2知,

最后證明{um(t)}m∈N0在J上是一致有界且擬等度連續(xù)的. 由式(18)知{um(t)}m∈N0在J上是一致有界的. 下面證明其在J上是擬等度連續(xù)的.

因?yàn)閡m(t)是式(12)的解,因此對(duì)x∈J′,有

-um″(x)=q(x)f(x,um(x))≤q(x)[f1(um(x))+f2(um(x))],

(20)

將式(20)從t(t∈J0)到1積分并由u′(t)的單調(diào)性可得

(21)

因此{(lán)um(t)}m∈N0在J上是擬等度連續(xù)的.

又由于

在式(22)中,令m→ ∞,m∈N*,由Lebesgue控制收斂定理可得

因此u(t)是邊值問(wèn)題(1)的正解,且u(t)≥Lt+L0,t∈J,‖u‖≤M. 證畢.

3 應(yīng)用實(shí)例

例1考慮邊值問(wèn)題:

(23)

邊值問(wèn)題(23)至少有一個(gè)正解.

從而(H4)成立,因此,由定理1可知,邊值問(wèn)題(23)至少存在一個(gè)正解. 證畢.

[1] Bainov D D,Simeonov P S. Systems with Impulse Effect: Stability,Theory and Applications [M]. Chichester: Ellis Horwood Ltd,1989.

[2] Lakshmikantham V,Bainov D D,Simeonov P S. Theory of Impulsive Differential Equations [M]. Singapore: World Scientific,1989.

[3] Bainov D D,Simeonov P S. Impulsive Differential Equations: Periodic Solutions and Applications [M]. Harlow: Longman,1993.

[4] Samoilenko A M,Perestyuk N A. Impulsive Differential Equations [M]. Singapore: World Scientific,1995.

[5] FENG Mei-qiang,PANG Hui-hui. A Class of Three-Point Boundary-Value Problems for Second-Order Impulsive Integro-Differential Equations in Banach Spaces [J]. Nonlinear Anal: Theor Meth Appl,2009,70(1): 64-82.

[6] Jankowski T. Positive Solutions of Three-Point Boundary Value Problems for Second Order Impulsive Differential Equations with Advanced Arguments [J]. Appl Math Comput,2008,197(1): 179-189.

[7] Jankowski T. Positive Solutions to Second Order Four-Point Boundary Value Problems for Impulsive Differential Equations [J]. Appl Math Comput,2008,202(2): 550-561.

[8] FENG Mei-qiang,XIE Dong-xiu. Multiple Positive Solutions of Multi-point Boundary Value Problem for Second-Order Impulsive Differential Equations [J]. J Comput Appl Math,2009,223(1): 438-448.

[9] LIU Bing,YU Jian-she. Existence of Solution form-Point Boundary Value Problems of Second-Order Differential Systems with Impulses [J]. Appl Math Comput,2002,125(2/3): 155-175.

[10] ZHANG Xue-mei,GE Wei-gao. Impulsive Boundary Value Problems Involving the One-Dimensionalp-Laplacian [J]. Nonlinear Anal: Theor Meth Appl,2009,70(4): 1692-1701.

[11] FENG Mei-qiang,DU Bo,GE Wei-gao. Impulsive Boundary Value Problems with Integral Boundary Conditions and One-Dimensionalp-Laplacian [J]. Nonlinear Anal: Theor Meth Appl,2009,70(9): 3119-3126.

[12] TIAN Yu,JI De-hong,GE Wei-gao. Existence and Nonexistence Results of Impulsive First-Order Problem with Integral Boundary Condition [J]. Nonlinear Anal: Theor Meth Appl,2009,71(3): 1250-1262.

[13] Agarwal R P,Franco D,O’Regan D. Singular Boundary Value Problems for First and Second Order Impulsive Differential Equations [J]. Aequationes Math,2005,69(1/2): 83-96.

[14] DAI Li-mei,LI Hong-yu. Positive Solutions of Singular Emden-Fowler Boundary Value Problem with Negative Exponent and Multiple Impulses [J]. Nonlinear Anal: Theor Meth Appl,2009,70(10): 3682-3695.

[15] LI Qiu-yue,CONG Fu-zhong,JIANG Da-qing. Multiplicity of Positive Solutions to Second Order Neumann Boundary Value Problems with Impulse Actions [J]. Appl Math Comput,2008,206(2): 810-817.

[16] ZU Li,JIANG Da-qing,O’Regan D. Existence Theory for Multiple Solutions to Semipositone Dirichlet Boundary Value Problems with Singular Dependent Nonlinearities for Second-Order Impulsive Differential Equations [J]. Appl Math Comput,2008,195(1): 240-255.

[17] Cannon J R. The Solution of the Heat Equation Subject to the Specification of Energy [J]. Quart Appl Math,1963,21(2): 155-160.

[18] Chegis R Y. Numerical Solution of a Heat Conduction Problem with an Integral Boundary Condition [J]. Litovsk Mat Sb,1984,24: 209-215.

[19] Ionkin N I. Solution of a Boundary Value Problem in Heat Conduction Theory with Nonlocal Boundary Conditions [J]. Diff Equ,1977,13: 294-304.