一類變號三點(diǎn)邊值問題正解的存在性

魏 嘉*，王 靜

(甘肅聯(lián)合大學(xué)師范學(xué)院數(shù)學(xué)系，中國 蘭州 730000)

一類變號三點(diǎn)邊值問題正解的存在性

魏 嘉*，王 靜

(甘肅聯(lián)合大學(xué)師范學(xué)院數(shù)學(xué)系，中國 蘭州 730000)

運(yùn)用錐上的Guo-Krasnoselskij’s不動點(diǎn)定理研究了一類非線性變號二階三點(diǎn)邊值問題至少一個正解的存在性.給出了與之相關(guān)聯(lián)的線性二階三點(diǎn)邊值問題的格林函數(shù)及格林函數(shù)的一些性質(zhì).最后，作為應(yīng)用，舉例證明了所得結(jié)論的正確性.

邊值問題；變號；格林函數(shù)；正解

近年來,由于常微分方程邊值問題在工程、物理方面的廣泛應(yīng)用，多點(diǎn)邊值問題受到了廣泛的關(guān)注,如文[1～12] . 在文[1]中,Sun和Wei通過運(yùn)用錐上的拉伸與壓縮不動點(diǎn)定理證明了半正二階三點(diǎn)邊值問題

至少一個正解的存在性, 其中0<α<β<1,λ>0為參數(shù),f:[0,1]×(0,+∞)→(-M,+∞).

本文考慮變號二階三點(diǎn)邊值問題

(1)

其中α≥0,0<β<1,η∈(0,1).h(t)可變號:h(t)≥0,t∈[0,η];h(t)≤0,t∈[η,1].為了討論(1)式正解的存在性，我們做如下假設(shè)：

(H1)f∈C([0,+∞),[0,+∞)),且非減.

(H2)h∈C([0,1],R)且h(t)≥0,t∈[0,η];h(t)≤0,t∈[η,1].另外，對于[0,1]上的任一子區(qū)間,h(t)?0.

(2)

本文所用的工具為如下的Guo-Krasnoselskij，s不動點(diǎn)定理.

(ⅰ) ‖Au‖≤‖u‖,u∈K∩?Ω1,且‖Au‖≥‖u‖,u∈K∩?Ω2；或

(ⅱ) ‖Au‖≥‖u‖,u∈K∩?Ω1,且‖Au‖≤‖u‖,u∈K∩?Ω2,

為了后面推理的需要，做如下記號：

1 預(yù)備知識

引理1如果y∈E,則邊值問題

(3)

存在唯一解

證由于邊值問題(3)的唯一解可表示為

(ⅰ)當(dāng)t≤η時，邊值問題(3)的唯一解能夠表示為

(ⅱ)當(dāng)t>η時，邊值問題(3)的唯一解能夠表示為

引理2格林函數(shù)G(t,s)滿足如下性質(zhì)：

(ⅰ)G(t,s)≥0,(t,s)∈[0,1]×[0,1];

證很容易證明G(t,s)≥0,(t,s)∈[0,1]×[0,1].下面分兩種情形來證明(ⅱ).

情形1, 0≤t≤η.

當(dāng)s1∈(t,η],s2∈[η,1]時，有

情形2,η

引理3設(shè)u∈P,則u(t)滿足如下性質(zhì)：

(ⅰ)u(t)≥γ(t)u(η),t∈[0,η]；u(t)≤γ(t)u(η),t∈[η,1],其中

證由于u∈P, 在[0,η]上u為凹的, 在[η,1]上u為凸的, 且u(0)=αu′(0),u(1)=βu(η).

(ⅰ),分下面兩種情況進(jìn)行討論：

當(dāng)t∈[0,η]時, 有

當(dāng)t∈[η,1]時, 有

從而u(t)≥γ(t)u(η),t∈[0,η];u(t)≤γ(t)u(η),t∈[η,1].

引理4設(shè)(H1), (H2), (H3) 成立. 則對于任意的D∈[0,+∞),有

證對任意的ε∈[0,1-η],有

由于f非減，對?ε∈[0,1-η],有

設(shè)s=η-δε,ε∈[0,1-η],對于?D∈[0,+∞),據(jù)引理2中(ⅱ)和(H3),可得

再令s=η+ε,ε∈[0,1-η],對于?D∈[0,+∞),有

引理5設(shè) (H1),(H2),(H3)成立. 則算子T是全連續(xù)的.

證由于T:P→P,函數(shù)f非減, 據(jù)引理3中的(ⅰ)和引理4可得

因而

(Tu)″(t)=-h+(t)f(u(t))≤0,t∈[0,η], (Tu)″(t)=h-(t)f(u(t))≥0,t∈[η,1].

那么由Arzela-Ascoli定理可知：T是全連續(xù)的.

2 主要結(jié)果

定理2設(shè) (H1)～(H3) 成立. 若如下條件之一成立，

(ⅰ)f0=0 且f∞=∞ (超線性),或

(ⅱ)f0=∞ 且f∞=0 (次線性),

則變號二階三點(diǎn)邊值問題(1)至少存在一個正解.

證(ⅰ)假設(shè)f為超線性.

當(dāng)f0=0, 存在H1>0,選取適當(dāng)?shù)摩?>0，使得對于0

令Ω1={u∈P:‖u‖

那么

(ⅱ)假設(shè)f為次線性.

當(dāng)f0=∞,存在H3>0,選取適當(dāng)?shù)摩?>0,使得對于0

令Ω3={u∈P:‖u‖

情況一,f有界.存在ρ>0使得f(u)≤ρ,u∈[0,∞).

令H4=max{2H3,ρΔ1},Ω4={u∈P:‖u‖

即對于u∈P∩?Ω4,有 ‖Tu‖≤‖u‖.

3 應(yīng)用舉例

考慮如下變號二階三點(diǎn)邊值問題

(4)

其中

[1] SUN J P, WEI J. Existence of positive solution for semipositone second-order three-point boundary-value problem [J]. Eletron J Diff Equ, 2008,41(1):1-7.

[2] BAI D L, FENG H Y. Eigenvalue for a singular second order three-point boundary value problem [J]. J Appl Math Comp, 2012,38(1-2):443-452.

[3] BAI Z B, DU Z J. Positive solutions for some second-order four-point boundary value problems [J]. J Math Anal Appl, 2007,330(1):34-50.

[4] BOHNER M, STEVIC S. Linear perturbations of a nonoscillatory second-order dynamic equation [J], J Diff Equ Appl, 2009,15(11):1211-1221.

[5] DU X S, ZHAO Z Q. A necessary and sufficient condition for the existence of positive solutions to singular sublinear three-point boundary value problems[J]. J Appl Math Comp, 2007,186(1):404-413.

[6] GUPTA C P. Solvability of a three-point nonlinear boundary value problem for a second order ordinary differential equations [J]. J Math Anal Appl, 1992,168 (2):540-551.

[7] KONG L J. Second order singular boundary value problems with integral boundary conditions [J]. Nonlinear Anal, 2010,31(5):2628-2638.

[8] KUN W, ZHI L. Positive solutions for a system of second-order boundary-value problems involving first-order derivatives [J]. Eletron J Diff Equ, 2012,135(1):1-17.

[9] LI G, LIU X, JIA M. Positive solutions to a type of nonlinear three-point boundary-value problem with sign changing nonlinearities [J]. Comput Math Appl, 2009,57(3):348-355.

[10] LIU B, LIU L, WU Y. Positive solutions for singular second order three-point boundary value problems [J]. Nonlinear Anal, 2007,66(12):2756-2766.

[11] MA R. Positive solutions of a nonlinear three-point boundary value problems [J]. Electron J Differential Equations, 1998,34 (1):1-8.

[12] 申騰飛,劉文斌,宋文耀.一類帶有p-Laplacian 算子分?jǐn)?shù)階微分方程邊值問題正解的存在性[J].湖南師范大學(xué)自然科學(xué)學(xué)報, 2012，35(5):9-14.

[13] GUO D, LAKSHMIKANTHAM V. Nonlinear problems in abstract cones [M]. New York:Academic Press, 1988.

[14] KRASNOSELSKIJ M A. Positive solutions of operator equations [M]. Groningen: Noordhoff, 1964.

(編輯 沈小玲)

The Existence of Positive Solutions to Three-point Boundary Value Problem with Change of Sign

WEIJia*,WANGJing

(Department of Mathematics, Normal College, Gansu Lianhe University, Lanzhou 730000, China)

This paper investigates the existence of at least one positive solution to the nonlinear second-order three-point boundary value problem with change of sign by the application of the Guo-Krasnoselskij’s fixed-point theorem in cones. The Green’s function and some of its properties of the linear second-order three-point boundary value problem related to the nonlinear boundary value problem are provided. At last, an example is given to demonstrate the results of this study.

boundary value problem; sign changing; Green’s function; positive solution

2012-11-05

甘肅省自然科學(xué)基金資助項(xiàng)目(3ZS042-B26-021)；甘肅省教育廳科研基金資助項(xiàng)目(1013B-03)

*

,E-mail：weijia_vick@163.com

O175

A

1000-2537(2014)02-0072-06