龍菲菲

(蘭州理工大學理學院,甘肅蘭州 730050)

一類帶積分邊界條件的三階微分方程邊值問題的解

龍菲菲

(蘭州理工大學理學院,甘肅蘭州 730050)

運用Banach壓縮映射原理以及Leray-Schauder連續(xù)性原理,在非線性項為L1-Caratheodory函數(shù)的條件下,研究了一類帶積分邊界條件的三階微分方程邊值問題解的唯一性、存在性以及解集的緊性.

邊值問題;解;唯一性;存在性;緊性

DO I：10.3969/j.issn.1008-5513.2013.04.013

1 引言

三階微分方程起源于應用數(shù)學和物理學的許多不同領(lǐng)域,譬如,帶有固定或變化橫截面的屈曲梁的撓度、三層梁、電磁波、地球引力吹積的漲潮等.帶積分邊界條件的邊值問題不僅包含兩點及三點邊值問題作為其特殊情形,而且還可以更精確地描述許多重要的現(xiàn)象,例如,在熱傳導、化學工程、地下水流、熱彈性、等離子物理等領(lǐng)域中,對許多問題的討論都可以歸結(jié)為對帶積分邊界條件的邊值問題的研究.目前對帶積分邊界條件的三階微分方程邊值問題的研究越來越多[18].特別地,2011年,文獻[8]研究了Banach空間E中的三階微分方程

下正解的存在性和不存在性,其中,f∈C([0,1]×P,P),這里,P是E中的錐.使用的主要工具是錐上的不動點定理和不動點指數(shù)理論.

2 預備引理

3 主要結(jié)果

參考文獻

[1]Wang Youyu,GeWeigao.Existence of solutions for a third order dif erential equation w ith integralboundary cond itions[J].Com pu t.M ath.A pp l.,2007,53:144-154.

[2]Graef JR,Yang Bo.Solution and positive solution of a sem ilinear third-order equation[J].Discrete Contin. Dyn.Syst.Ser.S,2008,1:89-97.

[3]Graef J R,Webb J R L.Third order boundary value p rob lem s with non local boundary conditions[J]. Non linear Anal.,2009,71:1542-1551.

[4]Boucherif A,Bouguima SM,A l-Malki N,et al.Third order dif erential equationsw ith integral boundary conditions[J].Non linear Anal.,2009,71:1736-1743.

[5]Sun Jianping,Li Haibao.Monotone positive solution of non linear third-order BVP w ith integral boundary cond itions[J].Bound.Value Prob l.,2010,20:1-12.

[6]孫建平,靳存程.帶積分邊界條件的三階邊值問題的單調(diào)正解[J].蘭州大學學報:自然科學版,2012,48:98-101.

[7]Zhang Haie,Sun Jianping.Existence and iteration ofmonotone positive solutions for third order non local BVPs involving integral cond itions[J].E lectron.J.Qual.Theory D if er.Equ.,2012,18:1-9.

[8]Zhao Jun fang,Wang Peiguang,Ge Weigao.Existence and nonexistence of positive solutions for a class of third order BVP w ith integral boundary conditions in Banach spaces[J].Commun.Nonlinear Sci.Numer. Simul.,2011,16:402-413.

[9]G ranas A,Dugund ji J.Fixed Point Theory,Sp ringer M onographs in M athem atics[M].New York:Sp ringer, 2003.

The solu tion for a class of th ird-order d if eren tial equation BVP w ith in tegral boundary cond itions

Long Feifei

(School of Science,Lanzhou University of Technology,Lanzhou 730050,China)

By using Banach contraction princip le and Leray-Schauder continuation p rincip le,the uniqueness and existence of solutions and the com pactness of solutions set are investigated for a class of third-order differential equation BVP with integral boundary conditions under the condition that the nonlinear term is an L1-Caratheodory function.

boundary value p rob lem,solution,uniqueness,existence,com pactness

O175

A

1008-5513(2013)04-0425-08

2013-04-10.

龍菲菲(1988-),碩士生,研究方向：應用微分方程.

2010 M SC:34B 15