姚慶六
(南京財經(jīng)大學(xué)應(yīng)用數(shù)學(xué)系,江蘇 南京 210003)
一類奇異半正三階兩點邊值問題的正解
姚慶六
(南京財經(jīng)大學(xué)應(yīng)用數(shù)學(xué)系,江蘇 南京 210003)
研究了一類奇異三階兩點邊值問題的正解存在性,其中非線性項可以在t=0,t=1處奇異,并且有一個函數(shù)型下界.通過考察非線性項在無窮遠處的極限增長函數(shù)的積分,并且利用錐上的Krasnosel'skii不動點定理證明了一個新的存在定理.
非線性常微分方程;邊值問題;正解;不動點定理
三階常微分方程與流體力學(xué)有著密切關(guān)系.例如它可以用于考察變動截面梁的形變,也可用于研究電磁波或者重力流等[1-2].當f(t,u)為連續(xù)函數(shù)時,問題(P)的可解性已經(jīng)被研究過[3].新近,文獻[4]在f(t,u)=a(t)g(u),g(u)連續(xù)而a(t)在t=0,t=1處奇異的情況下考慮過另一類三階兩點邊值問題的正解存在性.不過當f(t,u)在t=0,t=1處奇異時,現(xiàn)有文獻中尚無問題(P)的任何存在性結(jié)論.本文將在更為一般的假設(shè)(H1)—(H3)下考察問題(P)的正解存在性.非線性三階邊值問題的有關(guān)工作還可參見文獻[5-11].
[1] GREGUS M.Third order linear differen-tial equations[M].Dordrecht:Math Appl Reidel,1987:20-101.
[2] BERNIS F,PELETIES L A.Two problems from draining flows involving third-order ordi-nary differential equations[J].SIAM J Math Anal,1996,27:515-527.
[3] 姚慶六.三階常微分方程的某些非線性特征值問題的正解[J].數(shù)學(xué)物理學(xué)報,2003,23A:513-519.
[4] LI S.Positive solutions of nonlinear singu-Lar third-order two-point boundary valueproblem[J].J Math Anal Appl,2006,323:413-425.
[5] YAO Q.Successive iteration and positive solution for a discontinuous third-order boundary value problem[J].Comput Math Applic,2007,53:741-749.
[6] EL-SHAHED M.Positive solutions for nonli-near singular third order boundary value problem[J].Communications Nonlinear Science and Numerical Simulation,2009,14:424-429.
[7] 孫彥,劉立山.三階奇異邊值問題的正解[J].應(yīng)用數(shù)學(xué)學(xué)報:中文版,2009,32:50-59.
[8] 許曉婕,費祥歷.三階非線性奇異邊值問題正解的存在唯一性[J].系統(tǒng)科學(xué)與數(shù)學(xué),2009,29:779-785.
[9] 姚慶六.一類奇異三階兩點邊值問題的正解存在性與多解性[J].華東師范大學(xué)學(xué)報:自然科學(xué)版,2010(3):113-118.
[10] 姚慶六.非線性三階兩點特征值問題的一個注記[J].華中師范大學(xué)學(xué)報:自然科學(xué)版,2010,44:365-368.
[11] 姚慶六.奇異三階兩點邊值問題的相伴正解[J].山東大學(xué)學(xué)報:理學(xué)版,2010,45(12):24-27.
[12] 程其驤,張奠宙.實變函數(shù)與泛函分析基礎(chǔ)[M].第二版.北京:高等教育出版社,2003:123-162.
[13] ANURADHA V,HAI D D,SHIVAJI R.Exis-tence results for superlinear semiposi-tone BVP's[J].Proc Amer Math Soc,1996,124:757-763.
[14] 鐘承奎,范先令,陳文源.非線性泛函分析引論[M].蘭州:蘭州大學(xué)出版社,1998,150-154.
Positive solution to a class of singular semipositone third-order two-point boundary value problems
YAO Qing-liu
(Department of Applied Mathematics,Nanjing University of Finance and Economics,Nanjing 210003,China)
The existence of positive solution is studied for a singular third-order two-point boundary value problem,where the nonlinear term may be singular att=0,t=1and has a lower bound of function type.By considering integration of the limit growth function of nonlinear term at infinity and applying the Krasnosel'skii fixed point theorem on cone,a new existence theorem is proved.
nonlinear ordinary differential equation;boundary value problem;positive solution;fixed point theorem
O 175.8
110·44
A
1000-1832(2011)03-0023-05
2009-12-07
國家自然科學(xué)基金資助項目(11071109).
姚慶六(1946—),男,教授,主要從事應(yīng)用常微分方程研究.
陶 理)