具p-Laplacian算子型多点边值问题正解存在性
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摘要
非线性微分方程的奇异边值问题是微分方程中一个非常重要的研究领域,它出现在各种应用学科中,例如核物理、气体动力学、流体力学、边界层理论、非线性光学等。关于奇异p-Laplacian非线性边值间题正解存在性的研究日益受到人们的重视,所用到的方法有拓扑度理论、Schaude不动点定理、上下解方法、打靶法等。
本文主要研究具有变号非线性项的两点、三点边值问题正解的存在性以及具p-Laplacian算子型方程组正解的存在性。全文共分三章,主要内容如下:
第一章绪论介绍有关边值间题的正解的发展概况,并概述了本文的主要工作。
第二章研究依赖于一阶导数的两点及三点边值问题正解的存在性。
第三章研究具p-Laplacian算子型方程组多个正解的存在性。
The singular boundary value problem of nonlinear diferential equations is an important research field in diferential equations, which arises in all sorts of applied branch of learning, such as nuclear physics、gas dynamics、fluid mechanics、theory of boundary layer、nonlinear optics and so on. The research on the existence of positive solutions for singular p-Laplacian boundary value problems has brought people's attention and those results have been obtained by using topological degree theory、Schauder fixed point theorem、upper-lower solutions method and shooting method and so on.
In chapter one, we introduce a survey to the development of positive boundary value problems. We also summarize main results in the dissertation.
In chapter two, we discuss the existence of positive solutions for two-point and three-point boundary value problems with the nonlinearity depending on the first order derivative.
In chapter three, we discuss the multiplicity results of positive solutions for three-point boundary value systems with p-Laplacian.
本文主要研究具有变号非线性项的两点、三点边值问题正解的存在性以及具p-Laplacian算子型方程组正解的存在性。全文共分三章,主要内容如下:
第一章绪论介绍有关边值间题的正解的发展概况,并概述了本文的主要工作。
第二章研究依赖于一阶导数的两点及三点边值问题正解的存在性。
第三章研究具p-Laplacian算子型方程组多个正解的存在性。
The singular boundary value problem of nonlinear diferential equations is an important research field in diferential equations, which arises in all sorts of applied branch of learning, such as nuclear physics、gas dynamics、fluid mechanics、theory of boundary layer、nonlinear optics and so on. The research on the existence of positive solutions for singular p-Laplacian boundary value problems has brought people's attention and those results have been obtained by using topological degree theory、Schauder fixed point theorem、upper-lower solutions method and shooting method and so on.
In chapter one, we introduce a survey to the development of positive boundary value problems. We also summarize main results in the dissertation.
In chapter two, we discuss the existence of positive solutions for two-point and three-point boundary value problems with the nonlinearity depending on the first order derivative.
In chapter three, we discuss the multiplicity results of positive solutions for three-point boundary value systems with p-Laplacian.
引文
[1]Hale J.K..Theory of Functional Differential Equations.Now York:Springer Verlag,1977
[2]Wang Junyu.The existence of positive solutions for the one-dimensional p-Laplacian Proceedings of the American Mathematical Society,1997,125(8):2275-2283
[3]Guo Y.,Ge W..Positive solutions for three-point boundary value problems with depence on the first order derivative.J Math Anal Appl,2004,290:291-301
[4]刘斌.具p-Laplacian算子型奇异方程组边值问题正解的存在性.数学学报,2005,48(01):35-50
[5]Wang Junyu,Gao Wenjie.A singular boundary value problem for the the one dimensional β-Laplacian.J.Math.Anal.Appl.,1996,201:851-866
[6]廉立芳,葛渭高.高阶p-Laplacian算子的边值问题正解的存在性.数学学报,2005,48(02):267-264
[7]D.O-Regan.Some exintence principles and some general results for singular nonlinear two point boundary value problems.J.Math.Anal.Appl.,1992,166:24-40
[8]C.De.Coster.Pairs of positive solutions for the one-dimensional p-Laplacian.Nonlinear Analysis,Theory,Methods And Applications,1994,23(5):369-381
[9]Paul Manasevich and J.Mawhin.Periodic solutions for boundary value problem with singular and discontinuous nordinearity.Acts Mathematics Applicatae Sinica,2000,16(1):
[10]Alberto Cabada,Rodrigo L.Pouso.Existence result for the problem(φ(u'))'=of(t,u,u')with periodic and Neumann boundary conditions.Nonlinear Analysis,Theory,Methods And Applications,1997,30(3):1733-1742
[11]姚庆六,吕海探.一类奇异v-Laolace方程的正解.数学学报,1998,41(06):1255-126
[12]王宏洲,葛渭高.奇性边值问题的正解存在性.数学学报,1999,42(01):111-118
[13]刘炳文,黄立宏.一类n阶非线性常微分方程周期解的存在性.数学学报,2004,47(6):1133-1140
[14]鲁世平,葛渭高,郑祖庥.一类中立型泛函微分方程正周期解问题.数学年刊(A 辑),2004,25(5):645-652
[15]Wang H..On the number of positive solutions of nonlinear systems.J.Math.Anal.Appl.,2003,281:287-306
[16]Fu Hsing W'ong,Existence of positive solutions for m-Laplacian boundary value problems Applied Mathematics Leters,1999,V12:11-17
[17]Jean Mawhin.Some boundary value problems for Hartman-type perturbations of the ordinary vector p-Laplacian.Nonlinear Analvsis.2000,40:497-503
[18]Liu Bing.Positive periodic solution for a nonautonomous delay differential equation.Acta Mathematicae Applicatae Sinica,English Series.2003,19(2):307-316
[19]J.Mawhin,Boundary value problems for nonlinear second-order vector diferential equations J.Diferential Equations.1974,16:257-269
[20]李仁贵,刘立山.二阶奇异非线性微分方程边值问题的正解.应用数学和力学,2001,22(4):435-440
[21]Ma R.,N.Castaneda.Existence of solutions for nonlinear in-point boundary value problems.J.Math.Anal.Appl.,2001,(256):556-567
[22]Man Manjun,Yu Jianshe.Existence of multiple positive periodic solutions for nonlinear functional difference equations.J.Math.Anal.Appl.,2005,305:483-490
[23]Liu B..Positive solutions for Nonlinear three-point boundary value problems.Computers Math.Applie,2002,44(1/2):201-211
[24]Liu B..Positive solutions for a nonlinear three-point boundary value problem.Appl,Math.Comput.,2002,(132):11-28
[25]Wang H..Positive periodic solutions of functional differential equations.J.Diff.Equa.,2004,202:354-366
[26]Freedman H.I.,Wu Jianhong.Periodic solutions of single-species models with periodic delay.SIAM J.Math.Anal.,1999,23(3):689-701
[27]Yang Xiaojing.The method of lower and upper solutions for systems of boundary value problems.Appl.Math.Comp.,2003,144:169-172
[28]aiang Daqing,Nieto Juan a.,Zuo Wenjie.On monotone method for first and second order periodic boundary value problems and periodic solutions of functional differential equations.J.Math.Anal.Appl.,2004,289:691-699
[29]Hal Dang and Seth F.Oppenheimer.Existence and uniqueness results for some nonlinear boundary value problems.J.Math.Anal.Appl.,1996,198:35-48
[30]Raul Manasevich and Guido Sweets.A noncooperative elliptic system with p-Laplacians that preserves positivity.Nonlinear Analysis,1999,36:511-5
[31]翁佩茸,蒋达清.奇异二阶泛函微分方程边值问题的多重正解.应用数学学报, 2000,23(01):99-107
[32]R.Kannan and Donal O'Regan,A note on singular boudary value problem with solutions in weighted spaces.Nonlinear Analysis 1999,37:791-796
[33]R.1.Avery,J.Henderson.Three symmetric positive solutions for a second order boundary value problem.Appl.Math.Lett.,2000,13:1-7
[34]郭大钧.非线性泛函分析(第二版).济南:山东科学技术出版社,1985
[35]夏道行等.实变函数论与泛函分析(下册).北京:人民教育出版社,1979
[36]张恭庆,林源渠.泛函分析讲义(下册).北京:北京大学出版社,1987
[37]Krasnoselskii M..Positive Solutions of Operator Equations.Groningen Noorhoff,1964
[38]Deimling Klaus.Nonlinear Functional Analysis.New York:Springer-Verlag,1985
[39]郭大钧,孙经先.抽象空间常微分方程.(第二版).济南:山东科学技术出版社,2003
[40]郭大钧,孙经先,刘兆理.非线性常微分方程泛函方法.济南:山东科学技术出版社,1995
[41]马如云.非线性常微分方程非局部问题.北京:科学出版社,2004
[42]郑祖庥.泛函微分方程理论.合肥:安徽教育出版社,1994
[2]Wang Junyu.The existence of positive solutions for the one-dimensional p-Laplacian Proceedings of the American Mathematical Society,1997,125(8):2275-2283
[3]Guo Y.,Ge W..Positive solutions for three-point boundary value problems with depence on the first order derivative.J Math Anal Appl,2004,290:291-301
[4]刘斌.具p-Laplacian算子型奇异方程组边值问题正解的存在性.数学学报,2005,48(01):35-50
[5]Wang Junyu,Gao Wenjie.A singular boundary value problem for the the one dimensional β-Laplacian.J.Math.Anal.Appl.,1996,201:851-866
[6]廉立芳,葛渭高.高阶p-Laplacian算子的边值问题正解的存在性.数学学报,2005,48(02):267-264
[7]D.O-Regan.Some exintence principles and some general results for singular nonlinear two point boundary value problems.J.Math.Anal.Appl.,1992,166:24-40
[8]C.De.Coster.Pairs of positive solutions for the one-dimensional p-Laplacian.Nonlinear Analysis,Theory,Methods And Applications,1994,23(5):369-381
[9]Paul Manasevich and J.Mawhin.Periodic solutions for boundary value problem with singular and discontinuous nordinearity.Acts Mathematics Applicatae Sinica,2000,16(1):
[10]Alberto Cabada,Rodrigo L.Pouso.Existence result for the problem(φ(u'))'=of(t,u,u')with periodic and Neumann boundary conditions.Nonlinear Analysis,Theory,Methods And Applications,1997,30(3):1733-1742
[11]姚庆六,吕海探.一类奇异v-Laolace方程的正解.数学学报,1998,41(06):1255-126
[12]王宏洲,葛渭高.奇性边值问题的正解存在性.数学学报,1999,42(01):111-118
[13]刘炳文,黄立宏.一类n阶非线性常微分方程周期解的存在性.数学学报,2004,47(6):1133-1140
[14]鲁世平,葛渭高,郑祖庥.一类中立型泛函微分方程正周期解问题.数学年刊(A 辑),2004,25(5):645-652
[15]Wang H..On the number of positive solutions of nonlinear systems.J.Math.Anal.Appl.,2003,281:287-306
[16]Fu Hsing W'ong,Existence of positive solutions for m-Laplacian boundary value problems Applied Mathematics Leters,1999,V12:11-17
[17]Jean Mawhin.Some boundary value problems for Hartman-type perturbations of the ordinary vector p-Laplacian.Nonlinear Analvsis.2000,40:497-503
[18]Liu Bing.Positive periodic solution for a nonautonomous delay differential equation.Acta Mathematicae Applicatae Sinica,English Series.2003,19(2):307-316
[19]J.Mawhin,Boundary value problems for nonlinear second-order vector diferential equations J.Diferential Equations.1974,16:257-269
[20]李仁贵,刘立山.二阶奇异非线性微分方程边值问题的正解.应用数学和力学,2001,22(4):435-440
[21]Ma R.,N.Castaneda.Existence of solutions for nonlinear in-point boundary value problems.J.Math.Anal.Appl.,2001,(256):556-567
[22]Man Manjun,Yu Jianshe.Existence of multiple positive periodic solutions for nonlinear functional difference equations.J.Math.Anal.Appl.,2005,305:483-490
[23]Liu B..Positive solutions for Nonlinear three-point boundary value problems.Computers Math.Applie,2002,44(1/2):201-211
[24]Liu B..Positive solutions for a nonlinear three-point boundary value problem.Appl,Math.Comput.,2002,(132):11-28
[25]Wang H..Positive periodic solutions of functional differential equations.J.Diff.Equa.,2004,202:354-366
[26]Freedman H.I.,Wu Jianhong.Periodic solutions of single-species models with periodic delay.SIAM J.Math.Anal.,1999,23(3):689-701
[27]Yang Xiaojing.The method of lower and upper solutions for systems of boundary value problems.Appl.Math.Comp.,2003,144:169-172
[28]aiang Daqing,Nieto Juan a.,Zuo Wenjie.On monotone method for first and second order periodic boundary value problems and periodic solutions of functional differential equations.J.Math.Anal.Appl.,2004,289:691-699
[29]Hal Dang and Seth F.Oppenheimer.Existence and uniqueness results for some nonlinear boundary value problems.J.Math.Anal.Appl.,1996,198:35-48
[30]Raul Manasevich and Guido Sweets.A noncooperative elliptic system with p-Laplacians that preserves positivity.Nonlinear Analysis,1999,36:511-5
[31]翁佩茸,蒋达清.奇异二阶泛函微分方程边值问题的多重正解.应用数学学报, 2000,23(01):99-107
[32]R.Kannan and Donal O'Regan,A note on singular boudary value problem with solutions in weighted spaces.Nonlinear Analysis 1999,37:791-796
[33]R.1.Avery,J.Henderson.Three symmetric positive solutions for a second order boundary value problem.Appl.Math.Lett.,2000,13:1-7
[34]郭大钧.非线性泛函分析(第二版).济南:山东科学技术出版社,1985
[35]夏道行等.实变函数论与泛函分析(下册).北京:人民教育出版社,1979
[36]张恭庆,林源渠.泛函分析讲义(下册).北京:北京大学出版社,1987
[37]Krasnoselskii M..Positive Solutions of Operator Equations.Groningen Noorhoff,1964
[38]Deimling Klaus.Nonlinear Functional Analysis.New York:Springer-Verlag,1985
[39]郭大钧,孙经先.抽象空间常微分方程.(第二版).济南:山东科学技术出版社,2003
[40]郭大钧,孙经先,刘兆理.非线性常微分方程泛函方法.济南:山东科学技术出版社,1995
[41]马如云.非线性常微分方程非局部问题.北京:科学出版社,2004
[42]郑祖庥.泛函微分方程理论.合肥:安徽教育出版社,1994
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