非线性微分方程组多点边值问题正解的存在性
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摘要
近年来,三阶常微分方程边值问题受到了人们的广泛关注,许多作者已作过一系列的研究,主要的工具是不动点理论和上下解方法等。这篇硕士论文主要讨论非线性微分方程组多点边值问题正解的存在性,采用的工具主要是Leray-Schauder非线性抉择、Guo-Krasnoselskii不动点定理和不动点指数理论、Leggett-Williams不动点定理。全文由四部分组成。
第一章简述了问题产生的历史背景和本文的主要工作,并给出了本文用到的一些预备知识。
第二章主要讨论了下列非线性三阶微分方程组三点边值问题正解的存在性通过利用Leray-Schauder非线性抉择、Guo-Krasnoselskii不动点定理和Leggett-Williams不动点定理,建立了单个及多个正解存在的若干充分条件,所得结果是已有相关文献的推广和改进,且所用方法对非线性微分方程组边值问题正解的存在性的讨论具有一定的普遍意义。
第三章主要借助于Krasnoselskii不动点定理,讨论了下列三阶三点的微分方程组边值问题一个、两个正解的存在性,并举例说明。
第四章通过运用锥上不动点指数理论研究了下列含有参数的非线性微分方程组三点边值问题正解的存在性、多重性以及不存在性。这些结果改进和推广了一些已有的结果.
Recently, third-order boundary value problems for ordinary differential equations have received much attention, Many authors have made a series of studies, the main tools are the fixed point theorems and the upper and lower solution method. This thesis consists of four chapters,which mainly investigated the existence of positive solutions of three-pointboundary value problems for nonlinear differential equations.
In chapter 1 we introduce the historical background of problems which will be investigatedstate the main results of this thesis. In addition, we list some preliminary knowledge which is needed later.
In chapter 2 we study the existence of positive solutions of the following three-order three-point boundary value problems for nonlinear differential equationsand establish some suficient conditions for the existence of single, twin and three positivesolutions by using nonlinear alternative of Leray-Schauder type, Guo-Krasnoselskii's fixed point theorem and Leggett-Williams fixed point theorem.
Chapter 3 is concerned with the following three-order three-point boundary value problemsthe existence of one and two Positive Solutions.
In chapter 4 we deal with the following nonlinear boundary value problem with parameter.
The nonexistence, existence and multiplicity of positive solutions are obtained by means of fixed-point index theory on the cone. In particular, our results extend and improve some relate conclusions in recent literatures.
第一章简述了问题产生的历史背景和本文的主要工作,并给出了本文用到的一些预备知识。
第二章主要讨论了下列非线性三阶微分方程组三点边值问题正解的存在性通过利用Leray-Schauder非线性抉择、Guo-Krasnoselskii不动点定理和Leggett-Williams不动点定理,建立了单个及多个正解存在的若干充分条件,所得结果是已有相关文献的推广和改进,且所用方法对非线性微分方程组边值问题正解的存在性的讨论具有一定的普遍意义。
第三章主要借助于Krasnoselskii不动点定理,讨论了下列三阶三点的微分方程组边值问题一个、两个正解的存在性,并举例说明。
第四章通过运用锥上不动点指数理论研究了下列含有参数的非线性微分方程组三点边值问题正解的存在性、多重性以及不存在性。这些结果改进和推广了一些已有的结果.
Recently, third-order boundary value problems for ordinary differential equations have received much attention, Many authors have made a series of studies, the main tools are the fixed point theorems and the upper and lower solution method. This thesis consists of four chapters,which mainly investigated the existence of positive solutions of three-pointboundary value problems for nonlinear differential equations.
In chapter 1 we introduce the historical background of problems which will be investigatedstate the main results of this thesis. In addition, we list some preliminary knowledge which is needed later.
In chapter 2 we study the existence of positive solutions of the following three-order three-point boundary value problems for nonlinear differential equationsand establish some suficient conditions for the existence of single, twin and three positivesolutions by using nonlinear alternative of Leray-Schauder type, Guo-Krasnoselskii's fixed point theorem and Leggett-Williams fixed point theorem.
Chapter 3 is concerned with the following three-order three-point boundary value problemsthe existence of one and two Positive Solutions.
In chapter 4 we deal with the following nonlinear boundary value problem with parameter.
The nonexistence, existence and multiplicity of positive solutions are obtained by means of fixed-point index theory on the cone. In particular, our results extend and improve some relate conclusions in recent literatures.
引文
[1] R.P.Agarwal and D.O'Regan, A coupled system of difference equations, Appl. Math. Comput. 114 (2000), 39-49.
[2] R.P.Agarwal, D.O'Regan, P.J.Wong, Positive Solutions of Differential, Difference and Integral Equations.Dordrecht: Kluwer Acadenmic,1999.
[3] R.P.Agarwal, D.O' Regan,Twin solutions to singular boundary value problems, Proc. Amer.Math.Soc. 128(2000),2085-2094.
[4] R.P.Agarwal, M.Bohner and W. T. Li, Nonoscillation and Oscillation Theory for Functional Differential Equations, Pure and Applied Mathematics eries, Vol. 267, Marcel Dekker, 2004.
[5] R.P.Agarwal, D.O'Regan, Nonlinear boundary value problems on time scales, Nonlinear Anal. 44 (2001), 527-535.
[6] R.P.Agarwal and M.Bohner, Basic calculus on time scales and some of its applications, Results Math. 35 (1999),3-22.
[7] H.Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach space,SI AM Review, 18(1976),620-709.
[8] D.Anderson, Green's function for the third-order three-point generalized right focal problem, J. Math.nal. Appl.288(2003), 1-14.
[9] D.R.Anderson, Eigenvalue intervals for a two-point boundary value problem on a measure chain, J. Comput.Appl. Math.141(2002), 57-64.
[10] D.Anderson, J.Davis, Multiple solutions and eigenvalues for third order right focal boundary value problems, J.Math. Anal. Appl. 267 (2002), 135-157.
[11] D.Anderson and R.I.Avery, Multiple positive solutions to a third-order discrete focal boundary value problem,Computers Math.Applic. 42 (2001), 333-340.
[12] R.I.Avery and A.C.Peterson, Multiple positive solutions of a discrete second order conjugate problem,PanAmerician Math. J. 8 (1998), 1-t2.
[13] R.I.Avery and J.Henderson,Two positive fixed points of nonlinear operator on ordered Banachs paces,C omm.Appl.N onlinear Anal.8( 2001),27-3.
[14] R.Avery, J.Henderson, Three symmetric positive solutions for a second order boundary value problem, Appl.Math. Lett. 13 (2000), 1-7.
[15] R.Avery, Existence of multiple positive solutions to a conjugate boundary value problem, Math.Sci.Res. Hot-Line 2 (1998), 1-6.
[16] L.W.Cheng, Fu Hsiang Wong, Cheh Chih Yeh, On the existence of positive solutions of nonlinear second order differential equations, Proc. Amer. Math.Soc.124(4)(1996),1117-1126.
[17] K.Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985.
[18] P.W.Eloe, J.Henderson, Positive solutions for higher order ordinary differential equations, Electron. J. Differential Equations 3 (1995), 1-8.
[19] L.Erbe and A.Peterson, Eigenvalue conditions and positive solutions, J. Differ. Equations Appl. 6 (2000), 165-191.
[20] Y.Feng, S.Liu, Solvability of a third-order two-point boundary value problem, Appl. Math. Lett. 18 (2005), 1034-1040.
[21] D.O'Regan, Theory of Singular Boundary Value ProbIems,Scientific World, 1994.
[22] D.O'Regan, Existence Theory For Nonlinear Ordinary Differential Equations, Kluwer Acadenmic Publishers, 1997.
[23] D.Guo and V.Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic press, San Diego, (1988).
[24] 葛渭高.非线性常微分方程边值问题.北京:科学出版社.2007
[25] 郭大钧等.非线性常微分方程泛函方法(第二版).济南:山东科学技术出版社.2006
[26] 郭大钧.非线性泛函分析.济南:山东科学技术出版社.2003
[27] D.Jiang, Existence of positive solutions to third-order boundary value problem for ordinary differential equation,J. Northeast Normal Unversity (Natural Science), 4 (1996), 6-10.
[28] M.A.Jones, B.Song, D.M.Thomas, Controlling wound healing through debridement, Math. Comput. Modelling 40(2004), 1057-1064.
[29] M.Krasnoselskii, Positive Solutions of Operator Equations, Noordhoff, Groningen,(1964).
[30] B.Liu, Positive solutions of a nonlinear three-point boundary value problems,Comput.Math.Appl.44(2002), 201-211.
[31] R.W.Leggett and L.R.Williams, Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana University Math. J. 28 (1979), 673-688.
[32] W.T.Li and H.R.Sun, Multiple positive solutions for nonlinear dynamic systems on a measure chain, J. Comput. Appl. Math. 162 (2004), 421-430.
[33] X.Y.Liu, J.X.Sun, Calculation of the topological degree and its application to system of superlinear equations,J. Systems Sci. Math. Sci. 16 (1996), 51-59 (in Chinese).
[34] R.Ma, Positive solutions of a nonlinear three-point boundary value problems, Election.J.Differential Equations. 34 (1999),1-8.
[35] R.Ma, Multiple nonnegative solutions of second-order systems of boundary value problems, Nonlinear Anal. 42 (2000) 1003-1010.
[36] R.Ma, Positive solutions for second-order three-point boundary value problems, Comput.Math.Appl. 40 (2000), 193-204.
[37] R.Ma, Multiplicity results for a third order boundary value problem at resonance, Nonlinear Anal. TMA 32 (4) (1998), 493-499.
[38] 马如云.非线性常微分方程非局部问题.北京:科学出版社.2004
[39] 马如云.奇异二阶边值问题的正解.数学学报,1998;41(6):1225-1230
[40] Gregus. M, Third Order Linear Differential Equations.Math. Appl.Dordrecht: ReideJ,1987.
[41] W.C.Troy, Solutions of third order differential equations relevant to draining and coating flows, SIAM J. Math. Anal. 24 (1993) 155-171.
[42] H.R.Sun, Existence of Positive Solutions to Second-Order Time Scale Systems.Computers and Math. Appl.49 (2005), 131-145.
[43] Y.Sun, Positive solutions of singular third-order three-point boundary value problem, J. Math. Anal. Appl. 306 (2005), 589-603.
[44] H.R.Sun and W.T.Li, On the number of positive solutions of systems of nonlinear dynamic equations on time scales,Journal of Computational and Applied Mathematics 219 (2008), 123-133.
[45] W.C.Troy, Solutions of third order differential equations relevant to draining and coating flows, SIAM J. Math. Anal., 24(1993), 155-171.
[46] P.J.Wong, Triple positive solutions of conjugate boundary value problems, Computers Math. Applic. 36 (1998),19-35.
[47] P.J.Wong, Triple positive solutions of conjugate boundary value problems II, Computers Math. Applic. 40 (2000), 537-557.
[48] H.Wang, Positive periodic solutions of functional differential equations, J. Differ. Equ. 202(2004), 354-366.
[49] H.Wang, On the number of positive solutions of nonlinear systems, J. Math. Anal. Appl. 281 (2003), 287-306.
[50] P.J.Wong and R.P.Agarwal, Existence theorems for a system of difference equations with (n,p)-type conditions,Appl. Math. Comput. 123(2001),389-407.
[51] H.Wang, Multiple positive solutions of boundary value problems for systems of nonlinear second-order differential equations,J. Math. Anal. Appl. 335 (2007), 1052-1060
[52] Q.Yao, The Existence of Solution for a Three-Order Two-Point Boundary Value Problem, Math. Appl.Lett.15(2002), 227-232.
[53] Q.Yao, Positive solutions of eigenvalue problems for some nonlinear third-order ordinary differential equations, Acta Math. Scientific, 23 (2003),513-519.
[54] Z.L.Yang, J.X.Sun, Positive solutions of boundary value problems for systems of nonlinear second order ordinary differential equations, Acta Math. Sinica 47(1)(2004), 111-118 (in Chinese).
[55] W.Zhao, Singular perturbations for third order nonlinear boundary value problems, Nonlinear Anal. TMA, 23(1994), 1225-1242.
[56] H.M.Yu. H.Yan, Multiple positive solutions to third-order three-point singular semipositone boundary value problem, Math. Sci. 114 (2004), 409-422.
[57] Z.T.Zhang, Existence of non-trivial solution for superlinear system of integral equations and its applications, Acta Math. Sinica 15 (1999), 153-162 (in Chinese).
[58] W.Zhao, Singular perturbations for third order nonlinear boundary value problems, Nonlinear Anal. TMA, 23 (1994),1225-1242.
[59] 钟承奎等.非线性泛函分析引论.兰州:山甘肃人民出版社.1998.
[2] R.P.Agarwal, D.O'Regan, P.J.Wong, Positive Solutions of Differential, Difference and Integral Equations.Dordrecht: Kluwer Acadenmic,1999.
[3] R.P.Agarwal, D.O' Regan,Twin solutions to singular boundary value problems, Proc. Amer.Math.Soc. 128(2000),2085-2094.
[4] R.P.Agarwal, M.Bohner and W. T. Li, Nonoscillation and Oscillation Theory for Functional Differential Equations, Pure and Applied Mathematics eries, Vol. 267, Marcel Dekker, 2004.
[5] R.P.Agarwal, D.O'Regan, Nonlinear boundary value problems on time scales, Nonlinear Anal. 44 (2001), 527-535.
[6] R.P.Agarwal and M.Bohner, Basic calculus on time scales and some of its applications, Results Math. 35 (1999),3-22.
[7] H.Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach space,SI AM Review, 18(1976),620-709.
[8] D.Anderson, Green's function for the third-order three-point generalized right focal problem, J. Math.nal. Appl.288(2003), 1-14.
[9] D.R.Anderson, Eigenvalue intervals for a two-point boundary value problem on a measure chain, J. Comput.Appl. Math.141(2002), 57-64.
[10] D.Anderson, J.Davis, Multiple solutions and eigenvalues for third order right focal boundary value problems, J.Math. Anal. Appl. 267 (2002), 135-157.
[11] D.Anderson and R.I.Avery, Multiple positive solutions to a third-order discrete focal boundary value problem,Computers Math.Applic. 42 (2001), 333-340.
[12] R.I.Avery and A.C.Peterson, Multiple positive solutions of a discrete second order conjugate problem,PanAmerician Math. J. 8 (1998), 1-t2.
[13] R.I.Avery and J.Henderson,Two positive fixed points of nonlinear operator on ordered Banachs paces,C omm.Appl.N onlinear Anal.8( 2001),27-3.
[14] R.Avery, J.Henderson, Three symmetric positive solutions for a second order boundary value problem, Appl.Math. Lett. 13 (2000), 1-7.
[15] R.Avery, Existence of multiple positive solutions to a conjugate boundary value problem, Math.Sci.Res. Hot-Line 2 (1998), 1-6.
[16] L.W.Cheng, Fu Hsiang Wong, Cheh Chih Yeh, On the existence of positive solutions of nonlinear second order differential equations, Proc. Amer. Math.Soc.124(4)(1996),1117-1126.
[17] K.Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985.
[18] P.W.Eloe, J.Henderson, Positive solutions for higher order ordinary differential equations, Electron. J. Differential Equations 3 (1995), 1-8.
[19] L.Erbe and A.Peterson, Eigenvalue conditions and positive solutions, J. Differ. Equations Appl. 6 (2000), 165-191.
[20] Y.Feng, S.Liu, Solvability of a third-order two-point boundary value problem, Appl. Math. Lett. 18 (2005), 1034-1040.
[21] D.O'Regan, Theory of Singular Boundary Value ProbIems,Scientific World, 1994.
[22] D.O'Regan, Existence Theory For Nonlinear Ordinary Differential Equations, Kluwer Acadenmic Publishers, 1997.
[23] D.Guo and V.Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic press, San Diego, (1988).
[24] 葛渭高.非线性常微分方程边值问题.北京:科学出版社.2007
[25] 郭大钧等.非线性常微分方程泛函方法(第二版).济南:山东科学技术出版社.2006
[26] 郭大钧.非线性泛函分析.济南:山东科学技术出版社.2003
[27] D.Jiang, Existence of positive solutions to third-order boundary value problem for ordinary differential equation,J. Northeast Normal Unversity (Natural Science), 4 (1996), 6-10.
[28] M.A.Jones, B.Song, D.M.Thomas, Controlling wound healing through debridement, Math. Comput. Modelling 40(2004), 1057-1064.
[29] M.Krasnoselskii, Positive Solutions of Operator Equations, Noordhoff, Groningen,(1964).
[30] B.Liu, Positive solutions of a nonlinear three-point boundary value problems,Comput.Math.Appl.44(2002), 201-211.
[31] R.W.Leggett and L.R.Williams, Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana University Math. J. 28 (1979), 673-688.
[32] W.T.Li and H.R.Sun, Multiple positive solutions for nonlinear dynamic systems on a measure chain, J. Comput. Appl. Math. 162 (2004), 421-430.
[33] X.Y.Liu, J.X.Sun, Calculation of the topological degree and its application to system of superlinear equations,J. Systems Sci. Math. Sci. 16 (1996), 51-59 (in Chinese).
[34] R.Ma, Positive solutions of a nonlinear three-point boundary value problems, Election.J.Differential Equations. 34 (1999),1-8.
[35] R.Ma, Multiple nonnegative solutions of second-order systems of boundary value problems, Nonlinear Anal. 42 (2000) 1003-1010.
[36] R.Ma, Positive solutions for second-order three-point boundary value problems, Comput.Math.Appl. 40 (2000), 193-204.
[37] R.Ma, Multiplicity results for a third order boundary value problem at resonance, Nonlinear Anal. TMA 32 (4) (1998), 493-499.
[38] 马如云.非线性常微分方程非局部问题.北京:科学出版社.2004
[39] 马如云.奇异二阶边值问题的正解.数学学报,1998;41(6):1225-1230
[40] Gregus. M, Third Order Linear Differential Equations.Math. Appl.Dordrecht: ReideJ,1987.
[41] W.C.Troy, Solutions of third order differential equations relevant to draining and coating flows, SIAM J. Math. Anal. 24 (1993) 155-171.
[42] H.R.Sun, Existence of Positive Solutions to Second-Order Time Scale Systems.Computers and Math. Appl.49 (2005), 131-145.
[43] Y.Sun, Positive solutions of singular third-order three-point boundary value problem, J. Math. Anal. Appl. 306 (2005), 589-603.
[44] H.R.Sun and W.T.Li, On the number of positive solutions of systems of nonlinear dynamic equations on time scales,Journal of Computational and Applied Mathematics 219 (2008), 123-133.
[45] W.C.Troy, Solutions of third order differential equations relevant to draining and coating flows, SIAM J. Math. Anal., 24(1993), 155-171.
[46] P.J.Wong, Triple positive solutions of conjugate boundary value problems, Computers Math. Applic. 36 (1998),19-35.
[47] P.J.Wong, Triple positive solutions of conjugate boundary value problems II, Computers Math. Applic. 40 (2000), 537-557.
[48] H.Wang, Positive periodic solutions of functional differential equations, J. Differ. Equ. 202(2004), 354-366.
[49] H.Wang, On the number of positive solutions of nonlinear systems, J. Math. Anal. Appl. 281 (2003), 287-306.
[50] P.J.Wong and R.P.Agarwal, Existence theorems for a system of difference equations with (n,p)-type conditions,Appl. Math. Comput. 123(2001),389-407.
[51] H.Wang, Multiple positive solutions of boundary value problems for systems of nonlinear second-order differential equations,J. Math. Anal. Appl. 335 (2007), 1052-1060
[52] Q.Yao, The Existence of Solution for a Three-Order Two-Point Boundary Value Problem, Math. Appl.Lett.15(2002), 227-232.
[53] Q.Yao, Positive solutions of eigenvalue problems for some nonlinear third-order ordinary differential equations, Acta Math. Scientific, 23 (2003),513-519.
[54] Z.L.Yang, J.X.Sun, Positive solutions of boundary value problems for systems of nonlinear second order ordinary differential equations, Acta Math. Sinica 47(1)(2004), 111-118 (in Chinese).
[55] W.Zhao, Singular perturbations for third order nonlinear boundary value problems, Nonlinear Anal. TMA, 23(1994), 1225-1242.
[56] H.M.Yu. H.Yan, Multiple positive solutions to third-order three-point singular semipositone boundary value problem, Math. Sci. 114 (2004), 409-422.
[57] Z.T.Zhang, Existence of non-trivial solution for superlinear system of integral equations and its applications, Acta Math. Sinica 15 (1999), 153-162 (in Chinese).
[58] W.Zhao, Singular perturbations for third order nonlinear boundary value problems, Nonlinear Anal. TMA, 23 (1994),1225-1242.
[59] 钟承奎等.非线性泛函分析引论.兰州:山甘肃人民出版社.1998.