三阶微分方程三点边值问题及其应用
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摘要
本文主要运用微分不等式的技巧(或称为上、下解方法),在一定条件下证明了一类三阶非线性微分方程(不带小参数)三点边值问题解的存在性和唯一性,在此基础上研究了在实际应用中广泛出现的带有小参数的一类奇异摄动三点边值问题,利用积分算子和微分不等式技巧法,构造了其高阶渐近解并得到了解的一致有效估计;最后结合不动点定理,借助上、下解方法和微分不等式技巧,讨论了一类不满足Nagumo条件的三阶微分方程三点线性边值问题的微分不等式理论。
本文主要由四部分组成:
第一部分,介绍了常微分方程理论和方法的发展历程和奇异摄动理论的发展背景及前人的一些工作。给出上下解的概念及Nagumo条件,同时给出了微分不等式的基本结果,以及后面会用到的基本引理。
第二部分,利用上、下解方法和微分不等式技巧,引入积分算子在一定条件下研究了一类三阶非线性微分方程三点线性边值问题的解的存在性和唯一性。
第三部分,引入积分算子和微分不等式技巧,在一定条件下讨论了三阶非线性微分方程三点边值问题的奇摄动,得到了解的存在性和解的唯一性以及解的一致有效估计。
第四部分,结合不动点定理,借助上、下解方法和微分不等式技巧,研究了不满足Nagumo条件的三阶微分方程三点线性边值问题的解的存在性和唯一性。
In this paper, by the theory of differential inequalities(or upper and lower solutions meth- od),we study the existence and uniqueness solutions of some classes of three point boundary value problems for third-order nonlinear differential equations(without small parameter).At this basic, we study some calsses of singular perturbation of three-point boundary value problem which appear abroad in life and yield area.By making use of Volterra type integral operator and differential inequality techniques, existence and uniqueness and consistent availability estimation of solutions are obtained;Last,we discuss the theory of differential inequalities of some classes of three point linear boundary value problems for third-order differential equations without satisfying nagumo conditions.
This paper is made up of four parts:
The first part, introduce the circumstance of singular perturbation therom and the work of former,then give the therom of supper and lower solution and nagumo condition, at the same time the result of differential inequality.
The second part,At some conditions,by making use of supper and lower solution and dif- ferential inequality techniques integral operator study a class of linear three-point boundary value problems of third order nonlinear differential equation, existence and uniqueness and asymptotic estimmation of solutions are obtained.
The third part, at some conditions,by making use of Volterra type integral operator and differential inequality techniques, we study singular perturbation of three-point boundary value problem for third order nonlinear equation existence and uniqueness and consistent availability asymptotic estimmation of solutions are obtained.
The fourth part, in virtue of supper and lower solution and differential inequality techniques combined fixed points theory, we study some classes of differenttial system boundary value problem without satisfy nagumo conditions,the existence and uniqueness of solutions are obtained.
本文主要由四部分组成:
第一部分,介绍了常微分方程理论和方法的发展历程和奇异摄动理论的发展背景及前人的一些工作。给出上下解的概念及Nagumo条件,同时给出了微分不等式的基本结果,以及后面会用到的基本引理。
第二部分,利用上、下解方法和微分不等式技巧,引入积分算子在一定条件下研究了一类三阶非线性微分方程三点线性边值问题的解的存在性和唯一性。
第三部分,引入积分算子和微分不等式技巧,在一定条件下讨论了三阶非线性微分方程三点边值问题的奇摄动,得到了解的存在性和解的唯一性以及解的一致有效估计。
第四部分,结合不动点定理,借助上、下解方法和微分不等式技巧,研究了不满足Nagumo条件的三阶微分方程三点线性边值问题的解的存在性和唯一性。
In this paper, by the theory of differential inequalities(or upper and lower solutions meth- od),we study the existence and uniqueness solutions of some classes of three point boundary value problems for third-order nonlinear differential equations(without small parameter).At this basic, we study some calsses of singular perturbation of three-point boundary value problem which appear abroad in life and yield area.By making use of Volterra type integral operator and differential inequality techniques, existence and uniqueness and consistent availability estimation of solutions are obtained;Last,we discuss the theory of differential inequalities of some classes of three point linear boundary value problems for third-order differential equations without satisfying nagumo conditions.
This paper is made up of four parts:
The first part, introduce the circumstance of singular perturbation therom and the work of former,then give the therom of supper and lower solution and nagumo condition, at the same time the result of differential inequality.
The second part,At some conditions,by making use of supper and lower solution and dif- ferential inequality techniques integral operator study a class of linear three-point boundary value problems of third order nonlinear differential equation, existence and uniqueness and asymptotic estimmation of solutions are obtained.
The third part, at some conditions,by making use of Volterra type integral operator and differential inequality techniques, we study singular perturbation of three-point boundary value problem for third order nonlinear equation existence and uniqueness and consistent availability asymptotic estimmation of solutions are obtained.
The fourth part, in virtue of supper and lower solution and differential inequality techniques combined fixed points theory, we study some classes of differenttial system boundary value problem without satisfy nagumo conditions,the existence and uniqueness of solutions are obtained.
引文
[1]苏煜城,吴启光著.奇异摄动问题数值方法引论.重庆:重庆出版社1992:3-10
[2]林宗池,周明儒.应用数学中的摄动方法.江苏:江苏教育出版社,1995:7-30
[3] Johnny henderson and c. c.Tisdell,five-point boundary value problems for third order differential equations by solution matching.Mathematical and Computer Modelling , 2005vol.42(no.1-2)
[4]章国华,F.A.候斯;林宗池,倪守平,郑钦贵.非线性奇异摄动现象:理论和应用.福建:福建科学技术出版社,1989,4:1-15
[5]张凤然,梁俊平,马少仙.一类三阶三点边值问题的可解性.西北师范大学学报(自然科学版),1999:16-18
[6]余赞平.三阶微分方程的非线性三点边值问题.福建师范大学学报(自然科学版),1998:9-12
[7] M.Nagumo.über die .Differential gleichung y′′= f ( t , y ,y′).[J].proc.phys.Math.Soc. japan, 1937, 19:861-866
[8] L.K.jacson.Subfunctions and second-order ordinary differential inequalities.[J]Advance in Math, 1968, 2:308-363
[9] K.W.Chang,F.A.Howes.Nonlinear singular perturbation phenomenon:theory and Application.[M] New york,Springer Verlag,1984
[10] L.K.Jackson.Subfunctions and third-order ordinary differential inequalities.[J].J.Diff.Equa.,1970,8: 180-194
[11] F.A.Howes.Differential of higher order and the asymptotic solution of nonlinear Boundary value problems. [J].SIAM.J.Math.Anal.,1982,13(1):61-80
[12] Yao Qingliu.The Existence and Multiplicity of Positive Solutions for a Third-order Three-point Boundary Value Problem.Acta Mathematicae Applicatae Sinica, English Series(2003) : 117-122
[13]林宗池.伴有边界摄动的一类四阶非线性微分方程边值解的估计.数学年刊,1990,11A(6):683-690
[14]席进华.四阶两点常微分方程边值问题解的存在性.山东大学学报(理学版),2009,44(1):67-73
[15] Xu Guoan.Yu Zhangping,Zhou Zheyan, Singular perturbation of three—point boundary value problem for third order differential equations.[J].Ann.Diff.Equa.,2005,21(3):480-485
[16]林宗池,倪守平等译.非线性奇异摄动现象:理论和应用.[M].福州:福建科学技术出版社,1989:10-40
[17]周钦德,王怀忠.带有高阶转向点的奇摄动边值间题解的一致有效估计闭.吉大学学报,1955,4:19-26
[18]王国灿.某一类三阶非线性两点边值问题的解的存在性与唯一性.应用数学学报,1997:631-634
[19]李芳菲,贾梅,刘锡平等.三阶三点边值问题的三个解的存在性.应用泛函分析学报,2007,12
[20]余赞平.三阶微分方程的非线性三点边值问题.福建师范大学学报(自然科学版),1998,14 (3) :9-12
[21] Wang guocan.On the uniqueness of third order singularly perturbed boundary value problem.journal of mathematics for technology,2000,2:35-38
[22]沈建和,余赞平,周哲彦.非线性三阶常微分方程的非线性三点阶的存在性.纯粹数学与应用数学,12(3),1997:265-271
[23]廖健斌,余赞平.一类三阶微分方程的非线性三点边值问题.龙岩学院学报,2007,6:4-6
[24]孙永征,孙经先.一类周期边值问题解的存在性与唯一性.徐州师范大学学报,2005,23(4):28-31
[25]刘衍胜,綦建刚.不连续条件下二阶非线性微分方程的边值问题.工程数学学报,1998:72-78
[26]汪用征,周晓.一类三阶非线性积分微分方程三点边值问题的奇摄动解.南昌大学学报理科版,1995:147-153
[27]许国安,余赞平,周哲彦.奇摄动三阶半线性三点边值问题.福建师范大学自然科学学报, 22(3) 2006:6-9
[28] Xu Guoan,Yu Zhangping,Zhou Zheyan.Singular perturbation of three—point boundary value problem for third order differential equations.[J].Ann.Diff.Equa.,2005,21(3):480-485
[29]余赞平.三阶半线性微分方程的三点边值问题及奇异摄动.福建师范大学学报(自然科学版),1999,15 (1) :26- 29
[30]陈松林.一类三阶非线性方程三点边值问题奇摄动解的渐近性质.安徽师大学报,1988:8-14
[31]王光发,吴克乾,邓宋琦等.常微分方程.湖南:湖南教育出版,1988,8:198-218
[32]朱汉清,刘淑静.求解格林函数的本征函数法.大学物理,1995, 14(2):7-11
[33]李润菁.不具备Nagumo条件的二阶微分方程的微分不等式.三明学院学报,24(2),2007,12 :370-372
[34]傅仰耿.微分不等式理论与奇异摄动现像(硕士学位论文).福建师范大学,2008,4:3-37
[35]石磊王国灿.三阶非线性微分差分方程Robin边值问题的存在性与惟一性.纺织高校基础科学学报,2008,6:192-195
[36] Wang Jinzhi.Existence of solution of nonlinear two point value problem for third order nonlinear differential equation.Northeasrn Math,7(2),1991:181-189
[37]葛渭高.三阶常微分方程的两点边值问题.高校应用数学学报,12(3),1997:265-271
[38]沈建和,余赞平,周哲彦.非线性三阶常微分方程的非线性三点阶的存在性.纯粹数学与应用数学,23(3)2007:355-360
[39] Zhao weili.Singular perturbations for third order nonlinear boundary value problems. Nonlinear Analysis,Theory,Methods and Applications,44(10),1994:1225-1242
[40]丁同仁,李承治.常微分教程.[M].北京:高等教育出版社,1991:70-76
[41]时宝,张存德,盖明久.微分方程理论及其应用.[M].北京:国防工业出版社.2005:10-40
[42]周钦德,苗树梅.Volterra型积分微分方程的奇摄动.高校应用数学学报,3(3),1998:392-400
[43] S.R.Bernfeld and V.Lashmikanthan.An introduction to nonlinear boundary value problems.Academic press,New York,1974.
[44]张琼渊.微分方程中的奇异摄动现象.福建师范大学硕士论文,2008,4:1-10
[45]王国灿,金丽.三阶奇摄动非线性边值问题.应用数学与力学,23(6),2002:597-603
[46] Xie Feng.Singular perturbation of two points boundary value problem for a class of third order quasilinear differential equation.Chinese Quar.J.of Math,16(3),2001:69-74
[47]王国灿.奇摄动三阶非线性边值问题.吉林大学自然科学学报,112(4),1997:9-12
[48]许国安,余赞平,周哲彦.奇摄动三阶半线性三点边值问题.福建师范大学自然科学学报,22(3)2006 :6-9
[49]廖健斌.不具备Nagumo条件的边值问题的微分不等式.绍兴文理学院学报,27(8),2007,6:34-37
[50]黄柳铃,余赞平,周哲彦.不具备Nagumo条件的三阶微分方程两点边值问题.福建师范大学学报(自然科学版),24(2),2008,3:9-12
[2]林宗池,周明儒.应用数学中的摄动方法.江苏:江苏教育出版社,1995:7-30
[3] Johnny henderson and c. c.Tisdell,five-point boundary value problems for third order differential equations by solution matching.Mathematical and Computer Modelling , 2005vol.42(no.1-2)
[4]章国华,F.A.候斯;林宗池,倪守平,郑钦贵.非线性奇异摄动现象:理论和应用.福建:福建科学技术出版社,1989,4:1-15
[5]张凤然,梁俊平,马少仙.一类三阶三点边值问题的可解性.西北师范大学学报(自然科学版),1999:16-18
[6]余赞平.三阶微分方程的非线性三点边值问题.福建师范大学学报(自然科学版),1998:9-12
[7] M.Nagumo.über die .Differential gleichung y′′= f ( t , y ,y′).[J].proc.phys.Math.Soc. japan, 1937, 19:861-866
[8] L.K.jacson.Subfunctions and second-order ordinary differential inequalities.[J]Advance in Math, 1968, 2:308-363
[9] K.W.Chang,F.A.Howes.Nonlinear singular perturbation phenomenon:theory and Application.[M] New york,Springer Verlag,1984
[10] L.K.Jackson.Subfunctions and third-order ordinary differential inequalities.[J].J.Diff.Equa.,1970,8: 180-194
[11] F.A.Howes.Differential of higher order and the asymptotic solution of nonlinear Boundary value problems. [J].SIAM.J.Math.Anal.,1982,13(1):61-80
[12] Yao Qingliu.The Existence and Multiplicity of Positive Solutions for a Third-order Three-point Boundary Value Problem.Acta Mathematicae Applicatae Sinica, English Series(2003) : 117-122
[13]林宗池.伴有边界摄动的一类四阶非线性微分方程边值解的估计.数学年刊,1990,11A(6):683-690
[14]席进华.四阶两点常微分方程边值问题解的存在性.山东大学学报(理学版),2009,44(1):67-73
[15] Xu Guoan.Yu Zhangping,Zhou Zheyan, Singular perturbation of three—point boundary value problem for third order differential equations.[J].Ann.Diff.Equa.,2005,21(3):480-485
[16]林宗池,倪守平等译.非线性奇异摄动现象:理论和应用.[M].福州:福建科学技术出版社,1989:10-40
[17]周钦德,王怀忠.带有高阶转向点的奇摄动边值间题解的一致有效估计闭.吉大学学报,1955,4:19-26
[18]王国灿.某一类三阶非线性两点边值问题的解的存在性与唯一性.应用数学学报,1997:631-634
[19]李芳菲,贾梅,刘锡平等.三阶三点边值问题的三个解的存在性.应用泛函分析学报,2007,12
[20]余赞平.三阶微分方程的非线性三点边值问题.福建师范大学学报(自然科学版),1998,14 (3) :9-12
[21] Wang guocan.On the uniqueness of third order singularly perturbed boundary value problem.journal of mathematics for technology,2000,2:35-38
[22]沈建和,余赞平,周哲彦.非线性三阶常微分方程的非线性三点阶的存在性.纯粹数学与应用数学,12(3),1997:265-271
[23]廖健斌,余赞平.一类三阶微分方程的非线性三点边值问题.龙岩学院学报,2007,6:4-6
[24]孙永征,孙经先.一类周期边值问题解的存在性与唯一性.徐州师范大学学报,2005,23(4):28-31
[25]刘衍胜,綦建刚.不连续条件下二阶非线性微分方程的边值问题.工程数学学报,1998:72-78
[26]汪用征,周晓.一类三阶非线性积分微分方程三点边值问题的奇摄动解.南昌大学学报理科版,1995:147-153
[27]许国安,余赞平,周哲彦.奇摄动三阶半线性三点边值问题.福建师范大学自然科学学报, 22(3) 2006:6-9
[28] Xu Guoan,Yu Zhangping,Zhou Zheyan.Singular perturbation of three—point boundary value problem for third order differential equations.[J].Ann.Diff.Equa.,2005,21(3):480-485
[29]余赞平.三阶半线性微分方程的三点边值问题及奇异摄动.福建师范大学学报(自然科学版),1999,15 (1) :26- 29
[30]陈松林.一类三阶非线性方程三点边值问题奇摄动解的渐近性质.安徽师大学报,1988:8-14
[31]王光发,吴克乾,邓宋琦等.常微分方程.湖南:湖南教育出版,1988,8:198-218
[32]朱汉清,刘淑静.求解格林函数的本征函数法.大学物理,1995, 14(2):7-11
[33]李润菁.不具备Nagumo条件的二阶微分方程的微分不等式.三明学院学报,24(2),2007,12 :370-372
[34]傅仰耿.微分不等式理论与奇异摄动现像(硕士学位论文).福建师范大学,2008,4:3-37
[35]石磊王国灿.三阶非线性微分差分方程Robin边值问题的存在性与惟一性.纺织高校基础科学学报,2008,6:192-195
[36] Wang Jinzhi.Existence of solution of nonlinear two point value problem for third order nonlinear differential equation.Northeasrn Math,7(2),1991:181-189
[37]葛渭高.三阶常微分方程的两点边值问题.高校应用数学学报,12(3),1997:265-271
[38]沈建和,余赞平,周哲彦.非线性三阶常微分方程的非线性三点阶的存在性.纯粹数学与应用数学,23(3)2007:355-360
[39] Zhao weili.Singular perturbations for third order nonlinear boundary value problems. Nonlinear Analysis,Theory,Methods and Applications,44(10),1994:1225-1242
[40]丁同仁,李承治.常微分教程.[M].北京:高等教育出版社,1991:70-76
[41]时宝,张存德,盖明久.微分方程理论及其应用.[M].北京:国防工业出版社.2005:10-40
[42]周钦德,苗树梅.Volterra型积分微分方程的奇摄动.高校应用数学学报,3(3),1998:392-400
[43] S.R.Bernfeld and V.Lashmikanthan.An introduction to nonlinear boundary value problems.Academic press,New York,1974.
[44]张琼渊.微分方程中的奇异摄动现象.福建师范大学硕士论文,2008,4:1-10
[45]王国灿,金丽.三阶奇摄动非线性边值问题.应用数学与力学,23(6),2002:597-603
[46] Xie Feng.Singular perturbation of two points boundary value problem for a class of third order quasilinear differential equation.Chinese Quar.J.of Math,16(3),2001:69-74
[47]王国灿.奇摄动三阶非线性边值问题.吉林大学自然科学学报,112(4),1997:9-12
[48]许国安,余赞平,周哲彦.奇摄动三阶半线性三点边值问题.福建师范大学自然科学学报,22(3)2006 :6-9
[49]廖健斌.不具备Nagumo条件的边值问题的微分不等式.绍兴文理学院学报,27(8),2007,6:34-37
[50]黄柳铃,余赞平,周哲彦.不具备Nagumo条件的三阶微分方程两点边值问题.福建师范大学学报(自然科学版),24(2),2008,3:9-12