泛函方法在积—微分方程中的应用
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摘要
本文对实Banach空间中二阶混合型积-微分方程进行了研究，主要分为以下三个方面：二阶混合型积-微分方程的初值问题，二阶混合型积-微分方程的两点边值问题及二阶混合型积-微分方程的周期边值问题．
在对上述问题的研究过程中均采用了上下解方法和单调迭代方法，证明了相应问题最大解和最小解的存在性，并且得到了逼近解的单调迭代序列.在本文中建立了全新的比较定理，并且方程的形式也更加也丰富，扩大了方程的适用范围，推广了原有文献中的一些结果.
本文共分五章.第一章为前言.第二章、第三章分别讨论了二阶混合型积-微分方程的初值问题和两点边值问题.第四章研究了二阶混合型积-微分方程的周期边值问题，其中又分为不含微分项u和含有微分项u的周期边值问题.最后对本文全面总结.
In this thesis, we investigate the second order integro-differential equations. The main research includes three aspects: the initial value problem (IVP), the two-point boundary value problem (BVP) and the periodic boundary value problem (PBVP).
By using the upper and lower solutions method and monotone iterative technique, we obtain the existence of the maximal and minimal solutions and the iterative sequence of corresponding solutions for the Integro-differential equations. In this thesis, we establish new comparison results and the results obtained generalize and improve the results corresponding to those obtained by others.
The thesis is divided into five chapters according to contents. The first chapter is a preface. In the second and the third chapter, the IVP and BVP for second-order integro-differential equations are investigated. In the fourth chapter, we investigate the PBVP for second-order integro-differential equations and this problem includes two aspects: without u′item and with u′item. The last chapter is the summary of this thesis.
在对上述问题的研究过程中均采用了上下解方法和单调迭代方法，证明了相应问题最大解和最小解的存在性，并且得到了逼近解的单调迭代序列.在本文中建立了全新的比较定理，并且方程的形式也更加也丰富，扩大了方程的适用范围，推广了原有文献中的一些结果.
本文共分五章.第一章为前言.第二章、第三章分别讨论了二阶混合型积-微分方程的初值问题和两点边值问题.第四章研究了二阶混合型积-微分方程的周期边值问题，其中又分为不含微分项u和含有微分项u的周期边值问题.最后对本文全面总结.
In this thesis, we investigate the second order integro-differential equations. The main research includes three aspects: the initial value problem (IVP), the two-point boundary value problem (BVP) and the periodic boundary value problem (PBVP).
By using the upper and lower solutions method and monotone iterative technique, we obtain the existence of the maximal and minimal solutions and the iterative sequence of corresponding solutions for the Integro-differential equations. In this thesis, we establish new comparison results and the results obtained generalize and improve the results corresponding to those obtained by others.
The thesis is divided into five chapters according to contents. The first chapter is a preface. In the second and the third chapter, the IVP and BVP for second-order integro-differential equations are investigated. In the fourth chapter, we investigate the PBVP for second-order integro-differential equations and this problem includes two aspects: without u′item and with u′item. The last chapter is the summary of this thesis.
引文
[1]郭大均.非线性泛函分析[M].济南:山东科技出版社.1985
[2]Guo Dajun, Lakshmikantham V．Nonlinear problems in Abstract ConesAcademic[M].pres.Inc.1988
[3]郭大均，孙经先.非线性积分方程[M].济南:山东科技出版社. 1987
[4] Deimiling K.Nonlinear Functional Analysis[M]. NewYork:Spring Verlay. 1985
[5]郭大均,孙经先.抽象空间常微分方程[M].济南:山东科技出版社. 1989
[6] Lakshmikantham V, leela S. Nonlinear Differential Equation in Abstract Spaces[M].NewYork.1981
[7]郭大均，孙经先.Banach空间常微分方程理论的若干问题[J].数学进展.23(6)(1994).492-503
[8]郭大钧.非线性分析中的半序方法[M].济南:山东科学技术出版社，1995
[9]郭大钧,孙经先,刘兆理.非线性常微分方程泛函方法[M].济南:山东科学技术出版社. 1995
[10] Guo Dajun, V．Lakshmikantham. Nonlinear Integral Equation in Abstract Spaces[M].Dordrecht: Kluwer Academic Publisher,1996
[11]邢顺来,孙书荣. Banach空间中积分微分方程周期边值问题[J].山东大学学报(理学版), 2003(38):75--79.
[12] T.Jankowski.Boundary value problems for first order differential equations of mixedtype[J]. Nonlinear Analysis. 2006, 64: 1984-1997
[13] T.Jankowski.Solvability of three point boundary value problems for second orderdifferential equations with deviating arguments[J].J.math..Anal.appl. 2005,312:620-636
[14] Jiang Daqing, Wei Junjie. Monotone method for first and second-order periodic boundaryvalue problems and periodic solutions of functions differential equations[J].Nonlinear.Analysis.2002.50:885-898
[15] Song Guang-xing.Initial Value Problems for Systems of Integro-differential Equations inBanach Spaces[J]. Journal of Mathematical Analysis and Applications. 2001.264: 68–75
[16]宋光兴.Banach空间中二阶积-微分方程初值问题的解[J].数学学报，2004,47(1):71-78
[17]宋光兴.抽象空间二阶常微分方程两点边值问题解的存在唯一性[J].工程数学学报.1999,16(4): 133—136
[18]宋光兴.Banach空间二阶常微分方程两点边值问题迭代求解[J].应用数学.2000,13(2):9—13
[19]马德香,陈学刚. Banach空间微分方程两点边值问题[J].工程数学学报.2004,21(5): 817—820
[20]王文霞．一类二阶非线性积分-微分方程的边值问题[J].工程数学学报,2005,22(3):555-558
[21]宋光兴.Banach空间中两点边值问题解的存在唯一性[J].数学物理学报.1999,19(2):159-164
[22] Ding Wei, Han Maoan, Yan Jurang. Periodic problems for second Order functiondifferential equations[J]. Math Anal.2004,298:341-351.
[23] T.Jankowski. Boundary value problems for first order differentialequations of mixed type [J].Nonlinear Anal. 2006,64:1984-1997
[24] T.Jankowski. Monotone method for second-order delayed differential equationswithboundary value conditions[J].Appl. Math. Comput. 2004,149:589-598
[25] Jiang Daqing, JuanJ.Nieto, Zou Weijie. Monotone method for first and second orderperiodic boundary value problems and periodic solutions of functional equations [J].Math Anal. 2004,289:691-699
[26] Song Guang-xing, Zhu Xun-lin. Existence and Uniqueness of Solutions for the Systemsof Operator Equations in Banach Spaces[J].工程数学学报.2003,20(4):26-30
[27] Li Wang .The Unique Solution for Periodic Boundary Value Problems of theDiscontinuous Second Order Nonlinear Differential Equations[J].应用数学.2006,19(3):539-545
[28] Song Guang-xing,Zhu Xun-liu, Extremal solutions of periodic boundary value problemsfor first order integro-differential equations of mixed type[J]. Math AnalAppl.2004,300: 1-11
[29] Liang Yan-tang, Gao Xiao-fei. Li De-sheng. EXISTENCE OF SOLUTIONS FORPERIODIC BOUNDARY VALUE PROBLEMS OF SECOND2ORDERDISCONTINUOUS DIFFERENTIAL EQUATIONS[J].数学杂志.2004,24(2):156-16
[30]邢顺来,孙书荣. Banach空间中积分微分方程周期边值问题[J].山东大学学报(理学版). 2003,38:75-79
[31]刘庆荣,刘振栋.Banach空间中二阶积分微分方程周期边值问题[J].山东大学学报(理学版).2004,39(5):58-63
[32]张晓燕，孙经先，苏军.Banach空间中二阶非线性微分积分方程周期边值问题[J].工程数学学报.2004,21(6):1041-1044
[33]洪世煌,胡适耕.二阶积分微分方程周期边值问题的解的存在性[J].应用数学和力学.2000,21(3):315-322
[34]钱守国.Banach空间中二阶混合型积分微分方程的周期边值问题[J].青岛大学学报.2004,17(1):10-15
[35]崔玉军，周玉梅.不连续非线性二阶微分方程的周期边值问题[J].工程数学学报.2005,22(3):469-473
[36] Su Hua, Liu Lishan, Zhang Xianyan, Wu Yonghong, Global solutions of initial valueproblems for nonlinear second-order integro-differential equations of mixed type inBanach spaces[J]. J.Math.Anal.Appl. 2007,330:1139-1151
[2]Guo Dajun, Lakshmikantham V．Nonlinear problems in Abstract ConesAcademic[M].pres.Inc.1988
[3]郭大均，孙经先.非线性积分方程[M].济南:山东科技出版社. 1987
[4] Deimiling K.Nonlinear Functional Analysis[M]. NewYork:Spring Verlay. 1985
[5]郭大均,孙经先.抽象空间常微分方程[M].济南:山东科技出版社. 1989
[6] Lakshmikantham V, leela S. Nonlinear Differential Equation in Abstract Spaces[M].NewYork.1981
[7]郭大均，孙经先.Banach空间常微分方程理论的若干问题[J].数学进展.23(6)(1994).492-503
[8]郭大钧.非线性分析中的半序方法[M].济南:山东科学技术出版社，1995
[9]郭大钧,孙经先,刘兆理.非线性常微分方程泛函方法[M].济南:山东科学技术出版社. 1995
[10] Guo Dajun, V．Lakshmikantham. Nonlinear Integral Equation in Abstract Spaces[M].Dordrecht: Kluwer Academic Publisher,1996
[11]邢顺来,孙书荣. Banach空间中积分微分方程周期边值问题[J].山东大学学报(理学版), 2003(38):75--79.
[12] T.Jankowski.Boundary value problems for first order differential equations of mixedtype[J]. Nonlinear Analysis. 2006, 64: 1984-1997
[13] T.Jankowski.Solvability of three point boundary value problems for second orderdifferential equations with deviating arguments[J].J.math..Anal.appl. 2005,312:620-636
[14] Jiang Daqing, Wei Junjie. Monotone method for first and second-order periodic boundaryvalue problems and periodic solutions of functions differential equations[J].Nonlinear.Analysis.2002.50:885-898
[15] Song Guang-xing.Initial Value Problems for Systems of Integro-differential Equations inBanach Spaces[J]. Journal of Mathematical Analysis and Applications. 2001.264: 68–75
[16]宋光兴.Banach空间中二阶积-微分方程初值问题的解[J].数学学报，2004,47(1):71-78
[17]宋光兴.抽象空间二阶常微分方程两点边值问题解的存在唯一性[J].工程数学学报.1999,16(4): 133—136
[18]宋光兴.Banach空间二阶常微分方程两点边值问题迭代求解[J].应用数学.2000,13(2):9—13
[19]马德香,陈学刚. Banach空间微分方程两点边值问题[J].工程数学学报.2004,21(5): 817—820
[20]王文霞．一类二阶非线性积分-微分方程的边值问题[J].工程数学学报,2005,22(3):555-558
[21]宋光兴.Banach空间中两点边值问题解的存在唯一性[J].数学物理学报.1999,19(2):159-164
[22] Ding Wei, Han Maoan, Yan Jurang. Periodic problems for second Order functiondifferential equations[J]. Math Anal.2004,298:341-351.
[23] T.Jankowski. Boundary value problems for first order differentialequations of mixed type [J].Nonlinear Anal. 2006,64:1984-1997
[24] T.Jankowski. Monotone method for second-order delayed differential equationswithboundary value conditions[J].Appl. Math. Comput. 2004,149:589-598
[25] Jiang Daqing, JuanJ.Nieto, Zou Weijie. Monotone method for first and second orderperiodic boundary value problems and periodic solutions of functional equations [J].Math Anal. 2004,289:691-699
[26] Song Guang-xing, Zhu Xun-lin. Existence and Uniqueness of Solutions for the Systemsof Operator Equations in Banach Spaces[J].工程数学学报.2003,20(4):26-30
[27] Li Wang .The Unique Solution for Periodic Boundary Value Problems of theDiscontinuous Second Order Nonlinear Differential Equations[J].应用数学.2006,19(3):539-545
[28] Song Guang-xing,Zhu Xun-liu, Extremal solutions of periodic boundary value problemsfor first order integro-differential equations of mixed type[J]. Math AnalAppl.2004,300: 1-11
[29] Liang Yan-tang, Gao Xiao-fei. Li De-sheng. EXISTENCE OF SOLUTIONS FORPERIODIC BOUNDARY VALUE PROBLEMS OF SECOND2ORDERDISCONTINUOUS DIFFERENTIAL EQUATIONS[J].数学杂志.2004,24(2):156-16
[30]邢顺来,孙书荣. Banach空间中积分微分方程周期边值问题[J].山东大学学报(理学版). 2003,38:75-79
[31]刘庆荣,刘振栋.Banach空间中二阶积分微分方程周期边值问题[J].山东大学学报(理学版).2004,39(5):58-63
[32]张晓燕，孙经先，苏军.Banach空间中二阶非线性微分积分方程周期边值问题[J].工程数学学报.2004,21(6):1041-1044
[33]洪世煌,胡适耕.二阶积分微分方程周期边值问题的解的存在性[J].应用数学和力学.2000,21(3):315-322
[34]钱守国.Banach空间中二阶混合型积分微分方程的周期边值问题[J].青岛大学学报.2004,17(1):10-15
[35]崔玉军，周玉梅.不连续非线性二阶微分方程的周期边值问题[J].工程数学学报.2005,22(3):469-473
[36] Su Hua, Liu Lishan, Zhang Xianyan, Wu Yonghong, Global solutions of initial valueproblems for nonlinear second-order integro-differential equations of mixed type inBanach spaces[J]. J.Math.Anal.Appl. 2007,330:1139-1151