非线性脉冲积分—微分方程的解
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摘要
非线性泛函分析是现代分析数学的一个重要分支,它能够清楚地解释自然界中很多自然现象,因而受到了越来越多的数学家与数学工作者的关注.其中,非线性问题来源于应用数学和物理的多个分支,是目前分析数学中研究最为活跃的领域之一.本论文主要讨论了带有casual算子的一阶混合型脉冲积-微分方程积分边值问题,混合型脉冲积分-微分方程积分边值问题,混合型脉冲积-微分方程的非线性边值问题解的存在性,全文共分四章.
第一章,前言部分,主要介绍了选题来源、研究意义、国内外研究现状,以及论文的主要研究内容和目标.
第二章,利用新的比较原理和上下解方法,讨论了具有casual算子的一阶混合型脉冲方程积分边值问题,并改进了某些已有的结果.
第三章,利用单调迭代方法,研究了一阶混合型脉冲积分-微分方程积分边值问题极值解和唯一解的存在性.
第四章,利用上下解方法,研究了一阶混合型脉冲积-微分方程的非线性边值问题唯一解的存在性,对某些已有结果作了推广和改进.
Nonlinear functional analysis is an important branch of modern analysis mathematics. It can explain a lot of natural phenomena clearly, so more and more mathematical researchers are devoting their time to it. Among them, the nonlinear problem comes from a lot of branches of applied mathematics and physics, it is at present one of the most active fields that is studied in analysical mathematics.
The present thesis mainly discusses the problems for solutions of integral boundary value problems with causal operators , integral boundary value problems for first order impulsive integro-differential equations of mixed type and nonlinear boundary value problems for first order impulsive integro-differential equations of mixed type.
In chapter one, we mainly introduce background, research meaning and current situations of this study, and the main conclusions and motive of this thesis.
In chapter two, we discuss the existence of solutions of f integral boundary value problems with causal operators, by using a comparison result and partial method. It generalizes and improves some former corresponding results.
In chapter three, by using the cone theory and monotone iterative technique, we investigate the existence of extremal solutions and unique solutions of integral boundary value problems for first order impulsive integro-differential equations of mixed type. Our results improve and extend many recent results.
In chapter four, by using lower and upper solutions, we investigate the existence of unique solution of nonlinear boundary value problems for first order impulsive integro-differential equations of mixed type. Our results improve and extend some recent results.
第一章,前言部分,主要介绍了选题来源、研究意义、国内外研究现状,以及论文的主要研究内容和目标.
第二章,利用新的比较原理和上下解方法,讨论了具有casual算子的一阶混合型脉冲方程积分边值问题,并改进了某些已有的结果.
第三章,利用单调迭代方法,研究了一阶混合型脉冲积分-微分方程积分边值问题极值解和唯一解的存在性.
第四章,利用上下解方法,研究了一阶混合型脉冲积-微分方程的非线性边值问题唯一解的存在性,对某些已有结果作了推广和改进.
Nonlinear functional analysis is an important branch of modern analysis mathematics. It can explain a lot of natural phenomena clearly, so more and more mathematical researchers are devoting their time to it. Among them, the nonlinear problem comes from a lot of branches of applied mathematics and physics, it is at present one of the most active fields that is studied in analysical mathematics.
The present thesis mainly discusses the problems for solutions of integral boundary value problems with causal operators , integral boundary value problems for first order impulsive integro-differential equations of mixed type and nonlinear boundary value problems for first order impulsive integro-differential equations of mixed type.
In chapter one, we mainly introduce background, research meaning and current situations of this study, and the main conclusions and motive of this thesis.
In chapter two, we discuss the existence of solutions of f integral boundary value problems with causal operators, by using a comparison result and partial method. It generalizes and improves some former corresponding results.
In chapter three, by using the cone theory and monotone iterative technique, we investigate the existence of extremal solutions and unique solutions of integral boundary value problems for first order impulsive integro-differential equations of mixed type. Our results improve and extend many recent results.
In chapter four, by using lower and upper solutions, we investigate the existence of unique solution of nonlinear boundary value problems for first order impulsive integro-differential equations of mixed type. Our results improve and extend some recent results.
引文
[1]郭大钧.非线性泛函分析[M].济南:山东科技出版社,1985
[2] Guo D.J., Lakshmikantham V. Nonlinear Problems in Abstract Cones[M]. Boston and New York, Academic Press, 1988
[3]郭大钧,孙经先.非线性积分方程[M].济南:山东科技出版社,1987
[4] C. Corduneanu, Functional Equations with Causal Operators, Taylor and Francis, London, 2002
[5] Fengjie Geng, Differential equations involving causal operators with nonlinear periodic boundary conditions, Math. Comput. Modelling 48 (2008)859-866
[6] T. Jankowski, Boundary value problems with causal operators, Nonlinear Anal. 68 (2008) 3625-3632
[7] Z. Drici, F.A. McRae, J. Vasundhara Devi, Monotone iterative technique for periodic boundary value problems with causal operators, Nonlinear Anal.64 (2006) 1271-1277
[8] Juan J. Nieto, Rosana Rodriguez-Lopez, new comparison results for impulsive integro-differential equations and applications, J. Math. Anal. Appl. 328(2007) 1343-1368
[9] Ruixi Liang, Jianhua Shen, Periodic boundary value problem for the first order impulsive functional differential equations, J. Comput. Appl. Math. 202(2007) 498-510
[10] V. Lakshmikantham, D.D. Bainov, P.S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989
[11] Wei Ding, Yepeng Xing, Maoan Han, Anti-periodic boundary value problems for first order impulsive functional differential equations, Appl. Math.Comput. 186 (2007) 45-53
[12] Zhimin He, Xiaoming He, Periodic boundary value problems for first order impulsive integro-differential equations of mixed type, J. Math. Anal. Appl.296 (2004) 8-20
[13] G.S. Ladde, V. Lakshmikantham, A.S. Vatsala, Monotone Iterative Techniques for Nonlinear Differential Equations, Pitman, London, 1985
[14] Lishan Liu, Iterative method for solutions and coupled quasi-solutions of nonlinear integro-differential equations of mixed type in Banach space,Nonlinear Anal. 42 (2000) 583-598
[15] Daniel Franco, Juan J. Nieto, Donal O'Regan, Existence of solutions for first order ordinary differential equations with nonlinear boundary conditions,Appl. Math.Comput. 153 (2004) 793-802
[16] G. Wang, G. Song, L. Zhang, Integral boundary value problems for first order integro-differential equations with deviating arguments, J. Comput. Appl.Math. (2008) doi:10.1016/j.cam.2008.08.030
[17] G.-X. Song, X.-L. Zhu, Extremal solutions of periodic boundary value problems for first order integro-differential equations of mixed type, J. Math.Anal. Appl. 300 (2004) 1-11
[18] T. Jankowski, Differential equations with integral boundary conditions, J. Comput. Appl. Math. 147 (2002) 1-8
[19] Dajun Guo, Initial value problem for integro-differential equations of volterra type in Banach space, J. Appl. Math. Stoch. Anal 7, Number 1, Spring 1994, 13-23
[20] D.J.Guo,V.Lakshmikantham,X.Liu,Nonlinear Integral Equations in Abstract Space, Kluwer Academic,Dordrecht,1996
[21] T. Jankowski,Existence of solutions of boundary value problems for differential equations with delayed arguments,J. Comput. Appl. Math. 156 (2003) 239-252
[22] J.J. Nieto, Periodic boundary value problems for first-order impulsive ordinary differential equations, Nonlinear Anal. 51 (2002) 1223-1232
[23] X. Yang, J. Shen, Nonlinear boundary value problems for first order impulsive functional differential equations, Appl. Math. Comput. 189 (2007) 1943-1952
[24] T. Jankowski, Ordinary Differential Equations with Nonlinear Boundary Conditions of Antiperiodic Type, Comput. Math. Appl. 47 (2004) 1419-1428
[25] T. Jankowski, Boundary value problems for first order differential equations of mixed type, Nonlinear Anal. 64 (2006) 1984-1997
[26] W. Li, G. Song, Nonlinear boundary value problem for second order impulsive integrodifferential equations of mixed type in Banach space, Comput. Math. Appl. 56 (2008)1372-1381
[27]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
[28] Du S. W., Lakshmikantham V.. Monotone iterative technique for differential equations in Banach spaces[J]. J. Math. Anal. Appl., 1982,87: 454-459
[29]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: 159-164
[30] Wang Wenxia, Zhang Lingling, Liang Zhandong. Initial value problems for nonlinearimpulsive integro-differential equations in Banach space[J]. J.Math.Anal.Appl., 2006, 320: 510-527
[31] Chen Lijing, Sun Jitao. Nonlinear value problem for first order impulsive integro-differential equations of mixed type[J]. J.Math.Anal.Appl,2007,325: 830-842
[32]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
[33]郭大钧,孙经先.抽象空间常微分方程[M].济南:山东科技出版社,1989
[34] Lakshamikantham V., Leela S. Nonlinear Differential Equations in Abstract Spaces[M]. NewYork, 1981
[35]郭大钧,孙经先. Banach空间常微分方程理论的若干问题[J].数学进展,1994, 23(6): 492-502
[2] Guo D.J., Lakshmikantham V. Nonlinear Problems in Abstract Cones[M]. Boston and New York, Academic Press, 1988
[3]郭大钧,孙经先.非线性积分方程[M].济南:山东科技出版社,1987
[4] C. Corduneanu, Functional Equations with Causal Operators, Taylor and Francis, London, 2002
[5] Fengjie Geng, Differential equations involving causal operators with nonlinear periodic boundary conditions, Math. Comput. Modelling 48 (2008)859-866
[6] T. Jankowski, Boundary value problems with causal operators, Nonlinear Anal. 68 (2008) 3625-3632
[7] Z. Drici, F.A. McRae, J. Vasundhara Devi, Monotone iterative technique for periodic boundary value problems with causal operators, Nonlinear Anal.64 (2006) 1271-1277
[8] Juan J. Nieto, Rosana Rodriguez-Lopez, new comparison results for impulsive integro-differential equations and applications, J. Math. Anal. Appl. 328(2007) 1343-1368
[9] Ruixi Liang, Jianhua Shen, Periodic boundary value problem for the first order impulsive functional differential equations, J. Comput. Appl. Math. 202(2007) 498-510
[10] V. Lakshmikantham, D.D. Bainov, P.S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989
[11] Wei Ding, Yepeng Xing, Maoan Han, Anti-periodic boundary value problems for first order impulsive functional differential equations, Appl. Math.Comput. 186 (2007) 45-53
[12] Zhimin He, Xiaoming He, Periodic boundary value problems for first order impulsive integro-differential equations of mixed type, J. Math. Anal. Appl.296 (2004) 8-20
[13] G.S. Ladde, V. Lakshmikantham, A.S. Vatsala, Monotone Iterative Techniques for Nonlinear Differential Equations, Pitman, London, 1985
[14] Lishan Liu, Iterative method for solutions and coupled quasi-solutions of nonlinear integro-differential equations of mixed type in Banach space,Nonlinear Anal. 42 (2000) 583-598
[15] Daniel Franco, Juan J. Nieto, Donal O'Regan, Existence of solutions for first order ordinary differential equations with nonlinear boundary conditions,Appl. Math.Comput. 153 (2004) 793-802
[16] G. Wang, G. Song, L. Zhang, Integral boundary value problems for first order integro-differential equations with deviating arguments, J. Comput. Appl.Math. (2008) doi:10.1016/j.cam.2008.08.030
[17] G.-X. Song, X.-L. Zhu, Extremal solutions of periodic boundary value problems for first order integro-differential equations of mixed type, J. Math.Anal. Appl. 300 (2004) 1-11
[18] T. Jankowski, Differential equations with integral boundary conditions, J. Comput. Appl. Math. 147 (2002) 1-8
[19] Dajun Guo, Initial value problem for integro-differential equations of volterra type in Banach space, J. Appl. Math. Stoch. Anal 7, Number 1, Spring 1994, 13-23
[20] D.J.Guo,V.Lakshmikantham,X.Liu,Nonlinear Integral Equations in Abstract Space, Kluwer Academic,Dordrecht,1996
[21] T. Jankowski,Existence of solutions of boundary value problems for differential equations with delayed arguments,J. Comput. Appl. Math. 156 (2003) 239-252
[22] J.J. Nieto, Periodic boundary value problems for first-order impulsive ordinary differential equations, Nonlinear Anal. 51 (2002) 1223-1232
[23] X. Yang, J. Shen, Nonlinear boundary value problems for first order impulsive functional differential equations, Appl. Math. Comput. 189 (2007) 1943-1952
[24] T. Jankowski, Ordinary Differential Equations with Nonlinear Boundary Conditions of Antiperiodic Type, Comput. Math. Appl. 47 (2004) 1419-1428
[25] T. Jankowski, Boundary value problems for first order differential equations of mixed type, Nonlinear Anal. 64 (2006) 1984-1997
[26] W. Li, G. Song, Nonlinear boundary value problem for second order impulsive integrodifferential equations of mixed type in Banach space, Comput. Math. Appl. 56 (2008)1372-1381
[27]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
[28] Du S. W., Lakshmikantham V.. Monotone iterative technique for differential equations in Banach spaces[J]. J. Math. Anal. Appl., 1982,87: 454-459
[29]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: 159-164
[30] Wang Wenxia, Zhang Lingling, Liang Zhandong. Initial value problems for nonlinearimpulsive integro-differential equations in Banach space[J]. J.Math.Anal.Appl., 2006, 320: 510-527
[31] Chen Lijing, Sun Jitao. Nonlinear value problem for first order impulsive integro-differential equations of mixed type[J]. J.Math.Anal.Appl,2007,325: 830-842
[32]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
[33]郭大钧,孙经先.抽象空间常微分方程[M].济南:山东科技出版社,1989
[34] Lakshamikantham V., Leela S. Nonlinear Differential Equations in Abstract Spaces[M]. NewYork, 1981
[35]郭大钧,孙经先. Banach空间常微分方程理论的若干问题[J].数学进展,1994, 23(6): 492-502