非线性常微分方程正解的存在性
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摘要
本文利用锥上的不动点指数定理,范数形式的锥拉伸与锥压缩不动点定理，研究了几类非线性常微分方程边值问题的正解.
本文共分为四章：
第一章主要叙述了非线性分析的背景以及本文的创新之处.
在第二章中,我们利用范数形式的锥拉伸压缩不动点定理,考虑了下述四阶奇异半正边值问题其中λ>0,f∈C((0,1)×[0,+∞),(-∞,+∞)),且f在t=0,t=1处均可具有奇性.ω∈C((0,1),(-∞,+∞)),αi(s)(i=1,2)在[0,1]上非负可积,且∫01αi(s)ds∈(0,1).我们得到了该奇异半正边值问题正解的存在性.
在第三章中,我们利用锥上的不动点指数定理考虑了奇异边值问题其中α,β,γ>0,α>β/2,β>γ/2,γ>α/2,η∈(0,1),h(t)∈C((0,1),[0,+∞))可以在t=0,t=1点奇异.f(t,u)∈C((0,1)×(0,+∞),[0,+∞))可以在u=0点奇异.我们得到了该奇异边值问题对称正解的存在性.
在第四章中,我们考虑了下述n阶m点半正边值问题其中我们得到了该半正边值问题至少两个正解的存在性.
In this paper, we use the fixed point index theory as well as the fixed point theorem of generalized cone expansion and compression, to study the positive solution of the several kinds of boundary value problems for nonlinear differential equation.
The thesis is divided into four chapters according to contents.
The first chapter chiefly narrates the background of the nonlinear analysis and innovation of this paper.
In chapter 2, we use the fixed point theorem of generalized cone expansion and compression to investigate the following singular fourth-order boundary value problem with a sign-changing nonlinear term whereλ>0,f∈C((0,1)×[0,+∞),(-∞,+∞)), and f is allowed to be singular at t=0, t=1. moreover,ω∈C((0,1),(-∞,+∞)),αi(s)(i=1,2) is nonegative and integrable on [0,1], and∫01αi(s)ds∈(0,1). we obtain the existence of positive solution for the singular semipositone boundary value problem.
In chapter 3, By applying the fixed-point index theory, we investigate the existence of symmetric positive solutions of the following singular boundary value problem where a,α,β,γ>0,α>β/2,β>γ/2,γ>α/2,η∈(0,1) and ,h(t)∈C((0,1),[0,+∞)), is allowed to be singular at t=0, t=1. moreover, f(t,u)∈C((0,1)×(0,+∞), [0,+∞)) is allowed to be singular at u= 0. we obtain the existence of symmetric positive solution for the singular boundary value problem.
In chapter 4, we considered the existence of positive solutions for a class of n order m point boundary value problems with a sign-changing nonlinear term where f∈C([0,1]×[0,+∞), (-∞,+∞)),λ> 0,αi∈[0,+∞), i= 1,2,...,m-2, we obtain the existence of at least two positive solutions for the boundary value problem.
本文共分为四章：
第一章主要叙述了非线性分析的背景以及本文的创新之处.
在第二章中,我们利用范数形式的锥拉伸压缩不动点定理,考虑了下述四阶奇异半正边值问题其中λ>0,f∈C((0,1)×[0,+∞),(-∞,+∞)),且f在t=0,t=1处均可具有奇性.ω∈C((0,1),(-∞,+∞)),αi(s)(i=1,2)在[0,1]上非负可积,且∫01αi(s)ds∈(0,1).我们得到了该奇异半正边值问题正解的存在性.
在第三章中,我们利用锥上的不动点指数定理考虑了奇异边值问题其中α,β,γ>0,α>β/2,β>γ/2,γ>α/2,η∈(0,1),h(t)∈C((0,1),[0,+∞))可以在t=0,t=1点奇异.f(t,u)∈C((0,1)×(0,+∞),[0,+∞))可以在u=0点奇异.我们得到了该奇异边值问题对称正解的存在性.
在第四章中,我们考虑了下述n阶m点半正边值问题其中我们得到了该半正边值问题至少两个正解的存在性.
In this paper, we use the fixed point index theory as well as the fixed point theorem of generalized cone expansion and compression, to study the positive solution of the several kinds of boundary value problems for nonlinear differential equation.
The thesis is divided into four chapters according to contents.
The first chapter chiefly narrates the background of the nonlinear analysis and innovation of this paper.
In chapter 2, we use the fixed point theorem of generalized cone expansion and compression to investigate the following singular fourth-order boundary value problem with a sign-changing nonlinear term whereλ>0,f∈C((0,1)×[0,+∞),(-∞,+∞)), and f is allowed to be singular at t=0, t=1. moreover,ω∈C((0,1),(-∞,+∞)),αi(s)(i=1,2) is nonegative and integrable on [0,1], and∫01αi(s)ds∈(0,1). we obtain the existence of positive solution for the singular semipositone boundary value problem.
In chapter 3, By applying the fixed-point index theory, we investigate the existence of symmetric positive solutions of the following singular boundary value problem where a,α,β,γ>0,α>β/2,β>γ/2,γ>α/2,η∈(0,1) and ,h(t)∈C((0,1),[0,+∞)), is allowed to be singular at t=0, t=1. moreover, f(t,u)∈C((0,1)×(0,+∞), [0,+∞)) is allowed to be singular at u= 0. we obtain the existence of symmetric positive solution for the singular boundary value problem.
In chapter 4, we considered the existence of positive solutions for a class of n order m point boundary value problems with a sign-changing nonlinear term where f∈C([0,1]×[0,+∞), (-∞,+∞)),λ> 0,αi∈[0,+∞), i= 1,2,...,m-2, we obtain the existence of at least two positive solutions for the boundary value problem.
引文
[1]Li Gaoshang, Liu Xiping, Jia Mei. Positive solutions to a type of nonlinear three-point boundary value problem with sign changing nonlinearities[J]. Comput. Math. Appl.,2009, 57:348-355.
[2]Zhai Chengbo. Positive solutions for semi-positone three-point boundary value problem[J]. J. comput. Appl. Math.,2009,228:279-286.
[3]Yao Qingliu. Existence of n solutions and/or positive solutions to a semipositone elastic beam equation[J]. Nonlinear Anal.,2007,66:138-150.
[4]Lin Xiaoning, Li Xiaoyue, Jiang Daqing. Positive solutions to superlineear semipositone Periodic boundary value problems with repulsive weak singular forces[J]. Comput. Math. Appl.,2006,51:507-514.
[5]Xu Xian. Positive solutions for singular semi-positone three-point systems[J]. Nonlinear Anal.,2007,66:791-805.
[6]林晓宁,许晓婕.二阶时滞微分方程奇异半正边值问题[J].数学物理学报,2005,25：496-502.
[7]Zhang xinguang, Liu Lishan, Wu Yonghong. Existence of positive solutions for second-order semipositone differential equations on the half-line [J]. Appl. Math. Comput.,2007, 185:628-635.
[8]Zhang Mingchuan, Yin Yanmin, Wei Zhongli. Positive solution of singular higher-order m-point boundary value problem with nonlinearity that changes sign[J]. Appl. Math. Comput.,2008,201:678-687.
[9]赵增勤,孙忠民.一类四阶奇异半正Sturm-liouville边值问题的正解[J].系统科学与数学,2009,29(3)：378-388.
[10]Deimling K. Nonlinear Functional Analysis[M]. New York:Springer-Verlag,1985.
[11]Guo Dajun, Lakshmikantham V. Nonlinear Problems in Abstract Cones [M]. Boston:Aca-demic Press,1988.
(12]郭大均.非线性泛函分析[M].济南：山东科技出版社,2001.
[13]Zhang xuemei, Feng Meiqiang, Ge Weigao. Existence results for nonlinear boundary-value problem with integral boundary conditions in Banach spaces[J]. Nonlinear Anal.,2008, 69:3310-3321.
[14]韦忠礼.次线性奇异三点边值问题的正解[J].数学物理学报,2008,28：174-182.
[15]Chen Haibo, Li Peiluan. Three-point boundary value problem for second-order ordinary differential equations in Banach spaces[J]. Comput. Math. Appl.,2008,56:1852-1860.
[16]王淑丽,刘进生.二阶三点边值问题的正解[J].数学物理学报,2008,28：373-382.
[17]Sun Yan, Liu Lishan, Zhang Jizhou, Agarwal.R.P. Positive solutions of singular three-order differential equations[J]. J. Comput. Appl. Math.,2009,230:738-750.
[18]Liu Bingmei, Liu Lishan. Positive solutions for singular second order three-point boundary value problem[J]. Nonlinear Anal.,2007,66:2756-2766.
[19]Liu Bing. Positive solutions of second-order three-point boundary value problems with change of sign in Banach spaces[J]. Nonlinear Anal.,2006,64:1336-1355.
[20]Sun Yongping. Optimal existence criteria for symmetric positive solutions to a three-point boundary value problem[J]. Nonlinear Anal.,2007,66:1051-1063.
[21]Sun Yongping. Existence and Multiplicity of symmetric positive solutions for three-point boundary value problem[J]. J. Math. Anal.Appl.,2007,329:998-1009.
[22]Wang Shejun, Sun Hongrui. Optimal existence criteria for symmetric positive solutions to a songular three-point boundary value problem[J]. Nonlinear Anal.,2008,69:4266-4276.
[23]Krein. M. G, Rutman. M. A. Linear operators leaving invariant a cone in a Banach space[J]. Amer. Math. Soc. Transl.,1950,10:1-128.
[24]Guo Dajun, Lakshmikantham.V. Nonlinear Problem in Abstract cones[M]. San Diego: Academic Press,1998.
[25]Agarwal R. P, Agarwal.D. Note on existence of nonnegative solutions to singular semi-positone problem[J]. Nolinear Anal.,1999,36:615-622.
[26]Gupta. C. P. Solvability of a three-point nonlinear boundary value problem for a second order ordinary differential equation[J]. J. Math. Anal. Appl.,1992,168:540-551.
[27]Sun Yongping, Sun Yan. Positive solutions for second-order three-point boundary value problems[J]. Electron. J. Diff. Eqns.,2004,133:1-8.
[28]Xu Xian. Multiplicity results for positive solutions of some semi-positone three-point boundary value problems[J]. J. Math. Anal. Appl.,2004,291:673-689.
[29]Xu Xian. Existence and multiplicity of positive solutions for muli-parameter three-point differential equations system[J]. J. Math. Anal. Appl.,2006,324:472-490.
[30]Xu Xian. Positive solutions for singular semi-positone three-point systems[J]. Nonlinear Anal.,2007,66:791-805.
[31]Zhang Xinguang. Eigenvalue of higher-order semipositone multi-point boundary value problems with derivatives[J]. Appl. Math. Comput.,2008,201:361-370.
[32]Graef J. R, Kong Lingju. Positive solutions for third order semipositone boundary value problems[J]. Appl. Math. Comput.,2009,196:382-391.
[33]Zhang Mingchuan, Yin Yanmin, Wei Zhongli. Existence of positive solutions for singular semi-positive (k,n-k) conjugate m-point boundary value problem[J], Comput. Math. Appl., 2008,56:1146-1154.
[34]Zhang Mingchuan, Yin Yanmin, Wei Zhongli. Positive solution of singular higher-order m-point boundary value problem with nonlinearity that changes sign[J]. Appl. Math. Comput.,2008,201:678-687.
[2]Zhai Chengbo. Positive solutions for semi-positone three-point boundary value problem[J]. J. comput. Appl. Math.,2009,228:279-286.
[3]Yao Qingliu. Existence of n solutions and/or positive solutions to a semipositone elastic beam equation[J]. Nonlinear Anal.,2007,66:138-150.
[4]Lin Xiaoning, Li Xiaoyue, Jiang Daqing. Positive solutions to superlineear semipositone Periodic boundary value problems with repulsive weak singular forces[J]. Comput. Math. Appl.,2006,51:507-514.
[5]Xu Xian. Positive solutions for singular semi-positone three-point systems[J]. Nonlinear Anal.,2007,66:791-805.
[6]林晓宁,许晓婕.二阶时滞微分方程奇异半正边值问题[J].数学物理学报,2005,25：496-502.
[7]Zhang xinguang, Liu Lishan, Wu Yonghong. Existence of positive solutions for second-order semipositone differential equations on the half-line [J]. Appl. Math. Comput.,2007, 185:628-635.
[8]Zhang Mingchuan, Yin Yanmin, Wei Zhongli. Positive solution of singular higher-order m-point boundary value problem with nonlinearity that changes sign[J]. Appl. Math. Comput.,2008,201:678-687.
[9]赵增勤,孙忠民.一类四阶奇异半正Sturm-liouville边值问题的正解[J].系统科学与数学,2009,29(3)：378-388.
[10]Deimling K. Nonlinear Functional Analysis[M]. New York:Springer-Verlag,1985.
[11]Guo Dajun, Lakshmikantham V. Nonlinear Problems in Abstract Cones [M]. Boston:Aca-demic Press,1988.
(12]郭大均.非线性泛函分析[M].济南：山东科技出版社,2001.
[13]Zhang xuemei, Feng Meiqiang, Ge Weigao. Existence results for nonlinear boundary-value problem with integral boundary conditions in Banach spaces[J]. Nonlinear Anal.,2008, 69:3310-3321.
[14]韦忠礼.次线性奇异三点边值问题的正解[J].数学物理学报,2008,28：174-182.
[15]Chen Haibo, Li Peiluan. Three-point boundary value problem for second-order ordinary differential equations in Banach spaces[J]. Comput. Math. Appl.,2008,56:1852-1860.
[16]王淑丽,刘进生.二阶三点边值问题的正解[J].数学物理学报,2008,28：373-382.
[17]Sun Yan, Liu Lishan, Zhang Jizhou, Agarwal.R.P. Positive solutions of singular three-order differential equations[J]. J. Comput. Appl. Math.,2009,230:738-750.
[18]Liu Bingmei, Liu Lishan. Positive solutions for singular second order three-point boundary value problem[J]. Nonlinear Anal.,2007,66:2756-2766.
[19]Liu Bing. Positive solutions of second-order three-point boundary value problems with change of sign in Banach spaces[J]. Nonlinear Anal.,2006,64:1336-1355.
[20]Sun Yongping. Optimal existence criteria for symmetric positive solutions to a three-point boundary value problem[J]. Nonlinear Anal.,2007,66:1051-1063.
[21]Sun Yongping. Existence and Multiplicity of symmetric positive solutions for three-point boundary value problem[J]. J. Math. Anal.Appl.,2007,329:998-1009.
[22]Wang Shejun, Sun Hongrui. Optimal existence criteria for symmetric positive solutions to a songular three-point boundary value problem[J]. Nonlinear Anal.,2008,69:4266-4276.
[23]Krein. M. G, Rutman. M. A. Linear operators leaving invariant a cone in a Banach space[J]. Amer. Math. Soc. Transl.,1950,10:1-128.
[24]Guo Dajun, Lakshmikantham.V. Nonlinear Problem in Abstract cones[M]. San Diego: Academic Press,1998.
[25]Agarwal R. P, Agarwal.D. Note on existence of nonnegative solutions to singular semi-positone problem[J]. Nolinear Anal.,1999,36:615-622.
[26]Gupta. C. P. Solvability of a three-point nonlinear boundary value problem for a second order ordinary differential equation[J]. J. Math. Anal. Appl.,1992,168:540-551.
[27]Sun Yongping, Sun Yan. Positive solutions for second-order three-point boundary value problems[J]. Electron. J. Diff. Eqns.,2004,133:1-8.
[28]Xu Xian. Multiplicity results for positive solutions of some semi-positone three-point boundary value problems[J]. J. Math. Anal. Appl.,2004,291:673-689.
[29]Xu Xian. Existence and multiplicity of positive solutions for muli-parameter three-point differential equations system[J]. J. Math. Anal. Appl.,2006,324:472-490.
[30]Xu Xian. Positive solutions for singular semi-positone three-point systems[J]. Nonlinear Anal.,2007,66:791-805.
[31]Zhang Xinguang. Eigenvalue of higher-order semipositone multi-point boundary value problems with derivatives[J]. Appl. Math. Comput.,2008,201:361-370.
[32]Graef J. R, Kong Lingju. Positive solutions for third order semipositone boundary value problems[J]. Appl. Math. Comput.,2009,196:382-391.
[33]Zhang Mingchuan, Yin Yanmin, Wei Zhongli. Existence of positive solutions for singular semi-positive (k,n-k) conjugate m-point boundary value problem[J], Comput. Math. Appl., 2008,56:1146-1154.
[34]Zhang Mingchuan, Yin Yanmin, Wei Zhongli. Positive solution of singular higher-order m-point boundary value problem with nonlinearity that changes sign[J]. Appl. Math. Comput.,2008,201:678-687.