满足Dirichlet边界条件的2阶奇异微分方程的正解

doi:10.3969/j.issn.1007-2985.2012.06.001

journal6 ›› 2012, Vol. 33 ›› Issue (6): 1-9.DOI: 10.3969/j.issn.1007-2985.2012.06.001

• 数学 • 下一篇

满足Dirichlet边界条件的2阶奇异微分方程的正解

(南京财经大学应用数学系,江苏南京 210003)

出版日期:2012-11-25 发布日期:2013-01-14

Positive Solutions of Second-Order Singular Differential Equations with Dirichlet Boundary Condition

(Department of Applied Mathematics,Nanjing University of Finance and Economics,Nanjing 210003,China)

Online:2012-11-25 Published:2013-01-14
About author:YAO Qing-liu(1946-),male,was born in Shanghai City,professor;research area is applied differential equations.
Supported by:
National Natural Science Foundation of China (11071109)

摘要/Abstract

摘要：研究了非线性2阶Dirichlet 边值问题u″(t)-λu(t)+h(t)f(t,u(t))+g(t,u(t))=00<t<1,u(0)=u(1)=0的正解存在性与多解性，其中λ>-π²是常数，而g(t,u)可以在u=0处奇异.通过精确估计解的先验界并且利用锥拉伸-压缩的Guo-Krasnoselskii不动点定理，建立了几个存在定理.

关键词： 非线性常微分方程, 奇异边值问题, 正解, 存在性与多解性

Abstract: The existence and multiplicity of positive solutions are studied for the nonlinear second-order Dirichlet boundary value problem u″(t)-λu(t)+h(t)f(t,u(t))+g(t,u(t))=00<t<1,u(0)=u(1)=0,where λ>-π² is a constant and g(t,u) may be singular at u=0.By exactly estimating the priori bound of solution and applying the Guo-Krasnoselskii fixed point theorem of cone expansion-compression type,several existence theorems are established.

Key words: nonlinear ordinary differential equation, singular boundary value problem, positive solution, existence and multiplicity

姚庆六. 满足Dirichlet边界条件的2阶奇异微分方程的正解[J]. journal6, 2012, 33(6): 1-9.

YAO Qing-Liu. Positive Solutions of Second-Order Singular Differential Equations with Dirichlet Boundary Condition[J]. journal6, 2012, 33(6): 1-9.

参考文献

[1] JORDAN M,SMITH P.Nonlinear Ordinary Differential Equations [M].London:Oxford Univ. Press,1977.

[2] FUCIK S.Solvability of Nonlinear Equations and Boundary Value Problems [M].Boston:D. Reidel,1980.

[3] ASAKAWA H.Nonresonant Singular Two-Point Boundary Value Problems [J].Nonlinear Anal. TMA,2001,44:791-809.

[4] DANCER E N,GUPTA C P.A Liapunov-Type Result with Application to a Dirichle-Type Two-Point Boundary Value Problem at Resonance [J].Nonlinear Anal. TMA,1994,22:305-318.

[5] KUO C.Solvability of a Nonlinear Two Point Boundary Value Problem at Resonance [J].J. Differential Equations,1997,140:1-9.

[6] O’REGAN D.Singular Dirichlet Boundary Value Problem-1:Superlinear and Nonresonant Case [J].Nonlinear Anal. TMA,1997,29:221-245.

[7] PANG C.Positive Solutions of Singular Superlinear Dirichlet Boundary Value Problems at Nonresinance [J].J. Systems Science and Mathematical Sceinces,2002,22:78-84.

[8] YAO Q.A Note on the Nonlinear Dirichlet Problems [J].J. University Science and Technology of China,2004,34:307-314.

[9] YILDIZ B,SIMSEK H.On the Contraction Mapping [J].Appl. Math. Comput.,2000,108:91-101.

[10] AGARWAL R P,O’REGAN D.Multiplicity Results for Singular Conjugate,Focal and (n,p) Problems [J].J. Differential Equations,2001,170:142-156.

[11] ELOE P W,HENDERSON J.Singular Nonlinear (k,n-k) Conjugate Boundary Value Problems [J].J. Differential Equations,1997,133:136-151.

[12] LIU Z,UME J S,KANG S M.Positive Solutions of a Singular Nonlinear Third Order Two-Point Boundary Value Problem [J].J. Math. Anal. Appl.,2007,326:589-601.

[13] RACHUNKOVA I,STANEK S.Sturm-Liouville and Focal Higher Order BVPs with Singularities in Phase Variables [J].Georgian Mathematical Journal,2003,10:159-191.

[14] YAO Q.Local Existence of Multiple Positive Solutions to a Singular Cantilever Beam Equation [J].J. Math. Anal. Appl.,2010,363:138-154.

[15] YAO Q.Positive Solutions to a Singular Fourth-Order Two-Point Boundary Value Problem [J].Annales Polonici Mathematici,2011,100:1-12.

[16] YAO Q.Positive Solutions of Nonlinear Beam Equations with Time and Space Singularities [J].J. Math. Anal. Appl.,2011,374:681-692.

满足Dirichlet边界条件的2阶奇异微分方程的正解

Positive Solutions of Second-Order Singular Differential Equations with Dirichlet Boundary Condition

PDF (PC)

可视化

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 15

编辑推荐

Metrics

本文评价

[1]	王兰芳. Caputo-Fabrizio型分数阶微分方程两点边值问题正解的存在性[J]. 吉首大学学报（自然科学版）, 2023, 44(5): 1-5.
[2]	潘欣媛, 何小飞. 一类分数阶四点边值问题正解的存在性[J]. 吉首大学学报（自然科学版）, 2022, 43(2): 1-6.
[3]	施敏. 一类非线性分数阶微分方程正解的存在性[J]. 吉首大学学报（自然科学版）, 2018, 39(5): 8-13.
[4]	彭思梦，钟文勇. 分数阶微分包含三点边值问题正解的存在性[J]. 吉首大学学报（自然科学版）, 2016, 37(3): 10-13.
[5]	欧阳敏, 钟文勇. 一类含参数的分数阶微分方程边值问题[J]. journal6, 2015, 36(2): 16-19.
[6]	李君君. 奇异Sturm-Liouville边值问题的正解[J]. journal6, 2014, 35(6): 10-13.
[7]	杨甲山, 刘兴元. 具连续变量的高阶非线性变时滞差分方程的正解[J]. journal6, 2013, 34(3): 1-6.
[8]	曹银芳, 肖建中, 沈志默. 4阶微分方程3点边值问题3个正解的存在性[J]. journal6, 2011, 32(5): 26-31.
[9]	姚庆六. 变系数三阶周期边值问题的正解[J]. journal6, 2010, 31(6): 9-13.
[10]	刘倩, 孙钦福, 欧阳金刚. 含参数四阶奇异微分方程边值问题正解存在性[J]. journal6, 2010, 31(3): 15-20.
[11]	钟文勇. 分数阶微分方程非齐次边值问题的正解[J]. journal6, 2010, 31(3): 10-14.
[12]	钟文勇. 分数阶微分方程多点边值问题的正解[J]. journal6, 2010, 31(1): 9-12.
[13]	王兰芳. 二阶常微分方程的积分边值问题[J]. journal6, 2009, 30(6): 26-28.
[14]	王颖, 孙钦福, 李翠兰. 四阶奇异微分方程边值问题的多重正解[J]. journal6, 2009, 30(3): 11-16.
[15]	姚庆六. 奇异三阶三点边值问题解的存在性[J]. journal6, 2008, 29(2): 1-4.