具有变号非线性项的奇异三点边值问题正解的存在性和不存在性

数学物理学报 ›› 2011, Vol. 31 ›› Issue (1): 34-52.

具有变号非线性项的奇异三点边值问题正解的存在性和不存在性

闫宝强

山东师范大学数学科学学院济南 250014

收稿日期:2009-01-08 修回日期:2010-04-06 出版日期:2011-02-25 发布日期:2011-02-25
基金资助:
国家自然科学基金(10571111)和山东省自然科学基金(Y2008A06)资助.

The Nonexistence and Existence of Singular Three-point Boundary Value Problems |with Sign-changing Nonlinearities

YAN Bao-Qiang

School of Mathematics, Shandong Normal University, Jinan 250014

Received:2009-01-08 Revised:2010-04-06 Online:2011-02-25 Published:2011-02-25
Supported by:
国家自然科学基金(10571111)和山东省自然科学基金(Y2008A06)资助.

摘要/Abstract

摘要：

该文讨论奇异三点边值问题
y''(t)+a(t)f(t, y(t), y'(t))= 0, 0<t<1,
y'(0)= 0, y(1)= αy(η)
正解的存在与不存在性, 这里0<α<1, 0<η<1, f 变号且在y= 0和y'= 0具有奇异性.

关键词: 三点边值问题, 奇异性, 正解, 不动点指数

Abstract:

This paper discusses singular three-point boundary value problems
y''(t)+a(t)f(t, y(t), y'(t))= 0, 0<t<1,
y'(0)= 0, y(1)= αy(η)
where 0<α<1, 0<η<1, f changes sign and may be singular at y= 0 and y'= 0

Key words: Three-point boundary value problems, Singularity, Positive solutions, Fixed point index

中图分类号:

34B15

闫宝强. 具有变号非线性项的奇异三点边值问题正解的存在性和不存在性[J]. 数学物理学报, 2011, 31(1): 34-52.

YAN Bao-Qiang. The Nonexistence and Existence of Singular Three-point Boundary Value Problems |with Sign-changing Nonlinearities[J]. Acta mathematica scientia,Series A, 2011, 31(1): 34-52.

参考文献

[1] Agarwal R P, O'Regan D. Nonlinear superlinear singular and nonsingular second order boundary value problems. Journal of differential equations, 1998, 143: 60--95

[2] Agarwal R P, O'Regan D. Second order boundary value problems of singular type. Journal of Mathematical Analysis and
Applications, 1998, 226: 414--430

[3] Chen Y, Yan B, Zhang L. Positive solutions for singular three-point boundary-value problems with sign changing nonlinearities depending on x'. Electronic Journal of Differential Equations, 2007, 63: 1--9

[4] Gupta C P. Solvability of a three-point nonlinear boundary value problem for a second order ordinary differential equation. J Math Anal Appl, 1992, 168: 540--551

[5] Gupta C P, Trofimchuk S I. A sharper condition for the solvability of a three-point second order boundary value problem. J Math Anal Appl, 1997, 205: 586--597

[6] Guo D, Lakshmikantham V. Nonlinear Problems in Abstract Cones. New York: Cademic Press Inc, 1988

[7] Guo Y, Ge W. Positive solutions for three-point boundary value problems with dependence on the first order derivative. J Math Anal Appl, 2004, 290: 291--301

[8] Henderson J. Uniqueness implies existence for three-point boundary value problems for second order differential equations.
Applied Mathematics Letters, 2005, 18: 905--909

[9] Khan R A, Webb J R L. Existence of at least three solutions of a second-order three-point boundary value problem. Nonlinear Analysis, 2006, 64: 1356--1366

[10] Krasnoselskii M A. Positive Solutions of Operaor Equation. Groningen: Noordhoff, 1964

[11] Liu B. Positive solutions of a nonlinear three-point boundary value problem. Appl Math Comput, 2002, 132: 11--28

[12] O'Regan D. Existence Theory for Ordinary Differential Equations. Dordrecht: Kluwer, 1997

[13] O'Regan D. Theory of Singular Boundary Value Problems. Singapore: World Scientific, 1994

[14] Yan B, Liu Y. Positive solutions for superlinear second order singular three-point boundary value problems with derivative
dependence. Acta Mathematica Scientia, to appear.

[1]	李海侠. 一类食物链模型正解的稳定性和唯一性[J]. 数学物理学报, 2017, 37(6): 1094-1104.
[2]	廖家锋, 李红英. 带Sobolev临界指数的超线性Kirchhoff型方程正解的存在性与多解性[J]. 数学物理学报, 2017, 37(6): 1119-1124.
[3]	李清微, 高文杰, 韩玉柱. 一类带有退化强制项的奇异椭圆方程解的存在性[J]. 数学物理学报, 2017, 37(5): 877-894.
[4]	郝新安, 刘立山. 含参数四阶微分方程非局部边值问题的正解[J]. 数学物理学报, 2017, 37(2): 287-298.
[5]	张靖. 带有Hardy和Sobolev-Hardy临界指标项的扰动椭圆方程的解[J]. 数学物理学报, 2016, 36(3): 500-506.
[6]	黄祖达. 一类具反馈控制的时滞飞蝇方程的伪概周期解[J]. 数学物理学报, 2016, 36(3): 558-568.
[7]	冯育强, 王蔚敏, 李寿贵. 带有奇异非线性项的分数微分方程周期解的存在性与唯一性[J]. 数学物理学报, 2015, 35(6): 1059-1070.
[8]	杨芬. 全空间上一类半线性双调和方程正解的衰减[J]. 数学物理学报, 2015, 35(2): 282-287.
[9]	李波, 刘菡. 一类具有扩散和交错扩散的捕食模型的正解[J]. 数学物理学报, 2014, 34(6): 1578-1586.
[10]	姚宪忠, 穆春来, 张付臣. 带双参数的半线性椭圆方程正解存在与不存在性以及一个预值结论[J]. 数学物理学报, 2014, 34(5): 1144-1150.
[11]	张文丽, 钟立楠. R^N中一类(p, q)-Laplacian椭圆方程组的多重正解[J]. 数学物理学报, 2014, 34(5): 1264-1274.
[12]	谢华朝, 李素丽. 一类非齐次临界椭圆方程在R^N中的正解[J]. 数学物理学报, 2013, 33(6): 1099-1111.
[13]	彭艳芳, 汪继秀. 一类奇异椭圆型方程的多解性[J]. 数学物理学报, 2013, 33(4): 766-776.
[14]	张琦. 一类三阶两点边值问题的变号解[J]. 数学物理学报, 2013, 33(2): 216-223.
[15]	张新光, 潘传中. 一类半正非线性多点边值问题多个正解的存在性[J]. 数学物理学报, 2012, 32(5): 996-1010.