### Existence of positive solutions for a third-order three-point boundary value problem of nonlinear differential equations

GUO Li-jun

1. Department of Electro and Information Engineering, Lanzhou Jiaotong University Bowen College, Lanzhou 730101, Gansu, China
• Received:2016-05-31 Online:2016-12-20 Published:2016-12-20

Abstract: The third order differential equation has a wide application background and an important theoretical value, Green function plays an important role in the existence of positive solutions for the third order three point boundary value problems. This paper is concerned with the following boundary value problem{u(t)+a(t)f(u(t))=0, t∈(0,1),u(0)=u″(0)=0, u'(1)=αu(η),where 0<η<1 and 0<α<1. By establishing Green function for the associated linear boundary value problem, the solution for the above boundary value problem is obtained. Then some existence criteria of at least two positive solutions are obtained by using the fixed point index theorem.

CLC Number:

• O175.8
  SUN Jianping, GUO Lijun, PENG Junguo. Multiple nondecreasing positive solutions for a singular third order three point bvp[J]. Communications in Applied Analysis, 2008, 12:91-100. 孙建平,张小丽.非线性三阶三点边值问题正解的存在性[J].西北师范大学学报(自然科学版),2012, 48(3):29-31. SUN Jianping, ZHANG Xiaoli. Existence of positive solution for nonlinear third-order three-point boundary value problem[J]. Journal of Northwest Normal University(Natural Science), 2012, 48(3):29-31. 吴红萍.一类非线性三阶三点边值问题的多个正解[J].贵州大学学报(自然科学版),2014,31(2):4-6. WU Hongping. Multiple positive solutions for a third order three point boundary value problem[J]. Journal of Guizhou University(Natural Science), 2014, 31(2):4-6. 吕学哲,裴明鹤.一类三阶三点边值问题正解的存在性[J].北华大学学报(自然科学版),2014,15(5):577-580. LÜ Xuezhe, PEI Minghe. Existence of positive solution for a third-order three-point boundary value problem[J]. Journal of Beihua University(Natural Science), 2014, 15(5):577-580. 白婧,李永祥.含一阶导数项的三阶周期边值问题解的存在性[J].四川师范大学学报(自然科学版),2015, 6:834-837. BAI Jing, LI Yongxiang. Existence of solutions for the third order periodic boundary value problem with the first order derivative term[J]. Journal of Sichuan Normal University(Natural Science), 2015, 6:834-837. 姚志健.非线性三点边值问题正解的新的存在性定理[J].数学杂志,2014,34(1):173-178. YAO Zhijian. New existence of positive solutions for nonlinear third-order three-point boundary value problem[J]. Journal of Mathematics, 2014, 34(1):173-178. 张立新,孙博,张洪.三阶三点边值问题的两个正解的存在性[J].西南师范大学学报(自然科学版),2013, 38(10):30-33. ZHANG Lixin, SUN Bo, ZHANG Hong. Existence of two positive solutions for third-order three-point boundary value problem[J]. Journal of Southwest China Normal University(Natural Science), 2013, 38(10):30-33. 张立新.三阶边值问题的3个正解的存在性[J].四川师范大学学报(自然科学版),2011, 34(4):466-470. ZHANG Lixin. Existence of three positive solutions for the third order boundary value problem[J]. Journal of Sichuan Normal University(Natural Science), 2011, 34(4):466-470. 孙建平,曹珂.一类非线性三阶三点边值问题正解的存在性[J].兰州理工大学学报(自然科学版),2010, 36(2):123-124. SUN Jianping, CAO Ke. Existence of positive solution for a nonlinear third-order three-point boundary value problem[J]. Journal of Lanzhou University of Technology(Natural Science), 2010, 36(2):123-124. GUO Lijun, SUN Jianping, ZHAO Yahong. Existence of positive solution for nonlinear third-order three-point boundary value problem[J]. Nonlinear Analysis, 2008, 68:3151-3158. 郭大钧.非线性泛函分析[M].2版.济南:山东科学技术出版社,2004. GUO Dajun. Nonlinear functional analysis[M].2nd ed. Jinan: Shandong Science and Technology Press, 2004.
  WANG Su-yun, LI Yong-jun. Solvability of nonlinear second-order boundary value problems with nonlinearities which cross the resonance points [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(6): 53-56.  . Existence of positive solutions for a class of nonlinear second-order Dirichlet problem [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(6): 64-69.  XIAO Xin-ling. Forward-backward stochastic differential equations on Markov chains [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(4): 46-54.  YE Fu-mei. Existence results of a resonance problem with derivative terms [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(2): 25-31.  ZHEN Wei-wei, ZENG Jian, REN Jian-long. Time dependent parabolic inverse source problem based on variational theory [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(10): 61-71.  YAN Dong-liang. Positive solutions of a second order periodic problems with derivative terms [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(9): 69-75.  LI Tao-tao. Existence of radial positive solutions of second-order semi-positone elliptic differential equations [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(4): 48-55.  ZHANG Sha, JIA Mei, LI Yan, LI Xiao-chen. Existence and uniqueness of solutions for three point boundary value problems of impulsive fractional differential equations [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(2): 66-72.  FENG Hai-xing, ZHAI Cheng-bo. Multiple positive solutions of a system of high order nonlinear fractional differential equations [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(12): 48-57.  . Periodic solutions for second order singular damped differential equations with a weak singularity [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(10): 84-88.  XU Hai-li, LONG Pan-feng. Interaction between alkyl polyglycosides and other different surfactants [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(1): 9-14.  CHEN Bin. Third-order periodic boundary value problems with sign-changing Greens function [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2016, 51(8): 79-83.  SU Yan. Existence of solutions for second-order discrete Neumann problems at resonance [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2016, 51(6): 37-41.  CHEN Bin, Abuelgasimalshaby Elzebir. Existence and multiplicity results for a second-order multi-point boundary value problem at resonance [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2016, 51(4): 49-52.  CAI Chao. An inverse problem of identifying the coefficient in a Kolmogorov type equation [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2016, 51(4): 127-134.
Viewed
Full text

Abstract

Cited

Shared
Discussed
 No Suggested Reading articles found!