JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2021, Vol. 56 ›› Issue (12): 52-58.doi: 10.6040/j.issn.1671-9352.0.2021.191

Previous Articles    

Existence, nonexistence and multiplicity of positive solutions for a class of nonlinear third order three point boundary value problems

ZHANG Rui-yan   

  1. College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, Gansu, China
  • Published:2021-11-25

PDF (PC)

172

Abstract: This paper considers the existence, nonexistence and multiplicity of positive solutions for a class of nonlinear third-order three-point boundary value problems {u(t)+λf(t,u(t))=0, t∈［0,1］,u(0)=u'(0)=0, u'(1)=αu'(η),where λ>0 is a parameter, 0<η<1, 1<α<1/η, f:［0,1］×［0,∞)→(0,∞)is a continuous function. The proof of the main theorems is based on fixed point index theorems, Leray-Schauder degree and the method of upper and lower solutions.

Key words: third-order, multiplicity, existence, fixed-point index

CLC Number: 

  • O175.8
[1] MA Ruyun. Multiplicity results for a third order boundary value problem at resonance[J]. Nonlinear Analysis, 1998, 32(4):493-499.
[2] YAO Qingliu, FENG Yuqiang. The existence of solution for a third-order two-point boundary value problem[J]. Applied Mathematics Letters, 2002, 15(2):227-232.
[3] DOUGLAS A R. Greens function for a third-order generalized right focal problem[J]. Journal of Mathematical Analysis and Applications, 2003, 288(1):1-14.
[4] SUN Yongping. Positive solutions of singular third-order three-point boundary value problem[J]. Journal of Mathematical Analysis and Applications, 2005, 306(2):589-603.
[5] HOPKINS B, KOSMATOV N. Third-order boundary value problems with sign-changing solutions[J]. Nonlinear Analysis, 2007, 67(1):126-137.
[6] PALAMIDES A P, SMYRLIS G. Positive solutions to a singular third-order three-point boundary value problem with an indefinitely signed Greens function[J]. Nonlinear Analysis, 2008, 68(7):2104-2118.
[7] CABADA A, LUCIA L S, FELIZ M. Existence, non-existence and multiplicity results for a third order eigenvalue three-point boundary value problem[J]. Journal of Nonlinear Sciences and Applications, 2017, 10(10):5445-5463.
[8] GUO Lijun, SUN Jianping, ZHAO Yahong. Existence of positive solution for nonlinear third-order three-point boundary value problem[J]. Nonlinear Analysis, 2008, 68(10):3151-3158.
[9] SUN Jianping, REN Qiuyan, ZHAO Yahong. The upper and lower solution method for nonlinear third-order three-point boundary value problem[J]. Electronic Journal of Qualitative Theory of Differential Equtions, 2010, 26(8):1-8.
[10] 郭大钧. 非线性泛函分析[M]. 北京:高等教育出版社, 2015. GUO Dajun. Nonlinear functional analysis[M]. Beijing: Higher Education Press, 2015.
[1] ZHAO Jiao. Existence and multiplicity of positive periodic solutions for a class of nonlinear third-order difference equations [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2021, 56(9): 50-58.
[2] OUYANG Bai-ping, XIAO Sheng-zhong. Global nonexistence of solutions to a class of semilinear double-wave equations with space-dependent coefficients on the nonlinearity [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2021, 56(9): 59-65.
[3] YUAN Tian-jiao, LI Qiang. Existence of IS-asymptotically periodic mild solutions for a class of impulsive evolution equations [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2021, 56(6): 10-21.
[4] WU Ruo-fei. Existence of solutions for singular fourth-order m-point boundary value problems [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2021, 56(2): 75-83.
[5] WANG Tian-xiang, LI Yong-xiang. Existence and uniqueness of solutions for a class fourth-order periodic boundary value problems [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2020, 55(7): 16-21.
[6] LI Zhao-qian. Existence and uniqueness of solutions for a class of nonlinear fourth-order boundary value problem [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2020, 55(6): 93-100.
[7] YANG Hu-jun, HAN Xiao-ling. Existence of positive periodic solutions for a class of non-autonomous fourth-order ordinary differential equations [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2020, 55(6): 109-114.
[8] YANG Li-juan. Existence and uniqueness of solutions for a class of boundary value problems of nonlinear fourth-order ordinary differential equations [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2020, 55(6): 101-108.
[9] CHEN Rui-peng, LI Xiao-ya. Positive periodic solutions for second-order singular differential equations with damping terms [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(8): 33-41.
[10] MA Man-tang. Existence of positive solutions for a class of periodic boundary value problems of nonlinear second-order systems [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(6): 88-95.
[11] . Uniqueness of positive solutions of singular p-biharmonic equations with Hardy terms [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(6): 75-80.
[12] HE Yan-qin, HAN Xiao-ling. Monotone positive solutions of fourth-order boundary value problems with integral boundary conditions [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(12): 32-37.
[13] LUO Qiang, HAN Xiao-ling, YANG Zhong-gui. Existence of positive solutions for boundary value problems of third-order delay differential equations [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(10): 33-39.
[14] ZHU Xiao-lin, ZHAI Cheng-bo. Local existence and uniqueness of positive solutions for a Sturm-Liouville boundary value problem of second order differential equations [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(10): 91-96.
[15] . 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.
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
No Suggested Reading articles found!
Copyright 2018 © Editorial office of JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)
Supported by Beijing Magtech Co.Ltd