JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2020, Vol. 55 ›› Issue (8): 102-110.doi: 10.6040/j.issn.1671-9352.0.2019.725

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Global structure of the positive solution for a class of fourth-order boundary value problems with first derivative

ZHANG Ya-li   

  1. College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, Gansu, China
  • Published:2020-07-14

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Abstract: This paper considers the global structure of positive solutions for a fourth-order boundary value problem with first derivative{u(4)(t)=rf(t,u(t),u'(t)), t∈(0,1),u(0)=u'(0)=u″(1)=u(1)=0,where r is a positive parameter, f:［0,1］×［0,∞)×［0,∞)→［0,∞)is continuous, and f(t,0,0)=0. When the parameter r changes in a certain range, the global structure of positive solutions of the problem are obtained by using the Rabinowitz global bifurcation theorems. The conclusions in this paper generalize and improve the related results.

Key words: fourth-order, positive solution, principal eigenvalue, bifurcation theorem

CLC Number: 

  • O175.8
[1] BAI Zhanbing, WANG Haiyan. On positive solutions of some nonlinear fourth-order beam equations[J]. Journal of Mathematical Analysis and Applications, 2002, 270(2):357-368.
[2] LIU Bing. Positive solutions of fourth-order two point boundary value problems[J]. Applied Mathematics and Computation, 2004, 148(2):407-420.
[3] 姚庆六. 一类半线性四阶两点边值问题的可解性[J]. 西南大学学报(自然科学版), 2008, 30(4):15-19. YAO Qingliu. Solvability of a class of semilinear fourth-order two-point boundary value problems[J]. Journal of Southwest University(Natural Science), 2008, 30(4):15-19.
[4] YAO Qingliu. Solutions and positive solutions of a class of semilinear fourth-order elastic beam equations[J]. Journal of Mathematical Research Exposition, 2007, 27(2):385-390.
[5] YAO Qingliu. Positive solutions for eigenvalue problems of fourth-order elastic beam equations[J]. Applied Mathematics Letters, 2004, 17(2):237-243.
[6] MA Ruyun. Existence of positive solutions for a nonlinear fourth order boundary value problem[J]. Annales Polonici Mathematici, 2003, 81(1):79-84.
[7] MA Ruyun. Existence of positive solutions of a fourth-order boundary value problem[J]. Applied Mathematics and Computation, 2005, 168(2):1219-1231.
[8] MA Ruyun, XU Jia. Bifurcation from interval and positive solutions of a nonlinear fourth-order boundary value problem[J]. Nonlinear Analysis, 2010, 72(1):113-122.
[9] YAO Qingliu. Solutions and positive solutions of a class of semilinear fourth-order elastic beam equations[J]. Journal of Mathematical Research and Exposition, 2007, 27(2):385-390.
[10] DEIMLING K. Nonlinear functional analysis[M]. New York: Spring-Verlag, 1985.
[11] ZEIDLER E. Nonlinear functional analysis and its applications[M]. New York: Spring-Verlag, 1986.
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