JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2020, Vol. 55 ›› Issue (3): 113-120.doi: 10.6040/j.issn.1671-9352.0.2019.208

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Existence of optimal positive solutions for Neumann boundary value problems of second order differential equations

WANG Jing-jing, LU Yan-qiong*   

  1. College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, Gansu, China
  • Published:2020-03-27

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Abstract: By using the fixed point exponential theory of cone mapping, we show the optimal conditions for the existence of positive solutions for second-order continuous Neumann boundary value problems {u″(t)+a(t)u(t)=g(t)f(u(t)), t∈［0,T］,u'(0)=u'(T)=0with nonnegative Greens function, where f∈C(R+,R+), a(·)∈C(［0,T］,(0,+SymboleB@))satisfying the corresponding homogeneous linear problems have only trivial solutions, g∈C((0,T),R+), and g(t) is allowed to be singular at t=0 and t=T, R+:=［0,SymboleB@).

Key words: Neumann boundary value problem, positive solution, nonnegative Greens function, fixed point exponent

CLC Number: 

  • O175.8
[1] JIANG D Q, LIU H Z. Existence of positive solutions to second order Neumann boundary value problems[J]. Journal of Mathematical Research and Application, 2000, 20(3):360-364.
[2] 汤宇, 倪伟平. 二阶Neumann边值问题解的存在性[J].长春大学学报(自然科学版),2006,16(6):8-11. TANG Yu, NI Weiping. Existence of solutions to second order Neumann boundary value problems[J]. Journal of Changchun University(Natural Science), 2006, 16(6):8-11.
[3] SUN Yan, CHO Yeolje, OREGAN D. Positive solutions for singular second order Neumann boundary value problems via a cone fixed point theorem[J]. Applied Mathematics and Computation, 2009, 210(1):80-86.
[4] 闫东明. 变系数二阶Neumann边值问题正解的存在性[J].高校应用数学学报(A 辑), 2013,28(4):477-487. YAN Dongming. Existence of positive solutions to second order Neumann boundary value problems[J]. Journal of Applied Mathematics in Universities(Series A), 2013, 28(4):477-487.
[5] CABADA Aberto, ENGUICA Ricardo, LÓPEZ-SOMOZA Lucía. Positive solutions for second-order boundary value problems with sign changing Greens functions[J]. Electronic Journal of Differential Equations, 2017, 245:1-17.
[6] ZHANG G W, SUN J X. Positive solutions of m-point boundary value problems[J]. Journal of Mathematical Analysis and Applications, 2004, 291(2):406-418.
[7] CUI Y J, ZOU Y M. Nontrivial solutions of singular superlinear m-point boundary value problems[J]. Applied Mathematics and Computation, 2007, 182(2):1256-1264.
[8] WANG F L, ZHANG F. Positive solutions for a periodic boundary value problem without assumptions of monotonicity and convexity[J]. Bulletin of Mathematical Analysis and Applications, 2011, 3(2):261-268.
[9] GUO D J, LAKSHMIKANTHAM V. Nonlinear problems in abstract cones[M]. New York: Academic Press, 1988.
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