《山东大学学报(理学版)》 ›› 2020, Vol. 55 ›› Issue (3): 113-120.

• •

### 二阶微分方程Neumann边值问题最优正解的存在性

1. 西北师范大学数学与统计学院, 甘肃 兰州 730070
• 发布日期:2020-03-27
• 作者简介:王晶晶(1995— ), 男, 硕士研究生, 研究方向为差分方程及其应用. E-mail:WJJ950712@163.com*通信作者简介:路艳琼(1986— ),女,博士,副教授,研究方向为差分方程及其应用.E-mail: luyq8610@126.com
• 基金资助:
国家自然科学基金青年科学基金资助项目(11801453,11901464);甘肃省青年科技基金计划资助项目(1606RJYA232);西北师范大学青年教师科研能力提升计划项目(NWNU-LKQN-15-16)

### 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

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 fC(R+,R+), a(·)∈C(［0,T］,(0,+SymboleB@))satisfying the corresponding homogeneous linear problems have only trivial solutions, gC((0,T),R+), and g(t) is allowed to be singular at t=0 and t=T, R+:=［0,SymboleB@).

• 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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