JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2019, Vol. 54 ›› Issue (10): 7-12.doi: 10.6040/j.issn.1671-9352.0.2018.263

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Existence of positive solutions to a semipositone second-order boundary value problem

WEI Jin-ying, WANG Su-yun, LI Yong-jun   

  1. School of Mathematics, Lanzhou City University, Lanzhou 730070, Gansu, China
  • Published:2019-10-12

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Abstract: We consider the existence of positive solutions to the boundary value problemu″(t)+c(t)u+λf(t,u)=0, 0λ>0, c(·)∈C［0,1］ satisfies -∞π2 for t∈［0,1］, f:［0,1］×R+→R is continuous function and satisfies f≥-L, L>0 is a constant. By investigating the sign property of the Green function of the associated linear boundary value problem, we show the existence of positive solutions of semipositone problems. The proof of the main result is based on Krasnoselskii fixed point theorems in cone.

Key words: semipositone problem, boundary value problem, Green function, positive solution, fixed point theorem

CLC Number: 

  • O175.8
[1] AGARWAL R P, OREGAN D. Singular boundary value problems for superlinear second order Ordinary and delay differential equations[J]. Journal of Differential Equations, 1996, 130(2):333-355.
[2] JIANG Daqing. Multiple positive solutions to singular boundary value problems for superlinear second order ODES[J]. Acta Mathematica Scientia, 2002, 22(2):199-206.
[3] LIU Zhaoli, LI Fuyi. Multiple positive solutions of nonlinear two-point boundary value problems[J]. Journal of Mathematical Analysis and Applications, 1996, 203(3): 610-625.
[4] 程伟, 徐家发. 一类2n阶边值问题正解的存在性[J]. 重庆师范大学学报(自然科学版), 2018,35(1):66-69. CHENG Wei, XU Jiafa. Existence of positive solutions for a class of 2n-order boundary value problems[J]. Journal of Chongqing Normal University(Natural Science), 2018, 35(1):66-69.
[5] WALTER W. Ordinary differential equations[M]. New York: Springer, 1998.
[6] CASTRO A, SHIVAJI R. Nonnegative solutions for a class of nonpositone problems[J]. Proceedings of the Royal Society of Edinburgh, 1988, 108: 291-302.
[7] ANURADHA V, HAI D, SHIVAJI R. Existence results for superlinear semipositone BVPs[J]. Proceedings of the American Mathematical Society, 1996, 124(3):757-763.
[8] MA Ruyun, MA Qiaozhen. Positive solutions for semipositone m-point boundary value problems[J]. Acta Mathematica Sinica, 2004, 20(2):273-282.
[9] LI Yongxiang. Positive solutions of fourth-order boundary value problems with two parameters[J]. Journal of Mathematical Analysis and Applications, 2003, 281(2):477-484.
[10] LI Yongxiang. On the existence and nonexistence of positive solutions for nonlinear Sturm-Liouville boundary value problems[J]. Journal of Mathematical Analysis and Applications, 2005, 304(1):74-86.
[11] KRASNOSEL'SKII M A. Positive solutions of operator equations[M]. Gronigen: Noordhoff, 1964.
[12] COPPEL W A. Disconjugacy[M]. New York: Springer, 1971.
[13] MA Ruyun, LU Yanqiong. Disconjugacy and extremal solutions of nonlinear third-order equations[J]. Communications on Pure and Applied Analysis, 2014, 13(3):1223-1236.
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