JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2023, Vol. 58 ›› Issue (4): 89-96.doi: 10.6040/j.issn.1671-9352.0.2022.261

Previous Articles    

Existence of positive solutions for a class of second order semipositone problems

SHI Xuan-rong   

  1. College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, Gansu, China
  • Published:2023-03-27

PDF (PC)

156

Abstract: The existence of positive solutions for the second order semipositone problem{-u″(t)=λh(t)f(u(t)), t∈(0,1),αu(0)-b(u'(0))u'(0)=0, c(u(1))u(1)+δu'(1)=0 is studied, where λ is a positive parameter,α,δ>0 are constants,b,c∈C(［0,∞),［0,∞)),h∈C(［0,1］,［0,∞)), f∈C(［0,∞),R), f >-M(M>0)and f∞:=limx→∞(f(x))/x=∞。The proof of the main theorems is based on fixed point theorem of Krasnoselskii.

Key words: positive solution, semipositone problem, existence, Krasnoselskii fixed point theorem

CLC Number: 

  • O175.8
[1] HAGHSHENAS H, AFROUZI G A. Three solutions for a second-order Sturm-Liouville equation with impulsive effects[J]. Journal of Applied Mathematics and Informatics, 2020, 38(5/6):407-414.
[2] JIANG Jiqiang, LIU Lishan, WU Yonghong. Second-order nonlinear singular Sturm-Liouville problems with integral boundary conditions[J]. Applied Mathematics and Computation, 2009, 215(4):1573-1582.
[3] GE Weigao, REN Jingli. New existence theorems of positive solutions for Sturm-Liouville boundary value problems[J]. Applied Mathematics and Computation, 2004, 148(3):631-644.
[4] LI Zhilong. Existence of positive solutions of superlinear second-order Neumann boundary value problems[J]. Nonlinear Analysis, 2010, 72(6):3216-3221.
[5] RYNNE B P. Linear second-order problems with Sturm-Liouville-type multi-point boundary conditions[J]. Electronic Journal of Differential Equations, 2012, 146:1-21.
[6] LV Haiyan, SHI Yuming. Error estimate of eigenvalues of perturbed second-order discrete Sturm-Liouville problems[J]. Linear Algebra and its Applications, 2009, 430(8/9):2389-2415.
[7] PINASCO J P, SCAROLA C. A nodal inverse problem for second order Sturm-Liouville operators with indefinite weights[J]. Applied Mathematics and Computation, 2015, 256:819-830.
[8] LIU Yuji. On Sturm-Liouville boundary value problems for second-order nonlinear functional finite difference equations[J]. Journal of Computational and Applied Mathematics, 2008, 216(2):523-533.
[9] MA Ruyun, WANG Suyun. Positive solutions for some semi-positone problems with nonlinear boundary conditions via bifurcation theory[J]. Mediterranean Journal of Mathematics, 2020, 17(1):1-12.
[10] YANG Jingbao, WEI Zhongli. Existence of positive solutions of Sturm-Liouville boundary value problems for a nonlinear singular second order differential system with a parameter[J]. Journal of Applied Mathematics and Computing, 2010, 34(1/2):129-145.
[11] DRAME A K, COSTA D G. On positive solutions of one-dimensional semipositone equations with nonlinear boundary conditions[J]. Applied Mathematics Letters, 2012, 25(12):2411-2416.
[12] ERBE L H, WANG Haiyan. On the existence of positive solutions of ordinary differential equations[J]. Proceedings of the American Mathematical Society, 1994, 120(3):743-748.
[13] DHANYA R, MORRIS Q, SHIVAJI R. Existence of positive radial solutions for superlinear semipositone problems on the exterior of a ball[J]. Journal of Mathematical Analysis and Applications, 2016, 434(2):1533-1548.
[14] AMANN H. Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces[J]. SIAM Review, 1976, 18(4):620-709.
[1] XU Jing, GAO Hong-liang. Number of positive solutions for mean curvature problem with convex-concave nonlinearity [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2023, 58(4): 74-81.
[2] LEI Xiang-bing. Existence of positive solutions for a class of semipositone second order Neumann boundary value problems [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2023, 58(4): 82-88.
[3] ZHANG Ji-feng, ZHANG Wei, WEI Hui, NI Jin-bo. Existence and uniqueness of solutions for fractional Langevin type equations with dual anti-periodic boundary conditions involving p-Laplace operator [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2022, 57(9): 91-100.
[4] HAN Zhuo-ru, LI Shan-bing. Positive solutions of predator-prey model with spatial heterogeneity and hunting cooperation [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2022, 57(7): 35-42.
[5] REN Qian, YANG He. Existence of mild solutions for a class of Riemann-Liouville fractional evolution inclusions [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2022, 57(4): 76-84.
[6] DUAN Dui-hua, GAO Cheng-hua, WANG Jing-jing. Existence and nonexistence of blow-up solutions for a general k-Hessian equation [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2022, 57(3): 62-67.
[7] 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.
[8] YANG Li-juan. Existence of positive solutions for a class of fourth-order boundary value problems with parameter [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2021, 56(9): 35-41.
[9] 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.
[10] SU Xiao-xiao, ZHANG Ya-li. Existence of positive solutions for periodic boundary value problems of secondorder damped difference equations [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2021, 56(2): 56-63.
[11] 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.
[12] LIU Meng-xue, LI Jie-mei, YAO Yan-yan. Multiplicity of positive solutions for fourth-order boundary value problems with nonlinear boundary conditions [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2021, 56(2): 84-91.
[13] ZHANG Rui-yan. Existence, nonexistence and multiplicity of positive solutions for a class of nonlinear third order three point boundary value problems [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2021, 56(12): 52-58.
[14] ZHANG Ya-li. Global structure of the positive solution for a class of fourth-order boundary value problems with first derivative [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2020, 55(8): 102-110.
[15] 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.
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