JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2021, Vol. 56 ›› Issue (2): 64-74.doi: 10.6040/j.issn.1671-9352.0.2020.351

Previous Articles    

Ambrosetti-Prodi type results of the second-order discrete Neumann boundary value problem

YANG Xiao-mei, LU Yan-qiong, WANG Rui   

  1. College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, Gansu, China
  • Published:2021-01-21

Abstract: By using the method of the upper and lower solutions and topological degree, this paper obtains the relationship between s and the number of solutions for the second-order discrete Neumann boundary value problem{ Δ2u(t-1)+g(t,u(t))=s, t∈[1,T]Z,Δu(0)=Δu(T)=0,where s∈R is a real parameter, g:[1,T]Z×R→R is continuous,[1,TZ:={1, 2, …, T}, there exists s0∈R such that the problem has no solution if s0, at least one solution if s=s0 and at least two solutions if s>s0

Key words: Neumann boundary value problem, Ambrosetti-Prodi problem, lower and upper solutions method, topological degree theory

CLC Number: 

  • O175.7
[1] AMBROSETTI A, PRODI G. On the inversion of some differentiable mappings with singularities between Banach space[J]. Ann Mat Pura Appl, 1972, 93(4):231-246.
[2] 马陆一. 非线性二阶Neumann边值问题的Ambrosetti-Prodi型结果[J]. 山东大学学报(理学版), 2015, 50(3):62-66. MA Luyi. The Ambrosetti-Prodi type results of the nonlinear second-order Neumann boundary value problem[J]. Journal of Shandong University(Natural Science), 2015, 50(3):62-66.
[3] SOVRANO E. Ambrosetti-Prodi type result to a Neumann problem via a topological approach[J]. Discrete Conti Dyn Syst Ser S, 2018, 11(2):345-355.
[4] ERBE L H, WANG H Y. On the existence of positive solutions of ordinary differential equations[J]. Proc Amer Math Soc, 1994, 120(3):743-748.
[5] SOVRANO E, ZANOLIN F. Ambrosetti-Prodi periodic problem under local coercivity conditions[J]. Adv Nonlinear Stud, 2018, 18(1):169-182.
[6] FELTRIN G, ZANOLIN F. An application of coincidence degree theory to cyclic feedback type systems associated with nonlinear differential operators[J]. Topol Methods Nonliear Anal, 2017, 50(2):683-726.
[7] FABRY C, MAWHIN J, NKASHMA M N. A multiplicity result for periodic solutions of forced nonlinear second order ordinary differential equations[J]. Bull Lond Math Soc, 1986, 18(2):173-180.
[8] FETRIN G, SOVRANO E, ZANOLIN F. Periodic solutions to parameter-dependent equations with a φ-Lalacian type operator[J]. NoDEA Nonlinear Differential Equations Appl, 2019, 26(5):38.
[9] BEREANU C, MAWHIN J. Existence and multiplicity results for periodic solutions of nonlinear difference equations[J]. Difference Equ Appl, 2006, 12(7):677-695.
[10] BEREANU C, MAWHIN J. Boundary value problems for second-order nonlinear difference equations with discrete φ-Laplacian and singular φ[J]. Difference Equ Appl, 2008, 14(10/11):1099-1118.
[11] KELLEY W G, PETERSON A C. Difference equation an introduction with applications[M]. San Diego: Academic Press, 2001.
[12] BEREANU C, MAWHIN J. Existence and multiplicity result for some nonlinear problems with singular φ-Laplacian[J]. Differential Equ, 2007, 243(2):536-557.
[13] VILLARI G. Slouzioni periodiche di una classe di equazioni differenziali delterz'ordine quasi lineari[J]. Ann Mat Pura Appl, 1966, 73(1):103-110.
[14] 马如云. 非线性常微分方程非局部问题[M]. 北京: 科学出版社, 2004. MA Ruyun. Nonlocal problems of nonlinear ordinary differential equations[M]. Beijing: Science Press, 2004.
[15] MANASEVICH R, MAWHIN J. Periodic solutions for nonlinear systems with p-Laplacian-like operators[J]. Differ Equ, 1998, 145(2):367-393.
[16] OMARI P. Nonordered lower and upper solutions and solvability of the periodic problem for the Liénard and the Rayleigh equations[J]. Rend Istit Mat Univ Trieste, 1988, 20:54-64.
[1] WANG Jing-jing, LU Yan-qiong. Existence of positive solutions for a class of semi-positive nonlinear elastic beam equation boundary value problems [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2020, 55(6): 84-92.
[2] WANG Jing-jing, LU Yan-qiong. Existence of optimal positive solutions for Neumann boundary value problems of second order differential equations [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2020, 55(3): 113-120.
[3] ZHU Wen-wen. Existence of multiple of solutions of first order multi-point boundary value problem [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2016, 51(6): 42-48.
[4] ZHU Wen-wen. Existence and multiplicity of positive solutions of first order periodic boundary value problems with parameter [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2016, 51(12): 36-41.
[5] MA Lu-yi. The Ambrosetti-Prodi type results of the nonlinear second-order Neumann boundary value problem [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2015, 50(03): 62-66.
[6] YAO Qing-liu . Positive solutions of nonlinear second-order Neumann boundary value problems with a variable coefficient [J]. J4, 2007, 42(12): 10-14 .
[7] ZHANG Xing-qiu,WANG Shao-feng . The positive solution and multiplicity for second order differential [J]. J4, 2006, 41(4): 4-07 .
Viewed
Full text


Abstract

Cited

  Shared   
  Discussed   
No Suggested Reading articles found!