您的位置：山东大学 -> 科技期刊社 -> 《山东大学学报（理学版）》

《山东大学学报(理学版)》 ›› 2019, Vol. 54 ›› Issue (8): 33-41.doi: 10.6040/j.issn.1671-9352.0.2018.707

• • 上一篇    下一篇

带阻尼项的二阶奇异微分方程的正周期解

陈瑞鹏,李小亚   

  1. 北方民族大学数学与信息科学学院, 宁夏 银川 750021
  • 出版日期:2019-08-20 发布日期:2019-07-03
  • 作者简介:陈瑞鹏(1986— ), 男, 博士, 讲师, 研究方向为微分方程与动力系统. E-mail:ruipengchen@126.com
  • 基金资助:
    国家自然科学基金资助项目(11701012);北方民族大学重大专项资助项目(ZDZX201804);宁夏高等教育一流学科建设资助项目(NXYLXK2017B09)

Positive periodic solutions for second-order singular differential equations with damping terms

CHEN Rui-peng, LI Xiao-ya   

  1. College of Mathematics and Information Science, North Minzu University, Yinchuan 750021, Ningxia, China
  • Online:2019-08-20 Published:2019-07-03

PDF (PC)

321

摘要： 研究带阻尼项的二阶微分方程u″+p(t)u'+q(t)u=f(t,u)+c(t)正周期解的存在性, 其中 p,q,c∈L1(R/TZ;R), f为Carathéodory函数且在u=0处具有奇异性。运用不动点理论, 为该方程建立了若干正周期解的存在性结果, 所得结果推广并改进了已有文献的相关结论。

关键词: 正周期解, 存在性, 奇性, 不动点理论

Abstract: This paper studies the existence of positive periodic solutions of u″+p(t)u'+q(t)u=f(t,u)+c(t), where p,q,c∈L1(R/TZ;R), f is a Carathéodory function and is singular when u=0. By means of the fixed point theory, several existence theorems are established for the above equation, and some recent results in the literature are generalized and improved.

Key words: positive periodic solution, existence, singularity, fixed point theory

中图分类号: 

  • O175.8
[1] FORBAT F, HUAUX A. Détermination approchée et stabilité locale de la solution périodique dune équation différentielle non linéaire[J]. Mém Public Soc Sci, Artts Lettres Hainaut, 1962, 76(2):3-13.
[2] HUAUX A. Sur Lexistence dune solution pe'riodique de l équation différentielle non linéaire (¨overx)+0.2(·overx)+x/(1-x)=(0.5)cos ωt[J]. Bull Cl Sci Acad R Belguique, 1962, 48(5):494-504.
[3] MAWHIN J. Topological degree and boundary value problems for nonlinear differential equations[M] // FURI M, ZECCA P. Topological Methods for Ordinary Differential Equations. Berlin: Springer, 1993: 74-142.
[4] LAZER A C, SOLIMINI S. On periodic solutions of nonlinear differential equations with singularities[J]. Proceedings of the American Mathematical Society, 1987, 99(1):109-114.
[5] GORDON W B. Conservative dynamical systems involving strong forces[J]. Transactions of the American Mathematical Society, 1975, 204(1):113-135.
[6] BONHEURE D, FABRY C, SMETS D. Periodic solutions of forced isochronous oscillators at resonance[J]. Discrete and Continuous Dynamical Systems, 2002, 8(4):907-930.
[7] FONDA A, MANÁSEVICH R, ZANOLIN F. Subharmonics solutions for some second order differential equations with singularities[J]. SIAM Journal on Mathematical Analysis, 1993, 24(5):1294-1311.
[8] HABETS P, SANCHEZ L. Periodic solutions of some Liénard equations with singularities[J]. Proceedings of the American Mathematical Society, 1990, 109(2):1035-1044.
[9] JIANG Daqing, CHU Jifeng, ZHANG Meirong. Multiplicity of positive periodic solutions to superlinear repulsive singular equations[J]. Journal of Differential Equations, 2005, 211(2):282-302.
[10] DELPINO M, MANÁSEVICH R. Infinitely many T-periodic solutions for a problem arising in nonlinear elasticity[J]. Journal of Differential Equations, 1993, 103(2):260-277.
[11] DELPINO M, MANÁSEVICH R, MONTERO A. T-periodic solutions for some second order differential equations with singularities[J]. Proceedings of the Royal Society of Edinburgh(Section A), 1992, 120(1):231-243.
[12] RACHUNKOVÁ I, STANEK S, TVRDY M. Singularities and Laplacians in boundary value problems for nonlinear ordinary differential equations[M]. Amsterdam: Elsevier, 2006: 607-722.
[13] PEDRO J T. Bounded solutions in singular equations of repulsive type[J]. Nonlinear Analysis, 1998, 32(1):117-125.
[14] PEDRO J T, ZHANG Meirong. Twist periodic solutions of repulsive singular equations[J]. Nonlinear Analysis, 2004, 56(4):591-599.
[15] ZHANG Meirong. A relationship between the periodic and the Dirichlet BVPs of singular differential equations[J]. Proceedings of the Royal Society of Edinburgh(Section A), 1998, 128(5):1099-1114.
[16] RACHUNKOVÁ I, TVRDY M, VRKOC I. Existence of nonnegative and nonpositive solutions for second order periodic boundary value problems[J]. Journal of Differential Equations, 2001, 176(2):445-469.
[17] PEDRO J T. Existence of one-signed periodic solutions of some second order differential equations via a Krasnoselskii fixed point theorem[J]. Journal of Differential Equations, 2003, 190(2):643-662.
[18] FRANCO D, WEBB J K L. Collisionless orbits of singular and nonsingular dynamical systems[J]. Discrete and Continuous Dynamical Systems, 2006, 15(3):747-757.
[19] PEDRO J T. Weak singularities may help periodic solutions to exist[J]. Journal of Differential Equations, 2007, 232(1):277-284.
[20] CHU Jifeng, LI Ming. Positive periodic solutions of Hills equations with singular nonlinear perturbations[J]. Nonlinear Analysis, 2008, 69(1):276-286.
[21] LI Xiong, ZHANG Ziheng. Periodic solutions for second-order differential equations with a singular nonlinearity[J]. Nonlinear Analysis, 2008, 69(11):3866-3876.
[22] MA Ruyun, CHEN Ruipeng, HE Zhiqian. Positive periodic solutions of second-order differential equations with weak singularities[J]. Applied Mathematics and Computation, 2014, 232(1):97-103.
[23] HAKL R, PEDRO J T. Maximum and antimaximum principles for a second order differential operator with variable coefficients of indefinite sign[J]. Applied Mathematics and Computation, 2011, 217(19):7599-7611.
[24] CHU Jifeng, FAN Ning, PEDRO J T. Periodic solutions for second order singular damped differential equations[J]. Journal of Mathematical Analysis and Applications, 2012, 388(2):665-675.
[25] CABADA A, CID J. On the sign of the Greens function associated to Hills equation with an indefinite potential[J]. Applied Mathematics and Computation, 2008, 205(1):303-308.
[26] DEIMLING K. Nonlinear functional analysis[M]. Berlin: Springer-Verlag, 1985.
[1] 马满堂. 一类非线性二阶系统周期边值问题正解的存在性[J]. 《山东大学学报(理学版)》, 2019, 54(6): 88-95.
[2] 章欢,李永祥. 含时滞导数项的高阶常微分方程的正周期解[J]. 《山东大学学报(理学版)》, 2019, 54(4): 29-36.
[3] 王娇. 一类非线性二阶常微分方程 Dirichlet问题正解的存在性[J]. 山东大学学报（理学版）, 2018, 53(6): 64-69.
[4] 叶芙梅. 带导数项共振问题的可解性[J]. 山东大学学报（理学版）, 2018, 53(2): 25-31.
[5] 甄苇苇,曾剑,任建龙. 基于变分理论与时间相关的抛物型反源问题[J]. 山东大学学报（理学版）, 2018, 53(10): 61-71.
[6] 张莎,贾梅,李燕,李晓晨. 分数阶脉冲微分方程三点边值问题解的存在性和唯一性[J]. 山东大学学报（理学版）, 2017, 52(2): 66-72.
[7] 何志乾, 苗亮英. 带弱奇性的二阶阻尼微分方程正周期解的存在性[J]. 山东大学学报（理学版）, 2017, 52(10): 84-88.
[8] 王双明. 一类具有时滞的周期流行病模型的动力学分析[J]. 山东大学学报（理学版）, 2017, 52(1): 81-87.
[9] 陈彬. 格林函数变号的三阶周期边值问题[J]. 山东大学学报（理学版）, 2016, 51(8): 79-83.
[10] 苏艳. 共振离散二阶Neumann问题解的存在性[J]. 山东大学学报（理学版）, 2016, 51(6): 37-41.
[11] 陈彬,Abuelgasimalshaby Elzebir. 共振条件下的二阶多点边值问题解的存在性和多解性[J]. 山东大学学报（理学版）, 2016, 51(4): 49-52.
[12] 蔡超. 一类Kolmogorov型方程的系数反演问题[J]. 山东大学学报（理学版）, 2016, 51(4): 127-134.
[13] 郭丽君. 非线性微分方程三阶三点边值问题正解的存在性[J]. 山东大学学报（理学版）, 2016, 51(12): 47-53.
[14] 朱雯雯. 带参数的一阶周期边值问题正解的存在性及多解性[J]. 山东大学学报（理学版）, 2016, 51(12): 36-41.
[15] 吴成明. 二阶奇异耦合系统正周期解的存在性[J]. 山东大学学报（理学版）, 2015, 50(10): 81-88.
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
[1] 汤晓宏1，胡文效2*，魏彦锋2，蒋锡龙2，张晶莹2，. 葡萄酒野生酿酒酵母的筛选及其生物特性的研究[J]. 山东大学学报（理学版）, 2014, 49(03): 12 -17 .
[2] 易超群,李建平,朱成文. 一种基于分类精度的特征选择支持向量机[J]. J4, 2010, 45(7): 119 -121 .
[3] 孙亮吉,吉国兴 . 上三角形矩阵代数上的Jordan(α,β)-导子和广义Jordan(α,β)-导子[J]. J4, 2007, 42(10): 100 -105 .
[4] 孟祥波1,张立东1,杜子平2. 均值-方差标准下带跳的保险公司投资与再保险策略[J]. 山东大学学报（理学版）, 2014, 49(05): 36 -40 .
[5] 胡明娣1,2,折延宏1,王敏3. L3*系统中逻辑度量空间的拓扑性质[J]. J4, 2010, 45(6): 86 -90 .
[6] 刘战杰1,马儒宁1,邹国平1,钟宝江2,丁军娣3. 一种新的基于区域生长的彩色图像分割算法[J]. J4, 2010, 45(7): 76 -80 .
[7] 张京友,张培爱,钟海萍. 进化图论在知识型企业组织结构设计中的应用[J]. J4, 2013, 48(1): 107 -110 .
[8] 史开泉. 信息规律智能融合与软信息图像智能生成[J]. 山东大学学报（理学版）, 2014, 49(04): 1 -17 .
[9] 伍芸1,2, 韩亚蝶1. 一类临界位势非线性椭圆型方程非平凡解的存在性[J]. J4, 2013, 48(4): 91 -94 .
[10] 何海伦， 陈秀兰*. 变性剂和缓冲系统对适冷蛋白酶MCP-01和中温蛋白酶BP-01构象影响的圆二色光谱分析何海伦， 陈秀兰*[J]. 山东大学学报（理学版）, 2013, 48(1): 23 -29 .
版权所有 © 2018 《山东大学学报(理学版)》编辑部　
地址: 山东省济南市山大南路27号山东大学中心校区（250100）　电话: +86-0531-88366917　E-mail: xblxb@sdu.edu.cn
本系统由北京玛格泰克科技发展有限公司设计开发