《山东大学学报(理学版)》 ›› 2019, Vol. 54 ›› Issue (8): 33-41.

• •

### 带阻尼项的二阶奇异微分方程的正周期解

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

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.

• 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.
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