JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2025, Vol. 60 ›› Issue (12): 103-109.doi: 10.6040/j.issn.1671-9352.0.2023.463

Previous Articles     Next Articles

Periodic solutions of a second-order delay ordinary differential equation with derivative term

XI Xia, LI Yongxiang*   

  1. College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, Gansu, China
  • Published:2025-12-10

PDF (PC)

37

Abstract: In this paper, the existence and uniqueness of 2π-periodic solutions are discussed for the second-order delay ordinary differential equations with derivative term in the nonlinearity-u″(t)+a(t)u(t)=f(t,u(t),u(t-τ),u'(t)), t∈Rwhere a:R→(0,+∞)is a 2π periodic continuous function, f:R4→R is a continuous function and f(t,x,y,z) is 2π-periodic on t,τ>0 is a constant. The nonlinearity f satisfies some appropriate inequality conditions, the existence and uniqueness results of 2π-periodic solutions are obtained by using the Leray-Schauder fixed point theorem and the technique of prior estimates.

Key words: second order delay differential equations, 2π-periodic solution, existence and uniqueness, Leray-Schauder fixed point theorem

CLC Number: 

  • O175.8
[1] LI J W, CHENG S S. Periodic solutions of a second order forced sublinear differential equation with delay[J]. Applied Mathematics Letters, 2005, 18:1373-1380.
[2] WANG Y Y, LIAN H R, GE W G. Periodic solutions for a second order nonlinear functional differential equation[J]. Applied Mathematics Letters, 2007, 20:110-115.
[3] WU J, WANG Z C. Two periodic solutions of second-order neutral functional differential equations[J]. Journal Mathematical Analysis and Applications, 2007, 329:677-689.
[4] WU Y X. Existence nonexistence and multiplicity of periodic solutions for a kind of functional differential equation with parameter[J]. Nonlinear Analysis, 2009, 70:433-443.
[5] GUO C J, GUO Z M. Existence of multiple periodic solutions for a class of second-order delay differential equations[J]. Nonlinear Analysis: Real World Applications, 2009, 10:3285-3297.
[6] CHEUNG W S, REN J L, HAN W W. Positive periodic solution of second-order neutral functional differential equations[J]. Nonlinear Analysis, 2009, 71:3948-3955.
[7] LI Y X. Positive periodic solutions of second-order differential equations with delay[J]. Abstract and Applied Analysis, 2012, 13:829783.
[8] LI Q, LI Y X. On the existence of positive periodic solutions for second-order functional differential equations with multiple delays[J]. Abstract and Applied Analysis, 2012, 13:929870.
[9] LI Q, LI Y X. Monotone iterative technique for second order delayed periodic problem in Banach spaces[J]. Applied Mathematics and Computation, 2015, 270:654-664.
[10] GOU H D. Existence of ω-periodic solutions for second order delay differential equation in Banach spaces[J]. Filomat, 2022, 36(16):5347-5358.
[11] LI Y X, GUO L J. Odd periodic solutions of fully second-order ordinary differential equations with superlinear nonlinearities[J]. Journal of Function Spaces, 2017, 5:4247365.
[12] ZU J. Existence and uniqueness of periodic solution for nonlinear srcond-order ordinary differential equations[J]. Boundary Value Problems, 2011, 11:192156.
[13] 李永祥,张丽娟. 完全二阶常微分方程的奇周期解[J]. 数学学报(中文版),2022,65(2):287-300. LI Yongxiang, ZHANG Lijuan. Singular periodic solution of fully second order ordinary differential equations[J]. Acta Mathematicae(Chinese Version), 2022, 65(2):287-300.
[14] 刘晓明,李永祥. 具有非线性导数项的二阶常微分方程的正周期解[J]. 吉林大学学报(理学版),2023,61(6):1243-1250. LIU Xiaoming, LI Yongxiang. Positive periodic solutions of second-order ordinary differential equations with nonlinear derivative terms[J]. Journal of Jilin University(Natural Science), 2023, 61(6):1243-1250.
[15] LI Y X. Positive periodic solutions of first and second-order ordinary differential equations[J]. Chinese Annals of Mathematics(Series B), 2004, 25(3):413-420.
[1] CHEN Xiao, ZHOU Wenxue, HOU Zerong. A class of three-point boundary value problems for implicit impulsive fractional differential equations [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2025, 60(12): 121-129.
[2] Fangfang HU,Weimin HU,Yong ZHANG. Existence and uniqueness of positive solutions of integral boundary value problems for a class of fractional differential equations with Hadamard derivatives [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2024, 59(4): 53-61.
[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] 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.
[5] LI Zhao-qian. Existence and uniqueness of solutions for a class of nonlinear fourth-order boundary value problem [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2020, 55(6): 93-100.
[6] . Uniqueness of positive solutions of singular p-biharmonic equations with Hardy terms [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(6): 75-80.
[7] WANG Su-yun, LI Yong-jun. Solvability of nonlinear second-order boundary value problems with nonlinearities which cross the resonance points [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(6): 53-56.
[8] XIAO Xin-ling. Forward-backward stochastic differential equations on Markov chains [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(4): 46-54.
[9] ZHANG Sha, JIA Mei, LI Yan, LI Xiao-chen. Existence and uniqueness of solutions for three point boundary value problems of impulsive fractional differential equations [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(2): 66-72.
[10] WANG Xian-fei, JIANG Long, MA Jiao-jiao. Multidimensional backward doubly stochastic differential equations with generators of Osgood type [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2015, 50(08): 24-33.
[11] CHEN Qiang, JIA Mei, ZHANG Hai-bin. Existence and uniqueness of solutions for nonlinear fractional four-point boundary value problems [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2015, 50(04): 42-48.
[12] SHI Xue-jun1, JIANG Long2*. Lp-solutions for one-dimensional reflected backward stochastic differential equations with continuous generators [J]. J4, 2012, 47(11): 119-126.
[13] ZHANG Xiang, HUANG Shu-xiang. Monotone iterative technique, existence and uniqueness results for the nonlinear fractional reaction-diffusion equation [J]. J4, 2011, 46(2): 9-14.
[14] FANG Bao-yan1, WANG Zhi-gang2, TIAN Shuang-liang1*, SU Li-jun2. The existence and uniqueness of the solution of parabolic equations with wavelet collocation [J]. J4, 2010, 45(6): 65-69.
[15] WANG Zhi-Gang, QIN Xin-Qiang, DANG Fa-Ning, SU Li-Jun. The existence and uniqueness of the solution to meshless methodwith ridge basis functions [J]. J4, 2010, 45(2): 44-49.
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
[1] ZHANG Xue-feng1, LIU Peng1,2. An improved K-means algorithm by weighted distance based on maximum between-cluster variation[J]. J4, 2010, 45(7): 28 -33 .
[2] WANG Gui-Jie, WANG Wen-De, TAO De-Li. Study and application of CAD technology in grade crossing vertical design[J]. J4, 2009, 44(11): 79 -82 .
[3] NI Gui-ping, WANG Yan-ping, WANG Hua-tian*, HAN Ya-fei, SANG Ya-lin. Extraction and amplification of bacterial DNA from soils of poplar plantation[J]. J4, 2013, 48(05): 23 -28 .
[4] WANG Yi ,LIU Ai-lian . Cobweb models on time scales[J]. J4, 2007, 42(7): 41 -44 .
[5] LIU Bing, LIU Wei-Ji, YANG Guo-Sheng. QSRR of alkyl-nitrophenols in gas chromatography by genetic algorithms[J]. J4, 2009, 44(9): 8 -11 .
[6] GAO Zheng-hui, LUO Li-ping. Philos-type oscillation criteria for third-order nonlinear functional differential equations with distributed delays and damped terms[J]. J4, 2013, 48(4): 85 -90 .
[7] WANG Ji-qiang . AResearch on approximation algorithm for a kind of median problem[J]. J4, 2006, 41(4): 1 -03 .
[8] WANG Feng-Yu. On stochastic generalized porous media and fast diffusion equations [J]. J4, 2009, 44(10): 1 -3 .
[9] JIAO Yu-Juan, ZHANG Shen-Gui. S-Noetherianess of A+xB［［x］］[J]. J4, 2009, 44(8): 58 -61 .
[10] ZHAO Rui,HUO Gui-cheng* .

Diversity of lactic acid bacteria isolated in sour milk from Xinjiang

[J]. J4, 2008, 43(7): 18 -22 .
Copyright 2018 © Editorial office of JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)
Supported by Beijing Magtech Co.Ltd