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《山东大学学报(理学版)》 ›› 2025, Vol. 60 ›› Issue (12): 103-109.doi: 10.6040/j.issn.1671-9352.0.2023.463

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一类含导数项的二阶时滞微分方程的周期解

喜霞,李永祥*   

  1. 西北师范大学数学与统计学院, 甘肃 兰州 730070
  • 发布日期:2025-12-10
  • 通讯作者: 李永祥(1963— ),男,教授,博士,研究方向为非线性泛函分析. E-mail:liyx@nwnu.edu.cn
  • 作者简介:喜霞(1997— ),女,硕士研究生,研究方向为非线性泛函分析. E-mail:2711769909@qq.com*通信作者:李永祥(1963— ),男,教授,博士,研究方向为非线性泛函分析. E-mail:liyx@nwnu.edu.cn
  • 基金资助:
    国家自然科学基金资助项目(12061062)

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

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摘要： 讨论非线性项中含导数项的二阶时滞常微分方程-u″(t)+a(t)u(t)=f(t,u(t),u(t-τ),u'(t)), t∈R的2π-周期解的存在性与唯一性,其中a:R→(0,+∞)为以2π为周期的连续函数, f:R4→R连续, f(t,x,y,z)关于t以2π为周期,τ>0为常数。在非线性项f满足适当的不等式条件下,应用Leray-Schauder不动点定理与先验估计技巧,获得该方程2π-周期解的存在性与唯一性结果。

关键词: 二阶时滞常微分方程, 2π-周期解, 存在性和唯一性, Leray-Schauder不动点定理

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

中图分类号: 

  • O175.8
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