具有不定奇性的三阶微分方程周期正解的存在性
Existence of Positive Periodic Solutions for a Third-Order Differential Equation with an Indefinite Singularity
摘要: 不定奇性微分方程周期解的研究是微分方程中的一个重要组成部分，它在电子束模型、边界层理论和玻色–爱因斯坦凝聚体等多种学科中拥有广泛应用。近年来，许多研究关注的是排斥型三阶奇性微分方程周期正解的存在性问题。作为这一结果的延伸，本文讨论了一类具有不定奇性的三阶微分方程 周期正解的存在性，其中M是正常数，并且对任意有e(t)>0。函数h(t)在[0,T]上可变号的。利用Krasnoselskiĭ’s-Guo不动点定理和一些分析方法，我们证明该方程至少存在一个T-周期正解。
Abstract: The study of periodic solutions of indefinite singular differential equations is an important part of differential equations which has a wide range of applications in a variety of disciplines such as elec-tron beam focusing model, boundary layer theory and Bose-Einstein condensates. In recent years, much research has been concerned with the existence of positive periodic solutions of third-order differential equations with a repulsive singularity. As an extension of this result, in this paper, we consider the existence of positive periodic solutions to a class of third-order differential equation with an indefinite singularity , where M is a real constant and M>0, and is a positive. The weight function h(t) is allowed to change signon [0,T]. By using Krasnoselskiĭ’s-Guo fixed point theorem and some analysis skills, sufficient conditions for the existence of at least one positive periodic solu-tion of this equation are established.
文章引用：宋娟, 程志波. 具有不定奇性的三阶微分方程周期正解的存在性[J]. 应用数学进展, 2022, 11(11): 7688-7695. https://doi.org/10.12677/AAM.2022.1111814

参考文献

[1] Torres, P. (2015) Mathematical Models with Singularities—A Zoo of Singular Creatures. Atlantis Briefs in Differential Equations, Atlantis Press, Paris. [Google Scholar] [CrossRef] 
[2] Lazer, A. and Solimini, S. (1987) On Periodic Solutions of Nonlinear Differential Equations with Singularities. Proceedings of the American Mathematical Society, 99, 109-114. [Google Scholar] [CrossRef] 
[3] Cheng, Z. and Ren, J. (2018) Pe-riodic Solution for Second Order Damped Differential Equations with Attractive-Re- pulsive Singularities. Rocky Moun-tain Journal of Mathematics, 48, 753-768. [Google Scholar] [CrossRef] 
[4] Chu, J. and Torres, P. (2007) Applications of Schauder’s Fixed Point Theorem to Singular Differential Equations. Bulletin of the London Mathematical Society, 39, 653-660. [Google Scholar] [CrossRef] 
[5] Li, X. and Zhang, Z. (2009) Periodic Solutions for Damped Differential Equations with a Weak Repulsive Singularity. Nonlinear Analysis, Theory, Methods and Applications, 70, 2395-2399. [Google Scholar] [CrossRef] 
[6] Cheng, Z. and Ren, J. (2013) Studies on a Damped Differential Equation with Repulsive Singularity. Mathematical Methods in the Applied Sciences, 36, 983-992. [Google Scholar] [CrossRef] 
[7] Chu, J., Torres, P. and Zhang, M. (2007) Periodic Solution of Second Or-der Non-Autonomous Singular Dynamical Systems. Journal of Differential Equations, 239, 196-212. [Google Scholar] [CrossRef] 
[8] Jiang, D., Chu, J. and Zhang, M. (2005) Multiplicity of Positive Pe-riodic Solutions to Superlinear Repulsive Singular Equations. Journal of Differential Equations, 211, 282-302. [Google Scholar] [CrossRef] 
[9] Yu, X. and Lu, S. (2019) A Multiplicity Result for Periodic Solu-tions of Liénard Equations with an Attractive Singularity. Applied Mathematics and Computation, 346, 183-192. [Google Scholar] [CrossRef] 
[10] Lu, S., Guo, Y. and Chen, L. (2019) Periodic Solutions for Liénard Equation with an Indefinite Singularity. Nonlinear Analysis: Real World Applications, 45, 542-556. [Google Scholar] [CrossRef] 
[11] Sun, Y. (2005) Positive Solution of Singular Third-Order Three-Point Boundary Value Problem. Journal of Mathematical Analysis and Applications, 306, 586-603. [Google Scholar] [CrossRef] 
[12] Liu, Z., Ume, J. and Kang, S. (2007) Positive Solutions of a Sin-gular Nonlinear Third Order Two-Point Boundary Value Problem. Journal of Mathematical Analysis and Applications, 326, 589-601. [Google Scholar] [CrossRef] 
[13] Chu, J. and Zhou, Z. (2006) Positive Solutions for Singular Non-Linear Third-Order Periodic Boundary Value Problems. Nonlinear Analysis, Theory, Methods and Applications, 64, 1528-1542. [Google Scholar] [CrossRef] 
[14] Li, Y. (2010) Positive Periodic Solutions for Fully Third-Order Ordinary Differential Equations. Computers & Mathematics with Applications, 59, 3464-3471. [Google Scholar] [CrossRef] 
[15] Ren, J., Cheng, Z. and Chen, Y. (2013) Existence Results of Pe-riodic Solutions for Third-Order Nonlinear Singular Differential Equation. Mathematische Nachrichten, 286, 1022-1042. [Google Scholar] [CrossRef] 
[16] Cheng, Z. and Ren, J. (2015) Positive Solutions for Third-Order Variable-Coefficient Nonlinear Equation with Weak and Strong Singularities. Journal of Difference Equations and Ap-plications, 21, 1003-1020. [Google Scholar] [CrossRef] 
[17] Guo, D. and Lakshmikantham, V. (1988) Nonlinear Prob-lems in Abstract Cones. Academic Press, Boston.
[18] Ren, J., Siegmund, S. and Chen, Y. (2011) Positive Periodic Solutions for Third-Order Nonlinear Differential Equations. Electronic Journal of Differential Equations, 2011, 1-19.

为你推荐