Abstract

In this paper, we consider a sixth-order variable coefficient singular differential equation with indefinite weights

where 𝜔 is a positive constant, . By using the Krasnoselskii-Guo fixed point theorem, we prove the existence of positive periodic solutions to the above equation.

Keywords

References

[1] Jifeng Chu, “On a Differential Equation Arising in Geophysics,” Monthly Journals for Mathematics, vol. 187, pp. 499-508, 2018.
[CrossRef] [Google Scholar] [Publisher Link]

[2] Alberto Cabada, and José Ángel Cid, “On the Sign of the Green’s Function Associated to Hill’s Equation with an Indefinite Potential,” Applied Mathematics and Computation, vol. 205, no. 1, pp. 303-308, 2008.
[CrossRef] [Google Scholar] [Publisher Link]

[3] Zhibo Cheng, and Jingil Ren, “Positive Solutions for Third-Order Variable-Coefficient Nonlinear Equation with Weak and Strong Singularities,” Journal of Difference Equations and Applications, vol. 21, no. 11, pp. 1003-1020, 2015.
[CrossRef] [Google Scholar] [Publisher Link]

[4] Zhibo Cheng, and Jingil Ren, “Positive Solutions for Fourth-Order Singular Nonlinear Differential Equation with Variable Coefficient,” Mathematical Methods Applied Sciences, vol. 39, no. 9, pp. 2251-2274, 2016.
[CrossRef] [Google Scholar] [Publisher Link]

[5] Monica Conti, Susanna Terracini, and Gianmaria Verzini, “Infinitely Many Solutions to Fourth Order Superlinear Periodic Problems,” Transactions of the American Mathematical Society, vol. 356, pp. 3283-3300, 2003.
[Google Scholar] [Publisher Link]

[6] Jifeng Chu, and Zhongcheng Zhou, “Positive Solutions for Singular Non-Linear Third-Order Periodic Boundary Value Problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 64, no. 7, pp. 1528-1542, 2006.
[CrossRef] [Google Scholar] [Publisher Link]

[7] Yujun Cui, and Yumei Zou, “Existence and Uniqueness Theorems for Fourth-Order Singular Boundary Value Problems,” Computers & Mathematics with Applications, vol. 58, no. 7, pp. 1449-1456, 2009.
[CrossRef] [Google Scholar] [Publisher Link]

[8] Dajun Guo, and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, 2014.
[Google Scholar]

[9] Jicai Huang, Shigui Ruan, and Jing Song, “Bifurcations in a Predator-Prey System of Leslie Type with Generalized Holling Type III Functional Response,” Journal of Differential Equations, vol. 257, no. 6, pp. 1721-1752, 2014.
[CrossRef] [Google Scholar] [Publisher Link]

[10] Robert Hakl, and Pedro J. Torres, “On Periodic Solutions of Second-Order Differential Equations with Attractive-Repulsive Singularities,” Journal of Differential Equations, vol. 248, no. 1, pp. 111-126, 2010.
[CrossRef] [Google Scholar] [Publisher Link]

[11] Yongxiang Li, “Positive Periodic Solutions for Fully Third-Order Ordinary Differential Equations,” Computers & Mathematics with Applications, vol. 59, no. 11, pp. 3464-3471, 2010.
[CrossRef] [Google Scholar] [Publisher Link]

[12] Jie Liu, Zhibo Cheng, and Yi Wang, “Positive Periodic Solution or Second-Order Nonlinear Differential Equation with Singularity of Attractive Type,” Journal of Applied Analysis and Computation, vol. 10, no. 4, pp. 1636-1650, 2020.
[CrossRef] [Google Scholar] [Publisher Link]

[13] Shiping Lu, “A New Result on the Existence of Periodic Solutions for Liénard Equations with a Singularity of Repulsive Type,” Journal of Inequalities and Applications, vol. 37, pp. 1-13, 2017.
[CrossRef] [Google Scholar] [Publisher Link]

[14] Jie Liu et al., “Positive Periodic Solutions for a Sixth-Order Variable Coefficient Differential Equation with Repulsive Singularity,” Authorea, vol. 43, pp. 100-114, 2023.
[CrossRef] [Google Scholar] [Publisher Link]

[15] A.C. Lazer, and S. Solimini, “On Periodic Solutions of Nonlinear Differential Equations with Singularities,” Proceedings of the American Mathematical Society, vol. 99, no. 1, pp. 109-114, 1987.
[Google Scholar] [Publisher Link]

[16] F. Merdivenci Atici, and G.Sh. Guseinov, “On the Existence of Positive Solutions for Nonlinear Differential Equations with Periodic Boundary Conditions,” Journal of Computational and Applied Mathematics, vol. 132, no. 2, pp. 341-356, 2001.
[CrossRef] [Google Scholar] [Publisher Link]

[17] Jingxian Sun, and Yansheng Liu, “Multiple Positive Solutions of Singular Third-Order Periodic Boundary Value Problem,” Acta Mathematica Scientia, vol. 25, no. 1, pp. 81-88, 2005.
[CrossRef] [Google Scholar] [Publisher Link]

[18] Pedro J. Torres, “Existence of One Signed Periodic Solutions of Some Second-Order Differential Equations via a Krasnoselskii’s-Guo Fixed Point Theorem,” Journal of Differential Equations, vol. 190, no. 2, pp. 643-662, 2003.
[CrossRef] [Google Scholar] [Publisher Link]

[19] Pedro J. Torres, Mathematical Models with Singularities-A Zoo of Singular Creatures, Atlantis Briefs in Differential Equations, Atlantis Press, Paris, 2015.
[Google Scholar]

[20] Youyu Wang, Hairong Lian, and Weigao Ge, “Periodic Solutions for a Second Order Nonlinear Functional Differential Equation,” Applied Mathematics Letters, vol. 20, no. 1, pp. 110-115, 2007.
[CrossRef] [Google Scholar] [Publisher Link]

[21] Yun Xin, Xuefeng Han, and Zhibo Cheng, “Multiplicity Results of Fourth-Order Singular Nonlinear Differential Equation with a Parameter,” Journal of Applied Analysis and Computation, vol. 7, no. 2, pp. 455-477, 2017.
       [CrossRef] [Google Scholar] [Publisher Links]