三阶非线性微分方程周期解的非退化和存在唯一性

Abstract

Abstract:

In this paper, we study the nondegeneracy of periodic solution for a third-order linear differential equation $x'''(t)+a_{2}x''(t)+a_{1}x'(t)=a_{0}(t)x(t).$ By the Writinger inequality, we give a nondegeneracy condition of the above equation. By the nondegeneracy of third-order linear differential equation, we consider the existence and uniqueness of periodic solution third-order nonlinear differential equation with semilinear and superlinear terms.

Key words: Periodic solution, Nondegeneracy, Semilinear, Superlinear, Existence and uniqueness

CLC Number:

O175.2

Shaowen Yao,Wenjie Li,Zhibo Cheng. Nondegeneracy and Uniqueness of Periodic Solution for Third-Order Nonlinear Differential Equations[J].Acta mathematica scientia,Series A, 2022, 42(2): 454-462.

Trendmd

References 20

1	Chow S . Existence of periodic solutions of autonomous functional differential equations. J Differential Equations, 1974, 15: 350- 378 doi: 10.1016/0022-0396(74)90084-9
2	Hale J. Ordinary Differential Equations. Florida: Krieger Publishing Company, 1980
3	Chu J , Torres P . Applications of Schauder's fixed point theorem to singular differential equations. Bull Lond Math Soc, 2007, 39: 653- 660 doi: 10.1112/blms/bdm040
4	Gaines R, Mawhin J. Coincidence Degree and Nonlinear Differential Equation. Berlin: Springer-Verlag, 1977
5	Lu S , Yu X . Periodic solutions for second order differential equations with indefinite singularities. Adv Nonlinear Anal, 2020, 9: 994- 1007
6	Godoy J , Zamora M . Periodic solutions for a second-order differential equation with indefinite weak singularity. Proc Roy Soc Edinburgh Sect A, 2019, 149: 1135- 1152 doi: 10.1017/prm.2018.79
7	张芷芬, 丁同仁, 黄文灶, 董镇喜. 微分方程定性理论. 北京: 科学出版社, 1985
	Zhang Z , Ding T , Huang W , Dong Z . Qualitative Theory of Differential Equation. Beijing: Science Press, 1985
8	Kuznetsov Y. Elements of Applied Bifurcation Theory. New York: Springer-Verlag, 1995
9	Castelli R , Garrione M . Some unexpected results on the Brillouin singular equation: Fold bifurcation of periodic solutions. J Differential Equations, 2018, 265: 2502- 2543 doi: 10.1016/j.jde.2018.04.042
10	Li J , Christie J , Gopalsamy K . Perturbed generalised Hamiltonian systems and some advection models. Bull Austral Math Soc, 1998, 57: 1- 24 doi: 10.1017/S0004972700031373
11	Battelli F , Fečkan M . On the Poincaré-Adronov-Melnikov method for the existence of grazing impact periodic solutions of differential equations. J Differential Equations, 2020, 268: 3725- 3748 doi: 10.1016/j.jde.2019.10.014
12	Hakl R , Zamora M . Periodic solutions to second-order indefinite singular equations. J Differential Equations, 2017, 263: 451- 469 doi: 10.1016/j.jde.2017.02.044
13	Nunez D , Perdomo O , Rivera A . On the stability of periodic solutions with defined sign in MEMS via lower and upper solutions. Nonlinear Anal: Real World Appl, 2019, 46: 195- 218 doi: 10.1016/j.nonrwa.2018.09.010
14	Lasota A , Opial Z . Sur les solutions périodiques deséquations différentielles ordinaires. Ann Polon Math, 1964, 16: 69- 94 doi: 10.4064/ap-16-1-69-94
15	Fonda A , Mawhin J . Quadratic forms, weighted eigenfunctions and boundary value problems for nonlinear second order ordinary differential equations. Proc Roy Soc Edinburgh Sect A, 1989, 112: 145- 153 doi: 10.1017/S0308210500028213
16	Ortega R , Zhang M . Optimal bounds for bifurcation values of a superlinear periodic problem. Proc Roy Soc Edinburgh Sect A, 2005, 135: 119- 132 doi: 10.1017/S0308210500003796
17	Li W , Zhang M . Non-degeneracy and uniqueness of periodic solutions for some superlinear beam equations. Appl Math Lett, 2009, 22: 314- 319 doi: 10.1016/j.aml.2008.03.027
18	Torres P , Cheng Z , Ren J . Non-degeneracy and uniqueness of periodic solutions for $2n$-order differential equation. Discrete Contin Dyn Syst, 2013, 33: 2155- 2168 doi: 10.3934/dcds.2013.33.2155
19	Zhang M . An abstract result on asymptotically positively homogeneous differential equations. J Math Anal Appl, 1997, 209: 291- 298 doi: 10.1006/jmaa.1997.5383
20	Ward J . Asymptotic conditions for periodic solutions of ordinary differential equations. Proc Amer Math Soc, 1981, 81: 415- 420 doi: 10.1090/S0002-9939-1981-0597653-2

[1]	Lan Jun. Existence and Multiplicity of Anti-Periodic Solutions for a Class of Second Order Duffing Equation [J]. Acta mathematica scientia,Series A, 2022, 42(2): 463-469.
[2]	Chenyang Xia,Zhenhui Wang,Zhibo Cheng. Positive Periodic Solutions for a Damped Duffing Equation with Singularity of Attractive Type [J]. Acta mathematica scientia,Series A, 2022, 42(1): 131-138.
[3]	Changyou Wang,Nan Li,Tao Jiang,Qiang Yang. On a Nonlinear Non-Autonomous Ratio-Dependent Food Chain Model with Delays and Feedback Controls [J]. Acta mathematica scientia,Series A, 2022, 42(1): 245-268.
[4]	Hongwu Zhang. One Regularization Method for a Cauchy Problem of Semilinear Elliptic Equation [J]. Acta mathematica scientia,Series A, 2022, 42(1): 45-57.
[5]	Changmu Chu,Lu Meng. Existence of Positive Solutions for Semilinear Elliptic Equation with Variable Exponent [J]. Acta mathematica scientia,Series A, 2021, 41(6): 1779-1790.
[6]	Baiping Ouyang,Shengzhong Xiao. Nonexistence of Global Solutions for a Semilinear Double-Wave Equation with a Nonlinear Memory Term [J]. Acta mathematica scientia,Series A, 2021, 41(5): 1372-1381.
[7]	Qingye Zhang,Bin Xu. Multiple Periodic Solutions for a Class of Stationary Dirac Equations with Local Nonlinearity [J]. Acta mathematica scientia,Series A, 2021, 41(4): 1013-1023.
[8]	Jinglei Zhao,Jiacheng Lan,Shanshan Yang. Lifespan Estimate of Damped Semilinear Wave Equation in Exterior Domain with Neumann Boundary Condition [J]. Acta mathematica scientia,Series A, 2021, 41(4): 1033-1041.
[9]	Jing Yang,Changcheng Ke,Zhouchao Wei. Study on Periodic Solutions of a Class of Continuous and Discontinuous Piece-Wise Linear Systems [J]. Acta mathematica scientia,Series A, 2021, 41(4): 1053-1065.
[10]	Shiping Lu,Shile Zhou,Xingchen Yu. Periodic Solutions for a Singular Liénard Equation with Sign-Changing Weight Functions [J]. Acta mathematica scientia,Series A, 2021, 41(3): 686-701.
[11]	Tingting Jiang,Zengji Du. Periodic Solutions of a Neutral Impulsive Predator-Prey Model with Holling-Type IV Functional Response [J]. Acta mathematica scientia,Series A, 2021, 41(1): 178-193.
[12]	Changwang Xiao,Fei Guo. Global Existence and Blowup Phenomena for a Semilinear Wave Equation with Time-Dependent Damping and Mass in Exponentially Weighted Spaces [J]. Acta mathematica scientia,Series A, 2020, 40(6): 1568-1589.
[13]	Shoujun Huang,Xiwang Meng. Improved Ordinary Differential Inequality and Its Application to Semilinear Wave Equations [J]. Acta mathematica scientia,Series A, 2020, 40(5): 1319-1332.
[14]	Shoujun Huang,Juan Wang. Blow up of Solutions to Semilinear Wave Equations with Variable Coefficient for Nonlinearity in an N-Dimensional Exterior Domain [J]. Acta mathematica scientia,Series A, 2020, 40(5): 1259-1268.
[15]	Zhongwei Cao,Xiangdan Wen,Wei Feng,Li Zu. Dynamics of a Nonautonomous SIRI Epidemic Model with Random Perturbations [J]. Acta mathematica scientia,Series A, 2020, 40(1): 221-233.

Nondegeneracy and Uniqueness of Periodic Solution for Third-Order Nonlinear Differential Equations

RichHTML

PDF (PC)

Abstract

Cite this article

share this article

References 20

Related Articles 15

Metrics

Comments

Recommended 10