Periodic Solutions of Certain Higher Order Autonomous Differential Equations via Topological Degree Theory
Keywords:
Periodic solution, Autonomous Differential Equation, Homotopy invariant, Topological degreeAbstract
In this paper, we give some sufficient conditions for the existence of periodic solutions for some autonomous nonlinear ordinary differential equations of order n. The proposed method is based on the use of Brouwer’s degree and especially the homotopy invariant of the topological degree.
References
M. Asadi, Exact Solutions of Singular IVPs Lane-Emden Equation, Communications in Nonlinear Analysis, 6 (2019), 60–63.
M. Asadi and E. Karapınar, Coincidence Point Theorem on Hilbert Spaces via Weak Ekeland Variational Principle and Application to Boundary Value Problem, Thai Journal of Mathematics 19 (2021), 1–7.
F.W. Bates, Y.R. Ward, Periodic solutions of high order systems, Pacific J. Math. 84 (1979) 275–282.
M. Bayat, On the Existence of Periodic Solutions of a Certain Fifth-order Autonomous Differential Equations via Topo logical Degree Theory, Journal of Advanced Mathematical Modeling (JAMM), 13 (2023), 233–244.
M. Bayat and Z. Khatami, The existence of periodic solutions for the nonlinear fifth order autonomous ordinary differential equations , 43rd Annual Iranian Mathematics Conference, 27-30 August 2012.
M. Bayat and B. Mehri, Periodic Solutions of First-order Autonomous Quasi-linear Systems, Linear Algebra and its Applications, 432 (2010), 485–492.
M. Bayat and B. Mehri, A Necessary Condition for the Existence of Periodic Solutions of Certain Three Dimensional Autonomous Systems, J. Appl. Math. Letters, 22 (2009), 1292–1296.
F.Z. Cong, Periodic solutions for 2kth order ordinary differential equations with nonresonance, Nonlinear Anal. 32 (1998) 787–793.
F.Z. Cong, Q.D. Huang, S.Y. Shi, Existence and uniqueness of periodic solutions for (2n+ 1)th order differential equations, J. Math. Anal. Appl. 241 (2000), 1–9.
J. Cronin, Fixed Points and Topological Degree in Nonlinear Analysis, Math. Survey II, American Math. Society, Provi dence, R.I. (1964).
G. Dinca and J. Mawhin, Brouwer Degree, The Core of Nonlinear Analysis-Birkhuser (2021).
H.A. Emamirad and B. Mehri, On The Existence of Periodic Solutions for Autonomous Second Order Systems, Nonlinear Analysis, 9 (1979), 577–582.
M. Fˇckan, Topological Degree Approach to Bifurcation Problems, Springer, (2008).
G. Jiao, Periodic solutions of (2n + 1)th order ordinary differential equations, J. Math. Anal. Appl. 272 (2002) 303–317.
N.G. Lloyd , Degree Theory, Cambridge University Press, Cambridge (1978).
J. Mawhin, Degree, quaternions and periodic solutions, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 379 (2021).
B. Mehri, A Note on Periodic Solution for Certain Nonlinear Third Order Autonomous Differential Equation, Rev. Roumaine Math. Pures Appl., 26 (1981), 1211–1215.
B. Mehri and M. Niksirat, On the Existence of Periodic Solutions for The Quasi-Linear Third-Order Differential Equation, J. Math. Anal. Appl., 261 (2001), 159–167.
L. Perko, Differential Equations and Dynamical Systems, Springer, 2001.
