On periodic solutions of \(n\)th order ordinary differential equations. (English) Zbl 0953.34028

In this paper, the author studies the existence and uniqueness of a periodic solution to the following equations \[ u^{(n)} = \sum_{k=1}^np_k(t)u^{(k-1)} + q(t),\tag{1} \] and \[ u^{(n)} = f(t,u,\cdots,u^{(n-1)}),\tag{2} \] where the functions \(p_k(t)\), \(k=1,\cdots,n,\) and \(f\) are \(\omega\)-periodic in \(t\). Under the condition \[ \int_0^\omega p_1(t)\neq 0 \] and other conditions, he obtains the result that the linear equation (1) has only one \(\omega\)-periodic solution. For equation (2), he also establishes a similar result. The proof is based on the topological degree and properties of the \(\omega\)-periodic solutions to the following differential inequalities \[ u^{(n)}\text{sgn}(\pm u(t))\geq g_0(t)|u(t)|^\lambda - g_1(t, |u^\prime(t)|,\cdots, |u^{(n-1)}(t)|), \]
\[ |u^{(n)}(t)|\leq g(t, |u(t)|,\cdots, |u^{(n-1)}(t)|). \]
Reviewer: Bin Liu (Beijing)


34C25 Periodic solutions to ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
Full Text: DOI


[1] Bates, F. W.; Ward, Y. R., Periodic solutions of higher order systems, Pacific J. Math., 84, 2, 275-282 (1979) · Zbl 0424.34042
[2] Bernfeld, S. R.; Lakshmikantham, V., An Introduction to Nonlinear Boundary Value Problems (1974), Academic Press, Inc: Academic Press, Inc New York · Zbl 0286.34018
[3] Gaines, R. E.; Mawhin, J. L., Coincidence Degree and Nonlinear Differential Equations (1977), Berlin: Berlin Springer · Zbl 0326.34021
[4] Gegelia, G. T., On boundary value problems of periodic type for ordinary odd order differential equations, Arch. Math., 20, 4, 195-204 (1984) · Zbl 0589.34030
[5] Gegelia, G. T., On bounded and periodic solutions of nonlinear ordinary differential equations, Differentsial’nye Uravneniya, 22, 3, 390-396 (1986), (Russian) · Zbl 0635.34035
[6] Gegelia, G. T., On a boundary value problems of periodic type for ordinary differential equations, Trudy Inst. Prikl. Mat. im. I. N. Vekua, 7, 60-93 (1986), (Russian) · Zbl 0632.34010
[7] Gegelia, G. T., On periodic solutions of ordinary differential equations, (Colloq. Math. Soc. János Bolyai. 53. Qual. Th. Diff. Eq. Szeged (1986)), 211-217 · Zbl 0705.34039
[8] Hartman, P., Ordinary Differential Equations (1964), Wiley: Wiley New York · Zbl 0125.32102
[9] Kiguradze, I. T., On bounded and periodic solutions of linear higher order differential equations, Mat. Zametki, 37, 1, 48-62 (1985), (Russian) · Zbl 0572.34032
[10] Kiguradze, I., Initial and Boundary Value Problems for Systems of Ordinary Differential Equations. I (1997), Publishing House “Metsniereba”: Publishing House “Metsniereba” Tbilisi, (Russian) · Zbl 0956.34505
[11] Kiguradze, I. T.; Kusano, T., On periodic solutions of higher order nonautonomous ordinary differential equations, Differentsial’nye Uravneniya, 35, 1, 72-78 (1999), (Russian) · Zbl 0936.34033
[12] Kiguradzee, I.; Půẑa, B., On boundary value problems for functional differential equations, Mem. Differential Equations Math. Phys., 12, 106-113 (1997) · Zbl 0909.34054
[13] Kipnis, L. A., On periodic solution of higher order nonlinear differential equations, Prikl. Mat. Mekh., 41, 2, 362-365 (1977), (Russian)
[14] Lasota, A.; Opial, Z., Sur les solutions périodiques des équations defférentielles ordinaires, Ann. Polon. Math., 16, 1, 69-94 (1964) · Zbl 0142.35303
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