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Nonexistence of Nontrivial Periodic Solutions to a Class of Nonlinear Differential Equations of Eighth Order

Tunç, Cemil, (2009) Nonexistence of Nontrivial Periodic Solutions to a Class of Nonlinear Differential Equations of Eighth Order. Bulletin of the Malaysian Mathematical Sciences Society, 32 (3). pp. 307-311. ISSN 0126-6705

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Official URL: http://math.usm.my/bulletin/pdf/v32n3/v32n3p4.pdf

Affiliations

Yuzuncu Yil University, Turkey. Faculty of Arts and Sciences. Dept. of Mathematics.

Abstract

By constructing a Lyapunov function, a new result is given, which guarantees the non-existence of nontrivial periodic solutions to nonlinear vector differential equation of eighth order:

$X^{(8)} + AX^{(7)} + BX^{(6)} + CX^{(5)} + DX^{(4)} + E \dot{\ddot{X}} + F( \dot{X}) \ddot{X} + G(X) \dot{X} + H(X) = 0$.

An example is also established for the illustrations of topic. By this way, our findings raise a new result for the nonexistence of nontrivial periodic solutions related to this nonlinear vector differential equation of eighth order.

Item Type:Journal
Keywords:Nonlinear differential equation, eighth order, periodic solution, Lyapunov’s second (or direct) method.
Subjects:Q Science, Computer Science
ID Code:9183

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[2] S. A. Iyase, Periodic solutions of certain eighth order differential equations, J. Nigerian Math. Soc. 14/15 (1995/96), 31–39.

[3] C. Tunc, Instability of solutions of a certain non-autonomous vector differential equation of eighth-order, Ann. Differential Equations 22 (2006), no. 1, 7–12.

[4] Dzh. Tundzh and E. Tundzh, Differ. Uravn. 42 (2006), no. 1, 135–138, 143; translation in Differ. Equ. 42 (2006), no. 1, 150–154.

[5] C. Tunc, About uniform boundedness and convergence of solutions of certain non-linear differential equations of fifth-order, Bulletin of the Malaysian Mathematical Sciences Society (2) 30 (2007), no. 1, 1–12.

[6] C. Tunc, On the stability and boundedness of solutions of nonlinear vector differential equations of third order, Nonlinear Anal. 70 (2009), no. 6, 2232–2236.

[7] C. Tunc, On the existence of periodic solutions to a certain fourth-order nonlinear differential equation, Ann. Differential Equations 25 (2009), no. 1, 8–12.

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