##
**Stability and boundednessof the solutions of non autonomous third order differential equations with delay.**
*(English)*
Zbl 1317.34157

Summary: We establish sufficient conditions for the asymptotic stability and boundedness of solutions of a certain third order nonlinear non-autonomous delay differential equation, by using a Lyapunov function as basic tool. In doing so we extend some existing results. Examples are given to illustrate our results.

### MSC:

34K20 | Stability theory of functional-differential equations |

34K12 | Growth, boundedness, comparison of solutions to functional-differential equations |

### Keywords:

stability; Lyapunov functional; delay differential equations; third-order differential equations
PDFBibTeX
XMLCite

\textit{M. Remili} and \textit{L. D. Oudjedi}, Acta Univ. Palacki. Olomuc., Fac. Rerum Nat., Math. 53, No. 2, 139--147 (2014; Zbl 1317.34157)

Full Text:
Link

### References:

[1] | Burton, T. A.: Stability and periodic solutions of ordinary and functional differential equations. Mathematics in Science and Engineering 178, Academic Press, Orlando, FL, 1985. · Zbl 0635.34001 |

[2] | Burton, T. A., Makay, G.: Asymptotic stability for functional differential equations. Acta Math. Hung. 65, 3 (1994), 243-251. · Zbl 0805.34068 · doi:10.1007/BF01875152 |

[3] | Greaf, J. R., Remili, M.: Some properties of monotonic solutions of \(x^{\prime \prime \prime }+p(t)x^{\prime }+q(t)f(x)=0\). Pan. American Math. J. 22, 2 (2012), 31-39. · Zbl 1264.34100 |

[4] | Omeike, M. O.: New results on the stability of solution of some non-autonomous delay differential equations of the third order. Differential Equations and Control Processes 2010, 1 (2010), 18-29. |

[5] | Omeike, M. O.: Stability and boundedness of solutions of some non-autonomous delay differential equation of the third order. An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.) 55, 1 (2009), 49-58. · Zbl 1199.34390 |

[6] | Sadek, A. I.: On the Stability of Solutions of Some Non-Autonomous Delay Differential Equations of the Third Order. Asymptot. Anal. 43, 1-2 (2005), 1-7. · Zbl 1075.34075 |

[7] | Sadek, A. I.: Stability and Boundedness of a Kind of Third-Order Delay Differential System. Applied Mathematics Letters 16, 5 (2003), 657-662. · Zbl 1056.34078 · doi:10.1016/S0893-9659(03)00063-6 |

[8] | Swick, K.: On the boundedness and the stability of solutions of some non- autonomous differential equations of the third order. J. London Math. Soc. 44 (1969), 347-359. · Zbl 0164.39103 · doi:10.1112/jlms/s1-44.1.347 |

[9] | Tunç, C.: On asymptotic stability of solutions to third order nonlinear differential equations with retarded argument. Communications in applied analysis 11, 4 (2007), 515-528. · Zbl 1139.34054 |

[10] | Tunç, C.: On the asymptotic behavior of solutions of certain third-order nonlinear differential equations. J. Appl. Math. Stoch. Anal. 2005, 1 (2005), 29-35. · Zbl 1077.34052 · doi:10.1155/JAMSA.2005.29 |

[11] | Tunç, C.: On the stability and boundedness of solutions to third order nonlinear differential equations with retarded argument. Nonlinear Dynamics 57 (2009), 97-106. · Zbl 1176.34064 · doi:10.1007/s11071-008-9423-6 |

[12] | Tunç, C.: Some stability and boundedness conditions for non-autonomous differential equations with deviating arguments. E. J. Qualitative Theory of Diff. Equ. 2010, 1 (2005), 1-12. · Zbl 1201.34123 |

[13] | Tunç, C.: Stability and boundedness of solutions of nonlinear differential equations of third-order with delay. Differential Equations and Control Processes 2007, 3 (2007), 1-13. · Zbl 1299.34244 |

[14] | Yoshizawa, T.: Stability theory by Liapunov’s second method. The Mathematical Society of Japan, Tokyo, 1966. · Zbl 0144.10802 |

[15] | Zhu, Y. F.: On stability, boundedness and existence of periodic solution of a kind of third order nonlinear delay differential system. Ann. Differential Equations 8, 2 (1992), 249-259. · Zbl 0758.34072 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.