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**Stability and uniform boundedness in multidelay functional differential equations of third order.**
*(English)*
Zbl 1276.34058

Summary: We consider a nonautonomous functional differential equation of the third order with multiple deviating arguments. Using the Lyapunov-Krasovskiĭ functional approach, we give certain sufficient conditions guaranteeing the asymptotic stability and uniform boundedness of the solutions.

### MSC:

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

34K20 | Stability theory of functional-differential equations |

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\textit{C. Tunç} and \textit{M. Gözen}, Abstr. Appl. Anal. 2013, Article ID 248717, 7 p. (2013; Zbl 1276.34058)

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### References:

[1] | Chlouverakis, K. E.; Sprott, J. C., Chaotic hyperjerk systems, Chaos, Solitons & Fractals, 28, 3, 739-746 (2006) · Zbl 1106.37024 |

[2] | Cronin-Scanlon, J., Some mathematics of biological oscillations, SIAM Review, 19, 1, 100-138 (1977) · Zbl 0366.92001 |

[3] | Eichhorn, R.; Linz, S. J.; Hänggi, P., Transformations of nonlinear dynamical systems to jerky motion and its application to minimal chaotic flows, Physical Review E, 58, 6, 7151-7164 (1998) |

[4] | Friedrichs, K. O., On nonlinear vibrations of third order, Studies in Nonlinear Vibration Theory, 65-103 (1946), Institute for Mathematics and Mechanics, New York University |

[5] | Linz, S. J., On hyperjerky systems, Chaos, Solitons & Fractals, 37, 3, 741-747 (2008) · Zbl 1148.37026 |

[6] | Rauch, L. L., Oscillation of a third order nonlinear autonomous system, Contributions to the Theory of Nonlinear Oscillations. Contributions to the Theory of Nonlinear Oscillations, Annals of Mathematics Studies, no. 20, 39-88 (1950), Princeton, NJ, USA: Princeton University Press, Princeton, NJ, USA |

[7] | Ademola, T. A.; Ogundiran, M. O.; Arawomo, P. O.; Adesina, O. A., Boundedness results for a certain third order nonlinear differential equation, Applied Mathematics and Computation, 216, 10, 3044-3049 (2010) · Zbl 1201.34055 |

[8] | Afuwape, A. U.; Castellanos, J. E., Asymptotic and exponential stability of certain third-order non-linear delayed differential equations: frequency domain method, Applied Mathematics and Computation, 216, 3, 940-950 (2010) · Zbl 1200.34081 |

[9] | Chukwu, E. N., On the boundedness and the existence of a periodic solution of some nonlinear third order delay differential equation, Atti della Accademia Nazionale dei Lincei. Rendiconti. Classe di Scienze Fisiche, Matematiche e Naturali 8, 64, 5, 440-447 (1978) · Zbl 0424.34071 |

[10] | Ezeilo, J. O. C., A stability result for a certain third order differential equation, Annali di Matematica Pura ed Applicata 4, 72, 1-9 (1966) · Zbl 0143.11602 |

[11] | Ezeilo, J. O. C.; Tejumola, H. O., Boundedness theorems for certain third order differential equations, Atti della Accademia Nazionale dei Lincei. Rendiconti. Classe di Scienze Fisiche, Matematiche e Naturali 8, 55, 194-201 (1973/1974) · Zbl 0295.34022 |

[12] | Hara, T., On the uniform ultimate boundedness of the solutions of certain third order differential equations, Journal of Mathematical Analysis and Applications, 80, 2, 533-544 (1981) · Zbl 0484.34024 |

[13] | Mehri, B.; Shadman, D., Boundedness of solutions of certain third order differential equation, Mathematical Inequalities & Applications, 2, 4, 545-549 (1999) · Zbl 0943.34022 |

[14] | Ogundare, B. S.; Okecha, G. E., On the boundedness and the stability of solution to third order non-linear differential equations, Annals of Differential Equations, 24, 1, 1-8 (2008) · Zbl 1164.34444 |

[15] | Omeike, M. O., Stability and boundedness of solutions of some non-autonomous delay differential equation of the third order, Analele Ştiinţifice ale Universităţii Al. I. Cuza din Iaşi. Serie Nouă. Matematică, 55, 49-58 (2009) · Zbl 1199.34390 |

[16] | Reissig, R.; Sansone, G.; Conti, R., Non-Linear Differential Equations of Higher Order (1974), Leyden, The Netherlands: Noordhoff International, Leyden, The Netherlands · Zbl 0275.34001 |

[17] | Swick, K. E., Asymptotic behavior of the solutions of certain third order differential equations, SIAM Journal on Applied Mathematics, 19, 96-102 (1970) · Zbl 0212.11403 |

[18] | Tejumola, H. O., On the boundedness and periodicity of solutions of certain third-order non-linear differential equations, Annali di Matematica Pura ed Applicata 4, 83, 195-212 (1969) · Zbl 0215.14806 |

[19] | Tejumola, H. O., A note on the boundedness and the stability of solutions of certain third-order differential equations, Annali di Matematica Pura ed Applicata 4, 92, 65-75 (1972) · Zbl 0242.34046 |

[20] | Tunç, C., Uniform ultimate boundedness of the solutions of third-order nonlinear differential equations, Kuwait Journal of Science & Engineering, 32, 1, 39-48 (2005) · Zbl 1207.34043 |

[21] | Tunç, C., Boundedness of solutions of a third-order nonlinear differential equation, Journal of Inequalities in Pure and Applied Mathematics, 6, 1, article 3 (2005) · Zbl 1082.34514 |

[22] | Tunç, C., On the asymptotic behavior of solutions of certain third-order nonlinear differential equations, Journal of Applied Mathematics and Stochastic Analysis, 1, 29-35 (2005) · Zbl 1077.34052 |

[23] | Tunç, C., Stability criteria for certain third order nonlinear delay differential equations, Portugaliae Mathematica, 66, 1, 71-80 (2009) · Zbl 1166.34329 |

[24] | Tunç, C., A new result on the stability of solutions of a nonlinear differential equation of third-order with finite lag, Southeast Asian Bulletin of Mathematics, 33, 5, 947-958 (2009) · Zbl 1212.34226 |

[25] | Tunç, C., On the stability and boundedness of solutions to third order nonlinear differential equations with retarded argument, Nonlinear Dynamics, 57, 1-2, 97-106 (2009) · Zbl 1176.34064 |

[26] | Tunç, C., The boundedness of solutions to nonlinear third order differential equations, Nonlinear Dynamics and Systems Theory, 10, 1, 97-102 (2010) · Zbl 1300.34077 |

[27] | Tunç, C., On the stability and boundedness of solutions of nonlinear third order differential equations with delay, Filomat, 24, 3, 1-10 (2010) · Zbl 1299.34244 |

[28] | Tunç, C., Some stability and boundedness conditions for non-autonomous differential equations with deviating arguments, Electronic Journal of Qualitative Theory of Differential Equations, 1 (2010) · Zbl 1201.34123 |

[29] | Tunç, C., Bound of solutions to third-order nonlinear differential equations with bounded delay, Journal of the Franklin Institute, 347, 2, 415-425 (2010) · Zbl 1193.34139 |

[30] | Tunç, C., Stability and bounded of solutions to non-autonomous delay differential equations of third order, Nonlinear Dynamics, 62, 4, 945-953 (2010) · Zbl 1215.34079 |

[31] | Tunç, C., Existence of periodic solutions to nonlinear differential equations of third order with multiple deviating arguments, International Journal of Differential Equations, 2012 (2012) · Zbl 1251.34087 |

[32] | Tunç, C., Qualitative behaviors of functional differential equations of third order with multiple deviating arguments, Abstract and Applied Analysis, 2012 (2012) · Zbl 1261.34055 |

[33] | Tunç, C., On the qualitative behaviors of solutions of some differential equations of higher order with multiple deviating arguments · Zbl 1293.34052 |

[34] | Yoshizawa, T., Stability Theory by Liapunov’s Second Method. Stability Theory by Liapunov’s Second Method, Publications of the Mathematical Society of Japan, No. 9 (1966), Tokyo, Japan: The Mathematical Society of Japan, Tokyo, Japan · Zbl 0144.10802 |

[35] | Krasovskiì, N. N., Stability of Motion. Applications of Lyapunov’s Second Method to Differential Systems and Equations with Delay (1963), Stanford, Calif, USA: Stanford University Press, Stanford, Calif, USA · Zbl 0109.06001 |

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