On the stability and boundedness of solutions of a class of nonautonomous differential equations of second order with multiple deviating arguments.

*(English)*Zbl 1266.34117Summary: This paper is devoted to the mathematical analysis of stability, boundedness and uniform boundedness of solutions of a class of nonlinear differential equations of second order with multiple constant deviating arguments. We use Lyapunov functionals to verify the stability and boundedness of the solutions, and some examples are given to illustrate the theoretical analysis.

##### MSC:

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

34K20 | Stability theory of functional-differential equations |

##### Keywords:

differential equation; second order; multiple constant deviating arguments; stability; boundedness; uniform boundedness
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##### References:

[1] | Ahmad S., Rama Mohana Rao M.: Theory of ordinary differential equations. With applications in biology and engineering. Affiliated East-West Press Pvt Ltd, New Delhi (1999) |

[2] | Antosiewicz H.A.: On non-linear differential equations of the second order with integrable forcing term. J. London Math. Soc. 30, 64–67 (1955) · Zbl 0064.08404 |

[3] | Burton T.A.: On the equation x” + f(x)h(x’)x’ + g(x) = e(t). Ann. Mat. Pura Appl. 85(4), 277–285 (1970) · Zbl 0194.40203 |

[4] | Burton T.A.: Stability and periodic solutions of ordinary and functional differential equations. Academic Press, Orlando (1985) · Zbl 0635.34001 |

[5] | Burton T.A., Hering R.H.: Liapunov theory for functional-differential equations. 20th Midwest ODE Meeting (Iowa City, IA, 1991), Rocky Mountain. J. Math. 24(1), 3–17 (1994) · Zbl 0806.34067 |

[6] | Burton T.A., Townsend C.G.: On the generalized Liénard equation with forcing function. J. Differ. Equ. 4, 620–633 (1968) · Zbl 0174.13602 |

[7] | Caldeira-Saraiva F.: The boundedness of solutions of a Liénard equation arising in the theory of ship rolling. IMA J. Appl. Math. 36(2), 129–139 (1986) · Zbl 0658.34026 |

[8] | Constantin A.: A note on a second-order nonlinear differential system. Glasg. Math. J. 42(2), 195–199 (2000) · Zbl 0960.34027 |

[9] | Čžan, P.: On stability with arbitrary initial perturbations of the solutions of a system of two differential equations. (Chinese) Acta Math. Sinica 9, 442–445 (1959) |

[10] | Gao S.Z., Zhao L.Q.: Global asymptotic stability of generalized Liénard equation. Chin. Sci. Bull. 40(2), 105–109 (1995) · Zbl 0828.34040 |

[11] | Graef J.R.: On the generalized Liénard equation with negative damping. J. Diff. Equ. 12, 34–62 (1972) · Zbl 0254.34038 |

[12] | Hatvani L.: On the stability of the zero solution of nonlinear second order differential equations. Acta Sci. Math. 57, 367–371 (1993) · Zbl 0790.34046 |

[13] | Heidel J.W.: Global asymptotic stability of a generalized Liénard equation. SIAM J. Appl. Math. 19(3), 629–636 (1970) · Zbl 0203.39901 |

[14] | Huang L.H., Yu J.S.: On boundedness of solutions of generalized Liénard’s system and its application. Ann. Differ. Equ. 9(3), 311–318 (1993) · Zbl 0782.34036 |

[15] | Jin Z.: Boundedness and convergence of solutions of a second-order nonlinear differential system. J. Math. Anal. Appl. 256(2), 360–374 (2001) · Zbl 0983.34021 |

[16] | Jitsuro S., Yusuke A.: Global asymptotic stability of non-autonomous systems of Lienard type. J. Math. Anal. Appl. 289(2), 673–690 (2004) · Zbl 1047.34062 |

[17] | Kato J.: On a boundedness condition for solutions of a generalized Liénard equation. J. Differ. Equ. 65(2), 269–286 (1986) · Zbl 0612.34042 |

[18] | Liu B., Huang L.: Boundedness of solutions for a class of retarded Liénard equation. J. Math. Anal. Appl. 286(2), 422–434 (2003) · Zbl 1044.34023 |

[19] | Liu Z.R.: Conditions for the global stability of the Liénard equation. Acta Math. Sinica 38(5), 614–620 (1995) · Zbl 0838.34064 |

[20] | Luk W.S.: Some results concerning the boundedness of solutions of Lienard equations with delay. SIAM J. Appl. Math. 30(4), 768–774 (1976) · Zbl 0347.34055 |

[21] | Malyseva I.A.: Boundedness of solutions of a Liénard differential equation. Differ.’niye Uravneniya 15(8), 1420–1426 (1979) |

[22] | Nápoles Valdés J.E.: Boundedness and global asymptotic stability of the forced Liénard equation. Rev. Un. Mat. Argentina 41(4), 47–59 (2000) · Zbl 1028.34050 |

[23] | Omari P., Zanolin F.: On the existence of periodic solutions of forced Liénard differential equations. Nonlinear Anal. 11(2), 275–284 (1987) · Zbl 0644.34030 |

[24] | Qian C.X.: Boundedness and asymptotic behaviour of solutions of a second-order nonlinear system. Bull. London Math. Soc. 24(3), 281–288 (1992) · Zbl 0763.34021 |

[25] | Qian C.X.: On global asymptotic stability of second order nonlinear differential systems. Nonlinear Anal. 22(7), 823–833 (1994) · Zbl 0801.34055 |

[26] | Sugie J., Amano Y.: Global asymptotic stability of non-autonomous systems of Liénard type. J. Math. Anal. Appl. 289(2), 673–690 (2004) · Zbl 1047.34062 |

[27] | Tunç C.: Some stability and boundedness results to nonlinear differential equations of Liénard type with finite delay. J. Comput. Anal. Appl. 11(4), 711–727 (2009) · Zbl 1176.34089 |

[28] | Tunç C.: Some new stability and boundedness results of solutions of Liénard type equations with deviating argument. Nonlinear Anal. Hybrid Syst. 4(1), 85–91 (2010) · Zbl 1182.34099 |

[29] | Tunç C.: Stability and boundedness of solutions of second order non-autonomous and nonlinear differential equations. J. Comput. Anal. Appl. 13(6), 1067–1074 (2011) · Zbl 1227.34054 |

[30] | Tunç C., Tunç E.: On the asymptotic behavior of solutions of certain second-order differential equations. J. Franklin Inst. 344(5), 391–398 (2007) · Zbl 1269.34057 |

[31] | Yoshizawa T.: Asymptotic behavior of solutions of a system of differential equations. Contributions to Differential Equations 1, 371–387 (1963) · Zbl 0127.30802 |

[32] | Yoshizawa, T.: Stability theory by Liapunov’s second method. Publications of the Mathematical Society of Japan, no. 9, The Mathematical Society of Japan, Tokyo (1966) · Zbl 0144.10802 |

[33] | Zhou X., Jiang W.: Stability and boundedness of retarded Liénard-type equation. Chin Q. J. Math. 18(1), 7–12 (2003) · Zbl 1058.34098 |

[34] | Zhou J., Xiang L.: On the stability and boundedness of solutions for the retarded Liénard-type equation. Ann. Differ. Equ. 15(4), 460–465 (1999) · Zbl 0964.34064 |

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