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Stability and boundedness results for solutions of certain third order nonlinear vector differential equations. (English) Zbl 1132.34328

Consider the nonlinear vector differential equation
\[ \dddot x+F(x, \dot x,\ddot x)\ddot x+B(t)\dot x+h(\dot x)=p(t,x,\dot x,\ddot x), \quad t\geq 0, \tag{*} \] where \(F\) and \(B\) are symmetric \(n\times n\)-matrices depending continuously on their arguments, the functions \(h\) and \(p\) mapping into \(\mathbb{R}^n\) are also continuous functions. Using Lyapunov’s direct method, the authors present conditions guaranteeing the asymptotic stability of the zero solution of (*) and the boundedness of all solutions.

MSC:

34D20 Stability of solutions to ordinary differential equations
34C11 Growth and boundedness of solutions to ordinary differential equations
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[1] Reissig, R., Sansone, G., and Conti, R., Nonlinear Differential Equations of Higher Order, Noordhoff, Groningen, 1974. · Zbl 0275.34001
[2] Ezeilo, J. O. C., ’Periodic solutions of certain third order differential equations’, Atti della Accademia Nazionale dei Lincei. Rendiconti. Classe di Scienze Fisiche, Matematiche e Naturali. Serie 57(8), 1974, 1–2; 1975, 54–60.
[3] Ezeilo, J. O. C., ’Periodic solutions of certain third order differential equations of the nondissipative type’, Atti della Accademia Nazionale dei Lincei. Rendiconti. Classe di Scienze Fisiche, Matematiche e Naturali. Serie 63(8), 1977, 3–4; 1978, 212–224. · Zbl 0405.34041
[4] Feng, C. H., ’On the existence of almost periodic solutions of nonlinear third-order differential equations’, Annals of Differential Equations 9(4), 1993, 420–424. · Zbl 0792.34042
[5] Feng, C. H., ’The existence of periodic solutions for a third-order nonlinear differential equation’, (Chinese) Gongcheng Shuxue Xuebao 11(2), 1994, 113–117. · Zbl 0925.34009
[6] Mehri, B., ’Periodic solution for certain nonlinear third order differential equations’, Indian Journal of Pure and Applied Mathematics 21(3), 1990, 203–210. · Zbl 0708.34030
[7] Mehri, B. and Shadman, D., ’Periodic solutions of certain third order nonlinear differential equations’, Studia Scientiarum Mathematicarum Hungarica 33(4), 1997, 345–350. · Zbl 0908.34031
[8] Qian, C., ’On global stability of third-order nonlinear differential equations’, Nonlinear Analysis: Theory, Methods & Applications Ser. A 42(4), 2000, 651–661. · Zbl 0969.34048
[9] Qian, C., ’Asymptotic behavior of a third-order nonlinear differential equation’, Journal of Mathematical Analysis and Applications 284(1), 2003, 191–205. · Zbl 1054.34078
[10] Shadman, D. and Mehri, B., ’On the periodic solutions of certain nonlinear third order differential equations’, Zeitschrift für Angewandte Mathematik und Mechanik 75(2), 1995, 164–166. · Zbl 0828.34029
[11] Tejumola, H. O., ’Periodic solutions of certain third order differential equations’, Atti della Accademia Nazionale dei Lincei. Rendiconti. Classe di Scienze Fisiche, Matematiche e Naturali. Serie 66(8), 1979, no. 4, 243–249. · Zbl 0482.34029
[12] Vladimir, V., ’To existence of the periodic solution of a third-order nonlinear differential equation’, Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica 29, 1990, 123–163.
[13] Yoshizawa, T., ’Stability theory and the existence of periodic solutions and almost periodic solutions’, Applied Mathematical Sciences, Vol. 14, Springer-Verlag, New York-Heidelberg, 1975. · Zbl 0304.34051
[14] Zhang, X. Z., ’Stability in the large of a class of third-order nonlinear ordinary differential equations’, Kexue Tongbao (English edn.) 28(7), 1983, 999–1000.
[15] Wang, L. and Wang, M. Q., ’On the construction of globally asymptotically stable Lyapunov functions for a type of nonlinear third-order system’, Acta Mathematicae Applicatae Sinica 6(3), 1983, 309–323. · Zbl 0517.34042
[16] Abou-El-Ela, A. M. A., ’Boundedness of the solutions of certain third-order vector differential equations’, Annals of Differential Equations 1(2), 1985, 127–139. · Zbl 0603.34030
[17] Afuwape, A.U., ’Ultimate boundedness results for a certain system of third-order nonlinear differential equations’, Journal of Mathematical Analysis and Applications 97(1), 1983, 140–150. · Zbl 0537.34031
[18] Afuwape, A. U., ’ Further ultimate boundedness results for a third-order nonlinear system of differential equations’, Unione Matematica Italiana Bollettino C. Serie VI 4(1), 1985, 347–361. · Zbl 0592.34024
[19] Edziwy Spolhk, S., ’Stability of certain third order vector differential equations’, Bulletin de l’Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques 23, 1975, 15–21.
[20] Ezeilo, J. O. C. and Tejumola, H. O., ’Boundedness and periodicity of solutions of a certain system of third-order non-linear differential equations’, Annali di Matematica Pura ed Applicata 74(4), 1966, 283–316. · Zbl 0148.06701
[21] Feng, C., ’On the existence of periodic solutions for a certain system of third order nonlinear differential equations’, Annals of Differential Equations 11(3), 1995, 264–269. · Zbl 0880.34040
[22] Kalantarov, V. and Tiryaki, A., ’On the stability results for third order differential-operator equations’, Turkish Journal of Mathematics 21(2), 1997, 179–186. · Zbl 0892.35026
[23] Meng, F. W., ’Ultimate boundedness results for a certain system of third order nonlinear differential equations’, Journal of Mathematical Analysis and Applications 177(2), 1993, 496–509. · Zbl 0783.34042
[24] Tiryaki, A., ’Boundedness and periodicity results for a certain system of third order nonlinear differential equations’, Indian Journal of Pure and Applied Mathematics 30(4), 1999, 361–372. · Zbl 0936.34041
[25] Tunç, C., ’On the boundedness and periodicity of the solutions of a certain vector differential equation of third-order’, Applied Mathematics and Mechanics (English edn.) 20(2), 1999, 163–170. · Zbl 0933.34058
[26] Tunç, C., ’Uniform ultimate boundedness of the solutions of third-order nonlinear differential equations’, Kuwait Journal of Science & Engineering 32(1), 2005, 39–48. · Zbl 1207.34043
[27] Tunç, C., ’On the asymptotic behavior of solutions of certain third-order nonlinear differential equation’, Journal of Applied Mathematics and Stochastic Analysis 2005(1), 2005, 29–35. · Zbl 1077.34052
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