Ultimate boundedness of the solutions of certain third order nonlinear non-autonomous differential equations. (English) Zbl 1274.34103
Summary: We establish sufficient conditions for the uniform ultimate boundedness of solutions of a certain third order nonlinear non-autonomous 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:
| 34C11 | Growth and boundedness of solutions to ordinary differential equations |
Keywords:
third order; nonlinear differential equation; non-autonomous; stability; uniform ultimate boundednessReferences:
| [1] | 1. Afuwape, A.U. - Ultimate boundedness results for a certain system of third-order nonlinear differential equations, J. Math. Anal. Appl., 97 (1983), 140-150. · Zbl 0537.34031 · doi:10.1016/0022-247X(83)90243-3 |
| [2] | 2. Afuwape, A.U. - Uniform Ultimate boundedness results for some third order nonlinear differential equations, ICTP, Trieste, preprint IC/90/405. · Zbl 1299.34127 |
| [3] | 3. Afuwape, A.U. - Further ultimate boundedness results for a third-order nonlinear system of differential equations, Boll. Un. Mat. Ital. C (6), 4 (1985), 347-361. · Zbl 0592.34024 |
| [4] | 4. Afuwape, A.U.; Omeike, M.O. - Further ultimate boundedness of solutions of some system of third order nonlinear ordinary differential equations, Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math., 43 (2004), 7-20. · Zbl 1068.34052 |
| [5] | 5. Afuwape, A.U.; Omari, P.; Zanolin, F. - Nonlinear perturbations of differential operators with nontrivial kernel and applications to third-order periodic boundary value problems, J. Math. Anal. Appl., 143 (1989), 35-56. · Zbl 0695.47044 · doi:10.1016/0022-247X(89)90027-9 |
| [6] | 6. Andres, J. - Recent Results on Third-Order Nonlinear ODEs, J. Nigerian Math. Soc., 14/15 (1995/96), 41-66. |
| [7] | 7. Bai, Z. - Existence of solutions for some third-order boundary-value problems, Electron. J. Differential Equations, 25 (2008), 6 pp. · Zbl 1141.34307 |
| [8] | 8. Bernis, F.; Peletier, L.A. - Two problems from draining flows involving thirdorder ordinary differential equations, SIAM J. Math. Anal., 27 (1996), 515-527. · Zbl 0845.34033 · doi:10.1137/S0036141093260847 |
| [9] | 9. Burton, T.A. - Lyapunov functions and boundedness, J. Math. Anal. Appl., 58 (1977), 88-97. · Zbl 0386.34050 · doi:10.1016/0022-247X(77)90230-X |
| [10] | 10. Burton, T.A. - Stability and Periodic Solutions of Ordinary and Functional-Differential Equations, Mathematics in Science and Engineering, 178, Academic Press, Inc., Orlando, FL, 1985. · Zbl 0635.34001 |
| [11] | 11. Chukwu, E.N. - On the boundedness of solutions of third order differential equations, Ann. Mat. Pura Appl., 104 (1975), 123-149. · Zbl 0319.34027 · doi:10.1007/BF02417013 |
| [12] | 12. Ezeilo, J.O.C. - A note on a boundedness theorem for some third order differential equations, J. London Math. Soc., 36 (1961), 439-444. · Zbl 0104.06501 · doi:10.1112/jlms/s1-36.1.439 |
| [13] | 13. Ezeilo, J.O.C. - On the stability of solutions of certain differential equations of the third order, Quart. J. Math. Oxford Ser., 11 (1960), 64-69. · Zbl 0090.06603 · doi:10.1093/qmath/11.1.64 |
| [14] | 14. Ezeilo, J.O.C. - A generalization of a boundedness theorem for a certain third-order differential equation, Proc. Cambridge Philos. Soc., 63 (1967), 735-742. · Zbl 0163.10507 |
| [15] | 15. Giacomelli, L. - Non-linear higher-order boundary value problems describing this viscous flows near edges, J. Math. Anal. Appl., 345 (2008), 632-649. · Zbl 1184.76618 · doi:10.1016/j.jmaa.2008.04.038 |
| [16] | 16. Haddad, W.M.; Chellaboina, V. - Nonlinear Dynamical Systems and Control. A Lyapunov-based approach, Princeton University Press, Princeton, NJ, 2008. · Zbl 1142.34001 |
| [17] | 17. Hara, T. - On the asymptotic behavior of the solutions of some third and fourth order non-autonomous differential equations, Publ. Res. Inst. Math. Sci., 9 (1973/74), 649-673. · Zbl 0286.34083 · doi:10.2977/prims/1195192447 |
| [18] | 18. Hara, T. - On the asymptotic behavior of solutions of certain non-autonomous differential equations, Osaka J. Math., 12 (1975), 267-282. · Zbl 0357.34049 |
| [19] | 19. Hara, T. - On the uniform ultimate boundedness of the solutions of certain third order differential equations, J. Math. Anal. Appl., 80 (1981), 533-544. · Zbl 0484.34024 · doi:10.1016/0022-247X(81)90122-0 |
| [20] | 20. Lyapunov, A.M. - Stability of Motion, Academic Press, London, 1966. |
| [21] | 21. Macheras, P.; Iliadis, A. - Modeling in Biopharmaceutics, Pharmacokinetics, and Pharmacodynamics, Homogeneous and heterogeneous approaches. Interdisciplinary Applied Mathematics, 30, Springer, New York, 2006. · Zbl 1103.92023 · doi:10.1007/0-387-31910-7 |
| [22] | 22. Murray, J.D. - Mathematical Biology, Second edition. Biomathematics, 19, Springer-Verlag, Berlin, 1993. |
| [23] | 23. Qian, C. - On global stability of third-order nonlinear differential equations, Nonlinear Anal., 42 (2000), Ser. A: Theory Methods, 651-661. · Zbl 0969.34048 · doi:10.1016/S0362-546X(99)00120-0 |
| [24] | 24. Qian, C. - Asymptotic behavior of a third-order nonlinear differential equation, J. Math. Anal. Appl., 284 (2003), 191-205. · Zbl 1054.34078 · doi:10.1016/S0022-247X(03)00302-0 |
| [25] | 25. Reissig, R.; Sansone, G.; Conti, R. - Non-Linear Differential Equations of Higher Order, Noordhoff International Publishing, Leyden, 1974. · Zbl 0275.34001 |
| [26] | 26. Tunç, C. - Global stability of solutions of certain third-order nonlinear differential equations, Panamer. Math. J., 14 (2004), 31-35. · Zbl 1065.34047 |
| [27] | 27. Tunç, C.; Tunç, E. - New ultimate boundedness and periodicity results for certain third-order nonlinear vector differential equations, Math. J. Okayama Univ., 48 (2006), 159-172. · Zbl 1138.34322 |
| [28] | 28. Tunç, C. - Boundedness of solutions of a third-order nonlinear differential equation, JIPAM. J. Inequal. Pure Appl. Math., 6 (2005), Article 3, 6 pp. · Zbl 1115.34332 |
| [29] | 29. Tunç, C. - Uniform ultimate boundedness of the solutions of third-order nonlinear differential equations, Kuwait J. Sci. Engrg., 32 (2005), 39-48. · Zbl 1207.34043 |
| [30] | 30. Tunç, C. - On the boundedness of solutions of certain nonlinear vector differential equations of third order, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 49(97) (2006), 291-300. |
| [31] | 31. Zhou, X.Y.; Song, X.; Shi, X. - A differential equation model of HIV infection of CD4+T-cell with cure rate, J. Math. Anal. Appl., 342 (2008), 1342-1355. · Zbl 1188.34062 · doi:10.1016/j.jmaa.2008.01.008 |
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