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Annals of the Alexandru Ioan Cuza University - Mathematics

The Journal of "Alexandru Ioan Cuza" University from Iasi

Editor-in-Chief: Oniciuc, Cezar

2 Issues per year


CiteScore 2016: 0.34

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1221-8421
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Ultimate boundedness of the solutions of certain third order nonlinear non-autonomous differential equations

Anthony Uyi Afuwapẹ
  • Corresponding author
  • Departmento de Matemáticas, Universidad de Antioquia, Calle 67, No. 53-108, Medellín AA 1226, COLOMBIA
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  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Yamir Elena Carvajal
  • Corresponding author
  • Departmento de Matemáticas, Universidad de Antioquia, Calle 67, No. 53-108, Medellín AA 1226, COLOMBIA
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  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2012-11-06 | DOI: https://doi.org/10.2478/v10157-012-0011-1

Abstract

In this paper, we shall 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.

Keywords: third order; nonlinear differential equation; non-autonomous; stability; uniform ultimate boundedness.

  • 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.CrossrefGoogle Scholar

  • 2. Afuwape, A.U. - Uniform Ultimate boundedness results for some third order nonlinear differential equations, ICTP, Trieste, preprint IC/90/405.Google Scholar

  • 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.Google Scholar

  • 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.Google Scholar

  • 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.Google Scholar

  • 6. Andres, J. - Recent Results on Third-Order Nonlinear ODEs, J. Nigerian Math. Soc., 14/15 (1995/96), 41-66.Google Scholar

  • 7. Bai, Z. - Existence of solutions for some third-order boundary-value problems, Electron. J. Differential Equations, 25 (2008), 6 pp.Google Scholar

  • 8. Bernis, F.; Peletier, L.A. - Two problems from draining flows involving thirdorder ordinary differential equations, SIAM J. Math. Anal., 27 (1996), 515-527.Google Scholar

  • 9. Burton, T.A. - Lyapunov functions and boundedness, J. Math. Anal. Appl., 58 (1977), 88-97.CrossrefGoogle Scholar

  • 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.Google Scholar

  • 11. Chukwu, E.N. - On the boundedness of solutions of third order differential equations, Ann. Mat. Pura Appl., 104 (1975), 123-149.Google Scholar

  • 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.Google Scholar

  • 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.CrossrefGoogle Scholar

  • 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.Google Scholar

  • 15. Giacomelli, L. - Non-linear higher-order boundary value problems describing this viscous flows near edges, J. Math. Anal. Appl., 345 (2008), 632-649.Google Scholar

  • 16. Haddad, W.M.; Chellaboina, V. - Nonlinear Dynamical Systems and Control. A Lyapunov-based approach, Princeton University Press, Princeton, NJ, 2008.Google Scholar

  • 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.CrossrefGoogle Scholar

  • 18. Hara, T. - On the asymptotic behavior of solutions of certain non-autonomous differential equations, Osaka J. Math., 12 (1975), 267-282.Google Scholar

  • 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.CrossrefGoogle Scholar

  • 20. Lyapunov, A.M. - Stability of Motion, Academic Press, London, 1966.Google Scholar

  • 21. Macheras, P.; Iliadis, A. - Modeling in Biopharmaceutics, Pharmacokinetics, and Pharmacodynamics, Homogeneous and heterogeneous approaches. Interdisciplinary Applied Mathematics, 30, Springer, New York, 2006.Google Scholar

  • 22. Murray, J.D. - Mathematical Biology, Second edition. Biomathematics, 19, Springer-Verlag, Berlin, 1993.Google Scholar

  • 23. Qian, C. - On global stability of third-order nonlinear differential equations, Nonlinear Anal., 42 (2000), Ser. A: Theory Methods, 651-661.Google Scholar

  • 24. Qian, C. - Asymptotic behavior of a third-order nonlinear differential equation, J. Math. Anal. Appl., 284 (2003), 191-205.Google Scholar

  • 25. Reissig, R.; Sansone, G.; Conti, R. - Non-Linear Differential Equations of Higher Order, Noordhoff International Publishing, Leyden, 1974.Google Scholar

  • 26. Tunç, C. - Global stability of solutions of certain third-order nonlinear differential equations, Panamer. Math. J., 14 (2004), 31-35.Google Scholar

  • 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.Google Scholar

  • 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.Google Scholar

  • 29. Tunç, C. - Uniform ultimate boundedness of the solutions of third-order nonlinear differential equations, Kuwait J. Sci. Engrg., 32 (2005), 39-48.Google Scholar

  • 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.Google Scholar

  • 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.Google Scholar

About the article

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Published Online: 2012-11-06

Published in Print: 2012-10-01


Citation Information: Annals of the Alexandru Ioan Cuza University - Mathematics, ISSN (Print) 1221-8421, DOI: https://doi.org/10.2478/v10157-012-0011-1.

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