On the asymptotic behaviour of solutions of certain differential equations of the third order
DOI:
https://doi.org/10.4067/S071609172014000100008Keywords:
Third order, differential equations, uniform ultimate boundedness, asymptotic behaviour of solutions, complete Lyapunov function, función completa de Lyapunov, comportamiento asintótico de soluciones, acotamiento uniforme, tercer orden.Abstract
In this article, Lyapunov second method is used to obtain criteria for uniform ultimate boundedness and asymptotic behaviour of solutions of nonlinear differential equations of the third order. The results obtained in this investigation include and extend some well known results on third order nonlinear differential equations in the literature.References
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[16] Lakshmikantham, V. and Leels, S.; Ordinary differential and integral inequalities: Theory and applications Vol. 1 Academic Press New York, (1969).
[17] Nakashima, M.; Asymptotic behaviour of solutions of some third order differential equations. Rep. Fac. Sci. Kagoshima Uni. (Math. Phys. Chem.) No.4, pp. 715, (1971).
[18] Ogundare, B. S.; On the convergence of solutions of certain third order nonlinear differential equation, Math.Sci. Res. J. 9 (11), pp. 304312, (2005).
[19] Omeike, M. O.; New results on the asymptotic behaviour of a third order nonlinear differential equations, Differential Equations & Applications, pp. 113, (2008).
[20] Qian, C.; Asymptotic behaviour of a third order nonlinear differential equation, J. Math. Anal. Appl., 284, no.1, pp. 191205, (2003).
[21] Reissig, R., Sansone, G. and Conti, R.; Nonlinear differential equations of higher order, Noordhoff International Publishing Leyeden, (1974).
[22] Rouche, N., Habets, N. and Laloy, M.; Stability theory by Liapunov’s direct method. Applied Mathematical Sciences 22, SpringerVerlag New York. Heidelberg. Berlin, (1977).
[23] Swick, K. E.; Asymptotic behaviour of the solutions of certain third order differential equations, SIAM J. Appl., 19, N0. 1, pp. 96102, (1970).
[24] Swick, K. E.; On the boundedness and the stability of solutions for some nonautonomous differential equations of the third order, J. London Math. Soc.,, 44, pp. 347359, (1969).
[25] Tejumola, H. O.; A note on the boundedness of solutions of some nonlinear differential equations of the third order, Ghana J. of Science, 11 (2), pp. 117118, (1970).
[26] Tunc, C.; Boundedness of solutions of a third order nonlinear differential equation, J. Inequal. Pure and Appl. Math., 6 (1), pp. 16, (2005).
[27] Tunc, C.; On the asymptotic behaviour of solutions of certain third order nonlinear differential equations, J. Applied Math. and Stochastic Anal., 1, pp. 2935, (2005).
[28] Tunc, E.; On the convergence of solutions of certain third order differential equations. Discrete Dyn. Nature and Society., pp. 112, (2009).
[29] Yoshizawa, T.; Stability theory by Liapunov’s second method, The Mathematical Society of Japan (1966).
[2] Ademola, A. T. and Arawomo, P. O.; On the stability and ultimate boundedness of solutions for certain third order differential equations. Journal of Mathematics and Statistic., 4, pp. 202208, (2008).
[3] Ademola, A. T. and Arawomo, P. O.; Stability and ultimate boundedness of solutions to certain third order differential equations. Applied Mathematics ENotes., 10, pp. 6169, (2010).
[4] Ademola, A. T. and Arawomo, P. O.; Stability and uniform ultimate boundedness of solutions of some third order differential equations. Acta Mathematica Academiae Paedagogicae Nyiregyhaziensis, 27, pp. 5159, (2011).
[5] Ademola, A. T., Ogundiran, M.O., Arawomo, P.O. and Adesina, O.A.; Boundedness results for a certain third order nonlinear differential equations. Appl. Math. Comput. 216, pp. 30443049, (2010).
[6] Ademola, A. T., Ogundiran, M.O., Arawomo, P.O. and Adesina, O.A.; Stability results for the solutions of a certain third order nonlinear differential equation. Mathematical Sciences Research Journal MSRJ. Vol. 12, no. 6, pp. 124134, (2008).
[7] Afuwape, A. U. and Adesina, O.A.; On the bounds for meanvalues of solutions to certain third order nonlinear differential equations, Fasciculi Mathematici, Vol. 36, pp. 514, (2005).
[8] Afuwape, A. U. and Omeike, M.O.; Convergence of solutions for certain nonhomogeneous third order differential equations. Kragujevac J. Math., 31, pp. 516, (2008).
[9] Chukwu, E. N.; On boundedness of solutions of third order differential equations, Ann. Mat. Pura. Appl., 104, (4), pp. 123149, (1975).
[10] Ezeilo, J. O. C.; A note on a boundedness theorem for some third order differential equations, J. London Math. Soc., 36, pp. 439444, (1961).
[11] Ezeilo, J. O. C.; A boundedness theorem for a certain third order differential equation, Proc. London Math. Soc. (3), 13, pp. 99124, (1963).
[12] Ezeilo, J. O. C.; A generalization of a boundedness theorem for the equation x + áx + ö2(x) + ö3(x) = ψ(t, x, x, x), Atti. Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., (13), 50, pp. 424431, (1971).
[13] Ezeilo, J. O. C.; A generalization of some boundedness results by Reissig and Tejumola, J. Math. Anal. Appl., 41, pp. 411419, (1973).
[14] Ezeilo, J. O. C. and Tejumola, H.O.; Boundedness theorems for certain third order differential equations, Atti. Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., (10), 55, pp. 194201, (1973).
[15] Ezeilo, J. O. C.; Further results for the solutions of a third order differential equation, Proc. Camb. Phil. Soc., 59, pp. 111116, (1963).
[16] Lakshmikantham, V. and Leels, S.; Ordinary differential and integral inequalities: Theory and applications Vol. 1 Academic Press New York, (1969).
[17] Nakashima, M.; Asymptotic behaviour of solutions of some third order differential equations. Rep. Fac. Sci. Kagoshima Uni. (Math. Phys. Chem.) No.4, pp. 715, (1971).
[18] Ogundare, B. S.; On the convergence of solutions of certain third order nonlinear differential equation, Math.Sci. Res. J. 9 (11), pp. 304312, (2005).
[19] Omeike, M. O.; New results on the asymptotic behaviour of a third order nonlinear differential equations, Differential Equations & Applications, pp. 113, (2008).
[20] Qian, C.; Asymptotic behaviour of a third order nonlinear differential equation, J. Math. Anal. Appl., 284, no.1, pp. 191205, (2003).
[21] Reissig, R., Sansone, G. and Conti, R.; Nonlinear differential equations of higher order, Noordhoff International Publishing Leyeden, (1974).
[22] Rouche, N., Habets, N. and Laloy, M.; Stability theory by Liapunov’s direct method. Applied Mathematical Sciences 22, SpringerVerlag New York. Heidelberg. Berlin, (1977).
[23] Swick, K. E.; Asymptotic behaviour of the solutions of certain third order differential equations, SIAM J. Appl., 19, N0. 1, pp. 96102, (1970).
[24] Swick, K. E.; On the boundedness and the stability of solutions for some nonautonomous differential equations of the third order, J. London Math. Soc.,, 44, pp. 347359, (1969).
[25] Tejumola, H. O.; A note on the boundedness of solutions of some nonlinear differential equations of the third order, Ghana J. of Science, 11 (2), pp. 117118, (1970).
[26] Tunc, C.; Boundedness of solutions of a third order nonlinear differential equation, J. Inequal. Pure and Appl. Math., 6 (1), pp. 16, (2005).
[27] Tunc, C.; On the asymptotic behaviour of solutions of certain third order nonlinear differential equations, J. Applied Math. and Stochastic Anal., 1, pp. 2935, (2005).
[28] Tunc, E.; On the convergence of solutions of certain third order differential equations. Discrete Dyn. Nature and Society., pp. 112, (2009).
[29] Yoshizawa, T.; Stability theory by Liapunov’s second method, The Mathematical Society of Japan (1966).
Published
20170323
How to Cite
[1]
A. T. Ademola and P. O. Arawomo, “On the asymptotic behaviour of solutions of certain differential equations of the third order”, Proyecciones (Antofagasta, On line), vol. 33, no. 1, pp. 101122, Mar. 2017.
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