Previous |  Up |  Next

Article

Keywords:
ultimate boundedness; complete Lyapunov function; differential equation of third-order
Summary:
We prove the ultimate boundedness of solutions of some third order nonlinear ordinary differential equations using the Lyapunov method. The results obtained generalize earlier results of Ezeilo, Tejumola, Reissig, Tunç and others. The Lyapunov function used does not involve the use of signum functions as used by others.
References:
[1] Ademola, T. A., Ogundiran, M. O., Arawomo, P. O, Adesina, O. A: Boundedness results for a certain third order nonlinear differential equation. Appl. Math. Comput. 216 (2010), 3044-3049. DOI 10.1016/j.amc.2010.04.022 | MR 2653118 | Zbl 1201.34055
[2] Afuwape, A. U.: Frequency-domain criteria for dissipativity of some third order differential equations. An. Stiint. Univ. Al. I. Cuza Iasi, n. Ser., Sect. Ia 24 (1978), 271-275. MR 0533755 | Zbl 0405.34053
[3] Afuwape, A. U.: An application of the frequency-domain criteria for dissipativity of a certain third order non-linear differential equation. Analysis 1 (1981), 211-216. MR 0660717 | Zbl 0488.34043
[4] Afuwape, A. U., Omeike, M. O.: Further ultimate boundedness of solutions of some system of third order nonlinear ordinary differential equations. Acta Univ. Palacki. Olomuc., Fac. Rerum Nat., Math. 43 (2004), 7-20. MR 2124598
[5] Afuwape, A. U., Omeike, M. O.: Convergence of solutions of certain third order systems of nonlinear ordinary differential equations. J. Nigerian Math. Soc. 25 (2006), 1-12. MR 2376821
[6] Afuwape, A. U., Omeike, M. O.: Convergence of solutions of certain non-homogeneous third order ordinary differential equations. Kragujevac J. Math. 31 (2008), 5-16. MR 2478598 | Zbl 1199.34246
[7] Bereketoglu, H., Gyori, I.: On the boundedness of the solutions of a third-order nonlinear differential equation. Dynam. Systems Appl. 6 (1997), 263-270. MR 1461442
[8] Chukwu, E. N.: On the boundedness of solutions of third order differential equations. Ann. Mat. Pur. Appl. 104 (1975), 123-149. DOI 10.1007/BF02417013 | MR 0377180 | Zbl 0319.34027
[9] Ezeilo, J. O. C.: An elementary proof of a boundedness theorem for a certain third order differential equation. J. Lond. Math. Soc. 38 (1963), 11-16. DOI 10.1112/jlms/s1-38.1.11 | MR 0166450 | Zbl 0116.06902
[10] Ezeilo, J. O. C.: Some results for the solutions of a certain system of differential equations. J. Math. Anal. Appl. 6 (1963), 389-393. MR 0153926 | Zbl 0116.29405
[11] 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. MR 0213657
[12] Ezeilo, J. O. C.: New properties of the equation $x'''+ax'' + bx' + h(x) = p(t,x,x',x'')$ for certain special values of the incrementary ratio $y^{-1}\{h(x+y)-h(x)\}$. Equations differentielles et functionalles non-lineares, Hermann Publishing, Paris P. Janssons, J. Mawhin, N. Rouche (1973), 447-462. MR 0430413
[13] Ezeilo, J. O. C.: A generalization of some boundedness results by Reissig and Tejumola. J. Math. Anal. Appl. 41 (1973), 411-419. DOI 10.1016/0022-247X(73)90215-1 | MR 0316821 | Zbl 0253.34016
[14] Ezeilo, J. O. C.: A further result on the existence of periodic solutions of the equation $\stackrel{...}x+\Psi(\dot{x})\ddot{x}+\phi(x)\dot{x}+v(x,\dot{x},\ddot{x})=p(t)$ with a bound $\nu$. Atti. Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 55 (1978), 51-57. MR 0571030
[15] Haddad, W. A., Chellaboina, V. S.: Nonlinear Dynamical Systems and Control---A Lyapunov-Based Approach. Princeton University Press, Princeton (2008). MR 2381711 | Zbl 1142.34001
[16] Hara, T.: On a uniform ultimate boundedness of the solution of certain third order differential equations. J. Math. Anal. Appl. 80 (1981), 533-544. DOI 10.1016/0022-247X(81)90122-0 | MR 0614848
[17] Qian, C.: Asymptotic behavior of third-order nonlinear differential equations. J. Math. Anal. Appl. 284 (2003), 191-205. DOI 10.1016/S0022-247X(03)00302-0 | MR 1996127
[18] Reisssig, R.: Über die Existenz periodischer Lösungen bei einer nichtlinearen Differentialgleichung dritter Ordnung. Math. Nachr. 32 (1966), 83-88. DOI 10.1002/mana.19660320109 | MR 0216096
[19] Reisssig, R., Sansone, G., Conti, R.: Nonlinear Differential Equations of Higher Order. Noordhoff, Groningen (1974).
[20] Swick, K. E.: Boundedness and stability for a nonlinear third order differential equation. Atti. Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 56 (1974), 859-865. MR 0399597 | Zbl 0326.34062
[21] Tejumola, H. O.: On the boundedness and periodicity of solutions of certain third-order non-linear differential equations. Ann. Mat. Pura Appl., IV Ser. 83 (1969), 195-212. DOI 10.1007/BF02411167 | MR 0262597
[22] Tunç, Cemil: Boundedness of solutions of a third order nonlinear differential equation. J. Inequal. Pure Appl. Math. 6 (2005), Article 3, 6 pp. (electronic). MR 2122950 | Zbl 1082.34514
[23] Yoshizawa, T.: Stability Theory by Lyapunov's Second Method. Mathematical Society of Japan, Tokyo (1966). MR 0208086
Partner of
EuDML logo